CONCHA MARTÍNEZ VIDAL

DPTO. DE LÓGICA Y FILOSOFÍA MORAL

UNIVERSIDAD DE SANTIAGO DE COMPOSTELA

lf lpcmav@usc.es

Is Logic Objective?***

Abstract. The purpose of this research is to assess the possibility of arguing for the

objectivity of logic from the point of view of two methodologies for determining which logic is

the right logic. Those methodologies are the one proposed by the view of logic as model

(Corcoran, 1973) and that of wide reflective equilibrium (Resnik, 1996). The basic idea of the

view of logic as model is that historically the logician's task consisted (and consists)—mainly—

in developing models to account for the underlying logic for mathematical practice.

Mathematical practice is supposed to play the role scientific facts play in the case of natural

science. Resnik’s methodology of wide reflective equilibrium establishes that we start with a

set of initial beliefs and logical intuitions and work on the adjustment between them (from

data to theory and from theory to data) until a wide equilibrium is reached. The implications

that follow from these analyses in relation to the objectivity of logic, and the difficulties they

face, are contrasted and evaluated in the light of Wright's (Wright, 1992) criteria to appraise

the objectivity of a discourse.

I. The view of logic as model and the normativity of logic

The basic idea underlying the view of logic as model (see Corcoran, 1973) is that

logicians build models of mathematical practice very much like scientists build mathematical

models of a given research field.1 In other words, logicians build models of the logics

underlying mathematical discourse, which is the way in which mathematical practice manifests

itself. Those propounding it intend this analogy to respond to the way the process of building

logical theories actually started with Aristotle, but they do not claim his analogy to be a totally

* This work was supported by the Spanish Ministry of Science and Technology under project BFF2002-00920. ** I have published a previous paper on this very subject (Martínez Vidal, 2004). This paper expands on some issues discussed on that publication. 1 I would say that logical models are also built in order to address other practices. For instance, non-monotonic logics were developed in order to account for common sense reasoning and such sort of reasoning existed long before non-monotonic logics were developed in order to model it.

faithful or adequate description of what a logical theory is. However—in Corcoran words

(Corcoran, 1973, p. 23)— “[a]t the least it is a useful myth, a Wittgenstenian ladder”.2

There are different kinds of model. For instance, Falguera (Falguera 1994) distinguishes

four kinds: analogical models, scale-models, mathematical models and theoretical models.3

The sense of the term ´model´ that underlies Corcoran’s view of logic as model (as well as

that of Burgess 1992 or Shapiro 1998) is that of ‘mathematical model’:

“A mathematical model is a mathematical construct by means of which we intended to

explain and make possible an inquiry into a given field of research.”4

According to Shapiro:

“…mathematical models provide a tool that is easy to work with, and easy to study. […]

a model should be ‘realistic’ in that some of its features do correspond, more or less, to

features of what it is a model of—natural languages (of mathematics) in the case at hand and

there should be a balance between simplicity and closeness of fit. Notice that, typically, there

is no question of ‘getting it exactly right’. For a given purpose, there may be bad models —

models that are clearly incorrect—and there may be good models, but it is unlikely that one

can speak of the ‘correct model’.” (Shapiro, 1998, p. 138)

Moreover, adopting the view of any subject as model implies:

⎯ Accepting that we elaborate representations;

⎯ Accepting that something, ‘the given’, is being represented;

⎯ Accepting it is possible to determine whether our representations (in this case our

logical theories) are or are not adequate;

⎯ Accepting that hypothesis (those that follow from our model) can be tested locally

(non-holistically).

Logical models

Following Corcoran, a logical model is a triple, <L, C, S>, composed of a formal

language, L, a calculus or deductive mechanism, C, and an appropriate semantics, S (Corcoran,

1973, p.28). It is important to realize that the notion of 'model' Corcoran is referring to here is

not the technical notion of 'model' that features in model-theoretic semantics; yet, ‘models’,

2 Other authors share with Corcoran the view of logic as model, though their exact formulations may not be identical: Burgess 1992, Shapiro 1998, etc. 3 See Falguera 1994. 4 My translation from Falguera 1994, p. 8.

in the technical sense of the term characteristic of formal semantics, are part of the model

(part of the semantics, S) in the sense in which Corcoran is using the term above.

