Towards a represenation-based theory of meaning

UNIWERSYTET WARSZAWSKIINSTYTUT FILOZOFII

Towards a representation-based theory ofmeaning

ROZPRAWA DOKTORSKA

Opracowana pod przewodnictwemprof. UW dra hab. Stanis lawa Krajewskiego

Piotr WilkinWarszawa, maj 2012

Contents

Introduction 5

1 Cognitive representations 91.1 Defining cognitive representations . . . . . . . . . . . . . . . . 91.2 Sidenote on formalization . . . . . . . . . . . . . . . . . . . . 141.3 Formalizing representations . . . . . . . . . . . . . . . . . . . 16

1.3.1 The effect of representing and the paradox of the elu-sive representation . . . . . . . . . . . . . . . . . . . . 25

1.3.2 Representations formalized . . . . . . . . . . . . . . . . 271.3.3 The definition of representations . . . . . . . . . . . . . 33

1.4 Philosophical discussion . . . . . . . . . . . . . . . . . . . . . 331.4.1 Adequacy . . . . . . . . . . . . . . . . . . . . . . . . . 341.4.2 Profoundness . . . . . . . . . . . . . . . . . . . . . . . 39

2 Concepts, meaning and coordination 412.1 From representations to concepts . . . . . . . . . . . . . . . . 412.2 World-knowledge and coordination . . . . . . . . . . . . . . . 422.3 Representations abstracted away . . . . . . . . . . . . . . . . 502.4 Concepts, representations and notions . . . . . . . . . . . . . . 522.5 Tests introduced . . . . . . . . . . . . . . . . . . . . . . . . . 562.6 Concepts as tests . . . . . . . . . . . . . . . . . . . . . . . . . 592.7 Tests and language . . . . . . . . . . . . . . . . . . . . . . . . 63

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3 Proper names and reference 673.1 The mystery of proper names . . . . . . . . . . . . . . . . . . 673.2 Cognitive world-maps . . . . . . . . . . . . . . . . . . . . . . . 683.3 The meaning of proper names . . . . . . . . . . . . . . . . . . 703.4 The categorial content of names . . . . . . . . . . . . . . . . . 793.5 The solution to philosophical puzzles . . . . . . . . . . . . . . 81

4 Modalities and the cognitive structure 854.1 The importance of cognitive structure . . . . . . . . . . . . . . 854.2 Modalities and the philosophical tradition . . . . . . . . . . . 874.3 Analyzing possibility and necessity . . . . . . . . . . . . . . . 884.4 Types of modal constraints . . . . . . . . . . . . . . . . . . . . 954.5 Modal semantics and truth conditions . . . . . . . . . . . . . . 994.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5 Mathematical objects and proofs 1055.1 The nature of mathematical objects . . . . . . . . . . . . . . . 1055.2 Infinity and the grounding of axioms . . . . . . . . . . . . . . 1075.3 Proofs and proof theory . . . . . . . . . . . . . . . . . . . . . 1125.4 Mathematical concepts . . . . . . . . . . . . . . . . . . . . . . 1165.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6 The social aspects of language 1236.1 Sapir-Whorf’s hypothesis and the linguistic reality . . . . . . . 1236.2 Linguistic influences on cognition . . . . . . . . . . . . . . . . 1256.3 Language and attitudes . . . . . . . . . . . . . . . . . . . . . . 1276.4 Analyzing linguistic influence . . . . . . . . . . . . . . . . . . 1306.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Ending notes and further research 139

Introduction

‘Yes, all his horses and all his men,’ Humpty Dumpty went on. ‘They’d pickme up again in a minute, THEY would! However, this conversation is going on alittle too fast: let’s go back to the last remark but one.’

‘I’m afraid I can’t quite remember it,’ Alice said very politely. ‘In that casewe start fresh,’ said Humpty Dumpty, ‘and it’s my turn to choose a subject.’

Lewis Carroll, Through the Looking Glass

Chasing a theory of meaning is a difficult task indeed. Many philosophers havespent their lives developing such a theory; others have spent their lives arguingthat such a task is, in fact, impossible. Other half spent their life on the former andthe second half on the latter. Therefore, proposing a new approach to a theory ofmeaning seems a daunting task - especially considering over 100 years of advancesin this very topic. The proper approach to this would probably be going down oneof the many beaten paths. This is certainly a safe approach - even if you do notget far, you are still in good company. However, this does not seem to be the rightapproach.

Most of the paths in the study of meaning have been started quite some timeago - back in the days of the early XXth century, when logic was at its triumphantstage of development. Psychology - not so much. While the logical universe seemedto unveil all its secrets at the time, the land of the human psyche remained wrappedin shrouds of mystery, with the works of Freud on one hand and behaviorists onthe other only setting the foundations for what would become one of the mostrapidly developing sciences of the late XXth century.

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For a long time, we knew little to none of language acquisition. All we hadwas thought experiments - again, multiple philosophers used those to feed theirtheories of language and even to speak of language acquisition, but we had noempirical data to verify those theories. It was the emergence of developmentalpsychology as an important branch of psychological research that finally broughtus new data.

As was mentioned before, creating a theory of meaning is a daunting task.Therefore, this is not a task we will perform in this work. Rather, we will try tocreate the foundations for such a theory. We will develop some definitions, proposea conceptual framework, then try to analyze some philosophically interesting caseswith the use of those tools. The main goal in mind is to propose a frameworkthat, while internally consistent and applicable to multiple known philosophicalconundrums, is also consistent with the psychological data we have.

We will start by attempting to properly define, formalize and discuss the con-cept of representations. The term has been used in the literature in manycontexts, but we will use it in the narrow sense usually connected with mentalor cognitive representations. Around this term we will then attempt to build aframework for talking about language. We will not treat language as an abstractformal system in its own right, but rather as an abstraction of our cognitive capa-bility to represent reality and communicate those representations to others, as wellas a means to coordinate our actions in the world. Then, we will propose a wayto talk about concepts in language and about meanings as the contents of thoseconcepts. Finally, we will attempt to examine a couple of interesting philosophicalproblems connected to language and see whether our approach can shed some newlight on the debates.

The methodology used in our work is pragmatically-oriented conceptual anal-ysis. The latter is the cornerstone methodology of analytic philosophy, the formerpoints at its American flavor. The term conceptual analysis is slight misnomerhere, since, while we will start with the analysis of certain concepts, most im-portantly the concepts of meaning, language and representation, we will need toperform quite a bit of synthesis to form a conceptual framework that encompassesthose terms. The pragmatically-oriented part, on the other hand, is a key fac-

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tor here - our main aim will be to provide a functional and consistent conceptualframework regarding meaning, not to capture the essence of philosophical issues,some of them millennia old. The quest for a good conceptual framework involvingmeaning is demanding enough as it is.

The claim that we will want to defend is that adding a cognitive level ofdescription provides the means for properly analyzing the semanticsof natural language - meaning that we will adopt a psychologist attitude tosome degree and argue that the mental states of language users (or, at least,some assumptions about the structure of those mental states) are in fact neededto explain various semantic aspects of language. To that end, we will explorethe problems of proper names (together with a few philosophical puzzles thatare connected with the topic), trying to show that it is possible to have both acausal and informative theory of names. Next, we will attempt to provide a properanalysis of modal terms, giving a unified semantics for various cases of necessityand providing credible truth conditions for natural-language uses of modalitieswhile at the same time not endorsing a possible-worlds ontology. Afterwards wewill discuss the status of mathematical objects, showing the possibility of unifyingmathematical and non-mathematical induction and explaining the groundednessissue with axioms. Finally, we will talk about the social aspects of language,showing that with a proper framework, they can be analyzed in a bottom-upmanner, without the need to adopt a holistic theory of the social world. We willargue that in all those areas, having a cognitive level allows us to propose botha good theory, and that the very absence of this cognitive level is likely to havecaused the difficulty in solving various problems earlier on.

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Chapter 1

Cognitive representations

1.1 Defining cognitive representations

The concept of cognitive representation is quite troubling to both philoso-phers and psychologists alike. First of all, two terms for representations are used:psychologists (especially of the cognitive variety) prefer the term “cognitive rep-resentation”, while philosophers of mind prefer to use “mental representation”. Itis debatable whether the two terms are actually coreferring since, as I will try toshow, there is little clarity as to what the meanings of the respective terms actuallyare.

The term “cognitive representation” in psychological literature appears as earlyas the works of Piaget [Pia51]. For Piaget, who is considered the founding figure ofmodern developmental psychology, cognitive representations were a stepping stonein the process of the child’s social development - they followed sensory-motor as-similation during the formation of more complex mental entities. In this sense,out of the two components of the phrase “cognitive representations”, more empha-sis was placed on the former part - the function of representation was consideredprimitive and the consecutive, more complex forms of representing were studied.

Indeed, within contemporary developmental psychology the trend of focus-ing on the various types of representation - rather than representation itself - isquite strong. An explicit definition of representations is rarely given; for example,

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10 CHAPTER 1. COGNITIVE REPRESENTATIONS

[Mar07], which deals largely with representations, states only that “a representa-tion is an information state within the brain of the organism that contributes toadaptive behavior within a given environment”. Instead of defining representationsfurther, the work focuses on different aspects of representation. For enactivism andembodied cognition, representation is strongly tied to action, much in the spirit ofPiaget’s works.

On the other hand, the nativist tradition within psychology treats representa-tions as the building block for a higher order theory of concepts [Car09]. What iscommon to the different psychological traditions is that representation is treated asa basic concept - it is left undefined and used as a component in the theory. Again,Carey discusses various types of representations and representational systems, butthe ground notion is left undefined.

One cannot even refer to a textbook approach to representations: in fact, astudy of a cognitive psychology textbook ([Mar01]) shows the same methodologicalproblem, with mentions of representations in fragments about perception, mem-ory, imagination and concepts, a typology of representations and even the role ofvarious theories of representation in the formation of concepts, but no definitionof representations per se.

It might not be very surprising that psychological literature does not pro-vide a definition of representations: after all, one might consider providing properdefinitions (especially of the ontological nature) to be the work of philosophers.However, the philosophical literature is also not very helpful here, for a multitudeof other reasons.

First of all, it is hard to determine the relation between “cognitive representa-tions” as understood by psychologists and “mental representations” as understoodby philosophers. Psychologists use the term to with relation to human cogni-tive processes, both of the linguistic and pre-linguistic category. On the otherhand, the use of the term “mental representations” by philosophers is largely in-fluenced by the Representational Theory of Mind, which, despite what the namesuggests, is largely an abstract theory of cognitive systems, owing more to formaltheories of language and the attempt to construct a theory of arbitrary systemsimplementing those formal languages than to low-level psychological theories of

1.1. DEFINING COGNITIVE REPRESENTATIONS 11

cognition. What might be symptomatic of the problem: among the multitudeof bibliographical entries listed on the Stanford Encyclopedia of Philosophy un-der “mental representation” ([Pit08]), there is not one that refers to Piaget. Thephilosophical discussion has strong ties to philosophy of language and to texts onthe philosophy-psychology interface, but has little to do with psychology itself.

The above might be a moot point if we can somehow use the philosophical dis-cussion as a useful source of definitions, but unfortunately that proves not to be thecase. For one, the pragmatic goals are different: we want to use representations ina theory of language as tied to overall cognition, while for example Fodor ([Fod75],[Fod08]) wants to create an abstract theory of a cognitive system that implementslanguage as understood traditionally within philosophy of language. To underlinethe distinction - we want to change the philosophical picture of how language looksbecause we believe both that the picture faces serious internal difficulties and thatpsychological data undermines the picture from within, while Fodor and other pro-ponents of the computational theory of mind want to construct an abstract modelthat saves the traditional, structured view of language shared by formal logic andanalytic philosophy.

This statement might be surprising given the apparent strong ties of the com-putational theory of mind with artificial intelligence - our objections might explainwhy we don’t want to use existing formal approaches to symbol processing suchas [New80], since they precisely lack the features we require (a connection to awider cognitive framework), but they seem to fall short when faced with recentadvances in the study of robotics. For example, Clark and Grush in [Cla99] seemto provide exactly what we want - a theory of representation within the frameworkof robotics, i.e. within the context of an acting individual. Why do we not wantto use such an approach?

The answer is as follows: those approaches are still too narrow for our needs.In this respect, Newell’s symbolic processing system from 1980 and the recent workby Clark and Grush show one quality of artificial intelligence - it still simulates onlysmall parts of human cognition and this scarcity of parts makes it insufficient forour needs. This is not an attempt to denigrate robotics in any way - the attemptto build an entity that possesses at least a fraction of the capabilities a human has


is praiseworthy. Still, a theory of how such an entity functions is insufficient toexplain how a human functions, given our claims (which will be further elaboratedin the text) that language is largely obtained via social coordination mechanisms.Unless we create socially functioning robots (and a good formal framework ofthem), the study of robotics or the study of mechanisms that abstract from thevarious functions robots are not intended to handle will be simply insufficient forour needs.

This does not mean that we will not borrow various aspects both from thepsychological and the philosophical discussions about representations. However,instead of compiling a synthetic definition from the uses of the term in philosophicaland psychological literature (which might be both beyond the scope of this workand not very useful for my endeavors), we will instead try to construct a plausibledefinition for our needs (as stated in the introductory chapter, the plausability ofthe definition will be judged based on its usefulness, as per pragmatist standards[Jam02]). To do that, we have to first decide on some background assumptions,which will be fundamental to our further theory. Those assumptions are:

• All humans possess a certain apparatus for processing input from the out-side world - moreover, we can assume that at least some functions of thatapparatus are shared among humankind.

• There is a common outside world that all of us process and that can be usedto help define the contents of at least some of our mental structures.

• All humans possess a certain apparatus for processing internal input, bothfor probing their own mental states and processing internal feedback suchas emotions. Unless proven otherwise, we will not assume that we are givendirect access to our mental states, i.e. we will assume that introspection isas error-prone and indirect as perception.

• We are able to discern patters in both external and internal input with acertain regularity and structures are formed in our brain (and thus, in ourmind, which is a functional abstraction of our brain) that are responsible forthis discerning process.

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• Our cognitive structures can be complex, that is, built upon simpler struc-tures. This implies that there must also be basic cognitive structures, al-though producing their list is a job for cognitive and developmental psychol-ogy.

The first two assumptions are very natural - the first one being a methodologi-cal assumption for the possibility of psychology as a science in general, the second- a general anti-skeptical assumption of a pragmatic nature. It is worth mentioninghere that we are very well aware of the fact that skeptical arguments are far frombeing considered dismissed in philosophical literature (see eg. [Ung75]), however,the discussion of skeptical and anti-skeptical arguments is beyond the scope ofthis work. Therefore, we do indeed make the second assumption as a full-fledgedassumption rather than an established fact.

The third assumption is certainly more problematic. Certainly, there are manyphilosophical positions that defend introspection granting direct access to our men-tal states ([Sch10]). On the other hand, there is a growing amount of data, bothempirical and theoretical, suggesting that such an approach to introspection isimplausible (for one discussion of the objections, see [Sch08]). We believe that, inlight of all the evidence, assuming that introspection is subject to similar criteriaas our extraspective cognition is the rational assumption to make.

The fourth assumption consists of accounting for the existence of an intuitivemechanism of induction as observed by Hume ([Hum48]), coupled with a very weakassumption about the inner-workings of our brain.

Finally, the fifth assumption is again problematic, as, under a certain under-standing, it seems to imply a nativist approach to human cognition. While this isnot necessarily a bad thing, as the nativist approach to developmental data canresult in some interesting theories about cognition (see eg. [Car09], [Blo04]), wedo not want this work to be considered as taking a stance in the nativism debate.Therefore, we will adopt an understanding of this assumption which does not im-ply that the basic cognitive structures are inborn - instead, we will settle on aweaker assumption that there are certain basic cognitive structures common to all(or almost all) humans, regardless of whether the source of this regularity lies ingenetic or developmental factors.


The above-mentioned list of features allows us to produce the following pre-liminary definition of cognitive representations:

A cognitive representation is a structure in our mind which is responsible forconsistently recognizing a pattern in our internal or external input.

Note that this definition cannot be deemed satisfactory from a theoreticalperspective - there are too many terms here that have to be considered as basic,among them “structure”, “consistently recognizing” and “pattern”. Also, withdefinition, the term “representation” is kind of a misnomer, as the definition itselfdoes not entail that a representation actually represents anything. Indeed, if weimagine a “brain-in-a-vat” type situation ([Put81]), where our brain is connectedto machinery that simulates internal and external inputs, we will form cognitiverepresentations but not of any external or internal world (although one might arguethat we will then have cognitive representations of algorithms which the machineryuses to feed us the data). Further in this chapter, we will attempt to provide amore refined, proper definition of representations.

1.2 Sidenote on formalizationThroughout the text, I will be attempting to formalize the different notions thatI introduce. During the formalization, I will be making a few assumptions, whichI will now enumerate and justify.

First of all, I do not assume any specific formal system in which the formal-izations are made (which might be considered as a shortcut for saying I actuallyassume an arbitrary-order logic). This might be a surprising claim for many whoare acquainted with logical literature, but this is actually a common practice withinmathematical texts. The reason for this decision is that restricting oneself to aspecific framework limits the expressivity and requires one to focus on details ofvarious constructions more than on the formalization itself, while the properties ofthe framework are important mainly when dealing with metamathematical prop-erties such as the existence or length of proofs or with properties of infinite objects,which we will usually not do here. Since I do not use the formalisms to talk about

1.2. SIDENOTE ON FORMALIZATION 15

infinities, nor do I rely on any types of foundational arguments (of whom the mostcommon are probably the various diagonal-type arguments involving computabil-ity or cardinality), I do not feel the need to be very rigid in establishing a logicalsystem.

I am fully aware of the objections raised by various philosophers (most notablyQuine [Qui70]) to the effect that logical systems higher than first order logic areinadequate as tools for describing the world. However, since my text is neitherontological nor logical in nature, responding to those claims is beyond the scopeof my study and the only answer I can offer in limited space is that Quine’scriticism is far from being universally considered as valid and there are in factserious ontological projects (one example being that of Edward Zalta [Zal97]) whichfreely utilize higher order logic.

Now we will present some preferences wrt. to the formalizations used in thisstudy, which are mostly of a purely aesthetic nature. I usually assume that Iwork within a typed environment, i.e. an environment in which each object has adistinct type associated with it. I assume that higher order function types can befreely constructed without any restrictions. Also, we will sometimes use lambdanotation, with the typical semantics ([Bar85]) for beta reductions being:

(λx.M)N →β M [x := N ]

for example:

(λx.x+ 6)5→β (x+ 6)[x := 5] = 5 + 6

We will use lambda notation simply as a way of realizing functional abstraction- we will not rely on computational properties of lambda calculus, notably, we willtry to avoid composing two lambda abstractions with each other without firstdemonstrating that this does not lead to an infinite computation.


1.3 Formalizing representationsFirst, a question that might be considered important: why start with formaliza-tion? Our answer is that formalization is an important dialectic process - whenone formalizes a concept, the concept’s bounds become clear and thus new areasfor analysis open which might be omitted when using a purely verbal description.To put it in other words, formalizations are quite unforgiving - they don’t give theauthor the luxury of waving away certain problems with verbal tricks.

Therefore, this section deals not only with formalizing representations, butalso with problems arising during the conceptual analysis of the concept of rep-resentation which the formalization outlines. As such, the chapter needs not beinterpreted as containing definitions only - in fact, much of the contents of thechapter are devoted to building a toolset to be utilized in further research, notnecessarily within the scope of this study.

To formalize representations, we first have to decide on what is the most im-portant aspect of our formalization. The apparent choice seems to be betweenanalyzing representations in an internal or an external manner. An internal for-malization would focus on the features of the representations themselves and theirinternal structure, while an external formalization would describe the role of repre-sentations in a wider system. Since, as we mentioned before, this is a philosophicaltext, we do not want to study the internal structure of representations, therefore,the external view must be preferred. More importantly, we want to analyze repre-sentations as involved in language acquisition and language use - we are interestedin the functionality of representations. Therefore, with our formalizations wewill focus on the functional side of representations, namely - what inputs do theyprocess and how the processing of those inputs affects our cognitive states.

We will borrow a page from the book of a semantic approach to natural lan-guage called dynamic semantics ([Gro91]), which in turn borrows heavily from thesemantics of programming languages. In those semantic approaches, the basic the-oretical object is not a static entity but instead a transformation: something thatmodifies a given state into another state (for a textbook approach to denotationalsemantics, see eg. [Gor87]).

To put it formally, if we have a domain of states S, transformations are func-

1.3. FORMALIZING REPRESENTATIONS 17

tions S → S. If we want to mimic the behavior of a single, isolated statement, weapply our transformation to a given initial state.

This approach has been used for analyzing natural language semantics becauseit provides an elegant way to account for side effects of language - for example,using this approach, one can model the effect that consequent sentences within adiscourse have on the context that is set within that discourse (this is especiallyimportant for fictional discourse). The possibilities are not limited to modellingintralinguistic phenomena, however - one can also model performative functions oflanguage in this way. For example, imagine a negotiation session. For the purposeof modelling this session, the truth-conditional semantics of the sentences utteredmight as well be irrelevant - what matters is how the steps modify the negotiatingcondition. We will now present a token “negotiation semantics” to show how thistype of framework functions.

First, let us define negotiations states (I will use the standard computer sciencenotation for enumerating possible members of an algebraic domain, with variousconstructor types separated by a vertical bar, constructor functions or constantsin lowercase and variables in uppercase):

State ::= fail | s1(Num) + s2(Num) | succ(Num)

where Num is a variable from the domain of numbers in which the offer ispresented (which represent for example dollars). In this notation, the negotiationis either in a failed state, in a state where the two sides stand on their offers, orin a success state where the two sides negotiated a common amount.

Now, for the negotiation steps. A single step is defined as follows:

Step ::= Side [ Proposal Response ]Side ::= s1 s2

Proposal ::= ...

Response ::= ...

Since this is just a mock theory, we will not be providing a full list of proposalsand responses, we’ll focus on a few examples instead.


Let us assume for simplicity that the during the negotiation process, sides taketurns taking the initiative: first, side A makes their proposal and side B responds,then side B makes a proposal and side A responds and so on. So, a valid negotiatingprocess is a sequence of negotiation steps in which, for any two successive steps sand t, side(s) 6= side(t) (we assume that side(x) is a function that returns the sidefragment from a negotiation step structure) and that ends in step f = ? [finish ? ](we use ? to denote that the given part of a structure is irrelevant, this is justsyntactic sugar for (∃s ∈ Side)(∃r ∈ Response) f = s [ finish r ] ; also note thatin this fragment, we will sometimes use bold font to separate variables from othersyntactic objects, this is only a stylistic measure that is employed in this fragmentdue to the potential clarity issues with the syntax).

Now, let us start with a simple negotiation. Since, as we remember, the wholenegotiation is a process (a transformation of states), we need an initial state. Let’ssay the initial state is s1(100) + s2(200). Now, imagine the following negotiatingprocess:

s1 [ (yield 20) (concede10) ] ; s2 [ (yield 40) (accept) ] ; s1 [ finish accept ]

How do we interpret this? When we start, side A wants to settle for 100 dollars,while side B wants 200. As a first step, side A yields by 20, that is, moves theiroffer by 20 towards the opponent’s one (to make it 120). The other side respondsby conceding 10 from their offer (making it 190). Now, it’s the second side’s turnto make a proposal. Now, side B yields by 40 (making their offer 150), which sideA accepts as a compromise. Then, side A ends the negotiation.

This description seems a plausible interpretation of the formal notation pre-sented above, but no semantics has so far been proposed. To do this, we have todescribe how the given steps contribute to the final result. For the simplicity ofnotation, we adopt a convention where s2(y) + s1(x) is a state description equiv-alent to s1(x) + s2(y) and o(x) is a function that returns 1 for x = 2 and 2 forx = 1. Now, for example, the semantics of yield + concede is the following:


Jsi [ (yield x) (concede y) ] K(si(n) + so(i)(m)) ={

si(n+ x) + so(i)(m− y) if n < m

si(n− x) + so(i)(m+ y) otherwise

First, let us explain the notation, as it might be a bit confusing.The main functor is the J·K functor, which takes as an argument a certain

negotiation state together with the description of a negotiation step (thesubsequent actions of the two sides, enclosed in single square brackets) andreturns a result state. Therefore, the type of this functor can be recognizedas follows: J·K : (Side× (Step× Step))→ (State→ State)

Note that the semantic meaning function might be a partial function -for example, the semantics of a concede operation in a fail or success statemight not make sense at all.

Here, the sides negotiate gradually. However, some negotiations contain“take-it-or-leave-it” offers. For example, a modification of the previous ne-gotiation might look like this:

s1 [ (yield 20) (concede10) ] ; s2 [ (threaten 0) (refuse) ] ; s1 [ finish accept ]

How do the semantics of threaten look like?

Jsi [ (threaten x) r ] K(si(n) + so(i)(m)) =

succ(n+ x) if n < m ∧ r = acceptsucc(n− x) if n ≥ m ∧ r = acceptfail otherwise

(here, r is a variable containing the reaction of the other side: eitheraccept or refuse)

Note that negotiations end with a finish statement and not wheneverthe result is success or failure. This is due to the fact that, as anyone whohas seen the progress of real negotiations knows, failure within a negotiatingprocess need not be final - there might be reset statements:


Jsi [ (reset x) (reset y) ] K(fail) = si(x) + so(i)(y)

How do we add side effects to this story? For example, suppose thatduring the negotiations, sides issue press releases (the 2011 NBA collectivebargaining agreement negotiation process was a good example of such a ne-gotiating process [Ash11]). How do we factor this into our semantics? Theanswer is: we need a broader state domain plus selective state transitions -operations that modify only parts of the state. An extreme case of this isthat every process semantics can be considered a semantics for “world trans-formation”, where various operations transform fragments of the world insome way. In fact, while this case seems extreme, it might be the only viablesemantics for natural language with performative effects added (how wouldwe delimit the states of a semantics that could analyze a sentence “I declarewar on Russia” as uttered by the rightful President of the United States?)

While this semantics seems to handle everything well, one thing that itstruggles with is nonlinearity. Suppose that, during the negotiation process,one of the sides (say, side A) secretly obtains a legal document that entitlesit, no matter what, to force an agreement on certain terms (say, the amountX). Now, side A might want to introduce a step as follows: if, during furthernegotiations, the end result is either a failure or a concession of more than 20from our current proposal, use the legal leverage to enforce a deal of succ(X).How do we factor that into our semantics?

The problem is, there is no way we can know how the future negotiationswill progress. What we could do is pass a marker in the state saying that suchlegal leverage is possessed by side A and force each instruction to respect it (inthis case, it would probably be sufficient to amend the semantics of finish).However, this way the semantics for all instructions that might appear mustbe aware of this special marker and, in fact, of all such special markersthat might occur. This would be a very cumbersome way to construct suchsemantics.


Fortunately, there is a solution, again taken from programming languagesemantics and adopted for natural language. That solution is continuationsemantics ([Bar02]). Continuation semantics adopts the dialectic strategy oftransforming the abovementioned problem into the solution: under this ap-proach, instructions are no longer simple transformations, they are functionsthat take as arguments transformation continuations (which are themselvesfunctions from transformations to transformations) and yield a transforma-tion as a result. Why the name “continuation”? If we imagine a process thathas multiple steps, one can divide it into the current step and the remainder.Now, this remainder can be considered a “transformation of transformations”- a method of getting from the initial transformation to the final transfor-mation. In other words, this remainder describes how the process continues.Now, the idea of continuation semantics is to reify this very continuationand pass it as an argument. The only caveat is that now we have to waituntil the process ends to compute the semantics for the entire process - butremember that continuation semantics is supposed to remedy nonlinearitysituations where we have to wait for the process to end anyway.

So, how would our semantics for this leverage operation look like?

Jsi [ (leverage x) (pass) ] K(κ)(s) ={

succ(x) if badi(κ(s), s)κ(s) otherwise

where s = si(n) + so(i)(m), badi(s, t) is a predicate saying that result tis bad compared to result s for side i and κ is the continuation variable.Hence, the type of the J·K functor changes - it is now J·K : (Side × (Step ×Step)) → (Cont → (State → State)), where Cont is the state continuationtype State→ State.

This semantics does exactly what we needed - if the further negotiations(as read from κ) do not yield an expected result, the end result of the nego-tiations is the leveraged result - otherwise, the end result is the continuationapplied to s (in other words, the uninterrupted result of the further negoti-ation process, starting from state s).


How do we modify other statements to support continuations? The sec-ond branch of the semantics for leverage showcases the process: we applythe continuation to the non-continuized semantics. So, for example, one ofour previous rules would be rewritten as follows:

Jsi [ (reset x) (reset y) ] K(κ)(fail) = κ(si(x) + so(i)(y))

How do we compute the final semantics? We apply the identity con-tinuation to the end result. This way, we get the same semantics as forthe non-continuized version, apart from the non-linear side effects that wewanted. Imagine we have a negotiating process with only one step and failureas the initial state. What we would get is the following:

Js1 [ (reset 100) (reset 200) ] K (λx.x)(fail) =(λx.x)(s1(100) + so(1)(200)) = (s1(100) + s2(200))

Now that we have roughly sketched the outlines of denotational and con-tinuational semantics, we still have to answer the question: how does it applyto our analysis of mental representations? Does it, in fact, apply to it at all?To answer that question, two key observations are needed:

• cognitive processing has side effects - as well as recognizing some frag-ments of the world, it can alter the contents of our memory, change ouremotional states or focus our attention on certain aspects of the world

• there are cases where cognitive processing is non-linear - for example,we might want to perform some simulations internally; also, emotionalinput and/or attention shifts might influence further cognitive process-ing

Those observations, together with our previously established focus onfunctional, rather than structural, aspects of representations, tend to support


a view of the semantics that (a) deals with state transformations and (b) iscontinuational.

However, now we have to deal with a categorial problem. With our “ne-gotiation semantics”, semantics was provided for a negotiating step. Undera typical ontological categorization, a step is an event. Meanwhile, a rep-resentation is an object, not an event. How do we provide semantics forrepresentations if we just established that we want to construct a dynamic,event-based semantics?

The answer might seem surprising at first - we don’t. At least, not di-rectly. Recall that we defined a representation as “a structure in our mindwhich is responsible for consistently recognizing a pattern in our internal orexternal input”. What this definition leaves open is what actions are per-formed once the recognition is completed. For example, recognizing objectscan be part of a process of imagination, where we simulate the perceptualdata normally associated with the given representation. Certainly, if weimagine a mountain, our representation of a mountain is used in the process,but it’s a different use than the one where, upon seeing Mount Everest, weutter the sentence “This is a huge mountain!”. We use the representation inyet another way when we try to reason about the possibility of mountainsbeing made of glass.

Still, there is a common element present in my account of all the processesabove - the representation being used. However, one might argue that I amnow operating within a vicious circle here - I postulated the formalizationof representations to help make the notion precise, but now I am claimingthat I will instead provide semantics for cognitive processing and that repre-sentations are invariant components of some cognitive processes. It is hardto refute that claim without providing the semantics itself, which I will at-tempt in a moment, but before that can be started, one more backgroundassumption has to be added here: the assumption of the unity of cognitiveprocesses.


An explanation of the phrase is in order. In the account of the variouscognitive processes involving mountains two paragraphs above, I claimedthat there is a common element within our mind that is involved in thoseprocesses. It isn’t a priori certain that this is indeed the case. After all, atotally different part of the brain might be involved in touching parts of amountain, a completely different one might be used in imagining mountains,and yet a third part might be involved in reasoning about mountains. Onecould argue that the fact that those are different brain fragments doesn’tentail that they are different mind fragments, but to decide the other waywould be question-begging. The only thing that we have to connect thosethree is the linguistic label “mountain” - but that only works when we havelinguistic representations, not with cognitive representations in general. Infact, the claim that language is required to form coherent representations isnot that preposterous at all.

However, empirical evidence suggests that we form coherent representa-tions long before we attach any linguistic labels to them (in fact, our pre-linguistic understanding of semantics might be the key to explaining ouracquisition of syntax, see [Blo99]), thus supporting the unity assumption.Therefore, we will in fact assume that there is a single object in our mindresponsible for all the processes involving mountains. To add a philosophicalrationale to the empirical one - even if there wasn’t any single part of thebrain responsible for the processing (and in fact, it does seem that cognitiveprocessing is scattered among multiple parts of the brain for even the simplestcognitive processes), one can still postulate that, when considering the mind,which is a functional abstraction of the brain, we can still unify those partsunder a single entity (we would only have to tackle the claim that there areno cognitive representations without language, as without the unifying label,if there is no functional connection between the different processes involvingmountains, there is no way to tie them within a single representation).


1.3.1 The effect of representing and the paradox of theelusive representation

In order to formalize representations in the manner described above, wemust first answer an important question: what is the effect of the process ofrepresenting?

In the token semantics for negotiations above, the effect of a negotiat-ing process was simple to describe: either a failed negotiation, a successfulnegotiation, or both sides keeping their respective positions. On the otherhand, if, instead of representations by themselves, we consider the process ofrepresenting, what is the effect of such a process?

Note that the question we are asking is the theoretical one: how do wedelineate a representation process?, not an empirical psychological one: whatmental actions are associated with representing?. Since the problem mightseem very abstract, here’s a more down-to-earth example.

Suppose that John sees a car (imagine that John is standing on a parkinglot in front of a red Cadillac). We want to hypothesize that a representationof a car was used somewhere along the way, or, to put it in other terms, thata car was represented at some point during John’s stay at the parking lot.However, what do we actually mean by the car being represented?

John supposedly has many things on his mind during his stay on theparking lot. He might be thinking about what plans he has for the evening,how much fuel does he need for his own car, he might be thinking aboutwhat to write for his PhD thesis, he might be emotionally distraught by abad day at work - all of those contribute to his mental processes for thegiven period. Moreover, him seeing the Cadillac might be accompanied bydifferent actions: he might think how he himself would like to own a Cadillac,he might say: “What a classy car!”, he might contemplate the global warningproblems caused by gas-guzzlers such as said Cadillac, or he might ignore theCadillac whatsoever, passing it on the way to his own car. The question is:can we single out any pattern within the different cases outlined above that


can be considered representing the Cadillac?

While we might be tempted to answer the question in a positive manner,we quickly run into a problem of being able to say what that pattern is. Afterall, even in an introspective account we can’t really single out an action ofrepresenting an object. The closest that we might be to such a descriptionis when we say we’re thinking about an object, but clearly there are instanceswhere we aren’t thinking about an object and still representing it (mostlybecause “thinking about an object” is a partially exclusive action, so whenwe’re thinking about John loving to climb mountains, we’re certainly thinkingabout John, but probably not thinking about a mountain, but we’re mostlikely using a representation of a mountain).

Here we face one of the key paradoxes of representations - there is no dis-tinct action of representing. It is therefore no surprise that some accounts ofhuman cognition, especially those that focus on distributed cognition, wantto do away with representations at all. However, we want to keep representa-tions as a useful abstraction - not because of the action of representing, butbecause of the existence of objects in the world and our ability to distinguishthose objects reliably.

Let me once more recapitulate the methodological paradox (which we willcall the paradox of the elusive representation): the most useful accountof representations, as with most human cognitive activities, is a transforma-tional, state-based one, but the objects themselves (the representations) arebest singled out due to objective, static criteria. Therefore, it is hard to bothargue for representations and provide an account that makes their usefulnessclear in one clean sweep, which might be the reason why a good account ofrepresentations has been eluding philosophers and psychologists alike.

To summarize: there is no action of representing, but there are repre-sentations and those representations are best described in an action-basedsemantics, but it’s not a semantics of representing - it’s a semantics of cog-nition in which representations take part. Later on, we will try to provide a


definition of representations within that semantics, but one has to rememberthat, due to the reasons mentioned above, it will not be an essential definition- more a criterion of what it means to possess a given representation.

1.3.2 Representations formalized

Now that we’re finished with the preliminaries, the final task awaits - for-malizing representations. We will start with the domain of states for thesemantics. What we will have here is cognitive states, which represent thestate the mind is in currently with respect to cognition. Since cognition isa very complex process, we will not be able to provide a complete descrip-tion of those states. Also, since there are aspects of our mental states whichare not normally viewed as cognitive, but can nevertheless influence cogni-tion (for example emotions, as was outlined in the philosophical literature asearly as Spinoza [Spi77]), we won’t be even able to restrict those states topurely cognitive notion (it seems impossible to even give a clear account ofwhat “purely cognitive notions” are). Therefore, we will settle for a sketchof the domain, filling it in as new data is provided. Our key restriction hereis consistency - although the states are bound to be complex, we want ourdescriptions to be consistent, which by itself is no small task and providing aconsistent description of mental or cognitive states seems to be a reasonablegoal on the way to providing a complete description.

Since we’ve already established we’re not providing a complete accountof mental states, let us start with an inclusive approach - what do we want inour states in the first place? Certainly one thing that is needed is memory -after all, in traditional psychological approaches, representations are a part ofmemory. Speaking more abstractly, memory is our personal storage for morepermanent data (psychologists usually distinguish long-term and short-termmemory and only the first one is actually responsible for permanent storage,but for our description here, we will not worry about this distinction toomuch). Another important thing is attention - whatever our mental state


is, we want it to be intentional, in the sense that, unless we are sleeping, weare conscious and focused on at least one specific topic. Next, we want plansor goals and attitudes, which are the drives for all our activity, includingpurely cognitive activity. Finally, we want sensory input, which is thedata our cognition operates upon. In our description, the term “sensoryinput” will be used very broadly, as we will also include data that comesfrom introspection (considered “internal” sensory input) and simulation(all sorts of internal processes that we evoke to provide us with first-ordersensory data, for example such processes as imagination or empathy).

The other big question is how do we describe the respective elements ofthe state. For memory, the answer seems relatively simple - arbitrarily typedobjects hidden behind labels. Note that we do not postulate here that theactual mechanism of memory recall works on labels of any sort. The labelsused here are purely metalevel ones - the label itself is simply an abstractionfor whatever memory recall mechanism is used to select the proper memoryfragment, a memory address, if we may use such a computational analogy(one important distinction here is that the labels are certainly not linguistic,even if they look so). For attention, we basically want an object (either anexternal one such as a mountain or an internal one such as an emotionalstate). For plans or goals, we want a certain state-of-the-world (again, theexternal world or an internal world, such as 50,000 USD in my possession orme being happy), finally, for sensory input we want raw data (by raw datawe mean something that is in “binary”, unstructured format, similar to howa computer processes raw chunks of bytes), however, due to the difficultyin describing raw data we will sometimes be describing it in a structuredform (however, one has to remember that, similar to the labels on memory,those structural descriptions are metalevel in nature and do not intend todescribe any sort of structure on the object level). The objects mentionedhere are all easily typable on the formal level, short of memory, which hasto be represented either as an indexed product or a (partial) function from


labels to variably typed objects (in other words, a dependently typed func-tion). Also, our description of memory inside various states can, for practicalreasons, never be full, so instead we will only provide relevant fragments ofmemory in the semantics, noting changes and restrictions on the full mem-ory whenever applicable (for example, to describe how someone corrects anerroneous representation, one needs to assume that this person already hasin her memory a certain representation and that it’s different from the oneintended).

Finally, and this is certainly a philosophical decision that goes beyondpurely formal concerns, at least some representations have to be externalistin a sense that their existence depends on the existence of objects external tothe representation itself (and sometimes on the existence of objects externalto the mind itself, although that need not be the case for eg. representationsof emotions). Morever, that externalism will be represented in the formallayer, as some parts of the notation will reference objects within an assumedreal world (for the purposes of this text, we will use a rich “hunter-gathererontology”, assuming objects exist when they are needed for theoretical rea-son, although we will only apply this process to objects that are beyonddoubt physical in nature - chairs are in, justice is out).

After establishing all the abovementioned conditions, it’s time to moveon to the formalization (note that the State here is not the same State asthe one used in our token semantics in the previous section, it is the domainof mental states). We will not treat State as an algebraic type, instead, wewill describe its structure by the use of accessor functions.

First, a couple of symbol introductions for the accessors themselves:

• A : State → P (Object) is a functor that takes a state and returns itsattention focus (Object is a domain that captures objects of all types,similar to how the Object class behaves in object-oriented programminglanguages). We will here assume that attention is described externalis-tically, or that the objects returned are the actual objects being focused


on rather than parts of the mental state. This might lead to certainproblems (what is the attention focus of hallucinations?), but the as-sumption of an external (in the wide sense mentioned above in whichemotions are also “external”) world being the focus of our attentionis needed in order to meaningfully talk about representations withoutfalling into a vicious circle.

• M : State → Memory is a functor that takes a state and returns itsassociated memory (where Memory = Label→ Object)

• I : InputType→ State→ Input is a functor that takes an input typeand a state and returns its sensory input of the given type (auditory,visual etc.)

• G : State→ P (Goal) is a functor that takes a state and returns its setof goals and attitudes (P is the standard powerset functor).

Next, we want to decide how to formalize the respective components ofthe state. The Object domain is heterogenous, so we won’t be providinga single template for it, the same applies for Memory (as a domain thatconsists of function from Labels to Objects). The two domains that deservemore attention are Input and Goals.

As we stated before, we will consider Input to contain raw data (we arewell aware that the very question whether our perception yields raw data isa matter of philosophical and psychological debate, for a summary, see eg.[Sie11]; our account will not depend on any specific stance in this debateother than the very weak assumption that our perceptual data is not pre-structured linguistically, the adjective “raw” here has to be treated as simplymeaning “not processed”). Thus, we will treat it as an atomic type, althoughwe will describe inputs in a structured manner (the description here is, asin the label cases above, metalevel in nature). To mark the fact that inputsare considered raw, we will enclose the descriptions in angle brackets, so asample might look as follows:


Ivisual(s) = 〈a room with a wooden chair standing in the middle〉

As for the goals, the most reasonable way to write them down seems to bewith sentences expressing propositional attitudes: want, desire and the like.Therefore, a single goal will be a proposition containing such operators, andthe target of the G function will be a set of such propositions, for example,a set of materialist goals would be:

G(s) = {want(I have a red Ferrari), want(I have 200,000 USD in my bank account)}

Due to the sheer size of mental states, we will usually not describe thementirely, instead, we will use the abovementioned functors to set some neces-sary and sufficient conditions for those states.

Let us move to the topic of representations. What would be a sufficientcondition to say that somebody has a representation of a chair?

First of all, somebody must have a memory fragment that correspondsto the representation (or, one could even say, is the representation). Second,that fragment must be empirically adequate - they must be able to recog-nize chairs. Recognizing chairs is obviously a process - however, possessinga representation is a quality of states, not of processes. Nevertheless, we de-scribe the constraint within a process-oriented semantics, as justified abovein section 1.2.

Now, we also need a predicate that allows us to say that a certain object ofthe mental realm (in this case, a memory fragment) was involved in a certainmental process. We will use inv(p, s, o) for this, where p is the process, sis an input state and o is the object in question. The main premises forthe existence of this predicate are again non-trivial when we remember thatmental processes are basically state-transforming functions. The process canbe very long, but a function in the traditional set-theoretical sense is just a


set of ordered argument-value pairs, where the intermediate steps are lost.In order to make the inner workings of the predicate more explicit, we willnot be treating it as a simple predicate. Instead, we will use a more primitivenotion of “involve” for states (invS(s, o), where s is a state and o is the objectin question) as well as the notion of a process trace.

The trace of a process is a function from states to finite sequences (tuples)of primitive process - we assume that there is a subset of the domain of men-tal processes that consists of primitive process parts that cannot be decom-posed into more fine-grained steps. The function Tr(p, s) outputs a sequence〈p1...pn〉 such that p = p1 ◦ p2... ◦ pn, or p is a composition of the sequenceof primitive process parts (for example, a cognitive process of imagining amountain might involve process parts that recall the proper representationsfrom memory, create a mental image of a mountain, add properties to theperceived mental image, form judgements about the mountain and so on).Note that the trace is a function of the input state because we do not as-sume the homogeneity of processes irrelevant of the input - different processcompositions may be responsible for processing different input states.

A derivative helper notion that we will need is that of a state trace. Thestate trace is simply a “motion capture” of a process in progress on a certainstage. If, for an input state s and process p, Tr(p, s) has n elements, thenfor k ≤ n, StTr(p, s, k) is the result of applying the traced sequence up to ksteps (in other words, it’s (p1 ◦ ... ◦ pk)(s)).

Now, we are ready to explicitly define the inv predicate:

inv(p, o) =def ∀s∃k∃t(StTr(p, s, k) ∧ invS(s, o)

The definition states explicitly the intended semantics for inv - for aprocess to involve a certain object it has to be the case that for every inputstate the process has to handle, there is a decomposition (trace) of thatprocess that at some point, has a state that involves that object.

Note that we are keeping one thing implicit here - the quantification

1.4. PHILOSOPHICAL DISCUSSION 33

domain for s. Right now we tacitly assume that s ranges over all inputstates - we will revisit that in a moment. For now, let us also assume that wemight need a predicate invD, where D is the restriction of the quantificationdomain on states.

1.3.3 The definition of representations

Now, we are ready to state our definition of representations: a memoryfragment m is a representation of x if and only if, for all processesp, for the domain D of input states s such that x ∈ A(s), it is thecase that invD(s,m). In less formal terms: m is a representation of x if andonly if m is involved in all processes where in the input state s, attention isfocused on x.

Note that we might be tempted to provide an alternative definition ofrepresentations, one that is linguistically connected - m is a representation forx if n is a name for x and m will be recalled from memory whenever attentionis focused on a (spoken, written) phrase n. However, this definition would notallow us to state how we obtain representations in the first place. Moreover,it would not be possible to state the criteria for adequate and inadequate(or correct and erroneous) representations without a vicious circle. Like lateWittgenstein suggested ([Wit53], [Kri82]), correctness is a group matter -there is no correctness outside a society that enforces convention. So, onecan speak of a correct representation whenever it is a convention that nis a name for x and one’s memory fragment evoked by the name n is therepresentation for x, but for such a criterion to be meaningful, you have tohave a notion of representation that does not involve language.

1.4 Philosophical discussion

The definition provided in the section above might seem a bit arbitrary - afterall, why select this and not some other definition? One reason for picking


the definition above the linguistically loaded one has been already providedwhen discussing our preliminary definition, and in this section I will focuson the reasoning that will show the definition to be adequate for our varioustheoretical purposes.

1.4.1 Adequacy

First of all, the definition has to be adequate - it has to actually describe whatour common sense would label as representations. The problem with the def-inition, due to the general quantifier at the start, is that it’s an intersection-type construction. Intersection-type constructions are subject to emptinessissues - it might turn out there are simply no representations whatsoever ofcertain object types. Two things are done here to ensure this is not the case- one is a background assumption which we will now make clear, another isa contextual relaxation of the definition that has already been hinted in theprevious section.

The background assumption is that of regularity of mental processes. Inorder for us to postulate representations, we have to assume that it’s not thecase that a different portion of our mind is responsible for the same cognitivefunctions each time. If we assume said regularity (that is, that processes of asimilar type are handled by the same modules), we can also safely assume thatthere are representations - it is almost a deductive consequence. I say almostbecause it is impossible to provide an adequate explication of what it meansfor processes to be “of a similar type” without creating into a full-fledgedtypology of mental processes. For our needs, it’s sufficient to state a weakerclaim - that processes focusing attention of items of the same ontologicalkind are handled by the same parts of the mind each time.

The contextual relaxation involves the types of input states in which weare supposed to recognize objects. The definition of representations makesno claims about the difficulty of distinguishing an object within some sensedata. In fact, we have skimmed over the topic of the relation between sen-


sory input data (which is internalistic and not associated with any externalcontent) and atttention (which is externalistic and thus picks out objects inthe world). In doing so, we have implicitly assumed that we are capable ofperfect discrimination - whenever an object is present in the outside worldand we are in its vicinity, we can select it from its surroundings. This is ob-viously not the case - if we are in a completely dark room with the proverbialblack cat playing with a black ball of wool, we will not be able to tell the catapart from the rest of the room (at least not until we step on the cat’s tail).We have two ways out of this - either rely on attention only being able toselect objects we can reasonably discriminate (but this can lead to circularityproblems again, as it might be up to our representations to say which objectswe can discriminate and if we restrict our attention a priori, we can neverexplain how new, more fine-grained representations can be forged), or we canrelax the condition on the input states. Recall that the original definitionrestricted the domain of quantification to states in which the object repre-sented is present in the attention set; we can restrict that domain furtherby requiring that the object be present in the attention set and the sensoryinput in that state be sufficient to discriminate the object in question. De-pending on this, we can have adequate representations of different quality,where the quality is denoted by how little we want to restrict the domain ofstates (perfect representations obviously involve no restrictions).

One might also wonder why the definition restricts involvement of theobject in the attention focus of the input state, instead of all the states in thestate trace of the object. The reason for is the is the existence of parasiticmental processes and side effects. One example would be the process ofsomeone imagining what it would be like to have cars being named chairs.There are likely to be states during the execution of that process in whichour attention is being focused on a specific car, but the representation evokedduring that state is that of a chair (or vice versa). Various simulation stateswithin certain thought experiments are also a likely candidate. That’s why


the definition is restricted to input states - expanding the range of stateswe quantify over would possibly lead to the intersection construction beingempty, so we have to balance out precision with the risk of having overlyrestrictive criteria.

This restriction raises another issue - if we consider a process trace of p,every subsequence of that trace can also be considered a process ps. Withinthat process ps, the intermediate state of p is the input state. Now, ourrestriction on input states is meaningless - every state that appears duringthe execution of process p is an input state for some subprocess ps. To makethe restriction meaningful again, we have to force a restriction on processpart composition - not every composition of process parts leads to the for-mation of a process. This is also in line with the principle of heterogeneityof decomposition for processes described during the formal discussion - westated that different input states might yield different process traces, so itmight not make actual sense to consider all possible combinations of interme-diate process steps and input states for those intermediate steps. However,determining what the actual domain of processes consists of is a matter forthe respective sciences (psychology, neuroscience) and will not be furtherresearched here.