Notice that Corcoran in his 1973 paper establishes the gaps between a given type of

logical model (the one provided by the model-theoretic account of the notion of logical

consequence underlying mathematical practice) and mathematical practice. But other logical

theories can also be conceived as 'logical models'. For instance, Gentzen's logical systems can

be understood as logical models of the underlying logics of mathematical practice. If the sort

of account provided by Gentzen’s sequent calculus is considered, then the model will consist

only of a formal language and a deductive calculus, and the semantics involved will be

implicitly given by the introduction and elimination rules for logical constants in the deductive

apparatus. In the case of first-order logic both the model-theoretic account and Gentzen's

account characterize the same set of logical truths. Thus, logical models provide explanations

about what a proof is, why a proof allows us to obtain mathematical truths and when our

premises are true; they also allow us to conclude whether all mathematical truths will be

deducible using a given logic, and so forth.

Models and Normativity

Both Burgess (Burgess, 1992) and Resnik (Resnik, 1985, 1996) distinguish between

theories that provide descriptions of a given practice (whether normative or prescriptive) and

theories that are prescriptive, that is to say, that tell us what our practice should be.

The view of logic as model, as it is understood in this paper, considers logical theories

as descriptions of a prescriptive practice.5 Our reason for so understanding the task of the

logician results from reflecting on the problem posed by Shapiro (in relation to the view of

logic as model) by means of his ‘toy’ example:

“...suppose that a mathematician violates the conditions on free and bound variables,

but still insists that her practice contains no mistakes. She claims that she has made no false

conclusions, and we have to admit that we cannot find any in her practice. The logician points

out that by ignoring the rules for parameters or free variables which the mathematician seems

to be doing, she might reason from true premises to false conclusion. The mathematician

retorts that the pedantic logician has mis-described the practice, and that she is not following

the flawed rule he has attributed to her, nor is she incorrectly following what the logician says

is the correct rule. The logician asks her to say what rule she is following. She declines the

5 For a discussion of the different uses of logical theories (prescriptive and descriptive) see Burgess 1992, Resnik 1985, Martínez Vidal 2004.

invitation, claiming to have no patience for logic, but she insists that she knows what she is

doing—suggesting we butt out. The logician might feel like Berkeley complaining about the

infinitesimal calculus. And the mathematician might feel like the analysts who apparently felt

safe in ignoring Berkeley's critique, for a time, anyway.

I presume that most of us sympathize with the logician in the above dispute. But the

perspective of logic-as-model-building makes the logician’s critique problematic. Why think

that the practice is in error? Would it not be more natural to think that it is our model that is

not correct, as the imaginary mathematician claims? In general, when do we blame a supposed

gap on the practice, favoring the model over the practice, and when do we blame the gap on

the model?” (Shapiro, 2001, pp.161-2).

Shapiro’s point is that the view of logic as model does not allow us to account for the

normativity involved in logic because it does not let us fix those cases in which we are in front

of a gap between the model and the practice and those in which we are facing a mistake. His

point seems to be that the view of logic as model does not provide any privileged stance to

settle the dispute in favor of the logician.

The problem for the mathematician is that the logician is applying the wrong logical

model to explain her practice, while the logician claims that the rule the mathematician is

using is not truth-preserving, and so she makes an incorrect inference. Our intuition is that

either the mathematician proves her rule is correct or the logician proves it is incorrect. We

seem to think that one of the two things must be the case. Shapiro’s point is that since

mathematical practice is what is to be modeled, the logician is not legitimated to correct the

mathematician.

But if the person who is disputing the correctness of the rule is a second mathematician

rather than a logician, then the burden of the proof is on the side of the first mathematician

and she has to convince her colleague that her intended proof is actually a proof. This seems

to imply that, at least in many cases, the prescriptive role is assumed by the mathematical

community itself. Therefore, logical models are to be descriptive in relation to (prescriptive)

mathematical practice.

Providing evidence that a given mathematical model is adequate to account for a given

practice is an example of what Corcoran calls an external question.6 That is to say, a question

6 Note that distinguishing between internal and internal aspects of a logical model seems to compromise Corcoran with a non-holist view of logic.

“…concerning the relationship of the model to things outside of itself.” (Corcoran, 1974,

pp. 119-20).

These are determined empirically. Once a model proves appropriate, it is used in order

to account for the given mathematical practice and can be used prescriptively. 7 Thus, given

that the model-theoretic account is what Corcoran calls ‘our current model of mathematical

practice’, the elucidation of what a logical truth is or of what the validity of a given argument

consists in will be the one postulated by that model.8 Therefore, in order to determine whether

a given logical formula is a logical truth, or whether a given argument is valid, there is no need

to go and see what happens in practice. The property of ‘being a logical truth according to the

model’ will amount to ‘being a logical truth of the core logic of the given mathematical

practice’. As a result, questions of validity, soundness, and so forth, are internal to the model:

“…internal questions are perfectly definite mathematical questions concerning the logic

as a mathematical object. And if they are answered, then they are answered by the same

means used to answer any mathematical question—viz. by logical reasoning from the

definitions of the systems together with the relevant mathematical laws” (Corcoran, 1974, p.