Another problem the definition faces is the problem of representations ofcomplex objects. How good is the representations for objects such as numbersor abstract entities such as justice, especially considering our restriction onthe ontology mentioned earlier? Unfortunately, there is no simple answer tothis question and it is impossible to provide a simple answer without delvingdeeply into the ontology of the objects in question. For example, representa-tions of numbers are perfectly valid for me because I consider numbers to betranscendental objects in the Kantian sense ([Kan81]), therefore somethingthat is and external object in the sense in which introspection yields access toexternal data (and in which Kantian intuitions are external objects). Speak-ing in non-Kantian terms, numbers are an introspectively observed aspect of


our method of reasoning which is accessed via introspective means. Justiceis more complicated - I would be inclined to say we have a notion of justice,but not a representation of it, with notions being mental constructs thathave structural as well as representational fragments (I use the term notionas opposed to concept, because the term concept has a linguistic connotationto it which I will further exploit - so unless otherwise noted, notions aresubjective, while concepts are intersubjective). When talking about complexentities later on, I will elaborate more on notions; for now, it will suffice tosay that when I talk about representations, I use the term to denote men-tal constructs which directly select base ontological entities. Of course, onecould have an ontology in which all objects are base (first-order) entities andthus everything, even justice has a representation, but I do not consider sucha solution plausible and will distinguish between representations and notions.

Finally, there is a purely logical, formal problem with the definition. Wehave said that m represents x, but what exactly is x in this formula? Surely,when we mean someone has a representation of a chair, we do not mean hehas a representation of a specific chair, but of every chair out there, or ofchairs in general. Therefore, the formula should be understood as “for everyobject x which is a P”, where P is implicit in the selection of x. Therefore,the full definition should read as follows: a memory fragment m is arepresentation of P if and only if, for all processes p, for the domainD of input states s such that there exists x ∈ A(s) such that P (x),it is the case that invD(s,m). However, we will use the original definition,keeping the general/individual name ambiguity implicit unless the need fordisambiguation arises.

We have talked about various problems with the definition, but one keyquestion remains to be asked - does the definition work for even simple ob-jects? That alone does not seem obvious. After all, imagine a scenario wherewe see a burning building and a car standing outside the building. We mightas well ignore the car and focus on the burning building and the represen-


tation of the car will therefore never be present in our mental processingof that state - but if that’s the case, it won’t fit the definition either. Thekey to overcoming the difficulty is the use of attention in the definition -we use attention in such a way that it is not possible for an object to be inthe attention span of a state, yet not be processed cognitively. The object,once it enters attention, must either be explicitly filtered out or processedin some way. Note that this is a theoretical construct which does not entailthat actual neurological mechanisms involving attention work in that way -it is for our convenience that we model attention in this specific way. Let usmake explicit our assumptions that govern the understanding of attention:

• attention is focused on certain objects singled out from the environ-ment, i.e. is externalistic (in the wide sense described previously)

• whenever we talk of an object being within the attention span of acertain input state, we assume that the object will have to be furtherprocessed. Thus, our understanding of attention is, in a sense, con-tinuational - it does not make sense to single out attention withoutknowledge of the processes that will further process the input state.This creates a problem with our understanding of processes as statetransformation functions - it might be hard to imagine how a processthat transforms a state with an empty attention set will look like. Thereare two solutions to this problem - we either agree that processes arepartial functions and some states do not make sense for certain processtypes, or we add explicit process parts such as “attention focusing” toour process model.

• attention in our model is a primitive concept, there is nothing elsewe can reduce it to. Therefore, it follows that we base our definitionof representations on the concept of attention, although it does notfollow that we consider representations an attention-based phenomenon(remember that our definition does not aim to explain the nature of


representations or provide its essence, it simply aims at offering propercriteria to single out representations).

• for our model to be plausible cognitively, attention has to precede rep-resentation on a temporal level. In fact, we assume that people are ableto focus attention on different objects before they form a representationof that object, i.e. attention focusing is more basic than representing.

Now that we have considered the possible controversies with the defini-tion, it remains to be seen whether the definition itself survives those con-troversies, i.e. does our definition of a representation of a chair single outthe representation of a chair. From our discussion, it follows that a mem-ory fragment is the representation of a chair if and only if, for all processeswith input states that focus their attention on a chair and provide sensorydata of sufficient quality to discriminate the chair, the memory fragment isinvolved in the processing of the input. Coupled with the assumption on theregularity of mental processes, this seems to indeed be a good way to singleout representations without resorting to linguistic means.

1.4.2 Profoundness

Analytic philosophy has often been described by critics as the chase after in-tricate and complex definitions that are nevertheless devoid of actual mean-ing. The definition of representations we came up with in this chapter willhave no significance if it cannot be employed for some theoretical purposesother than talking about representing chairs.

One important use of the definition, which has already been alluded to, isthe ability to talk about representations in relation to language without thelinguistic level being fundamental. Thus, we can actually provide a definitionof what it means for a representation to be correct: we will say that a(subject S’s) representation m of x is correct with respect to name n inlanguage L when n is the name of x in language L, n is used (by S) to evoke


m and m is a representation of x. By simple modification, we can show whatit means for a representation to be incorrect or mistaken: that happenswhen n is the name of x in language L, n is used (by S) to evoke m and m

is a representation of y such that y is not x (in the sense that x is a P , y isa Q and P 6= Q, even if P and Q are extensionally equivalent).

The above-mentioned definition is just a hint of what the intended useof representations in our theory is - they are supposed to provide a funda-mental level for a theory of meaning that allows to explain various linguisticphenomena on a more fine-grained level. While the level of detail and for-mal complexity presented in analyzing the definition of representations mightseem overly complicated, it is necessary in order to avoid various pitfalls thatsurround the subject matter. Circularity is one I have already addressed, an-other one is coarse-grainedness which makes the definition useless in certaintheoretical aspects (most parts of this text would be impossible to write if Iassumed at the start that representations were linguistically grounded).

Instead, what I aimed at was a definition of representations that was aslow-level as possible while still being philosophical in nature (i.e. not makingsignificant empirical presuppositions) and made all the assumptions behindthe definition explicit. In further chapters, I will build upon this definitionby using it to explain various linguistic phenomena and then I hope the needfor such a definition will become clear.

Chapter 2

Concepts, meaning andcoordination

2.1 From representations to concepts

In the previous chapter, the concept of representations was introduced anddiscussed. However, as the title of this work suggests, the focus of this textis not representations, but meanings - representations are just means to anend. It is therefore imperative to link our definition of representations to theintended theory of meaning.

Why did we start out theory with such a detailed discussion of repre-sentations in the first place? Unfortunately, it is not possible to give a fullanswer to that question without presenting various tidbits of the theory thatwill come later, so for now, a partial answer will have to suffice: the reasonfor this step is that a theory of meaning that omits the individual repre-sentational concept of language is unable to cope with various theoreticalproblems that have been presented over years of philosophical research.

The claim above can be considered as anti-Fregean, standing against onekey belief of the founder of modern logic and philosophy of language: anti-psychologism ([Fre84]). However, one must consider that belief in context.

41

42 CHAPTER 2. CONCEPTS, MEANING AND COORDINATION

Back in the days of Frege, psychology was at its infant stages and whenone thought of psychology, one meant the introspective accounts the likes ofwhich were presented by Mill ([Mil43]). Nowadays, psychology is an empiri-cal science, with an ever-improving methodology that allows us to recognizecertain truths about the inner workings of our mind (moreover, while psy-chological studies still largely rely on introspective data, qualitative tools areused to ensure that the data is as intersubjective as possible, eliminating thedependency on individual factors; also, the emergence of neurological studiesin the recent years has allowed us to correlate introspective data with objec-tive measures of brain activity). Meanwhile, the charge of “psychologism” isbeing thrown around nowadays in a similar manner as it was a century ago,often ignoring the fact that the surrounding conditions have largely changed.

Due to the reasons mentioned above, I will not respond to charges ofmy theory being “psychologist”, as I consider that a somewhat outdated ob-jection that might have been valid in Fregean times (although some philoso-phers at least considered the idea of linguistic terms evoking thoughts, see eg.[Ajd60]), but would need independent reasons to stay valid in contemporarydebates. However, that does not mean that there aren’t related objectionsthat I have to consider. One such objection is that language is, in fact, anobjective or at least an intersubjective phenomenon, and thus you cannotfully describe language without catering to its objective layer that has torise beyond purely subjective, individual content. It is to answering thatobjection that I devote most of this chapter.

2.2 World-knowledge and coordination

To establish the connection between representations and language, we firsthave to state our background assumptions on what language is and whatis the role of language. First of all, we will assume throughout this textthat language is an abstraction. To put it plainly, there’s no such thing

2.2. WORLD-KNOWLEDGE AND COORDINATION 43

as “natural language” by itself, you can only approximate it with certaintheoretical constructs, in a similar fashion that there are no material pointsin the physical world, even though physicists use material points in many oftheir working theories. That does not mean that language does not have atangible implementation - justice is an abstract idea, but when one speaksof justice within a given country’s legal system, one refers to the written codeof law and the inner workings of the court to judge how well the abstractis realized. We will often conflate the language as an abstract idea and itsmaterial implementations (such as sentences written and uttered), but it isimperative to keep the difference in mind as there are important theoreticalconclusions that rest upon it.

Still, when speaking of language as an abstraction, one has to answerthe impending question: what is it an abstraction of? Our answer will bethat language is an abstraction of our codified ways to communicateworld-knowledge and coordinate activities. This immediately pointsto the two key functions that language has: transferring world-knowledgeand coordinating activities (note that those functions account for the perfor-mative and expressive roles of language as much as they do for the descrip-tive role, eg. the performative role is largely how we coordinate actions usinglanguage). Neither of those functions can be omitted in a good theory of lan-guage, although historically those have been the subject of separate fields ofphilosophical enquiry: the transfer of world-knowledge has been traditionallythe subject-matter of semantics, while the aspect of coordination has mostlybeen handled by various theories of speech acts (eg. [Aus62]), with Griceanpragmatics ([Gri75]) combining the two to an extent. One of the aims of mywork is to show that to solve some philosophical puzzles regarding meaning,even those that seemingly are completely unrelated to the pragmatic aspectsof language, one has to incorporate both aspects.

First, let us consider world-knowledge. How does it tie into the topic ofrepresentations? The answer might seem obvious - after all, representations


are specific examples of world-knowledge. However, that is not really thecase. While representations allow us to reliably select certain objects fromthe environment, there is no normativity imposed on them that tells us whichobjects to select. Therefore, there is some information-processing involvedin representations, but not necessarily any knowledge. To talk of knowledge,we need some normativity involved - we have to be able to say that it isknowledge of a particular fragment of the world that is involved, and,by ourselves, we cannot both select a fragment of the world and say that weselected the correct fragment.

One could adopt a Wittgensteinian/Lewisian ([Lew69], [Lew79]) perspec-tive and say that the normativity lies in the social conventions and rules thatunderlie our use of language. We will return to this topic, however, right nowit should suffice to say that the normativity that is tied to world-knowledgeshould not be sought there. Instead, one should seek it in the purely in-dividual sphere of more complicated mind structures, namely those whichpsychologists call theories (see eg. [Car96]). Theories are mental constructsthat are responsible for predicting the behavior of various objects in the out-side world. In our picture, they might be considered complex notions thatcontain propositional (or predicative) content. For a simple example: when achild learns to reliably recognize chairs, she has the representation of a chair;when she develops a belief that chairs make a funny sound when knocked on,that constitutes a certain theory regarding chairs.

Now, there is obvious normativity within theories that does not requireany sort of social conventions to be present: theories have obvious criteria ofadequacy. A theory is adequate if it reliably predicts the behavior of objectswhich it is about. If I have a theory that glass cups break when droppedon concrete, it’s a good theory, if instead I have a theory that glass cupsbounce off harmlessly when dropped on concrete, it’s a bad theory. Note thatwhile the meaning of the linguistic token “glass cup” is bound conventionally,the theory that I just described does not rely on that binding, in the sense


presented in the previous chapter: I possess a representation of glass cupslong before I know that the objects I represent are named “glass cups”.

So, if the normativity of our mental constructs that are responsible forworld-knowledge is not driven by convention, what is the role of language indeveloping this world-knowledge? After all, one could argue on the groundspresented above that language plays merely a helper role here. However,there are key aspects of language that contribute to the construction of world-knowledge and that I will focus on in more detail in the following chapters,those being:

• coordinating and organizing the categorial structure of our thoughts

• providing a common anchor for our world-knowledge via proper names,thus ensuring the possibility of comparing world-knowledge among in-dividuals

• indirectly optimizing our formation of theories, by ensuring that it isrepresentations of objects that are intersubjectively recognized as salientthat are constructed and theorized upon

• embedding theories about various objects directly in their meaning viacategorial structure or association of complex notions with commonlyused terms

• stimulating the growth of abstract terms by the construction of notionsthat reference other notions instead of objects in the world directly(thus promoting a hierarchical structure of concepts)

Moreover, while the normative aspects (among others, truth) of some di-rectly referential fragments of language will not depend on convention, therewill be some aspects of language that do. Since world-knowledge conveyedby linguistic means contains both the referential and non-referential parts,sometimes closely intertwined, it is sometimes hard to separate the directly


referential fragment that is not subject to convention. Examples of suchentanglements will follow.

For now, let us tackle the second function of language: coordination ofactivities. For a long time, this has been a function of language neglectedamong analytic philosophers, although it has enjoyed much success in conti-nental philosophy, especially among the social philosophers concerned withthe power language has over human thoughts and activities. However, re-cently there has been a large movement concerned with uncovering this roleof language, going as far as to shift the focus of language studies on theaspect of social coordination.

The fact that language is largely used to coordinate activities has hugeimplications for our understanding of how it works in general, especially ifyou consider what conditions have to be met for it to be able to serve thispurpose. Without social coordination, we have a group of individuals eachwith their own set of mental representations, however, if we want them tocoordinate anything, both their representations and the higher-order notionsand theories must be at least partially in sync. While the first part of thestatement seems obvious, the second is less so and we will study it in moredetail throughout the text.

The phenomenon we set out to explain is as follows: how is it possiblethat people can use language to coordinate not only simple activities such asmoving chairs, but also very complex activities (such as doing philosophy ordebating legal standards)? On the surface, it might seem simple, but onceyou attempt to study the activities in more detail, the level of complexityquickly becomes obvious.

Let us start with the simpler case study: moving chairs. Of course, itmight be possible for two people to coordinate the action of moving chairswithout any use of language, simply by trial and error. One can point to thefact that in many situations, using language is strictly unnecessary in suchsituations - one can simply point to the chair, make a gesture and another


person will understand that as a instruction to move the chair. However, thisdescription is misleading: it tacitly presupposes a lot of background commonknowledge (in the technical sense of common knowledge: propositions p suchthat for all subjects S1, ..., Sn involved, Si knows that p (for any i), Sj knowsthat Si knows that p (for any combination of i, j), Sk knows that Sj knowsthat Si knows that p (for any combination of i, j, k) and so on). Moreover,even our explanation of why this situation is misleading is misleading ititself: it presupposes a notion of common knowledge that assumes a sharedspace of propositions that each agent in the room will understand. The sit-uation will get much more difficult to describe if we actually try to explainit in pre-linguistic terms. Then it turns out that to coordinate the activityof moving chairs, both sides have to be able to single out chairs from theirsurroundings, they must have a notion of moving them that is somehow nat-urally connected to the notion of chairs (as opposed to, for example, settingthem on fire, smashing them on someone’s head or using them as a mobiledoor-opener) and they must be able to come to a shared understanding ofthat based on the signals exchanged. Saying that they use language for thatpurpose doesn’t explain anything: the instruction to move a chair will beunsuccessful if the recipient considers the word “move” to describe an actioninvolving chopping an object to pieces with an axe (which might be espe-cially dangerous if there are further misunderstandings regarding the word“chair”).

This situation might be considered bizarre, but actually, misunderstand-ings like the ones suggested happen all the time and there is a large numberof documented differences eg. between meanings of seemingly obvious ges-tures in various parts of the world ([Ken97]). To make the problem moreexplicit, let us consider a thought-experiment of the time-travel variety - letus go backwards in time to the moment language was not yet fully devel-oped. Imagine a hunter that wanted to convey to his tribe a new strategyof hunting deer - one group would chase the deer, while another would set


traps and lie in wait at a designated spot. The problems with coordinatingsuch a strategy are enormous. First of all, you have to make sure everyoneunderstands what they are supposed to do - that the group chasing the deerhas to chase them in a specific direction, that they other group is supposed toset traps and wait (and not scare the deer off by charging towards them) andso on. The power of language is shown in the mere fact that I just describedthe strategy (at least its rough sketch) in two sentences. However, without atleast sharing the notions of “traps”, “chasing”, “direction”, “waiting” and,of course, “deer”, one cannot even start working on a successful commonstrategy.

Out of the terms mentioned in the paragraph above, at least “direction”deserves a special mention, as it is most certainly an abstract term. To havethe notion of “direction”, one must first be able to think about a group ofactivities that involve movement, then abstract the aspect of direction fromthem (as opposed to, say, their speed, intensity and so on). Obviously, oneneed not have the abstract notion of direction for all activities involvingmovement, but it seems to be necessary to formulate and execute strategicplans (the plan relies on the deer being herded in the right direction, notwith a certain speed, intensity and so on).

Now, the question remains: how important is language for developingthose more abstract notions? One could argue that at least some of themcould be created without the use of language, simply by trial and error andthrough worldly needs. However, language most certainly speeds up the pro-cess by one key mechanic: fixing certain terms and thus allowing for thehierarchical construction of more complex notions. In other words: we canfirst make sure we all agree on the notion of “movement” before we startworking on “direction”. For anyone who has worked on mastering the ab-stract apparatus of modern formal logic, it should be apparent how importantit is to build a good understanding of simple notions and then move upwardin the hierarchy, each time abstracting one level at a time. Saying that some-


one could come up with the notion of “direction” without first obtaining alinguistically coordinated notion of “movement” comes off as being aboutas likely as a student of logic understanding Cantor’s theorem without be-ing aware of what sets, powersets and various types of functions (includingbijections) are.

While some researchers (eg. [Bra94]) would want to consider language asa “hard” prerequisite for the formation of abstract notions, I believe that sortof approach is both not warranted and not needed. It is extremely unlikely(and thus systematically implausible) that people form complex abstractnotions without the use of language. On the other hand, for the ontogenyof language to be explainable without magical jumps, such complex notionformation must be indeed possible. The formation of language in the firstplace (and an adoption of language as the tool optimizing the formationof higher-order notions across societies) quickly follows from such a “soft”,probabilistic version of the abstract concept formation barrier - the firstlanguage-users had to form abstract notions on their own, but once languagebecame an established form of coordinating notions, more members of thecommunity were likely to form abstract ones (and further coordinate them,thus expediting the process of abstract concept formation).

So, once we assume that language is in place, how do we explain the for-mation of complex notions? How would we perform the case study of peopleworking on changes in the legal system, using terms such as human rights,retribution, resocialization and so on? Obviously, a detailed case study isbeyond the scope of a single paragraph and most of this work will indeedbe devoted to making such studies possible. Nevertheless, a sketch of whatsuch a study would require is in order. First of all, we will have to establishwhat does it mean for people to agree on the key notions used in the debate.Then, we have to figure out how to describe disagreements and how to de-tach material disagreements (such as ones over how the legal system lookslike) from conceptual disagreements (what does the term “human rights” ac-


tually mean?). We have to deal with the problem of the class of materialdisagreements being entangled with the class of conceptual disagreements (infact, some of the key political debates in the modern world involve repeateddebates over definitions, eg. the definition of human life). Certainly, thereis much more involved here than when moving chairs around. However, wecan’t ever start describing this process before we deal with the simpler issue:how do we go up from representations to language, concepts and meaning?

2.3 Representations abstracted away

The question of how the use of language ties to our other cognitive functionshas long been a subject of heated debate. The nativist position, which tracesits roots back to Chomsky (for a current recapitulation of the position, see[Cho06]), has been grounded in the assumption of humans having innatelinguistic capabilities, whereas its opponents argue that there is inadequatedata to support this claim and that language is better explained as part of thegeneral cognitive apparatus. In our work, we adopt a view that’s closest tothe anti-nativist position. While some aspects of linguistic interaction mightbenefit from certain structural features of our brain, it also seems plausi-ble that those features grant a quantitative, rather than qualitative, benefit.Some potential benefits of this type have already been hinted at in the previ-ous section - the mere presence of repeatable, firmly entrenched patterns ofthought (provided by the language acquisition process) encourages abstractreasoning based on those patterns and might indeed be responsible for theexplosion of abstract concepts which makes humankind so special. However,for understanding how language use works, treating it as a specific type ofcognitive activity seems the more plausible approach.

In the previous chapter, we mentioned representations - the way our brainreliably recognizes objects from the outside world. Now we want to tie repre-sentations to our theory of language. The simple and naive way to do it would

2.3. REPRESENTATIONS ABSTRACTED AWAY 51

be by some sort of identity theory - linguistic concepts are simply represen-tations, but with the added normativity of being the right representations.This would do justice to Wittgenstein’s remarks about the impossibility ofprivate language, while being a simple and elegant theoretical solution.

However, the problem with such a solution is that it’s too simple and el-egant. First of all, representations are an object within a single individual’smind. Taking concepts to be identical with representations requires either se-lecting a specific individual to be the “perfect language user” or else assuminga very strong (and empirically implausible) claim about the common struc-ture of human minds. However, both of those objections are secondary to themore important epistemic difficulty - we would have to assume that compe-tent language users build up exactly the same representational structurewhen they acquire a language and that any deviation from that structure,even if extensionally equivalent, produces a linguistic error. In a world wherepeople acquire language in very different circumstances, that would be toomuch to ask.

Are we therefore left with the skeptical results of Quine ([Qui51])? Shouldwe indeed avoid talking of meanings, as any talk of meanings raises problemswith radical interpretation? Fortunately, the situation is not as dire. Weneed not stick with behaviorism as our psychological theory of choice. Wecan safely assume that there are functional similarities in how the mind worksbetween various individuals that allow us to base a theory of language on.However, we have to strike a delicate balance - between an identity theoryon one side and the behavioristic skepticism of Quine on the other.

The solution to this conundrum comes from an improbable source, yetanother linguistic skeptic - the already mentioned Wittgenstein ([Wit53]). Tohim we owe the idea of language as a social activity that is driven by rules andhas meaning determined by use. Such an approach requires a shift in focus -instead of considering linguistic concepts to be objects, we better treat themas reified tests. While there might not be any prototypical representation


that corresponds to the linguistic label “chair”, there is certainly a feature ofrepresentations of chairs possessed by competent users of English - namely,they all reliably select chairs, and only chairs, from the outside world whenpresented with the word “chair”.

2.4 Concepts, representations and notions

Before we discuss the tests, however, we need to do some conceptual ground-work. Until now, we have been talking about two different types of objects:representations and notions, which are individual, mental objects and linguis-tic concepts (from now on, as previously noted, we will drop the “linguistic”part and unless otherwise noted, we will use the term “concept” to refer tolinguistic concepts and not individual concepts, for which we will use theterm “notion”; also, we will group representations and notions together un-der that term). What we have not yet established, however, is the ontologicalrelation between notions and concepts. While we have stated previously thatontology is not a key subject of this work, in this specific case certain onto-logical decisions have to be made, as without them, it will be impossible toanswer certain questions about concepts.

More precisely, what has to be determined is the relation between notionsand concepts. We have already opposed the idea of an identity theory, there-fore, notions and concepts cannot be identical. It seems reasonable to borrowa solution from another theory from philosophy of mind, namely the multiplerealization theory and talk about notions that realize certain concepts. Howdoes the realization relation work? On one level, we have notions that canbe tested for various properties, both internal (their relations to other no-tions) and external (their referential properties). On the other level, we haveconcepts that are purely abstract entities that we will represent by the tests.We will therefore say that notion n realizes (or implements) concept c whenn passes all the tests within c (which we will simply denote as c(n)). We will

2.4. CONCEPTS, REPRESENTATIONS AND NOTIONS 53

also develop various notions of partial realization, with the partiality possiblybeing qualitative (a notion n passing a test within c for the majority of cases,but not all of them, for example a notion of cars-and-buses as a realizationof the concept of car) or quantitative (a notion n passing some but not alltests within c, for example, a notion of red car passing the test within theconcept of blue car for being a car, but not for being blue).

What tests are we talking about? One obvious category is the deno-tational tests. We already discussed how the notion of representations isessentially externalistic, now it’s time to put that assumption to use. Sincelanguage is used to coordinate world-knowledge, the world itself should bean important source of input. After all, we do not want to end up with aperfectly internally consistent language with no empirical meaning whatso-ever.

On the surface, the denotational tests are obvious - a notion n passesthe test of realizing concept chair whenever n adequately represents chairs.However, the problem is more tricky than that and has been hinted as earlyas Quine’s rabbit dilemma ([Qui60]). Before, we did not make any decisionsabout the grainedness of representations, hiding behind a background ontol-ogy to do the job for us. However, now we have to give the matter furtherthought.

The reason for this is very subtle and thus we will explain it in more detail.Since representations are, by definition, the sole way our brain is able to selectobjects from our surroundings, the features of objects our representationsselect, plus our criteria of demarcation for objects are what ultimately decideshow the world is segmented in our cognition. From that point, we face theold Kantian dilemma: it is entirely possible that the world itself is structuredin a completely different manner, but we cannot detach ourselves from theway our cognitive apparatus works in determining that structures. Puttingit in Kantian terms: the world of pure noumena is completely closed to us.

Therefore, any ontology we assume in a theory that aims to describe the


real world has to incorporate the structure of the phenomenal world in asufficient manner. Translating that from Kantian to modern terms, it meansthat when we select an ontology, we have to take care to posit one that canbe reliably aligned with our phenomenal data.

On the level of representations, the base mental entities responsible forrecognizing objects in the world, this basically means we have to say not onlywhat objects we can reliably select, but also what it means for a specific typeof object to be selected. More precisely, we have to describe what it meansfor an object to be of a specific type. This is an area where we have to treadespecially carefully.

Consider our favorite category of analysis, chairs. We said that a rep-resentation of chairs adequately selects chairs from the environment - butwhat do we mean by that statement? Do we mean that there is a categorialproperty, in this case, chairness, that we are able to recognize in chairs andonly in chairs? How does that categorial property appear in our sense data?How many categorial properties are there and how do we determine whichare the right ones to select? This line of reasoning, although it has beenadopted by many philosophers with the notion of natural kinds, seems to mea dead end, leaving more questions than answers.

The line of reasoning we will adopt is the more Russellian ([Rus18]) ap-proach of certain properties being passed on from the outside world throughour cognitive facilities. Among those are the relational properties of someproper parts of the world which allow us to select objects. In other words,being an object is not a primitive notion of our background ontology, be-ing a proper part of the world is. Some proper parts may stand in certaincontrastive relations to other proper parts and that contrastive relation isenough to make the proper part appear an object. However, note that thisis an ontological decision (probably the most profound one in this text) -there is data that suggests that we are born with the ability to look at theworld as containing objects, we don’t seem to have a truly holistic view of the

2.4. CONCEPTS, REPRESENTATIONS AND NOTIONS 55

entire world as it is. Therefore, although in our ontology the notion of objectis derived, we withhold judgement on whether object selection is a derivedmechanism as well, since there is insufficient empirical evidence to decideeither way (with some data suggesting that it might indeed be a primitiveone).

In what follows, we will assume that our cognitive capabilities allow usto recognize certain properties of various world fragments. We will assumethat some types of those properties are commonly recognized by all humanbeings and thus form the basis of our cognition. This solution certainly has aKantian feel to it (although it must be noted that we restrict our solution tocognitive capabilities of humans, therefore giving the solution a pragmaticfeel - we do not aim to either concern ourselves with cognitive capabilitiesof animals nor to construct a more general transcendental subject for whomthe solution applies), although it is irrelevant for our discussion whether theability to recognize those properties is inborn or acquired very early in ourdevelopment.

The solution gets more complicated when we realize that humans arecapable not only of recognizing certain properties, but of abstract reason-ing - they are able to reflect upon certain qualities of their own cognitiveframework. In fact, certain categories seem to be not those of objects in theworld, but of our own representations - the category of numbers being oneof the more prominent. We will devote much attention further on to theepistemology of mathematical objects, but for now we need just note thatwhenever we speak of properties and categories, we can mean both the onesrecognized in the world and the ones recognized in our ways of thinking (infact, sometimes it might be hard to decouple one from the other). When weneed to disambiguate, we will speak about concrete properties or categoriesto denote the ones recognized in the world and about abstract ones to denotethose recognized in our cognitive apparatus.

Another topic that deserves more attention is our use of the worlds “prop-


erty” and “category”. Up until now we have used those almost interchange-ably, but those words have a long history both in the philosophical and psy-chological literature, so we must be careful to explain how we use them. Infact, philosophers and psychologists often differ greatly in the use of the world“category”, with the philosophical understanding often having a Kantian,transcendental sense while the psychological understanding is often ontolog-ical (psychologists tend to use the word ”category” for ontological categoriesand use the term ”sortal” in the way we use the word ”category” [Car99]).

Our understanding of categories is more on the philosophical side, withthe added assumption that, in our ontology, there is no such thing as built incategories. The concept of category is one of a grouping - it involves objectsthat are selected under a certain common notion, whether it be a primitiveone or a complex one. On the other hand, when we talk of properties, wemean either the basic ontological features we can recognize in world parts(then we talk of properties in the world) or of our respective cognitive capa-bilities to recognize them (note that in some cases, namely those of abstractproperties, there’s a subtlety involved here, with our mental capability torecognize an abstract property being of higher order than the abstract prop-erty itself being recognized). We will use those meanings interchangeablywherever no confusion arises. If we were to tie the distinction into the oldphilosophical debate, properties would simply be properties, while categorieswould be sets, albeit non-extensional ones.

2.5 Tests introduced

Now that we have discussed the foundational matters, we are ready to discussthe tests - the main components of concepts - themselves. When we say thatcertain notions implement concepts, we are actually talking of five types ofbasic tests in mind:

• formal tests

2.5. TESTS INTRODUCED 57

• demarcation tests

• property tests

• structural tests

• mental tests

Each of these types of tests will now be discussed in detail.Formal tests decide on what aspect of the world is used for determining

the object in question. The term is derived again from the Kantian notionof “forms of intuition”, of which Kant names time and space. The formaltests decide whether the object in question should be selected based on itstemporal or spatial merits (or possibly both), or whether the object is perhapsan abstract one (and thus lies outside the realm of material forms).

The demarcation tests have to do with the characteristic of our back-ground ontology which contains world parts (instead of just objects) as prim-itives. Although we have withheld judgement on whether object selection isa primitive or derived concept, it is still a fact that there are many objectsthat it is possible to select from the world. The demarcation tests determinewhich parts are exactly selected - eg. in case of chairs it is the chair itself andnot it immediate surroundings, nor the person currently sitting in it, whilefor example the notion of “garden” denotes the entire area together with theground, the plants, the decorations and so on.

Property tests have to do with the properties the selected world parthas. Note that it is very difficult to distinguish between demarcation testsand property tests and we will not put too much import on this distinction,leaving it on the intuitive level: demarcation tests have to do with how toselect the object from other objects in the world, while property tests have todo with making sure the object indeed fits the required notion. For example,the test of being able to take an object and move it without moving anythingin the surroundings is a demarcation test for many physical objects, whileits being a certain color and texture are property tests.


Structural tests have to do with other concepts and introduce a holis-tic element to the entire theory. Some concepts only make sense when theyare related to other concepts - for example, democracy is a form of govern-ment and it does not make sense to consider the concept of “democracy”when the concept of “government” has not been established. Sometimes, itmight be possible to reduce structural tests to the underlying tests of thefirst three types for the ingredient concepts, but in some cases, especiallythose of abstract concepts, it might not be possible (the concept of “lexicalmeaning” without the concept of “word” is impossible, although we can prob-ably explain the term “stool” to someone who doesn’t possess the concept of“chair”). However, while some structural tests could be reduced away, theirpresence is a necessary element of the economy of language, which is why wewill not do that even if it were possible.

Cognitive tests are related to what we talked about in the first chapter- they reference the properties of cognitive states that have to be associatedwith a given notion. Note that we run into a potential categorial difficultyhere - after all, representations are memory fragments, which are just com-ponents of mental states (and indirectly, of processes). However, since arepresentation is defined as a memory fragment present in a certain class ofprocesses, we can use a notational shortcut: we will say that some tests areperformed on a representation (or on a notion) whenever they are performedon states of each process in which the representation takes part (or on eachprocess in which the representation takes part). Note that these types oftests might restrict some notions in a very non-obvious way - we will explorethat part in more detail in our discussion of the social aspects of language.

Those five types of tests lay the groundwork for our theory of concepts,so now, we turn to exploring them in more detail.

2.6. CONCEPTS AS TESTS 59

2.6 Concepts as tests

For our formalization of tests, we will use lambda-calculus as the convenienttool. Since we have hinted before that we might have to use a measure ofpartial conformance to some tests, we cannot use standard logical operatorsto combine the tests - instead, we have to explicitly compose tests with eachother in a way that allows for their subsequent decomposition.

For the most part, simple tests will just be predicates - functions fromobjects to truth values. More precisely, they will be functions from repre-sentations to truth values - however, we will often rely on various features ofrepresentations that are tested instead of the representations themselves.

The easiest to formalize are the formal tests - they act upon notions anddetermine whether the notion selects temporal, spatial or spatio-temporalworld parts, or, as a separate case, abstract objects. Therefore, those tests arerestricted to four basic variants: λn.Sp(n), λn.Tmp(n), λn.(Sp(n)∧Tmp(n))and λn.Abs(n).

We will treat property and demarcation tests as sharing a common for-mal structure, in line with what we claimed earlier about the difficulty ofseparating the former from the latter. There are four basic types of teststo be considered here: applicability tests, degree tests, simple property testsand relational tests.

The applicability tests check whether a certain world fragment can bequeried for a certain type of property. For example, it does not make sensefor objects that are not human to test them for intentions. Those tests do notcheck whether an object possesses a certain property, they are categorial tests- they check whether a certain category applies to a certain object at all. Totest an object for the color it possesses, one has to know whether the objectcan be considered to have a color at all (magnetic fields, for example, do nothave a color). Therefore, applicability tests can be considered preconditiontests and must be applied before some successive tests can be applied further.They are formalized in a relational manner, with the categorial content being


the second part of the relation.

In the applicability tests, the concept of referent is used, and it willbe commonly applied in other tests as well - the referent of the notion be-ing a part of the world selected by the notion. Note that unless specifiedotherwise, the referent function returns an abstract object (world fragment)in the Berkeleyan sense - one whose specific features have been abstractedaway to leave only the features relevant for the notion itself. Therefore, thenotion of “chair” does not select a specific chair from the world - it selects anabstract chair, without any specific color, location, surroundings etc. When-ever a specific referent for a representation being used in context is needed,it will be mentioned explicitly. Applicability tests share the following pat-tern: λn.App(ref(n), C), where C is the categorial variable (for example,λn.App(ref(n), Colored)) and ref is the referent function.

Degree tests apply to properties whose intensity can very on a certainscale. Size is a very typical example of such a property, as is temperature,brightness and many others. The scale used will always be implicit in the test(provided by the context), unless there is ambiguity, then it will be mentionedexplicitly (for example, in a test of temperature, the implicit scale is one ofdegrees, although a specification - Fahrenheit, Celsius - might be needed).The tests here involve a tertiary relation, with the scalar property beingtested, the object and the range involved. Therefore, they will be of thefollowing shape: λn.Deg(ref(n), P,R), where P is the property variable andR is the range variable, for example: λn.Deg(ref(n),Width, [50cm; 150cm])tests whether the object has width between 50 cm and 150 cm, inclusive.

Simple property tests are just that: tests that determine whether anobject has a given property. They can be written using higher order notationas λn.Prop(ref(n), P ), but we will use the common first order notation ofλn.P (ref(n)), for example λn.Red(ref(n)).

Relational tests determine the relation between a given object and someother objects. Note that some quantification might be involved here, for ex-

2.6. CONCEPTS AS TESTS 61

ample the term “rest area” refers to a building complex that offers food,optionally a motel and is located in the vicinity of some highway. The rela-tional tests are the least patterned because of the multiple ways the object orobjects in relation to our notion’s referent might be selected. For example, thesaid rest area’s relational test might look as follows: λn.(∃x(Highway(x) ∧Near(x, ref(n))).

Now that we have described the subtypes of property and demarcationtests, it is time to turn to the structural tests. For those to have any senseat all, we have to assume a structure of notions. This is something we havenot done before - we did mention that notions are the more complex siblingsof representations, containing referential as well as logical components, butso far we have assumed nothing of their structure. However, now we have tomake one further, although plausible assumption: that notions do not form ina void, that our economy of cognition makes it easier for us to create notionswith the use of other notions. In fact, it is the economy of our cognition thatgives rise to the economy of language, although there is also feedback theother way - the economical way that language is structured gives rise to richcognitive capabilities.

The structural tests’ patterns are even harder to enumerate than thoseof relational tests, although the reason is quite different: there can be manystructural relations between notions and furthermore, establishing what onesactually hold (i.e. what are our actual cognitive capabilities in terms of reflec-tion and abstract reasoning) is an empirical matter better left to psychologistsand neuroscientists than philosophers. Therefore, any examples we providefor the structural tests need to be considered as sample material only andnot as having any epistemic pretenses.

Among the simplest structural tests are those that simply require othernotions to be involved in the construction of our chosen notion: those wouldfollow a pattern similar to λn.(∃m(V ehiclec(m)∧Inv(m,n)), where V ehiclecis the implementation predicate (from now, we will use the subscript c to dis-


tinguish implementation predicates from normal predicates) for the conceptof “vehicle”, whereas Inv is the involvement predicate. Others are categorialpredicates, like λn.(Cat(n, V ehicle)) tests whether n selects objects of thecategory “vehicle” (we do not use a referent because, as mentioned earlier,we consider categories to be something on the cognitive side and not on theontological side). Yet others require for a certain feature to be abstractedfrom a set of objects, eg. λn.(Common(n, {red, blue, green, orange})) wouldbe the structural abstraction test for the notion of “color”.

We are left with cognitive tests, which are a blanket term for any predi-cates on mental states associated with notions. For example, λn.(∃e(Joy(e)∧e ∈ A(input(n)))) is a test for one’s notion being associated with the stateof emotional joy, where A is the attention accessor for cognitive states, whileinput is the input selection function, like with the referent selection - return-ing an abstract object with the specific features removed.

Now that we have described all the basic types of tests, what remains isthe means of conjoining them. We have already seen, with formal tests, thata simple conjunction is not sufficient here - some tests have to be prerequisitesfor others, plus we need to be able to add weighed tests, alternative tests andperhaps also probabilistic tests.

The two main ways of combining the tests are sequential and parallel.In a sequential merge, one test or group of tests has to be fulfilled beforethe other can be even tested. In a parallel merge, all tests from the grouphave to be fulfilled irrespective of each other. Moreover, a test can be strictor relaxed, with different forms of relaxation available to different types oftests (with degree tests, for example, it can be the matter of distance fromthe required range). We will assume that tests are strict, with relaxationsprovided explicitly.

Thus, we have two types of merging operators: → and | , parentheses forgrouping, a binary rel(t,m) operator with t being the test and m being themeans of relaxation and a wg(t, w) weight operator, with w being the relative

2.7. TESTS AND LANGUAGE 63

weight of the test (normalized to a number between 0 and 1). Whenever wgis used, we are left with a gradable concept, however, for some tests, prereq-uisites might require a weighed alternative to be fulfilled (which introducesbinary comparison operators > and ≥), but the resulting concept will beabsolute. We will for now keep the domain for m underspecified, with ele-ments of the domain filled in during our further research. Thus, an examplecompound test would be t1 → (t2|t3|rel(t4,m)) → (t5|(t6 → t7)), meaningthat t1 is to be carried out first, then t2, t3 and a relaxed (according to m)t4 in parallel and then t5 and a sequence of t6 and t7, again in parallel. An-other example test would be (wg(t1, 0.4), wg(t2, 0.4), wg(t3, 0.2)) > 0.5→ t4,meaning that a weighed parallel test of t1 (with weight 0.4), t2 (with thesame weight) and t3 (with weight 0.2) has to yield a combined result of over0.5 for the test t4 to be then successively applied. Yet another example wouldbe (wg(t1, 0.2), wg(t2, 0.8)), resulting in a gradable concept, with weights 0.2and 0.8 applied to the component tests.

For example, if we consider the concept “red car”, we first have a formaltest t1 that checks for a spatial notion, then two structural tests t2 and t3

for the presence of concepts “red” and “car” performed in parallel, then asubsequent demarcation test t4 for selecting a car and finally a property testt5 checking for the property “red”. Alternately, we could have a concept“spicy”, which would first have a formal test t1 for spatial notions, then aweighed alternative for various forms of spicyness (sweet, sour, salty, hot andso on).

2.7 Tests and language

In the chapter, we have introduced the notion of concepts as tests for notions,as well as mentioned how those concepts are used for coordinating activitiesand world-knowledge. The rest of the work is devoted to showing examplesof how those functions are realized in practice. However, before we can


conclude, one last link must be forged - from tests to language itself. Untilnow, we have provided a formal definition for tests, which are supposed tostand for linguistic concepts themselves, however, we have not shown in anyway how those tests tie into real linguistic practice.

We already mentioned a Wittgensteinian perspective on language. Thismeans that we consider concepts themselves, as well as the underlying tests,to be rule-driven - the linguistic society determines the tests by enforcinga certain way to use the linguistic terms in question. Since we adopt herea vision of meaning being determined by use, it should be obvious that thenormativity behind meanings is never stated explicitly and, in fact, the lin-guistic society might not itself realize what the norms it enforces are. More-over, there might be many different subsocieties within the main linguisticsociety, enforcing its own complementary or contradictory norms, leading tothe emergence of various dialects. Since the linguistic norms are formed byuse, the only thing we can do is reconstruct them rather then construct them,although adequate knowledge of the mechanisms of language propagationcan, in fact, allow some agents to construct parts of our linguistic system.

The situation gets even more tricky when we consider the fact that lan-guage is intimately tied to our general cognitive activity, which means thatby influencing the linguistic norms which we use, certain people or groupscan in fact influence the way our cognition works - sometimes overtly, some-times covertly. We will explore the ways in which our language ties to ourcognition in many parts of this work, revisiting the Sapir-Whorf hypothesisand, at the end, looking at the way various modifications of language canhave an effect on our social life.

Creating a complete theory of meaning and thus, describing in detailhow the mechanism we posit works is beyond the scope of a single work -instead, in subsequent chapters we will focus on certain key examples of howvarious puzzles regarding meaning can be solved by adopting the viewpointwhich we presented in the opening fragments, thus showing the possibility of

2.7. TESTS AND LANGUAGE 65

developing such a complete theory of meaning in further research.

Chapter 3

Proper names and reference

3.1 The mystery of proper names

For a long time, proper names have been a puzzle to philosophers. Theyare one fragment of language for which a working theory has eluded capturefor years. The riddle starts with John Stuart Mill ([Mil43]), who suggestedthat names do not have a sense associated with them, only a referent. Thencame Frege’s puzzle about coreferential proper names ([Fre92]) - the founderof modern philosophy of language duly noted that if names had no content,then sentences establishing the equality of two coreferential names wouldcarry no weight - but yet it is obvious from our use of language that they doand that we learn something whenever two proper names are identified witheach other.

Russell ([Rus05]) proposed an elegant solution to this problem, by iden-tifying proper names with definite descriptions for which they were but analias. The solution was logically sound, explained the cognitive content ofnames, but quickly came under attack by Saul Kripke in his influential text,Naming and Necessity ([Kri80]). Kripke presented a barrage of argumentsagainst the descriptive theory of proper names, noticing that we often use

1Many of the ideas in this chapter are based on a theory presented in [Wil10]

67

68 CHAPTER 3. PROPER NAMES AND REFERENCE

proper names without the knowledge of any descriptive content associatedwith it and using modal intuitions to show that there is no single identify-ing description that we necessarily associate with a proper name. Kripke’sarguments have been a death blow to the pure descriptive theory of namesand Kripke indeed proposed an alternative, causal theory, but his solutionhad problems as well, as evidenced by Kripke’s another text, A Puzzle AboutBelief ([Kri79]), in which he notes that an individual can consistently haveconflicting beliefs about two coreferring names, as long as she doesn’t realizethe names are coreferring.

Since then, a multitude of theories have arisen which attempt to solve thepuzzle of proper names. One of the best known is a hybrid theory suggestedby Gareth Evans ([Eva82]), although one should also point to a lesser knownsolution presented by Peter Strawson in his text Proper names – and others([Str74]). Nevertheless, none of those theories actually solves the puzzle, dueto a phenomenon which we hinted at in the introduction to Chapter I - theantipsychologist attitude, which disallows the use of arguments from privatecognitive content in philosophy of language. Our aim here will be to presenta theory of proper names that deals away with that attitude and uses theframework presented within the first two chapters to solve most of the puzzleswhich have been present in the proper names debate.

3.2 Cognitive world-maps

Before we move onto proper names, however, we need to address anothertopic. How do we organize our knowledge about the world? Surely, most ofour knowledge is supposed to be reusable - it concerns object and event typesof which there is more than one token in the world. However, sometimes thatisn’t the case. Take the most obvious example - a person’s representation ofthemselves. There certainly isn’t any other individual in the world that canbe recognized by that representation and yet it is indeed quite useful to be

3.2. COGNITIVE WORLD-MAPS 69

able to single out oneself from others. Similarly, our closest family falls underthis category, as well as other people, objects and places which are somehowimportant and organize our lives.

When we move to language, individual concepts start to serve yet an-other purpose - they are landmarks which allow us to pick certain objects inthe world and organize our common lives around it. Thus the entire socialconvention of naming objects that are important to the linguistic community- whenever an object is named, it is somehow recognized as being importantfor the shared experience.

As we start to both learn language and acquire world knowledge, we startdeveloping something that I will refer to as the cognitive world-map. One canthink of the cognitive world-map as our theory (in the psychological sensedescribed previously) of the entire world, however, the term “map” here isused to place emphasis the individual concepts. We will therefore considerthe term “cognitive world-map” as referring to the minimal fragment of ourtheory of the world that includes all our individual notions.

An important issue here is one of the referential import of individualnotions. Before we start learning language, the individual notions we formare all ones that indeed have causal links at their core. However, as we learnlanguage and start acquiring linguistic labels from other language users, thatno longer remains the case. We replace the direct causality of our knowledgewith a phenomenon that, for a lack of a better name, I will call linguistictrust. Basically, the phenomenon can be described as follows: we believethat linguistic labels are not empty. In our terminology: we believe thatthere are concepts linked to linguistic labels. On a deeper level of linguistictrust, we believe that there are notions that realize those concepts. The firstlevel can be described as “words have meanings”, the second, as “words havecomprehensible meanings”.

In case of proper names, linguistic trust means that we assume that labelshave associated notions that single out certain objects in the world. However,


one must recognize that this is due to various assumptions we make aboutlanguage that is used to talk about the external world, not something inherentto the names themselves. This can be easily seen in case of fictional names,where we use the naming convention without the referential function, sincein that case, linguistic trust is consciously suspended.

3.3 The meaning of proper names

How do we handle the various puzzles regarding proper names? We alreadyestablished in the previous chapters that concepts are tests for notions -it should be no different for individual notions. However, we must exercisecaution if we are not to fall into the traps outlined by Kripke in his work. Thetools we described before are mostly descriptive ones - they would work witha descriptive theory of proper names, but there are various reasons to rejectsuch a theory. Therefore, we must attempt to construct a theory of propernames that uses the outlined tools while still deflecting Kripke’s criticism.

Since we are starting with a cognitive perspective, we must first providea meaning for individual concepts that is cognitively plausible. It might betempting to try to adopt the Russellian idea of proper names being associatedwith a definite description, since that would be easily formalized under ourframework - a definite description is simply a property test and thereforethe name would simply implement that test. However, the simplicity of thissolution masks its many faults, most of which have already been outlinedby Kripke: for one, a descriptive interpretation of names doesn’t pass modaltests, since a sentence

(1) Aristotle might not have been the most famous Greek philosopher.

seems to us intuitively true, even if the name “Aristotle” were ultimatelyassociated with the description “most famous Greek philosopher”. Kripke’sargument is that, whatever description we are faced with, we can always

3.3. THE MEANING OF PROPER NAMES 71

formulate a sentence like (1) that will modally denounce it. Even morealarming is the fact that often we are faced with proper names for whichwe have not been provided any appropriate description and we neverthelessaccept them and use them in our speech.

It seems that we are therefore left to seek out some type of causal theory ofproper names. However, this presents much more challenges for our account,simply because causal theories are exactly something our framework doesnot cope with very well.

The first reason for that is that the standard way of handling causaltheories, namely one that treats names to be directly referential, is closedto us here. We established that concepts are tests for notions - it wouldbe a categorial mistake to allow concepts to instead select objects directly.Another reason is that it would be cognitively implausible - there is no way wecan directly pass objects using language. Finally, it would not solve Kripke’spuzzle about belief, nor it would answer the apparent analycity of sentencessuch as

(2) Warsaw is a city.

In order to find the answer, we must gather what we know about propernames and their usage and try to incorporate it within our framework. Whatare some of the most important things we know about proper names?

(i) names lack descriptive content

(ii) on the other hand, they seem to have categorial content, as evidencedby (2)

(iii) truth conditions for sentences with names seem to rely on certain prop-erties of specific objects

(iv) names do have cognitive significance, sentences that identify two namesgive us knowledge about the world


(v) some names are, intentionally or unintentionally, empty, which is theresult of a suspension or misfire of linguistic trust

Reconciling some of those points might prove to be difficult, for they seemto contradict each other - for example, (i) seems to strongly conflict with (iv)and (iii) seems to contradict (v). We will focus on the second conflict first,as it requires us to address an issue that we have not yet discussed and thatseems to be essential to any theory of meaning - truth conditions.

Recall the notion of linguistic trust presented earlier - we mentioned thatthe fact of concepts having notions that implement them and that the notionshave reference is not part of the concept itself. Couldn’t we do a similar thingwith names - claim that names have reference as part of linguistic trust, butthe reference isn’t part of the meaning? After all, we already hinted at such asolution when we pointed out fictional names. However, this would contradictour important assumption about the cognitive significance of names - thatproper names are a way of circulating world-knowledge. In order to do that,they must in fact single out certain objects in a meaningful way, since thenthose very objects are the means of coordinating other meanings as well asnon-linguistic knowledge.

In fact, linguistic trust is a mechanism that underlies language acquisi-tion, not meaning itself. It relies on the assumption that there are learnablenotions that realize certain linguistic concepts - after all, it would be nonsen-sical to introduce a concept without a notion to realize it. However, once theproper notion is found, linguistic trust has no further part to play. It is onlyat the earlier stage, one of possessing a proto-notion, that the mechanismis important.

So, now that we have established that individual concepts do indeed re-quire certain referential content, how do we make sense of that referentialcontent within the theory? In other words, how does that referential contentcontribute to the truth conditions of propositions that contain individualconcepts (proper names)?


To answer that question, we must first consider the role of truth andtruth conditions in our framework for meaning. Those concepts cannotbe neglected - after all, they are central to philosophy of language and anytheory of meaning has to at least try to find a place them. In our case,truth is a construct derived from the normativity of concepts. Notions re-alize concepts, but they also select parts of the world. Without providinga full compositional definition of truth (which would be too ambitious anundertaking, since we have to account for the cognitive level as well as thelinguistic level), we will settle for the following loose definition: a propo-sition is true if the notions that realize concepts within it can becoordinated with respect to world fragments which they select. Thedefinition is vague by design, since a full account would require us to giveproper compositionality rules for various propositional components. How-ever, we will provide examples which should make this definition at least abit clearer for certain cases.

For concepts which stand in the place of predicates, it is easy to translatetheir test conditions to truth conditions - a concept is true of an object if anotion that passes the test conditions for the concept can be applied to theobject in question. For example, the sentence

(3) This chair is red.

is true if and only if a notion realizing the concept “red” can select thechair pointed out by the demonstrative “this chair”.

However, let us consider the sentence

(4) The Eiffel Tower is at least 50 feet high.

By an analogy with (3), this sentence should be true if and only if thenotion realizing the concept “at least 50 feet high” can select the objectpointed out by the proper name “The Eiffel Tower”. However, now we run


into the real problem: how does the proper name “The Eiffel Tower” pointout an object?

Running through our catalogue of tests, we can quickly dismiss formaltests as ones responsible for selecting the proper object. The obvious can-didate would be property and demarcation tests, but they are incorrect forthe very reasons we outlined earlier - they allow us to test objects for havingcertain properties and we already pointed out that this would tie us to adescriptive theory of proper names which we do not want. Structural testsare obviously incapable of performing any referential functions and thus arenot the right way to go. Only one category remains: the cognitive tests.

Cognitive tests are a very large bag to sift through, but in our theory,only one part of cognitive states holds referential import: the attention focus.Thus, there is only one way we can test for a notion having the properreference - we constrain the attention of the input state for the notion (aswe mentioned in our formalization in chapter I, the set of objects in theattention focus of the subject is one our accessible components of cognitivestates). However, doing that, we run into two problems - one, which I will,for lack of a better name, call the problem of magical attention and theother, less obscure: of a vicious circle.

We will first focus on the vicious circle. The difficulty might not beobvious, for it is now hidden under layers of definitions, but we will try tobring it to light. Representations are defined due to one common feature- having an object in the attention focus of its input state. Concepts arerealized by representations. Therefore, we risk explaining the nature of anindividual notion of x by appealing to the definition of a representation of x.This certainly does not look promising.

However, upon further examination, we should discover that the viciouscircle is, in fact, not a vicious circle at all. Rather, it is a curious case oftriviality - the nature of individual concepts reduces exactly to the referentialaspect of the notions which realize them. There is no further content to


individual notions. One could say that this realizes the maxim associatedwith causal theories of names, that of direct referentiality. In fact, namesare directly referential in the sense that, apart from exploiting the referentialcharacter of representations, they do nothing else (we will in fact claim thatthere is one more test that is to be taken into account, but that will beelaborated upon further).

In fact, there is one assumption we made earlier on that simplifies thereferentiality of proper names - that of the referentiality of representations.However, as with most theoretical assumptions, it is not without cost. Itintroduces another problem, namely that of the mechanisms of reference,which we will now have to tackle.

When we talked about the referentiality of attention, we made an as-sumption that our cognitive capabilities allow us to select objects from theworld via our senses. However, now we have to face another question - howfar does this ability reach? Surely, the case seems easy with demonstratives- we do not doubt that when we use the word “it”, we can access the objectthat it demonstrates. However, the Eiffel Tower from (4) is another mat-ter. One could argue that in fact, the name “Eiffel Tower” implements atest that restricts realizing notions to those that have the Eiffel Tower as adirect object of attention, but that would run contrary to our rule (iii), ormore precisely - to our intuition that we are able to meaningfully talk aboutobjects which we aren’t currently in contact with, or even ones that we nevercame into contact with (otherwise, all talk of Aristotle and other historicalfigures would most certainly be futile). This is precisely the problem of mag-ical attention that has been mentioned above - in order for this talk to bemeaningful and our definition of proper names to hold, we have to explainhow we can have distant objects in our attention focus.

Further investigation reveals a surprising result: not only is this not aproblem, but in fact, in a dialectic shift, it is a solution to our problems withempty names, missing referents and so on. However, we should not act like


bad book reviewers and spoil the conclusion of the story - before we makesuch claims, we should show how such a conclusion can come to be.

Here, Russell, whom we have been avoiding for the most of this chapter,can finally lend a hand ([Rus10]). Russell had a concept of “logically propernames”, which were exactly the indexicals - the direct reference selectors.However, he also had a whole theory of “knowledge by acquaintance” and“knowledge by description”, suggesting that, while we can directly know onlythe objects with which we are directly acquainted (and which we can selectby the pure indexicals), we know all other objects in an indirect manner. Itis precisely this type of epistemic theory that we are in need of here.

To solve our conundrum, we need but relax our constrains on how theattention focus works. Before, we only limited it to direct evidence from oursenses - however, such a constraint is both too wide (as evidenced by thevarious skeptical arguments against the possibility of knowing the externalworld) and too narrow (as evidenced by our Eiffel Tower example). Wetherefore have to relax our referentiality condition - from direct cognition toall cognition whatsoever. Due to the fact that linguistic activity is also a formof cognition, this type of relaxation seems to put us on impredicative grounds,but that is to be expected - after all, self-referentiality is deeply ingrainedinto the cycle in which we learn language to enrich our knowledge of theworld, which in turn allows us to enrich our language. Various studies intoself-referential theories of truth (among them, Kripke’s “Outline of a Theoryof Truth” [Kri75]) and the Liar Sentence suggest that self-referentiality is notnecessarily a bad thing it itself, provided one is wary of the processing loopsit creates.

What self-referentiality are we talking about? Suppose we extend ourrange of attention focus to allow us to focus attention on any object thatwe can know about. Now, since “knowing about” is a general cognitivedescription that involves also linguistic tools, we can have a situation wherewe try to select a reference for name n1 by appealing to the cognition provided


to us by the individual concept behind name n2 and we select a referencefor n2 based on the cognition provided by n1. That would certainly be anunwelcome situation.

Fortunately, however, while this sort of situation is logically possible,it is not really genetically possible. Well-foundedness of reference is gen-erally an inherited trait - as long as all concepts previously introduced arewell-founded (do not have infinite chains of reference), any newly introducedconcept will be well-founded as well. Since we are not born with potentiallymutually referential notions in our head, but we acquire them through lan-guage acquisitions, situations like the one described in the paragraph abovewill likely not happen.

That is not to say that we can feel totally safe. After all, our process oflanguage acquisition is not entirely monotonic. Sometimes, the contents ofsome concepts can change and then well-foundedness of certain notions can belost. However, this can occur in much more obvious cases then the extendednotion of reference in attention. For it to happen, it suffices that a concept c1

starts to rely on c2, while concept c2 already relied on c1. Conceptual changeis a tricky process and there will be no way to assure that it does not leadto vicious circles in definition - one can only bar it on the normative level byassuming that our system of linguistic concepts is in fact well-founded.

Returning to our “extended attention focus” topic, we will now simplyassume that we can focus attention on objects through any cognitive link.This cognitive link is what replaces the causal chains in our theory of propernames. Together with such a relaxation of our notion of attention, we have toadd two concepts: reference failure and degrees of cognitive directness.We will not formalize those concepts, but instead will explain them in somedetail.

Reference failure happens whenever our cognitive mechanisms which weconsider to be reliable fail to actually select objects from the world. Evendirect cognition is subject to reference failure, as evidenced by the skeptical


arguments from hallucination. The longer the chain of cognition, the morelikely reference failure is. A good example of a broken chain of cognition isthe case where we think a certain name references an object that a friendtold us about (for example, the person he saw last day at the beach), but ourfriend actually lied to us about seeing that person and he never saw anybody.We use a mechanism of indirect cognition (transferring a cognitive chain viaevidence from another person) and that mechanism failed.

Closely tied to this is the idea of degrees of cognitive directness. Generally,the longer the cognitive chain, the less reliable the cognition, although it isnot simply a quantitative issue - some forms of cognition are more reliablethan others. Certainly, the less direct the cognition, the bigger the risk offailure, although here language (or more generally, society) comes to help -we use the mechanism of sharing knowledge to strengthen certain aspectsof cognition, as in a group of fifty independent people all saying they saw acertain phenomenon is more reliable than one person making the same claim.

When it comes to cognitive chains, they can be in two states - they caneither be connected or dangling. A connected chain is one that actuallyselects a referent - a dangling chain is one that doesn’t due to cognitivereference failure. This is actually very important for our notion of propernames, since most cases of reference failure aren’t ones which we easily realize.While we might think that all our proper names select reference (and wecertainly assume that via linguistic trust when acquiring new names), itmight be the case that a reference chain associated with an individual notionis actually a dangling one, leaving us without a referent. This is somethingthe linguistic society might not realize for a long time, therefore, empty namescan function for quite some time in a society.

Now that we have described how attention focus acquires reference, weare ready to present the referential part of proper names: it is a referentialtest on the attention focus of an input state of the notion that realizes theindividual concept. In other words: λn.(o ∈ A(input(n))), where o is the

3.4. THE CATEGORIAL CONTENT OF NAMES 79

referenced object in question. However, to emphasize the cognitive chain, insome cases we will use a functional expression instead of a logical constant,for example λn.(seenBy(j, t1) ∈ A(input(n))) will refer to the object seen bythe person represented by the constant j (say, John) at time t1.

It is important to realize that in most cases, the functional expressionwill not be part of the meaning of the name. However, this is not a to-tally general rule. For one, the functional expression will be always presentin the meaning of empty names, showing the common dangling end of thecognitive chain. Furthermore, functional expressions will be the meanings offictional names, which differ from normal proper names in that they do notactually select any real objects, but instead refer to elements of a previousconstruction. Finally, functional expressions will be used in case of objectsfor which there is no social consensus that they have been clearly recognized,but which nevertheless have names - this corresponds to the awareness of thefact that the cognitive chain behind the name might be a dangling one.

3.4 The categorial content of names

It has already been mentioned that the referential test is not the only onethat contributes to the meaning of names. Another equally important oneis the categorial test. One could argue that the categorial test is alreadyimplicit in the attention focus - after all, to select an object, we have toknow what type of object to select. However, it is actually the other wayaround - the categorial test is the prerequisite to the reference test. In orderto be able to focus attention on an object, we first have to recognize the typeof the object.

This ties to a strong assumption regarding our cognition: that all ourobject selection is actually categorial in nature. There is no such thing as“objects per se” in the world which we recognize - each object that we selectalready has a category attached to it, if even the category is as wide as


“physical object”. Actually, there is a reasonably strong cognitive argumentbacking this assumption: the set of objects even in our direct field of cognitionis potentially infinite or at least very large and it does not seem possible thatwe select objects of all possible categories in parallel. We have to have afinite (and reasonably small) list of categories of objects that we focus on inany given situation.

Note that this tells us an important thing about the logic that should beused to describe natural language: to adequately reflect this categorizationmechanism, the logic should be typed. However, not any typing system willdo. In fact, the logic system should be one that recognizes subtyping (fora discussion of such typing systems, see eg. [Pie02]).

What is subtyping? Under typical logic, subtyping looks simply like setinclusion. Philosophers are a subset of the set of all people and Greek philoso-phers are a subset of the set of philosophers. However, subtyping is more - itis an essential relation between concepts. For example, the set of all peopleon a city square in Warsaw might currently be a subset of the set of citizensof Poland, however, the two types are not under a subtyping relation sincethe connection is not of a necessary (or, more precisely, analytical) nature.

In a linguistic community, subtyping has an important role to play -it can specify a minimal categorization level for certain names that allowthem to be used by the general population, while still allowing populationsof experts to use more fine-grained categorizations. For example, Aristotlecan be categorized as a person by the general populace, as a philosopher bythe more educated people and as a Greek philosopher by an even narrowergroup. The typing relation is a way of smuggling knowledge and contentin an otherwise content-less class of proper names. Note that this is not atrivial issue - not only can categories be quite rich semantically (and thusinsert quite a bit of content into the names), but the subtyping relation mightbe non-trivial as well.

3.5. THE SOLUTION TO PHILOSOPHICAL PUZZLES 81

3.5 The solution to philosophical puzzles

We have outlined our theory of proper names, now it is time to put the theoryto a test. How does it solve some of the most famous puzzles about propernames?

Let us start with the one that began it all: Frege’s puzzle about corefer-ence.

(5) Hesperus is Phosphorus.

How do we explain the cognitive significance of (5)? Under our theory,the answer is simple - since we allowed for names to select objects based onvery long cognitive chains, those chains interact with our cognitive world-maps on a very intricate level. Sentences like (5) serve to merge points onour cognitive world-map, therefore increasing our knowledge (by transferringsome bits of information from one node in the map to another) or evenintroducing conflicts.

The conflict introduction bit accounts for two other famous puzzles re-garding proper names - the lack of substitution in belief contexts and Kripke’spuzzle about belief.

(6) Pierre believes that London is ugly.

(7) Pierre believes that Londres is beatiful.

Belief contexts are a tricky concept that we will elaborate on further,but one thing should be obvious - whenever we report contents of individualbeliefs, we leave the safe normative world of linguistic concepts and enter themurky marsh of individual notions. The obvious conflict between (6) and(7) stems from the fact that, while the names “London” and “Londres” arecoreferential, Pierre might not know that and thus might have conflictingtheories about the object he places at the end of the cognitive chain labelled“London” and about the one at the end of the chain labelled “Londres”.


Since we cannot make any assumptions about the cognitive world-map of anysingle language user, we also cannot allow for coreferential name substitutionin belief contexts.

Since our theory is directly referential in the sense that it there is a non-descriptive reference selection mechanism behind object selection, it does notfall into any of the traps set by Kripke for the descriptive theory of names.We can still have language users that meaningfully use proper names theyhave just acquired - the legitimacy of such an activity is due not to the factthat they obtained some sort of description when first acquiring the name,but due to the fact that we believe linguistic reporting to be a viable way ofextending our cognitive chains (therefore, when a friend of ours utters

(8) Susie is a good basketball player.

we assume that there is actually a person called Susie that we can cogni-tively access). Note that, since we assumed meanings are actually determinedby use, the referent selected by a name depends on the way most languageusers actually use the name, or the cognitive chains they associate with it.This explanation is close to the one Gareth Evans’ proposes in his hybridtheory of names and does an equally good job of explaining the mechanismof reference change in case of proper names.

We did also note that sometimes, cognitive chains are left dangling andfail to select referents. This can be intentional or unintentional, and thusaccounts both for sentences of fiction like

(9) James Bond was an MI6 agent with a license to kill.

and for sentences with mistaken reference, like

(10) Nicolas Bourbaki was a bald mathematician.

where we might be convinced there is a referent, even though there isnone (in this case, because we were misled by a cognitive mechanism that

3.5. THE SOLUTION TO PHILOSOPHICAL PUZZLES 83

leads us to assume that books signed with a single name are indeed writtenby a single person).

Finally, the fact that there is a categorial test associated with propernames explains why sentences like (2) can be considered analytical. More-over, our discussion of subtyping and expert groups might suggest while forsome people the sentence

(11) Aristotle was a Greek philosopher.

might be analytical, while for others it might be a case of genuine knowl-edge, with neither group being incompetent language users (due to the lin-guistic competence being relativised to knowledge-dependent subgroups).

Moreover, our solution has the added benefit of being uniform. For ex-ample, we do not need to add special clauses to fictional names, treatingthem (like some causal theories do) as hidden descriptions - instead, we usea single cognitive mechanism to account for all grammatical cases of propername usage.

Overall, it does seem like our framework allows to handle proper namesin a way that is both internally consistent and allows us to solve multiplephilosophical puzzles about proper names. We will also see in further chap-ters how this solution ties in nicely to the handling of modalities, somethingthat Kripke has struggled with in Naming and Necessity. As we stated atthe beginning of the chapter - within the standard theories of language, itdoes not seem possible to solve many of the puzzles presented here withinone framework. It is only after one recognizes the cognitive aspect of lan-guage that those puzzles indeed become solvable. This dialectic result shouldreassure us that we are on the right track and further chapters should serveto strengthen this conviction even further.

Chapter 4

Modalities and the cognitivestructure

4.1 The importance of cognitive structure

In the previous chapter, we have discussed in much detail the importance ofcategories in proper names, as well as the role they play in natural language.In fact, categories are so important they deserve its independent treatment,and we will now turn to them specifically.

Recall that we have said how our “categorial lexicon” decides the waywe select objects from the world. This is a key point that must not beunderstated. Categories change the way we partition the world and thusdictate our ways of thinking. The way we think about various substancesinteracting with each other, for ¡example, is different now that we considerthem as subject to chemical laws and not as atomic objects in their ownright. Similarly, we have notions of objects interacting with others at adistance via powers that were not even considered 500 years ago. It is notjust our knowledge of those objects that matters - it is also the ways wecategorize them.

However, categories are not the only way by which we structure our cog-

85

86 CHAPTER 4. MODALITIES AND THE COGNITIVE STRUCTURE

nition. There are many other constraints that we impose on our cognitionand many others that are imposed by external factors (among others, theway our brain is structured and how it processes information). It would benaive to think that all of our theories about the world are of equal weight.Within our cognitive framework, there are fringe elements which are easilyrevised and there are some which we are almost absolutely unwilling to re-vise. The boundary between structural and non-structural elements of ourknowledge is therefore fuzzy and sometimes context-dependent. To pick anobvious example: in most contexts other than skeptical discussions duringphilosophical seminars, we do not really entertain the possibility of therebeing no external world. Even in context in which we do entertain such apossibility, we find it difficult to entertain it as a possiblity about our ownreality. However, we also tend to have different assumptions about the worldwhich are not as obvious, but just as strongly held: for example, most of usdo not believe it possible for people to randomly start killing people in thestreet without a reason. Such a belief ultimately rests on certain assumptionsabout human nature - ones that we are not realistically willing to relinquish,even if various events in the world sometimes prod us to.

Certainly, language has an important function to play when structuringour cognition. Since our world knowledge is largely shaped by linguisticmeans, it is hard to imagine that there would be no linguistic means ofcoordinating and communicating our cognitive structure. Indeed, in thischapter we will show that such linguistic means exist and in fact, have alreadybeen used in the very text of the previous paragraph. What we have in mindis the word possibility, which, along with its dual - necessity - are instancesof modalities.

4.2. MODALITIES AND THE PHILOSOPHICAL TRADITION 87

4.2 Modalities and the philosophical tradi-tion

Modalities, as opposed to representations, are not a new philosophical topic.Aristotle devoted quite a lot of attention to the topics of possibility and ne-cessity, in fact, it was one of the central topics of his enquiry. Many other an-cient philosophers devoted their entire studies to modalities, with stoic logicputting a noticeable emphasis on the notions of possibility and necessity.Many logical puzzles of the ancients involve modal notions, with probablythe most famous one being the “Master Argument”, attributed to DiodorusCronus ([Ohr11]). Medieval philosophy also contributed to the study, withprobably the most famous modal argument of all times, Anselm’s proof ofthe existence of God, later enhanced by one of the founding fathers of mod-ern logic, Kurt Godel ([God95]). Modality plays a central role in some of themost famous contemporary philosophical debates and in fact, Kripke’s workmentioned in the previous chapter is also full of modal arguments. Kripkehimself attempted to revise the way we think about modalities, however, hisview of necessity is still very much entrenched in the philosophical traditionwhich gives modalities a metaphysical reading. It is our intention to dis-pense with that tradition here and instead give modalities an almost purelycognitive reading - however, to do that, we must also explain the apparentmetaphysical connotations of modal terms.

It should be noted that in contemporary philosophical tradition, modalterms have received two main types of treatment. One was the metaphysicalone, compounding necessity with essentiality and sometimes with naturalkinds, the other was an eliminativist one, which proposed to dispose withmodalities whatsoever, claiming that modal terms cannot be given a clearsemantics. The founding father of the latter approach was Quine (his criti-cism of modalities can be found in multiple texts, including [Qui43], [Qui47],[Qui53], [Qui70] and [Qui76]). However, his arguments against modal con-


texts seem fatally flawed and have been quite thoroughly debunked over theyears (the most influential of the critics being probably Ruth Barcan-Marcus[Bar93]) - for an overview of the arguments and their flaws, see [Cie08]. Nev-ertheless, the essentialist treatment of modal terms is problematic too, andin recent years, various epistemic theories of modalities have also emerged,such as the ones developed by George Bealer ([Bea04]). While those theoriesshare some of our intuitions about modal terms, they lack some necessarycomponents to make them adequate, mainly due to the traditional treatmentof concepts as abstract objects devoid of cognitive content. Again, antipsy-chologism triumphs.

While in this study we build our modal theories mostly from scratch, wedo not want to build them in a void - therefore, we will try to refer our modaltheory to various examples and solutions present in the contemporary modaldebate. We will also try to show how certain traditional concepts of modalitycan be reconciled with our approach, also in cases when such reconciliationseems extremely problematic and unlikely (the most prominent example willbe Lewis’ modal realism [Lew86]).

4.3 Analyzing possibility and necessity

Let us take three sentences involving modalities. The first one involves acertain historical event:

(1) Napoleon could have won the battle of Waterloo.

The second one is a well known “natural kind” essentialist claim:

(2) Water is necessarily H2O.

The third one deals with mathematical entities:

(3) It is not possible for there to be a greatest natural number.

4.3. ANALYZING POSSIBILITY AND NECESSITY 89

Our goal is to find a common notion of possibility (and dually, necessity)that explains the meaning of those sentences and possibly provides the truthcondition, if any is to be found.

Let us start with (1). How would we explain that kind of sentence?One explanation would be that there was a possible turn of events in whichNapoleon won the battle. Here, we are indirectly appealing to the notionof possible worlds, which has been often proposed as a semantics for modalterms. However, an appeal to possible worlds does little to actually explainmodalities. The main problem with possible worlds is that they are possibleworlds. The adjective “possible” is still present. When we explain (1) bysaying that there was a possible situation in which Napoleon won, we cannotremove the term “possible” because, historically, there was no such situation- there was only the actual situation in which Napoleon lost.

One could try the epistemic way out - replacing the adjective “possible”with a more basic term to which possibility reduces. One such notion is thenotion of “conceivability” - we could say that there is a conceivable scenarioin which Napoleon won. However, there are at least two main problemswith such a solution. One is that conceivability seems to be a very privatenotion: what is conceivable to one might not be conceivable to another. Itis also a reasonably immutable notion: once something is conceivable to aperson, it is hard to persuade them that it’s not really the case. The notionof conceivability seems to lack normativity, which makes it a bad candidatefor explaining the linguistic phenomenon of modal terms.

Another problem is the apparent difficulty in applying the notion of con-ceivability to abstract examples such as (3) or even (2). The problem is thatit’s not just hard to imagine water not being H2O, it is hard to imaginewhat would it mean for water not to be H2O. One could then argue, likeBealer does, that one should not think about conceivability in the intuitivesense, but rather as some kind of normative proper understanding of concepts.However, this sort of explanation is actually an ignotum per ignotum, since


we would like to know what it is about the concept of water that requires itto be always associated with the concept of H2O, not that it is simply thecase that it must be. In other words - if we turn to the intuitive notion ofconceivability, we lose empirical adequacy with respect to the use of the word“possible”; if we turn away from the intuitive notion, we risk trivializing thetheory.

In fact, maybe we should trivialize the theory? Maybe we should acceptthat the terms “necessary” and “possible” are in fact primitive and no anal-ysis can be provided? While it is always hard to rule out that this mightbe the case, we must remember that treating certain concepts as primitive isthe last resort of the theorist - only used if an explanation has been soughtand has not been found. Clearly, we have not yet searched far enough.

If the conceptual way is not the right one, maybe the logical one can help?There is certainly a long tradition of treating sentences like (1) explained bya form such as

(4) The proposition “Napoleon won the battle of Waterloo” is notinconsistent.

This tradition dates at least a couple of centuries and has certainly en-joyed much success. However, it has also multiple pitfalls, one if which itshares with the conceivability solution: it extends the notion of possibilitymuch too far. Surely we might at least want to argue the point that it wasin fact impossible for Napoleon to have won the battle, even if the proposi-tion mentioned in (4) does not give any hints of logical inconsistency (whichshould be the only proof needed of it being possible). On the other hand,the logical consistency theory does little to actually explain empirical claimslike (2), since for it to be inconsistent, an axiom must be provided claimingthat water is indeed H2O, basically giving little advantage over the notionof modalities as primitive terms, unless we provide external grounds for thevalidity of such an axiom. Only with propositions such as (3) does such a


theory seem to work well, but that is hardly surprising giving that math-ematics is the one area where logic seems to actually serve an explanatoryfunction well (and even on the grounds of mathematics propositions such as(3) start being volatile when you consider eg. finite models).

So, if both the conceptual and the logical theories fail us, what else can werely on? We seem to be left with metaphysical theories. Of those, the mostdirect is David Lewis’ modal realism - a theory that claims that wheneverwe state a modal truth, we are actually speaking of a real possible worldin a multiverse of possible worlds and about individuals in there. In otherwords, (1) translates to the fact that there is some possible world in which acounterpart of Napoleon actually wins the battle of Waterloo.

There are certainly some benefits to Lewis’ theory. For one, it is extremelyconsistent internally and it does indeed provide an explanation to modalsentences. However, it does so at a great price. For one, it creates a veryrich ontology, one that suggests we are living in a multiverse of real possibleworlds. The more serious problem, however, is the epistemic problem - inorder for us to be able to meaningfully talk of truth conditions for modalsentences, we have to assume that we have some sort of epistemic connectionwith those possible worlds, while at the same time having no real connectionto them. Therefore, we can say things about the possible worlds, but thereis no way we could travel to one. It must be hard to accept such a scenariowithout asking about the nature of this epistemic relation. It certainly hasa magical feel to it, with the notion of our epistemic connection with thepossible worlds being left primitive. This does not rule out Lewis’ theoryas a bad one, but Occam’s Razor demands we continue searching for other(nomen omen) possibilities before accepting the Lewisian metaphysics.

Another option is to take some version of essentialism as the underlyingtheory for modality. It is hard to point one person responsible for that theory- it certainly owns much to another Lewis, Clarence Irving, but its mostfamous recent exposition is probably that by Kripke in Naming and Necessity.


In this view, necessity simply involves certain objects having essential featuresand modal discourse is all about asserting those features. Kripke providessome examples: for a human being, having certain parents is an essentialfeature, i.e. if Joe is Frank’s father, then the following sentence is true:

(5) It is not possible for someone else than Joe to have been Frank’s father.

Note that the difference between essentialism and Lewisian modal realismis akin to that between the old Platonian - Aristotelian distinction betweenessences. For Aristotle, the essences (forms) lie in the things themselves,whereas for Plato, they populated a separate realm of ideal objects. Similarly,for Kripke there are no possible worlds - possibilities are merely constructionsthat do not tamper with the essences of objects.

However, how do we discover the essences? In other words, what is theanswer to the epistemic problem? Kripke leaves that matter very muchopen, only connecting it vaguely to empirical research. However, it is farfrom obvious how would determining the essences of objects look like. Afterall, there does not seem to be any real empirical test to determine whethersomething is a necessary or contingent feature of an object. We do often fallback on “natures” of things when we want to argue modal claims, but thosediscussions about “natures” are certainly not empirical at its core.

First of all, let us notice that the truth of sentences like (5) is debatable.We could, after all, claim that Frank could’ve had another father (for exam-ple, Joe’s twin brother). Kripke would respond to this by saying that it’s justan illusion and in reality, that wouldn’t be Frank anymore, but that rebuttalcertainly seems to be question-begging - after all, we did in fact claim that westill believe it possible for Frank to have had different parents and our belief(or the intuition behind it) is supposed to show that (5) is false, therefore adefined parentage is not an essential property for being a human, therefore,such a rebuttal has no force.

We could then note that we have no real way of establishing the essencesand no amount of empirical data is going to help here. However, it still


does seem that we use modal sentences in a meaningful way. What does thismeaningfulness relate to?

In fact, the observation that modal sentences are meaningful goes a longway to refute the modal primitivism option. After all, if modal notions wereprimitive, they’d have to serve a purpose by themselves to be invoked inspeech (in a similar way to ethical terms under Moore’s theory [Moo03]).However, it does not seem like modal notions are used in that way. Instead,modal notions are used for epistemic reasons - to reassure other language-users that a certain piece of knowledge is certain.

So, maybe an epistemic version of modality is in order? In fact, some ofthe modalities we use are epistemic in nature; that is the case, for example,if we say something like:

(6) In fact, Mary might be at the university right now.

However, we run into immediate problems with sentences such as (1). DoI really want to state by (1) that I do not know enough about the battleof Waterloo to be certain of its outcome? Surely not, in fact, (1) might beinvoked by a historian studying battles from the Napoleon period in a well-informed, historical debate. If modalities are in fact epistemic, they are soin a subtler sense than just expressing certainty.

To ascertain that sense, let us try to think about how we describe con-ditions making (1) false. Why would someone argue that Napoleon couldn’thave won the battle of Waterloo? There are many reasons, but most involvehow we perceive certain laws regarding reality. In this case, there might notbe a plausible scenario in which Napoleon wins at Waterloo (because, forexample, the disparity in the number of troops was too great).

However, such analysis gives us next to nothing - we replaced the notionof possibility with the notion of plausibility. We need something more, some-thing that will tell us what this plausibility consists in. For some hints, let usturn towards (2). What would be our explanation of how we argue for (2)?


If we discount natures, the next thing that comes to mind is that we simplycannot imagine water as being anything other than H2O. Of course, somemight argue that this is not the case - if something is clear and drinkableand usually found in lakes on Earth, then it will be water regardless of itschemical composition. There are multiple arguments around that, among themost famous being that from Putnam’s Twin Earth experiment ([Put81]).

What way could we respond to such a counterargument? Well, for one,we could say that water as a chemical substance has to be H2O. Here, weare even consistent with Kripke’s view - this is not an analytic truth, this issomething we discover empirically. However, we added another specification:the “as a chemical substance” part. How could we understand such a speci-fication? Certainly, it cannot be part of the “essence” of water. The reasonis simple - the “chemical substance” part specifies by what aspect we selectwater from other entities in reality. We could try to argue that we designatewater by using demonstratives and connecting to water by causal chains, butwe are still left with the problem of multiple inhabitation: a pool of water(as a chemical substance) also contains a certain drinkable fluid, a certainlight-reflecting surface and so on. We still have to select a category even ifwe use demonstratives to point at a certain region of the physical world -this is tied to the demarcation problem we discussed in chapter 2.

Now it would be useful to recall an observation we made before - categoriesare not tied essentially to world fragments. Even if there are certain objectsin the world and thus material categories, the base categorial distinction isthe one we make internally to demarcate certain world fragments. Perhaps,then, modal discourse is not about things in the world at all? Perhaps it isabout the categorial structure of our language?

Let us recapitulate our hierarchy of entities. Linguistic concepts are testson notions. Notions possess categorial content. Since some tests withinconcepts are structural tests, concepts enforce a categorial structure on ourlinguistic system. This categorial structure is very important for coordinat-

4.4. TYPES OF MODAL CONSTRAINTS 95

ing our linguistic activity - after all, when synchronizing meaning betweenlanguage users, we want to be able to establish that type of objects a certainconcept denotes. It is reasonable to assume we have a separate way to dothat in language, and now modalities emerge as the apparent mechanism inwhich this is realized.

Of course, there isn’t any social convention saying “modalities are used totalk about the categorial structure of language”. After all, language itself doesnot have too many meta-level mechanisms. This, again, has sound cognitivegrounds - individuals which are most prone to learning language - infantsand young children - are notoriously incapable of understanding meta-levelspeech. Then again, this is not really needed - for learning language, we needa means to train our brains in recognizing the proper categorial structure,not a means to discuss the categorial structure by itself.

However, there’s a side effect to all this. Since modality is, as one couldargue, “cleverly disguised” as a way of talking about constraints on reality,it is used to in a parasitic manner to implicitly describe other constraints onour overall world-view, not all necessarily of a categorial structure. In whatfollows, we will try to provide a typology of modal contexts and argue thatthey fall under the same general category of providing cognitive constraints.

4.4 Types of modal constraints

First, let us ask the question whether it is possible to distinguish any trulyessential necessities. After all, our cognition is constrained not only by cul-tural factors. It is also constrained by the ways our brain is structured andhow it functions. In other words - there are certain constraints based onhow our mind works and owing to the Kantian tradition, we will call themtranscendental constraints.

One good example of transcendental constraints, which we will explorefurther in a separate chapter, is the expression of mathematical statements.


We adopt a Kantian perspective on mathematics - in such a perspective,mathematics is the study of the structure of our cognitive mechanisms relatedto counting, measurement and reasoning. Therefore, statements about math-ematics are really statements about fundamental properties of our thinking.Sentences such as (3) are therefore expressions of transcendental modalities.

Another type of modal constraints is the one encompassed by classicexamples such as (2). Those form the class of categorial constraints.However, we should distinguish two types of categorial constraints. The dualone to (2) would be

(7) Water is necessarily a chemical substance.

It should be apparent what the difference between (2) and (7) is. (7)is true due to a purely analytic fact of the concept “water” falling undercategory “chemical substance”. This can be established without any empiricalfindings, what’s more - no empirical data can determine propositions such asthe one expressed in (7). We will call such constraints categorial analytic.On the other hand, there are propositions such as (2), which, when coupledwith a founding definition such as

(8) We will call that chemical substance, whatever it is, water.

certainly leave room for empirical research. When we discover the chemi-cal composition of a chemical substance, we certainly come close to what thetraditional approach to modalities has referred to as discovering the essencesof things in the world. Therefore, we will call such constraints categorialessential.

Sentences such as (2) seem very close to sentences such as

(9) Water necessarily boils at 100 degrees Celsius under normal conditions.

However, they are of a different nature. The fact of boiling at a giventemperature is certainly not constitutive of water as a chemical substance.

4.4. TYPES OF MODAL CONSTRAINTS 97

On the other hand, we believe there is something final to this statement, as itowes its truth to certain regularities in the world. Those are not regularitiesthat are temporary - they are eternal laws that allow us to operate in theworld. Our cognition would be impossible without the implicit assumptionthat there are such regularities in the world. It is not something which we canargue, instead, it is to be a pragmatic necessity, a habit, as Hume named itin his famous critique of induction. Again, it is an ontological debate whichwe are not prepared to fully handle here whether there really are laws inthe world, but for our needs we will assume that indeed there are, withoutassuming any sort of metaphysical background for them.

Sentences expressing laws about reality will introduce a constraint typewhich we will call a nomic constraint. While in the paragraph above, wementioned mostly metaphysical laws, the term is wider - it can be used torefer to social or conventional laws, for example

(10) A metre necessarily has 100 centimetres.

We still have a final puzzle left: to explain the use of modalities in sen-tences such as (1). Neither of the types of constraints above really fit thepicture - surely there can be no law about Napoleon winning or losing thebattle of Waterloo, seeing as it was a one-time event, and there is certainlynot a categorial or transcendental constraint that can be applied here. Onecould try to argue that there are certain general laws involved that make thissentence true or false, but that would probably be stretching the notion of“law” as something universally applicable and recognizable too far. We donot want the nomic constraints to apply to “ad-hoc” laws. For that, we needa weaker category.

In most conversations, topics or settings, there are certain facts or state-ments that we consider fixed. For example, when we talk about what J.R.R.Tolkien might have changed in the world of Silmarillion (a viable debategiven as the work was never finished and was published posthumously by


Tolkien’s son), we do not want to stray from the assumption that the workwas a fantasy mythology. Thus, we might subscribe to statements such as

(11) It would not be possible for elves to fly in starships.

(11) is an example of a sentence that doesn’t even talk about real objects- however, in certain contexts we would recognize it as true. Sentences suchas (11) (but also such as (1)) form a large category that we will call topic-bounding constraints. They express context-dependent parts of our cognitiveworld-view that we want to consider fixed for a given discussion, similar toelements of Stalnaker’s context set ([Sta99]).

We just said that (1) expresses a topic-bounding constraint, but manyreaders might object - after all, it is not really clear that (1) expresses anyconstraint as all. After all, there are not many historical discussions that wewant to start with the assumption that Napoleon could have won the battleof Waterloo - there probably are many more that we might want to concludewith such a statement. This points to an important matter that we now haveto clarify.

Up to now, we have used modal propositions to illustrate the respectivetypes of modal constraints. However, there is no one-to-one correspondencehere. Most modal propositions will be true or false due to a combinationof constraints. (1) might be true due to certain topic-bounding constraintsabout the historical assumptions we make about Napoleonian times, but itmight also be true due to certain nomic constraints, for example disallow-ing the instantaneous transport of large groups of humans over distances ofmultiple miles.

It must be underlined that the typology presented here is a typology ofmodal constraints, not of true modal propositions. One true modalproposition might owe its truth to multiple constraints and might not evenexpress a constraint directly - instead, it might be a result of logical inferencesfrom a modal constraint. It is best to think about modal constraints asaxioms from which the theory of true modal sentences is derived.

4.5. MODAL SEMANTICS AND TRUTH CONDITIONS 99

4.5 Modal semantics and truth conditions

Now that we have established the typology of modal constraints, we need tolook at the semantics for modal terms such as “necessary” and “possible” andthe truth conditions for modal sentences. The reason for this is that whilewe have provided an intuitive explication for the various modal constrainttypes, it is still not clear from that account what the proper semantics formodal terms might be.

The big problem with modalities, as far as our framework is concerned, isthat they are operators. That means we will not be able to provide as simplean account for necessity as we do for chairs. Operators, when understoodas linguistic concepts, act upon propositions, which are abstract entities bythemselves. Thus, they do not seem to parts of the world in the same senseas material objects described before were.

However, that does not mean that they are not parts of the world at all.In fact, we could simply treat them as another type of objects and claimthat they should be handled in our semantics in the same way as materialobjects are. This wouldn’t be plausible in our approach, however, because wewant our semantics to be cognitively sound. We cannot make it so withoutaccounting for the epistemic problem with abstract objects - how do weactually access them? After all, one of the most prominent problems withPlatonism is the problem of accessing the abstract ideas with out cognitivefaculties.

Fortunately, we have an answer to that problem that both solves multi-ple problems and does not tie us deeply into ontological debates. We willassume that the world of abstract objects is simply an abstraction of theworld of individual mental entities. After all, we did the very same thingwith linguistic concepts - why not extend this approach to abstract objectsin general? Of course, the matter gets very tricky here - after all, when wedealt with linguistic concepts, we proposed a very specific way to abstractfrom notions - here, we need a more general way to handle things, seeing as


we deal with all arbitrary abstract entities. However, we can actually extendthe approach we formed with concepts, by saying that all abstract entitiescan be understood as tests on notions from which they are abstracted.

Now comes the tricky part. Let’s say we have a cognitive attitude towardsthe abstract object “freedom”. That abstract object is (by definition) anabstraction from some notion we have, namely the notion of “freedom” (notethat we name the notion and the abstract object in the same manner, butthey are not the same object, other than the notion of “freedom” implementsthe abstract object - in this case, concept - “freedom”). What does thatmean? The first answer could be that I have an introspectively accessednotion that we cognitively process. The notion is a mental object, it canbe subject of attention, so everything seems to be in order. However, thiswould lead to a very naive theory of transparent introspection, which is veryquickly falsified empirically (we use complex grammatical structures, but wecannot easily access the structures themselves). In order to be consistentwith this view, we have to assume that an additional object - a notion of aconcept - gets accessed here. Not via attention - but via memory recall, asthe notion of a concept is a notion like many others, but one used to selectabstract objects.

How does it select abstract objects? The answer might be striking at first- it does so directly. However, the answer is not as striking if we considerwhat abstract objects are - they are, after all, tests. To access an abstractobject means simply to perform certain tests - since those tests are, by theirnature, cognitive operations, there is no need to actually introduce anotherintermediary layer here. The “directness” here isn’t one of ontological iden-tity (after all, notions of abstract objects are concrete mental entities andabstract objects are, well... abstract objects), but merely one of immediaterealization - to select an abstract object means simply to perform the testswhich the object constitutes of.

Of course, we might have an erroneous notion of an abstract objects.

4.5. MODAL SEMANTICS AND TRUTH CONDITIONS 101

Why? We’ll study the previously introduced realm of concepts as an illus-tration. Remember that concepts are constructed in use and that it mightbe the case that no single member of the linguistic community actually real-ized what the concepts being used really are. Therefore, we might have ournotions of, say, the concept of “chair”, but neither of them has to correspondto the real concept of “chair”. We might simply be considering different teststhan those that are actually in place (a similar approach regarding vagueterms appears in [Wil97]).

We have put together an account of what it means for us to access ab-stract objects, but now we need to settle another matter - what types ofabstract objects are propositions? Since all abstract objects are abstractionsfrom certain notions, we need a base notion type for propositions. Let’scall that notion type statements. Statements are mental objects that cor-respond to the mental act of assertion. We will leave aside the questionwhether temporary mental objects (ones that use short-term memory) canbe still considered notions, as well as the related, but probably more complexdebate of whether assertions are in fact conscious, i.e. whether notions ofthe assertion process are automatically formed during assertion - while thatis no doubt an interesting debate, it is one slightly out of scope of this work.We will assume that assertions give rise to statements, that statements arenotions and can be tested for various properties.

Statements as notions are generally situation selectors. That means theyselect a spatio-temporal world fragments. In the case of many operators,we will be interested in the notion itself - in what world fragments it selects.However, we will not need to enter into that field while dealing with necessity,as this operator has a totally different function.

In case of necessity, as we already noted, the question is whether ourcognitive world-view would survive accepting the falsity of the statement inquestion. As we noted in our typology of modal constraints, not the entiretyof our world-view needs to be taken into account when considering such a


claim - we will assume a contextually filled-in scope variable that tells us howmuch of our world-view is to be taken into account by the modality.

How do we test whether a certain statement is necessary? We alreadyknow what a statement is - it is a notion of an assertion. An assertion isa cognitive process. We can query the start and end states of that process.The test for necessity looks as follows:

Nec := λ(n : stmt).¬Cons(wV iew(endState(proc(neg(n))), S)

where Cons is a consistency predicate (checking whether two cognitivestates are consistent with each other), wV iew is a function that returnsthe cognitive world-view from a cognitive state, endState is a function thatreturns the final state of a process, proc is the function that returns a processselected by a notion and neg is a function that returns a negated statement.S here is the contextually filled in scope variable. The notation n : stmt heremeans that n is of type stmt (statement), otherwise the function neg wouldnot make sense here.

The shape of the formula immediately shows us the reason for the dualitybetween necessity and possibility, as well as the intuition we might possessthat possibility is the simpler notion. Let us check whether this really works.Consider (2). Applying our definition to the statement brought forward by(2) tells us that the world-view produced by assuming that water is not H2O

would be inconsistent with our world-view. If our world-view assumes thatwater is categorially a chemical substance, then that would indeed be thecase. Note that another seemingly unintuitive conclusion arises here - if wedo not know, as a linguistic community, what the chemical constitution ofwater is, then this sentence is true as an epistemic modality. Therefore, wateris only possibly something other than H2O until we have established that itis, indeed, H2O.

Let us now consider (1) (or more precisely, its negation). Here, we say that

4.6. CONCLUSIONS 103

asserting the scenario of Napoleon winning the battle of Waterloo conflictswith a salient part of our world-view. If that salient part indeed containsfragments which contradict such a scenario, then that will indeed be thecase.

Finally, (3) is probably the simplest one to analyze. As long as we agreeon the notion of “natural number”, there is no possibility to reconcile the ideaof a greatest natural number with our cognitive world-view simply because itis not possible to reconcile it with our cognitive mechanisms. In other cases,we can have a process that asserts the relevant statement and returns aninternally consistent state that is simply inconsistent with our current world-view. Here, the function proc will simply return an internally inconsistentend state, which represents a processing error (which computer scientistsusually tend to denote with ⊥).

4.6 Conclusions

It does seem like we managed to reach a cognitively plausible and satisfyingdefinition of necessity, but it would be a shallow victory if that definitiondidn’t fit well into the debate about modalities which we previously men-tioned. Let us try to reconcile our position with that of the most famoushistorical standpoints.

It seems our solution is very far from that of David Lewis, but we alreadystated previously that we will aim to reconcile our approach with modalrealism and it is indeed possible to do so. Remember that our world-view is,through representations, referential in nature. When we change our world-view, it is because our cognition of the world changes. However, we couldtry reversing the process - changing our world-view and imagining how thechanged world would look like. In fact, we do things like that when readingall sorts of fiction literature. We can therefore think of Lewis’ modal realismas our idea reflected through the lens of referentiality - his other possible


worlds are simply worldly reflections of our changed world-views.Our approach seems very easily reconcilable with that of Kripke, espe-

cially with his idea of modalities as constructions on reality. Again, we donot actually perform constructions on reality, we perform them on our cog-nitive world-view. This is also why the problem of essences that so stronglypuzzled Kripke is gone from our account - there are no essences of thingsin our account, the thing closest to them being the categorial essences andadmitted laws.

Finally, our approach fits very nicely with various conceivability and con-ceptual approaches to modality. What we add here is our entire frameworkfor bridging conceptuality with the world through referentiality (the mecha-nisms described in chapters 2 and 3) and through social coordination, so inour case, when we speak of the possibility of changing certain notions, it nolonger has a purely individual feel of a thought experiment, instead carryingin it the responsibility of changing a working representation of the real worldthat is also shared with the entire linguistic community.

To summarize, the account presented in this chapter provides a unifiedaccount of modality which, again, exploits the cognitive layer to provideanswers to various philosophical conundrums as well as giving a satisfactorysemantics for modal sentences for many different types of modalities. In thecourse of the chapter, we have encountered the problem of abstract entitiesand of transcendental objects (abstract entities that are not contingent onany external world-fragments, but only on the structure of our cognition), towhich we will devote the following chapter.

Chapter 5

Mathematical objects andproofs

5.1 The nature of mathematical objects

On many previous occasions, we have stated that in this text, we do notwant to engage in ontological discussions nor make metaphysical statementsunless absolutely necessary. However, in this chapter we have to make anexception. The reason is simple - much of the discussion on mathematicalobjects is concerned with their nature and ontological status and we havealready made a very strong statement concerning that very status in the pre-vious chapter. Therefore, before we talk about mathematical objects more,we need to discuss the metaphysics.

The idea that mathematical objects are “in our head” is nothing new, infact, one of the three major contemporary metaphysical positions in philoso-phy of mathematics, intuitionism, stems largely from this view. In fact, ourposition will in many points be consistent with intuitionistic views and wewill borrow heavily from proof theory, which has also historically been devel-oped by mathematicians within the intuitionist tradition. However, since werely heavily on our cognitive level of description, we cannot simply describe

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106 CHAPTER 5. MATHEMATICAL OBJECTS AND PROOFS

mathematical objects as “mental”. In fact, the key idea in our entire researchso far has been abstracting from individual mental entities to account for in-tersubjective abstract objects - the area of mathematical discourse will be noexception.

Therefore, our approach will be more Kantian than intuitionistic in na-ture, as we will claim that mathematical objects are, in fact, objective, orat least intersubjective - if in a specifically convoluted way. We certainlydon’t want to subscribe to the view that mathematical objects are purelyconstructions, as intuitionists would want it. What we do want to claimis that mathematical objects are transcendental in nature - they representproperties of our cognitive apparatus that are common to all human beings(mental deficiencies aside) and as such exist objectively - even if in a non-direct way. They are mostly discovered rather than constructed, althoughthere is a specific entanglement involved here - as new mathematical theoriesare constructed to explain certain patterns in our cognition, they themselvescan be objects of enquiry and thus give rise to more complicated theoriesand more complicated mathematical entities. One can envisage this as apyramid-type construction - the basic blocks are the real properties of ourcognition, the first level is the theories constructed to explain them (them-selves structures of our cognition, although derived, not basic) and so on.We will later see why this pyramid-like structure is important.

To make this picture clearer, let us ask ourselves this question: what isthe reference of notions that are supposed to represent mathematical objects?Are mathematical notions similar to propositions in the sense that they referto abstract objects? Our answer to this question would be “no”, at least forthe base mathematical notions (eg. the natural numbers “one”, “two” and“three”). The reference is, as with “normal” notions, to external objects -in the wide sense of “external” which we described in chapter 1. Apart fromthe fact that our cognition of mathematical objects is focused inwards ratherthan outwards, there is nothing special about mathematical objects. They

5.2. INFINITY AND THE GROUNDING OF AXIOMS 107

do, however, provide an excellent sample of our capability for constructinghigh-order abstract concepts, for a multitude of reasons which we will furtherdiscuss.

Note that, in a way, this view about mathematical objects is very similarto Platonism. One could say that they key difference is that the objectsreally live within us, but that would be misleading - after all, mathematicsis not the study of any patterns in our individual cognition, but ratherthe study of common patterns of human cognition. Therefore, in a sense,mathematical objects do live in an ideal, Platonic world. However, the bigadvantage our approach has over Platonism is that it solves the epistemicproblem. We have access to this world of ideal objects due to the fact thatthose objects constitute our very mental construction.

5.2 Infinity and the grounding of axioms

The title of this section might seem odd indeed - what does one have todo with the other? At first glance, the concept of infinity is completelydisassociated from the grounding problem for axioms. However, as we explorethe matter further, we will find that they are, in fact, very closely related.

The XX century resulted in many results in the foundations of mathe-matics, the most famous of those being probably Godel’s theorem claimingthe incompleteness of first-order arithmetic ([God31]). Godel’s result showedthat for a sufficiently rich theory (one that is able to efficiently encode itsown formulas) it is possible to find a sentence within the language of thattheory that is neither proved nor disproved by it.

Godel’s proof was non-direct (it used the diagonal lemma to establish theexistence of such a Godelian sentence rather than show one), but quickly mul-tiple such sentences were discovered. Moreoever, a technique (called forcing)was created that allowed for proofs of independence in axiomatic set theory([Coh66]). This resulted in the discovery that, within Zermelo-Fraenkel’s Set


Theory, the famous Continuum Hypothesis (“there exists a cardinal numberκ such that ℵ0 < κ < c”) is independent from the axioms.

This result called quite a stir. After all, we were no longer consideringabstract Godelian sentences - we were talking about the, seemingly real,universe of sets. There either were sets of cardinality between ℵ0 and c - orthere weren’t. In the case of arithmetic, we still had the standard model to fallback to - first order logic, due to its natural limitations, was unable to pick thestandard model over other infinite models of arithmetic, but we all had theintuition what the genuine natural numbers were. In the world of infinite sets,however, the matter was not as simple - there was no corresponding intuitivestructure which could resolve the continuum hypothesis. How could we solvethis seemingly unsolvable dilemma?

It is worth taking a step back from this picture and asking ourselves thequestion: where did the problems start? After all, any and all sentencesabout specific natural numbers seem perfectly decidable (in fact, they aredecidable - any formula or sentence involving restricted quantifiers, whichcorresponds to a finite subset of natural numbers, is decidable). It is onlywhen we start talking about infinity that problems start arising. Why wouldthis happen?

Let us now consider our assumption regarding the nature of mathematicalobjects. When it comes to specific natural numbers, our grounds for identi-fying them are pretty obvious - all humans have an innate ability to count, atleast with respect to small numbers ([Sta80]). Moreover, all humans possessthe ability to “add one”, or perform what in arithmetic is called the successoroperation. However, our ability to perform infinite operations is somethingthat should be called in question here.

It should be obvious that we cannot perform an infinite count in a finiteperiod of time. However, another cognitive ability we have is induction -the capability of constructing a general statement based on a finite samplesize and an assumption of regularity. Therefore, once we have assumption


about the structure of natural numbers, we can formulate statements forall of them. Since mathematical objects, unlike worldly objects, are perma-nently available, we can test this induction principle any number of timeswe want, therefore, it seems to us much more infallible than the materialinduction principles used to reason about the outside world. However, andthis is probably the most controversial claim made in this text so far, wewill want to state that apart from the accessibility of mathematicalobjects, there is no essential difference between mathematical andnon-mathematical induction. The difference is of a quantitative, not aqualitative nature.

This claim might seem quite strong and counterintuitive, so we mustmake a clarification here. Where we mean that the difference is of a quan-titative nature, we simply mean that the mathematician has much more (infact, potentially infinitely many) opportunities to repeat his experiment, inexactly the same and fully controlled test conditions. For many philoso-phers of science (and for many scientists alike), this very fact (the possibilityof conducting the experiments in an “ideal laboratory”) would qualify as aqualitative difference. Therefore, it must be noted that the term “qualita-tive” refers to the nature of the induction itself, not the mechanisms used inthe context of discovery to generate the inductive hypothesis.

So, where does infinity appear? The answer is simple - it appears whenwe take our principle of induction and reify it. Note that in Zermelo-Frankelset theory, we do not propose the existence of an infinite set without anyproperies - in fact, the so-called axiom of infinity actually establishes theexistence of an inductive set.

However, note the subtle jump which we perform here. The finite naturalnumbers are indeed components of our cognitive structure. The ability toreason about properties of any number given the assumption of the structureof natural numbers also fits there. However, it requires a much strongerassumption to actually reify the entirety of natural numbers based on the


induction principle. Moreover, the induction principle is already a notionabout the natural numbers, so any notion built upon the induction principlewill be on the second level of our conceptual pyramid.

Now, the problem with reifying infinity is that we are no longer buildinga notion that is based on our cognitive structure. Instead, we are buildinga notion that’s based upon a notion of our cognitive structure - a higher-order notion, if one prefers that terminology. Higher-order notions refer tothe properties of the lower-order notions they are built upon - but, sincethey are not based on real physical properties, but on abstractions, they areincomplete when it comes to determining the properties of those lower-ordernotions that are not defined within them.

This is when the problem of grounding of axioms comes in. As long weuse axioms to describe a real structure of our cognition, they aren’t reallytotally arbitrary - in a way, they have referential content. However, once westart axiomatizing objects that aren’t really part of our cognitive structure,but are constructs devised to understand that structure, we are often leftwith a degree of choice.

For example, consider the notion of natural numbers. When we constructthe notion, we probably take two very basic phenomena (the ability to countsmall numbers and add to one), add the inductive hypothesis about thesuccessor... but then the choice to add zero to the mix is not as obvious (andcertainly wasn’t for many centuries). Therefore, even such a simple notionalready has a level of arbitrariness to it.

Note, however, that it would be incorrect to assume that all propertiesof higher-order mathematical objects are arbitrary and subject to randomchoice. There are still structural constraints involving those objects thathave be preserved and moreover, the lower level structure has to be adheredto. For example, let us consider the following axiom added to a theory ofsets (that includes the set of natural numbers):


(∀A)(∀x, y, z)(x ∈ A ∧ y ∈ A ∧ z ∈ A)→ (x = y ∨ y = z ∨ x = z)

This axiom establishes that any given set must have at most two elements.Ignoring for a moment the fact that this might be inconsistent with otheraxioms (in fact, adding such an axiom would raise inconsistency with multipleaxioms of ZFC, aside from the axiom of infinity also eg. the powerset axiom),such an axiom in any theory of sets would also make it impossible for us torepresent the actual set of natural numbers. Therefore, while this might be aset theory of some sort, it certainly would not be the set theory in the senseof describing our cognitive capability of collective abstraction.

What is interesting is the question what type of mathematics would bepossible if one were only to reason about the base cognitive capabilities, leav-ing higher order ones intact. Aside from the fact that it might be a difficultempirical question to separate the base capabilities and notions from the de-rived ones, it could be an interesting thought experiment the consequencesof such an approach. What would they be for the field of mathematics?

In fact, the history of philosophy knows at least one case where suchcriticism was forged. The critic was a firm opponent of abstract ideas in anyshape or form, deeming them illusions created by naming, so it is no wonderthat he also opposed such abstract objects in mathematics. The philosopherin question was Berkeley, who strongly opposed the introduction of calculusand the subsequent reification of infinitesimal limits ([Ber34]). His claimwas, in a certain way, similar to ours - he claimed that the mathematicalobjects proposed by Leibniz and Newton, the creators of calculus, did notreally exist.

History proved Berkeley wrong - not in an absolute metaphysical sense,but in a pragmatic sense, as calculus quickly turned out to be a useful tool.The metaphysical consequences of this pragmatic success are harder to mea-sure, as is always the case with metaphysical consequences of pragmatic


results. However, our approach differs from Berekley’s in that we do notoutlaw abstract objects - in fact, we consider them key elements of our cog-nition. Therefore, we do not want to ban higher-order mathematical objects- but we have to be aware that the tradeoff on introducing them is that welose completeness. We can use higher-order objects, reason about them,but some properties thereof will be undefined and subject to arbitrary, prag-matic decisions. In fact, the multitude of possible ways of formalizing anddescribing our base concepts and the branching of theories this spawns isa key component in the development of the foundations of mathematics, aswell as of mathematics itself.

5.3 Proofs and proof theory

Mathematical objects are one things, but mathematics is not just aboutstudying mathematical objects - it is about proving things about them.It is the proofs more than anything that set mathematics apart from othersciences. The Hilbert program of the early XX century set out to codifyall mathematics in a rigorous framework of proofs and it was only Godel’simpossiblity result that triggered a debate about the possibility of havingfully formalized mathematics.

Nowadays, the disparity between formal proofs and informal proofs issomething well established and widely discussed. However, the differenceis seldom considered from a cognitive point of view (this is largely due tothe fact that most mathematicians and philosophers of mathematics are notintuitionists with respect to mathematical objects, so they have no incentiveto consider proofs from such a perspective). Adopting a cognitive perspectivecan actually provide much insight into the problem of proofs.

We established in the previous section that mathematical objects, at thebase level, are simply patterns of our cognitive apparatus. How should wetreat proofs? If we follow this analogy, proofs should correspond to reason-

5.3. PROOFS AND PROOF THEORY 113

ing patterns. However, not all reasoning patterns can be associated withproofs. Certainly, various heuristics which we use all the time in our cognitiveprocessing cannot be considered here. For example, the availability heuris-tic ([Tve73]) is something we use to evaluate probability, but it cannot beconsidered a method of proving probability. Proofs can only be considered asrepresenting reasoning patterns if we restrict them to analytic reasoning,i.e. those reasoning patterns that analyze connections between notions andtheir internal structure. The term more widely used in literature is “deduc-tive reasoning”, however, I prefer to use the term “analytic” simply becauseit has a more cognitive feel to it. The term “deductive” tells us nothing ofwhat cognitive mechanisms are involved, while the term “analytic” impliesthat only the analysis of preexisting concepts is used.

Before we study proofs as something representing reasoning patterns, wehave to answer another question: what are reasoning patterns? We are notasking the low-level question of brain activities or structures associated withreasoning, what we are asking is the higher order question - what is the men-tal function of reasoning in our system? Generally speaking, our definitionof reasoning patterns is very much similar to one that is often connectedwith the logical consequence relation and reasoning rules in logic - reason-ing patterns are cognitive mechanisms that allow us to constructreference-preserving notions. Note that the term “reference-preserving”is the mirror of “truth-preserving” when it comes to logical concepts, whichis not surprising considering our adherence to the correspondence theory oftruth.

However, what differentiates our definition from the one known from logicis that our definition is not normative in nature. In fact, it is an assump-tion we make here that we are in possession of mechanisms that allow usto construct reference-preserving notions. Admittedly, this is a very well-founded assumption, with its negation entailing the acceptance of a veryspecific skeptical scenario: one in which we live in a world to whose structure


we are constantly ill-attuned, as all our mental functioning patterns causeus to shift from referential to non-referential notions and most of our cogni-tive activity is involved in rationalizing to ourselves this very fact. In otherwords - if we did not have reliable reasoning patterns, we would not have anyknowledge of the world, but we would have very extensive rationalizationsabout the world - however, those rationalizations would nevertheless allowus to function in this world properly. While this is a scenario that is notcompletely impossible (one can for example imagine benevolent beings thatinvisibly guide us in such a world while at the same time being unable to liftour cognitive curse or malevolent beings that use our cognitive disability totheir advantage and hide the true nature of the world from us via a brain-in-a-vat-turned-Matrix mechanism), it certainly seems as implausible as mostskeptical scenarios that we have to face and thus, has to be rejected if we areto perform any reasoning about the external world.

Now that we have established what reasoning patterns is, we come backto our notion of proofs. We said before that proofs correspond to reasoningpatterns. In fact, we have to amend this statement: logic represents reasoningpatterns. Proofs represent reasoning processes. Cue the famous tortoise-and-Achilles example by Lewis Carroll ([Car95]): if we are only in possessionof logical formulas, but unable to perform any logical steps, we cannot proveanything. Therefore, while logic is about uncovering reasoning patterns,proof theory uncovers our reasoning processes.

So, in light of this analysis, how do we close the gap between formal andinformal proofs? After all, it certainly seems that the two areas differ verylargely, although if one looks into constructive proof theory instead of formalproofs as understood by Hilbert, the gap is largely diminished. Formal proofshave rigid components that document each step of the reasoning process,while informal proofs largely gloss over some most of the details - but arestill considered proofs by those involved.

One part of the answer is that informal proofs are a sketch of what a for-

5.3. PROOFS AND PROOF THEORY 115

mal proof would be if it were fully realized. One phenomenon that fuels thisapproach is the introduction of computer-assisted proofs ([Wie03]) - proofsthat are based on a sound formalized proof-theory, but nevertheless rely onintermediate steps similar to those applied in real-life proofs and use auto-mated tools to bridge some tedious gaps in reasoning. Recently, there havebeen multiple projects to formalize large fragments of mathematics with theuse of such proof assistants and one of the criteria for developing the soft-ware is ease of use for the mathematicians involved, therefore, this serves asa pragmatic argument that the gap between formalized proofs and informalproofs might not be as large as one expects.

Another part of the answer is that perhaps our logic is simply not thatgood a theory of reasoning as we might think it is. Remember, physics as ascience is barely 500 years old and logic is 400 years younger, yet our physi-cal theories of the world constantly evolve and and certainly the Theories ofRelativity are much unlike the Newtonian physics that preceded them. Whyshould we assume that our internal world is that much less complicated thanthe external one? Moreover, there is also a serious demarcation problemthat might make creating adequate theories in mathematics more difficult -while we do not generally have any trouble with saying what physical objectsare, we might have a serious problem with determining what reasoning pat-terns are parts of our core cognitive functions and which ones are a result ofhabitual training or social development.

Finally, there is yet another problem that comes to haunt yet again here:that of the levels of abstraction. Note that reading informal proofs is moreor less a process of simulation - one simply needs to verify whether the steps“make sense”. On the other hand, checking formal proofs is a meta-leveloperation: one needs a certain theory (here in the logical sense of the word)of formal proofs and then one checks the formal proof for conformance tothat theory. This might explain why we are much better at producing andverifying informal proofs than we are at doing the same with formal proofs -


the former is simply a lower-level cognitive ability than the latter (and thusinvolves significantly less processing power). This might also explain why wecan consider informal proofs as carrying ideas ([Bye07]) - since we are notprovided with the reasoning steps explicitly, we are often left with figuringout a single concept that allows us to bridge the simulational gap, to enableus to make the proper step and ideas are the notions that make running thesimulation possible.

5.4 Mathematical concepts

Up until now, we have concerned ourselves with mathematics in a moremetaphysical sense. However, our work in this chapter would not be in linewith the general theme of this text if we did not ask the question: how domathematical objects relate to mathematical concepts?

Since concepts are tests, mathematical concepts have to be tests as well.However, the question remains: what kinds of tests are they? Here, we haveto focus on the distinction between ground-level and higher-level mathemat-ical objects that we made previously. Concepts that correspond to ground-level mathematical objects (and thus, to core components of our cognition)are simply representations - they are referential in nature and they do noth-ing else than single out specific external objects (again, as we feel the needto remind: in the wide sense of external). Higher order concepts are tests onnotions that are mostly structural in nature. A good example would be thedifference between our notion of natural numbers and the notion of successor(adding one). The latter is likely a representation that refers to a core cogni-tive mechanism. The former is a notion, which, as we described in a previoussection, likely groups together the notions of the basic numbers, the notionof the successor operation, induction applied to the notion of successor andthe notion of zero.

The study of mathematical object is a good place to delve into one topic

5.4. MATHEMATICAL CONCEPTS 117

we have been avoiding for most of the work: how do abstract concepts lookin our framework? We have been avoiding this topic for a reason. As wesaid in the beginning chapters of this study, it is nor our role to determinethe empirical mechanisms of cognitive functioning that underlie this frame-work; all the mechanisms mentioned here are provided as examples only.However, with respect to abstract thinking it is very hard to provide an ex-ample without actually answering an empirical question about the underlyingmechanisms - something we are not well prepared to handle.

However, there are cases of abstraction which are documented well enoughthat we can take them for granted, so we can try to illustrate our point withthem, even though the explanations can be sketchy at best. One such case ofabstraction is the (mentioned already in this chapter) collective abstraction,in which we take a couple of entities together and form an entity simply basedon the idea of grouping them together (hence the adjective “collective”). Thisabstraction forms the obvious grounds for set theories of various kinds.

Therefore, our concept sets will be based on the idea of collective ab-straction. The underlying test will be functional in nature, in fact, mirroringthe very mathematical mechanism of reasoning by abstraction: the con-cept of set is something that is tied to the concept (previously established)of membership (here, we assume that the cognitive capability of recognizing“objects in a container” is prior to the capability of collective abstraction, ina sense that collective abstract is abstraction from the “being in a container”basic notion) in such a way that whenever we apply the concept of set toany group of objects, we obtain an entity of which each of the objects is amember. In other words, there are a series of tests - among them certainlythe structural one:

t1 := λn.(∃m)(Memberc(m) ∧ Inv(m,n))

where Inv is the structural involvement predicate, and the functional one(which is, in fact, also a structural test):


t2 := λn.(∀a, x)(Apply(n, a, x)→Member(a, x))

which basically says “whenever x is a result of applying n to a, then a

has to be a member of x”.One immediately sees the problem with such a definition - the collectivity

has somehow vanished. We do not apply the collective abstraction operationto the set members individually, we do it “all at once”. However, doing sowould require us to produce an infinite number of tests, one for each (finite)cardinality, and even that would not account for infinite sets. Is there a wayout of this conundrum?

An obvious way would be to relax the formal constraints on the tests -after all, nothing says that we cannot use some sort of set theory or inductionprinciple within the tests themselves. In fact, that would be true whenwe would be talking of concepts in general - concepts are tests that aresupposed to be cognitively plausible, i.e. performable by a human being, butif the tests are themselves expressed in a sound logical theory - which, byinvoking the definition from earlier in this chapter, describes our patternsof reasoning - then they should be “good to go”. However, we might beinvolved in a circularity here: what if the formalism we want to use involvesa cognitive mechanism that isn’t there? In other words - what if the formaltheory we use in the tests doesn’t really refer to anything?

Here, again, we have to use a pragmatic criterion to reply to such apossibility. If we use a sound logical theory that people use, they have to useit somehow. Granted, it might be the case that the logical theory actuallydescribes a higher-order mechanism that we believe it describes (for example,we might believe that addition is a basic cognitive operation, when in realityit involves iterated repetitions of the successor operation, or we might believethat collective abstraction is a basic cognitive operation, while it is simplyan abstraction from the basic “is a member of” notion), but that is reallythe “worst case” scenario - which isn’t even that bad because hardly have

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any preconceptions about how high-order a mechanism described by a logicaltheory is, given the fact that it is not commonly accepted that logical theoriesdescribe any cognitive mechanisms at all.

Given all of that, we can revise t2 in the following manner:

t2′ := λn.(∀X, x)((Apply(n,X, x)→ ((∀a)a ∈ X →Member(a, x)))

Now our test looks surprisingly trivial. It should be expected, however,given that we are attempting to formalize a very basic cognitive structureand in the formal language for the tests, we are assuming the very cognitivestructure that we are formalizing. What remains to be explained is how theabove process is not circular. Remember that what we are describing arenormative tests on mathematical notions. Our assumption that we possessthe ability of collective abstraction is a mere empirical observation. Theformalization of this very abstraction might as well take into account thisobservation: after all, while the fact that we use our cognitive faculties todescribe how those very faculties work might seem paradoxical (and certainlysmells of self-referentiality), it is also a sheer empirical fact.

However, to avoid notational confusion, some things have to be said aboutt2′ . Note that we used a different notation for the logical membership relationand a different one for the cognitive membership relation. This isn’t a no-tational quirk. In fact, they have a different status. The predicate Member

is a member of our cognitive vocabulary (it is constrained earlier in t1, socan be considered a kind of bound variable as far as the entire set of tests isconcerned). On the other hand, the membership relation is part of our logi-cal vocabulary - the one in which the tests are expressed and that implicitlyassumes some cognitive capabilities.

These differences aren’t important from a cognitive point of view - afterall, both the cognitive and logical fragments ultimately refer to some cog-nitive mechanisms. They are important from the theoretical focus point of


view. When we speak about the tests, we want to distinguish between thethings being tested - and the ways used to test them. Although the tests areabstract entities - not really performed by individual beings - we providedreasons why we want them to be cognitively plausible and thus expressed ina way as if they were performed by individual beings. However, any way oftesting has to assume certain mechanisms for testing are available - in otherwords, it has to assume some sort of background logic is in place. If we wereto present a complete semantics for natural language as a complete systemof tests, we might run into well-foundedness problems (especially if we as-sumed the presence of higher-order mechanisms for some tests like we didabove). However, since we are not providing such a complete system, we willreserve ourselves the comfort of using this notational schemes and avoidingunnecessary complexity for the sake of theoretical clarity.

Now we should come back to our discussion about infinity. Note one pe-culiar quality about this concept of set - it tells us nothing of infinite sets. Infact, it does not tell us whether it is possible to have an infinite set at all. Dowe indeed use a single cognitive operation for normal collective abstractionsuch as grouping my computer, my car and my phone into one set and forabstract, induction-based collective abstraction such as grouping all naturalnumbers into one set? That is a question to be solved empirically. However,it seems at least plausible that the answer to such a question could be “no”.This would explain why intuitions about finite collections fail us when weconsider infinite sets - perhaps our cognitive mechanism for collection onlyworks on finite sequences and what we consider collective abstraction is in-deed functional abstraction wrapped in a reification layer? Or perhaps yetthe product of yet another cognitive mechanism which eludes us?

Certainly, we can add further tests on our concept of set, which willcause the linguistic community to request different properties of sets. Doingso, however, we change the reference of such a concept. It might still cor-respond to a viable mathematical object, but it might be a complex, higher

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order one as opposed to perhaps the primitive one we held earlier. Theonly problem is, we do not have a reliable way to tell. Introspection as ameans of determining primitive mental objects is notoriously unreliable - ifit weren’t, we’d already be in possession of a universal catalogue of basicmental categories and mechanisms.

Therefore, we have to settle for a pragmatic criterion with respect to thedifferent versions of mathematical concepts. We could try the grounding cri-terion, asking for concepts that are realized by our very simplest mathemat-ical notions, but this criterion might not really be worthwhile for anythingother than sheer metaphysical purism - after all, it turned out that Berkeleywas wrong and calculus, despite being “impure”, did in fact allow us to gainmuch knowledge about our reality.

5.5 Conclusions

In this chapter, we explored the realm of mathematical objects. Again, ouraim was to show how adding a cognitive level of analysis can enrich ourunderstanding of the philosophy of mathematics. Admittedly, our solutionto the origin of mathematical objects is nothing new - the idea should his-torically probably be attributed to Kant. However, combining it with thecognitive approach is what nets us some new and interesting results.

Two of the most important problems in the contemporary debate in phi-losophy of mathematics are the questions of mathematical knowledge andof the uniqueness of mathematical axiomatizations. Our approach gives an-swers to both questions. We explain how we access mathematical objects,claiming that this is not that different from how we access other abstract ob-jects, while at the same time accounting for the impression that this methodof access is somehow special and that mathematical objects are in a certainsense “different”. On the other hand, we answer the question of multiplepossible axiomatizations by showing why it is in most cases impossible to


settle for a single “correct” axiomatization. In a way, this is our own “im-possibility result”, and as we have seen, impossibility results are sometimesmore important to mathematics than concrete proofs and solutions.

Chapter 6

The social aspects of language

6.1 Sapir-Whorf’s hypothesis and the linguis-tic reality

At the beginning of the XX century, Edward Sapir and his student, BenjaminLee Whorf, made popular a maxim of linguistic relativity. According to thatmaxim, societies using different languages actually lived in different worlds,rather than simply attaching different linguistic labels to the same world.The claim that language influences the world we live in henceforth becameknown as the “Sapir-Whorf hypothesis” ([Who56]).

During the universalist phase of the development of linguistics, the hy-pothesis became discredited, but the appearance of cognitive linguistics hasrenewed interest in its claims among linguists ([Lak03]). However, linguisticrelativism has also triumphed in a much wider field - that of continentalphilosophy.

Continental philosophers have, for the most of the XXth century, beenfirm anti-realists, with much of the thinkers influenced by Heidegger andhis anti-metaphysical turn. However, it was the second half of the centurythat marked the most definite anti-realist turn, with both the influential

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124 CHAPTER 6. THE SOCIAL ASPECTS OF LANGUAGE

marxist movements following the Hegelian line of thought which discounteda static reality and opted for a reality-in-the-making viewed as a neverendinghistorical process, and the literary criticism French thinkers led by Derridawho questioned the possibility of establishing a referential relation to theworld ([Der72]) and thus suggested that our reality should be viewed in termsof intertextual relationships, with there being no reality, but only readingsof reality.

Exploring the connections between those two lines of thought (three, ifone counts the linguistic tradition stemming from Sapir and Whorf) wouldbe certainly too much to ask here, but certainly continental thought haslargely involved itself with studying the social reality in its linguistic aspects- either because there is apparently nothing else left to study or becauselanguage itself is a very important means of acting in the world and, in fact,describing the world is a very distinct way of shaping it (the idea going as faras Horkheimer’s essay describing the notion of a “critical theory” [Hor37]).

Unfortunately, analytic philosophy has largely ignored this area of re-search, mostly because of its bottom-up nature, which is not suited to theconstruction of huge, overarching theories of the social world. For sometime, only the speech-act theorists, led by Searle, tried to form some sortof counter-theory ([Sea95]), however, recently the emergence of post-griceantheories which merge semantics and pragmatics has renewed the interest ofanalytic philosophy in the latter and therefore slowly invited more researchinto the social aspects of language (see eg. [Jas05]). What we would like topropose here is somewhere on the border between the continental interest inlanguage as a means of social action and the analytic interest in pragmaticsas a means of supplementing semantics in the quest for meaning and truth.On one hand, we will very much be using the analytic methodology withits bottom-up approach and putting focus on specific linguistic phenomenainstead of looking top-down onto the linguistic community as a whole, but onthe other hand, we want to venture outside the purely truth-functional side

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of language and even outside the fields of speech act theory and show howsome cognitive mechanisms that underlie the acquisition and coordination oflanguage can have long-term social consequences.

We did mention in chapter 2 how language can have an impact on ourcognition of the world and in chapter 4 we outlined how the categorial struc-ture of our thoughts is very much influenced by the concepts we use withinlanguage. Now, we will try to look at how those and similar mechanismscan account for two roles of language: influencing our cognition and shapingsocial reality.

Before we delve any further into the topic, it should be made clear that ourwork is not of the “critical” variety - it does not aim to shape the world, it onlyaims to describe it. Since social and political topics are notoriously volatileand readers’ opinions on the validity of our theories could be influencedby the choice of topics (which would in itself be a rather self-referentialapplication of the theories presented here), we will refrain from using anyreal-life controversial examples in the chapter. Hopefully, the token examplespresented will be sufficient to illustrate the main concepts.

6.2 Linguistic influences on cognition

How does language influence our attitudes towards the world and our worldknowledge? It is hard to provide a general answer to that question. Onone hand, since we already established that language is a cognitive activityand part of our overall cognitive functioning, one could claim that there isno separation between language and cognition - thus, language can influencecognition in every possible way. However, that type of claim would likelybe too strong and it would certainly be ungrounded. After all, the reverseinclusion doesn’t work - not all of our cognition is linguistic in nature. Andeven for the types of cognition that are linguistic in nature, often language isa tool and not a determinant - when we say that language influences cogni-


tion, what we usually mean is that the structure of language influences thestructure of our cognition, or even, in a stronger version, that the structureof language influences the contents of our cognition.

Since we already established a framework for linguistic concepts that hasso far proven useful, we can query this very framework for the ways in whichlanguage can influence our cognition. Concepts are tests on notions andcognitive processes performed while using those notions. They are acquiredto make communicating things about the world easier or even possible. Thus,there are two different main ways in which the structure of language caninfluence cognition: the internal and external one.

Internal influence is the influence contained within the tests themselves.Remember that among the tests we included cognitive tests that constrainthe states and processes involved when processing the respective notions.By shaping the cognitive tests associated with some concepts, the linguisticcommunity can influence the cognition of the language users.

External influence is the influence exerted by the language acquisitionand coordination mechanisms themselves. Since we actually need languageto communicate within members of our society, there is a pressure on us toacquire the proper concepts.

While the difference between the two types of influences seems sharp,that isn’t exactly the case. This is because the tests within the conceptsinclude structural tests. Structural tests have a holistic nature - by changingthe structure of some concepts, one can change the requirements to acquirethose concepts. Say concept X requires (structurally) concepts Y and Z

in one language, but concepts Y and W in another. The need to acquireconcept X might be external in nature, but the need to acquire concept Z,granted the need to acquire concept X, is internal for the first language.

We have outlined the theory behind linguistic influence, but what is miss-ing here is the systematic picture in which such influence happens. First ofall, it doesn’t seem likely that all linguistic influences happen on a global

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scale in the official language. Therefore, in order to account for many casesof such influence we have to first establish the existence of linguistic sub-groups. However, doing that is certainly not as simple as it appears.

Linguistic subgroups can be selected based on many criteria. One ofthe most obvious is geographical location - regional dialects certainly fallunder this category. The other is the language user’s main area of activity- this accounts for all sorts of occupational jargons. However, the criteriado not have to be mutually exclusive and can, in fact, be overlapping - twopeople living in the same region, but having a different occupation can takesome concepts from the regional dialect and others from the occupationaldialect, while two people of the same occupation, but having different beliefsystems might use the occupational dialect for some words, but the belief-system dialect for others. The situation can be even more complicated - twopeople sharing the same region, occupation, belief-system and other linguisticsubgroups can differ with respect to the degree they select the concepts fromthe respective dialects. Finally, the mixing does not have to be exclusive- meaning that within a single notion, tests belonging to different dialectsmight be applied, resulting in a hybrid concept. All of those phenomena, aswe will soon see, are very important in the analysis of linguistic influence.

The other systemic aspect of linguistic influence is the division betweenintentional and non-intentional linguistic influence, or linguistic influenceand linguistic manipulation. Again, the division is not really clear, withdominating belief systems influencing the public language (especially in thefragments concerning ethics and politics) and the difficulty of distinguishingsocial engineering from social influence.

6.3 Language and attitudes

In chapter 1, when we introduced cognitive states, we mentioned a componentof the states that we haven’t used so far - the set of goals and attitudes. It


would be indeed weird to ignore the fact that our cognition is teleologicalin nature - we don’t aim to randomly grasp reality, we have our goals andattitudes that dictate which parts of reality we want to know, which we wantto avoid and how we want our reality to be shaped.

We conjoined goals and attitudes into one category, but they are entities ofa slightly different nature. Goals are what we ourselves want to obtain, whileattitudes are how we value different things in the world. One can have goalsthat are not in sync with the attitudes that we have (those would be the goalswe consider self-defeating, such as “get drunk so much that you pass out”when one ascribes to the attitude “drinking excessive amounts of alcohol is tobe avoided”. On the other hand, one can have attitudes that are not realizedby any goals, for example “people in Third-World countries should not beallowed to starve”. Basically, in an illustrative if a bit oversimplified manner,one could describe the difference in a slogan that says: goals determinewhat we act upon, attitudes determine how we feel about it.

Feelings, or emotions, are another important component of the influencelanguage has on our cognition. The idea that our consciously realized goalsand attitudes and our feelings and emotions towards some events are notnecessarily aligned is not a new idea - it has certainly been explored bySpinoza in “Ethics” and has been brought to the philosophical mainstreamby the works of Freud. Although Freud’s theories have since been largelydiscounted empirically, one thing he has permanently introduced into psy-chology is the need to distinguish between our consciously constructed andrealized theories on one hand and the subconscious mechanisms driving ouractions on the other.

One area in which this distinction becomes apparent is the distinctionbetween emotionally-laden concepts and attitude-laden concepts. Thisdistinction is sometimes blurred in the case of negatively loaded concepts,which are grouped under the common label “pejorative”. However, as we willtry to show here, there are actually two different types of pejorative terms

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corresponding to the distinction mentioned above.

Emotionally-laden concepts are concepts that contain a test which re-quires the presence of a certain emotion in the attention focus of the end-stateduring the processing of the implementing notion. In other words, those arethe concepts that evoke a certain emotional state in us.

Note that, since the tests in concepts are underdetermined, one can haveemotionally-laden notions which implement perfectly emotionally neutralconcepts. This can be a result of many factors, including intentional in-fluence, but as long as it’s not due to a systemic process, it’s a matter forsocial psychologists and beyond the scope of this work. Obviously, due tothe inherent difficulty in determining meanings within a certain language(and an even bigger difficulty in determining the emotional components ofthose meanings), the question whether the appearance of emotionally-ladennotions within some social group is the result of linguistic influence or notmight be difficult to answer, but regardless of that matter, it is an empiricalquestion. We only want to deal with those cases of emotionally-laden notionswhich are the result of emotionally-laden concepts.

Attitude-laden concepts are distinct from emotionally-laden concepts inthat they explicitly require a certain set of goals/attitudes to be presentin the output state of the cognitive process involved. Note that there is asimilar phenomenon here to the modal constraints from chapter 4 - manyattitude-laden concepts are also emotionally-laden and vice-versa, which isdue to the fact that concepts contain more than one test and they mightcontain the emotional test as well as the attitude test. However, the twoclasses are not mutually inclusive and some of the more interesting cases oflinguistic influence will happen outside their intersection.

It might be surprising that so far, we have been only been concerning our-selves with the output states of the respective cognitive processes. Is therenothing to be said about the constraints of input states? In fact, for both ofthe classes of concepts, one can find their mirrors that constrain input states.


We will call them attitude-driven and emotionally-driven concepts. Onthe surface, those two classes of concepts are much less interesting in termsof linguistic influence - if they require a certain emotional or attitude statein the first place, then they are not really suitable for shaping our cognition.However, that is not necessarily the case. One reason is that, again, thoseclasses can overlap with the two named before - for example, we can haveattitude-driven, emotionally-laden concepts. The other reason is due to theexternal influences associated with linguistic influence - if the only means toexpress some otherwise emotionally-neutral term is emotionally-driven, thenwe might in turn start associating the phenomenon with the emotion andthe concept might itself become emotionally-laden.

6.4 Analyzing linguistic influence

We have established a theoretical basis for talking about linguistic influence,now it’s time to put it to action. We will look at a couple of scenarios andtry to analyze them from this point of view.

For our first scenario, let us consider Jane. Jane is a member of a linguis-tic community which lives in a region inhabited by lots of Equitorians. TheEquitorians are a proud people that once used to rule an empire spanningmany miles of land, but have since been conquered by Jane’s people, theMerids. Nevertheless, nationalistic instincts in the population of Equitori-ans are strong and they have been strongly alienating themselves from theMerids, with radicals forming armed resistance groups and even resorting toacts of terror.

Let us consider the concept of “an Equitorian” in the language of theMerids. How would linguistic influence work through such a concept? Be-fore the Equitorians opposed the Merids in any way, the term was probablyneutral. Afterwards, due to the fighting between the two nations, it probablybecame emotionally-driven, and with the retelling of the stories of Equitorian

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violence within the Meridian community, it became emotionally-laden.Why is this a case of linguistic influence? In this case, it is due to a

phenomenon which we will call the linguistic emotional memory. Imaginethat after the war, a new generation of leaders came to power among theEquitorians which were entirely pacifist. Would the Merids believe in theirgood intentions? Probably not, given that the term “Equitorian” itself inthe Meridian language would now evoke negative emotions, regardless of theactions of the particular Equitorians themselves. Imagine the proposition

(1) The Equitorian delegation has come to discuss a peace treaty.

as uttered within a group of Meridians. The very act of uttering andcomprehending such a message already evokes negative emotions (possiblyof suspicion and distrust), therefore making the peace treaty negotiationsthat much more difficult.

Note that linguistic emotional memory is distinct from what might becalled linguistic attitude history. The latter is the case of moving fromattitude-driven concepts to attitude-laden concepts. It is again a question ofempirical research to determine which of the shifts happens more often (it isalso possible that there are cross-shifts, most notably from emotionally-drivento attitude-laden concepts). In the case of attitude history, the utterance of(1) would evoke an attitude expressed by

(2) The delegation that has come to discuss the peace treaty is not to betrusted.

Note that accepting (2) is different from feeling distrust, although thetwo might be connected. Certainly, overcoming (2) might be easier thanovercoming the emotional consequences mentioned before, as the attitudecan be consciously confronted. In this case, we can see the difference betweenimplicit and explicit linguistic influence. The latter involves cases in whichlanguage influences our cognition of the world and we realize it if we only pay


attention to it - the other involves cases in which our cognition is influencedindirectly and thus might not be immediately apparent.

The cases mentioned are simple situations - a concept becomes directlyattitude or emotionally-laden. However, most cases are more subtle. ImagineJoe, who is a friend of Jane, but is not a Merid. However, he also speaks theMeridian language. He and Jane discuss the phenomenon of “Equitophil-iacs” - people within the Meridian society who admire Equitorian cultureand customs. It is clear that the concept of “Equitophiliac” is structurallyconstrained to involve the concept of “Equitorian”. However, the emotionalcontent of the concept is different for Joe than it is for Jane. Since Joe isnot a Merid, his notion of “Equitorian” is not emotionally-laden. Therefore,his notion of “Equitophiliac” is not emotionally-laden either. On the otherhand, Jane’s notion of “Equitophiliac” is likely to be tainted with the similarnegative emotions as her notion of “Equitorian” is.

Here, the structural constraints cause the conditional transfer of emo-tional load over whole families of concepts. For a community where the baseconcepts do not carry emotional load, the concepts built upon them willnot either, but for ones that do, the emotional load will transfer over to thecomplex concepts.

The situation becomes more complicated when we consider situationswhere multiple linguistic subgroups mingle and their concepts mix. Then,structural constraints from one linguistic system might combine with theemotional load from another system to form a whole family of emotionally-laden concepts. Taking our example further, consider another linguistic com-munity on the other side of the globe which builds a whole system of values(called “Equiticism”) based on a system of Equitorian beliefs. However, thecommunity has a large Meridian minority. For that minority, everythingwithin Equiticism will carry a negative emotional connotation.

Of course, the linguistic community does not have to countenance actionsleading to the negative load of certain concepts. What it can do is counteract


some processes. For example, some concepts can be negatively emotionally-laden, but positively attitude-laden. Imagine that the Meridian governmentstarts a large campaign to promote the Equitorian culture and customs andto portray Equitorians in a positive light. Now, even though the concept of“Equitorian” might still convey negative emotional load, it also conveys apositive attitude, which might in time offset the negative emotions present.

However, the case is not as simple as it seems. This is due to the fact that(a) language cannot be separated from our other cognitive activities and (b)concepts are formed through use and not engineered via a conscious decision.Imagine that in the Meridian society there is a special derogatory term forEquitorians and most people use that term to refer to Equitorians. Now,to counteract the negative emotional load carried by the term, actions areundertaken to forge a new word that would be free of emotional load.

We previously mentioned how emotionally-driven concepts become emo-tionally-laden concepts within a linguistic society. This mechanism also playsan equilibrium-keeping role - as long as a concept is emotionally-driven, anyattempts to remove its emotional load will almost certainly fail. Therefore,the success in introducing a new term to refer to the Equitorians will largelydepend on the political and social situation - if, for example, radical groups ofEquitorians will keep performing terrorist attacks on Meridian targets, thenany term used to refer to Equitorians will likely be emotionally-driven andthe attempt to make it neutral in terms of emotional-load will fail. If we lookat our theory of concepts and recall how emotionally-laden and emotionally-driven concepts are linked, this will not be surprising - having a term that isemotionally-driven, but not emotionally-laden means that the concept itselfhas the power of erasing or nullifying an emotion - which is quite a lot to askof any language.

On the other hand, attempts to counteract other cognitive factors leadingto the emotional load can fail due to either the incorrect identification of thecorrect factors or the incorrect steps used to remedy those. This is the ground


of social engineering and social engineering is inherently tricky (to say theleast), as evidenced by many real-life examples. However, those attemptsare usually epistemic rather then linguistic in nature and as such they arebeyond the scope of our research.

Up until now, we have concerned ourselves mostly with direct linguisticinfluence based on emotions or attitudes. However, there is one other fun-damental sphere of influence that language can use, which has already beenhinted at in chapter 4 - categorial constraints.

Let us imagine that Meridians’ concept of Equitorians is tied to a categoryof “national spirit”. The Meridians believe that “national spirit” means a setof essential personal features being passed genetically from generation togeneration. Therefore, they subscribe to the following modal statement:

(3) It is not possible for an Equitorian to not be violent by nature.

Note that this is a categorial essential statement, so it is subject to empir-ical verification, however, the problem here lies in the features of the categoryof “national spirit”, since the following implication holds in this categorialstructure:

(4) If Equitorians are violent, then it is not possible for them to not beviolent by nature.

The problem with (4) is that once the antecedent is established, the con-sequent is true via categorial analytic deduction. Therefore, for example,within such a categorial system imposed on a society, politicians can arguefor death sentences for captured Equitorian criminals, stating that their re-socialization is not possible.

This example illustrates the potentially pernicious character of categorialstructure - without explicitly stating the case, it laces our cognitive systemwith a set of presuppositions that cannot be easily changed unless the cate-gorial structure itself changes. Therefore, the most malicious cases of social


engineering known to mankind exploit this type of linguistic influence, withprobably the most notable example being the process known as “dehuman-ization”, where the categorial structure of a language is changed to deny thecategory of “groups of human beings” to certain groups of people.

Another area in which linguistic influence is strong is the field of ethicaldiscourse. We have talked about attitude-laden concepts - but for ethicalterms, it is their main function to be attitude-laden. However, some termsstart as emotionally-driven, to then become emotionally-laden and finallyenter the ethical discourse as attitude-laden - it is questionable whether asystem of morality should actually be developed in such a way (i.e. shouldbe driven by emotional input).

The combination of categorial constraints with linguistic influence on eth-ical discourse also yields surprising results - when ethical judgements are cat-egorially influenced, it sometimes suffices to change the category of a givenobject to change its moral status whatsoever. The above-mentioned mecha-nism of dehumanization actually exploited this phenomenon - if certain rightsare given to groups of people based on their being human beings, then if theyare not considered human beings, those rights are automatically revoked forthem. Here, Sapir-Whorf’s thesis resurfaces in the worst possible manner- remember from chapter 4 that categorial structure is not something thatis embedded in the world, although we often act under a cognitive illusionas though it is so. Therefore, changing the categorial structure of languagerequires nothing more than a pragmatical decision of performing the change- however, the consequences in terms of the changed ways of thinking andthus the actions of the linguistic community can be overwhelming.

So far, we have discussed internal linguistic influence. Now, it’s time toturn to cases of external influence. Imagine that Jane wants to learn some-thing about Equiticism. However, it turns out that all concepts describingthe various beliefs and customs of Equiticism portray Equitorians in an over-whelmingly positive light. There are literally no neutral terms to be used:


for example, the term “equijust” means “as just as only an Equitorian canbe”.

If she is a lone scholar studying Equiticism from a theoretical point ofview, she can ignore the attitude-laden concepts in favor of a new set ofattitude-neutral ones. However, if she wants to participate in a group thatactively follows Equiticism, she has no choice but to actually adopt the con-cepts being used. If she is especially wary of being manipulated, she canprotect her cognitive framework with a set of safeguards intended to combatthis bias, however, she has no real way of both participating in Equiticistactivities and refusing to accept the concepts.

Why would she actually want to participate in Equiticist activities in thefirst place if she recognized the concepts used therein are biased? There aremultiple answers to that question. One is that in those types of situations, weface a certain tradeoff - a certain linguistic community offers us a cognitively(or otherwise) valuable set of concepts, but the set of concepts requires anattitude shift from us. We can at most try to reject this attitude shift and, ina sense, feign linguistic competence. Another is that she doesn’t recognize thebias - the “equijust” example presented above is an obvious case of a biasedconcept, but we already discussed how emotional load and attitude loadcan be conveyed through structural constraints, masking them from directobservation. This can be especially true of linguistic groups with maliciousintent, i.e. those actually set on manipulating their members.

Politics in a democratic society presents an excellent field for linguisticmanipulation. Due the various technological and social developments in therecent century, the entirety of a modern country’s functioning is outside thecognitive scope of possibly any single individual. On the other hand, electionscreate a cognitive demand to be able to judge the capabilities of different can-didates in order to pick the best one - after all, most people aim to performrational choices. Therefore, the very need to select a proper candidate leavesus with huge cognitive gaps that we are usually unable to quickly fulfil and


a strong desire to fill them. What follows is a scenario that closely resem-bles the above described Equiticism example - political parties themselvespropose their own linguistic systems in which the cognitive void needed tojudge a government’s performance in various areas are conveniently filledin. Sometimes those systems will be simply filled with emotionally-ladenand attitude-laden terms, but more sophisticated politicians will exploit cat-egorial structure constraints to derive the emotional and attitude load fromtheir own party’s strengths and their opponents’ weaknesses, rather thenintroducing them directly.

Imagine that a Meridian politican wants to discredit their opponents.What course of action can they take? The most obvious one is that they canpropose a language in which everything associated with their opponent iscolored negatively. However, that would be probably not be a very efficientstrategy. The simplest added level of subtlety is that they could introducea language that puts negative load on the various solutions the opponentswould propose. This approach might have varying degrees of success, butit might be too direct. Alternately, they could try associating their oppo-nents with Equitorians, using the emotional-load of the Equitorian-connectedconcepts to their advantage. For example, drawing a parallel between theopposing party’s ethical system and the beliefs of Equiticism (i.e. intro-ducing structural constraints between terms describing key elements of theopponents’ belief system and those of Equiticism) could reliably influencepotential voters.

It might be difficult to counteract this type of action in politics simplybecause of the huge gap between the cognitive needs produced by the ra-tionality constraint on our voting in a democratic system and our inabilityto rationally fill those gaps. Certainly, the increased presence of linguisticinfluence in the sphere of politics is something that has raised considerableinterest as an area of potential research. Hopefully, the framework presentedhere provides at least some theoretical groundwork for conducting such re-


search.

6.5 Conclusions

In this chapter, we have used the tools developed in the previous sectionsof this work to present an analysis of various social phenomena connectedwith linguistic coordination and language acquisition. Most notably, we havelooked at the various aspects of linguistic influence with respect to the lan-guage users’ cognition. This is certainly an area that largely requires empir-ical study, but this work aims to provide a theoretical framework for suchresearch that does not rely on vague and underspecified concepts.

The analysis presented here shows that a bottom-up approach to complexsocial issues involving language is certainly possible. The question whethersuch an approach is credible is one to be answered empirically, but certainlyit might be valuable to study the social aspects of language from a differentperspective than a top-down, holistic perspective of the society as a whole.

Ending notes and furtherresearch

Through the various chapters, we have explored how a representation-basedtheory of language can be used in the study of various aspects of language.The proposed set of issues studied is certainly arbitrary and there are manyothers that certainly could be studied instead. In fact, the main goal of thiswork, as mentioned in the introduction, is to lay a foundation - one thatwould allow us to study more aspects of language than just the select fewmentioned here.

Since the building is far from being complete, some fragments stand outmore than others as unfinished. One notable omission is the study of compo-sitionality. After all, compositionality is one of the core features of naturallanguage and we did hint at some pretty powerful tools (most notably com-positional semantics) that can be used to ensure compositionality with asemantics of cognitive state transformations. Also, some more spectacularcases of the cognitive level being required in the analysis of natural language,most notably: contextualism and propositional attitudes, are missing here.

This omission is, however, necessary to save the structural integrity ofthe work. To establish any sort of propositional semantics (which wouldbe needed to talk about compositionality in general and about the areasmentioned in particular), we would first need to develop a theory of syntax.Such a theory is indeed one of the most important plans of further research,

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however, creating a proper theory of syntax while preserving the composi-tional level is a significant undertaking by itself, certainly large enough for aseparate work of its own. We did hint at how we would compose individualconcepts to form complex meanings in chapter 4, but due to the added cogni-tive level (and thus, the need to move the composition from the concept levelto the notion level and back) plus the typing rules that were also mentionedwhen talking about proper names and categorial constraints would mean thatthe resulting theory would certainly be complex - thus the impossibility ofintroducing a chapter with even a token theory of this sort.

However, the analyses presented in the chapters 3-6 seem to show that arepresentational approach to language and meaning certainly has potential,as we were able to provide some new perspective on a group of very diversephilosophical topics, each with their own long history of solutions, objec-tions and debates. In chapter 3, we established a theory of proper namesthat allowed us to solve multiple key issues in the philosophical debate aboutnames. In chapter 4, we provided a uniform semantics for seemingly dif-ferent senses of the basic modalities (“possible”/“necessary”). In chapter5, we proposed a solution to the problem of groundedness of mathemati-cal axioms and also made a claim about the identity of mathematical andnon-mathematical induction. Finally, in chapter 6, we provided an analysisof various social phenomena concerning language. All of this was presentedwithin the same framework, established in chapters 1 and 2. Moreover, eachof those chapters provides a foundational block for the future compositionalsemantic/pragmatic theory of natural language, showing how certain partsof language can be analyzed and modelled.

As mentioned, the next step of the research will be to provide a theoryof syntax that will allow us to handle various propositional and discourse-level issues: propositional attitudes and contextualism among them, but alsoanaphora, discourse-level phenomena such as settings and so on. This stepwill be also much more formally demanding, as it will be less about de-

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veloping core ideas and more about fitting those ideas together - a phasethat generally demands more stringent control to restrict hand-waving awayimportant issues. As with most buildings, the one started here will proba-bly look more refined and the architectural concepts behind it more visiblewhen it is built more fully, but hopefully the problems tackled in this studyare sufficient to claim that at least its foundations show promise for a goodconceptual framework for natural language.

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Towards a represenation-based theory of meaning

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