119).

Thus there are properties of the model that are internally decided, while other aspects

are to be solved empirically. This division between internal and external questions seems to

be a peculiarity of logical and mathematical theories. Logicians (and mathematicians) develop

other activities apart from developing models to describe practice (or reality). For instance,

logicians construct mathematical models, not only for the actual underlying logics of the

mathematician but also for putative logics⎯i.e., for logics which are envisaged by various

theories of logic but which do not necessarily correspond in any very faithful way to actual

underlying logics.9 When logicians are involved in this task, their methodology is exclusively

that of deduction and the resulting systems provide norms for correct reasoning. In these

cases, when a logical or mathematical theory is proved to be consistent, truth-preserving or

complete, even if it turns out to be insufficient to account for any given mathematical practice,

7 I am not claiming that Corcoran supports the account of normativity provided above. I am only claiming it is compatible with Corcoran’s view of logic as model. 8 Moreover, some models become theoretical models. This seems to be the case with the model-theoretic account of logical consequence. In that case the underlying logic of a specified mathematical practice is identified with the theoretical model and it comes to constitute a standard of adequacy. (See Martínez Vidal, 2004). 9 I paraphrase Corcoran, 1973, p. 31. Corcoran indicates he uses the term 'putative' without prejudice, i.e., we do not take as necessarily implying non-correspondence with an actual underlying logic.

it is enough for it to be interesting, though not enough for it to be adequate to account for a

given practice. In short, the situation seems to be that no logical (or mathematical) theory is

falsified except by proving its inconsistency.10 Considering logical (and mathematical) theories

interesting independently of their application explains the occurrence of what Resnik calls

‘Euclidean rescues’. We perform a Euclidean rescue when we consider a given theory

interesting and useful even if it proves false as a description of whatever data; for instance, we

consider Euclidean geometry useful and interesting even if it proved inadequate as a

description of the structure of physical space. According to Resnik this is so because (logico-)

mathematical theories need no interpretation since they do not assert that the structures they

describe are realized in this world (Resnik 1997, p.130).

II. Wide reflective equilibrium and the no mativity of logi r c

Michael Resnik has developed a completely different account of the normativity of logic.

First of all it is important to say that he is interested in the normativity of logic understood as

our all-purpose informal logic. For him, mathematical logic is part of mathematics and we use

logico-mathematical theories to account for our informal logic, but that informal logic cannot

be identified with its mathematical models. In fact, Resnik is a realist about mathematics

(about mathematical logic, too) while he rejects logical realism, as we shall see below. 11

According to him, in order to develop a logical theory for normative purposes we should

use the methodology of wide reflective equilibrium:

“Developing a logical theory for normative purposes does require one to focus on

reasoning, and the work closely parallels developing one for descriptive ends. However,

instead of starting with another’s deductive practice, one starts with one’s own judgments of

logicality, that is, considered judgments of validity, consistency, implication, and equivalence.

It is unlikely that initial attempts will produce an exact fit between the theory and the data.

And frequently ‘theory’ will lead one to reject ‘data’. […] Moreover, in deciding what must

give, not only should one consider the merits of the logical theory per se, such as its

simplicity, fruitfulness or elegance, or the firmness of one’s logical beliefs, but one should

10 With the exception of paraconsistent logics that seem to succeed in characterising inconsistent non-trivial theories. 11Resnik rejects the possibility of being a realist about our informal logic while rejecting mathematical realism. Hartry Field and Charles Chihara maintain such a position: they are realists about our all-purpose logic while they discard mathematical realism. Resnik’s arguments against logical realism (understood as realism about some kind of informal logic) aim at showing positions such as those defended by Field and Chihara untenable. That is his reason for making a distinction between logical theories that use mathematical apparatus (those are objective since they are part of mathematics) and informal logic, our all-purpose logic.

also consider how the theory and one’s intuitions cohere with one’s other beliefs and

commitments, including philosophical ones. When the theory rejects no argument that one is

determined to preserve and countenances no argument that one is determined to reject, then

the theory and its terminal database (set of considered judgments) are, to use Rawls’s term, in

wide refle tive equilibrium.” (Resnik, 1996, pp. 492-493) c

He understands ‘narrow reflective equilibrium’ as the process seeking “fit between our

intuitions concerning cases and logical rules”. While, ‘wide reflective equilibrium’

“… goes beyond such an initial fit and uses either empirical or philosophical views to

reform either our logical principles or our judgments about particular cases” (Resnik, 1985,

pp.224-5).

Once wide reflective equilibrium is achieved, the logic that allows one to attain it will be

the correct logic. In other words, a certain logic will be accepted in as far as it allows us to

simultaneously hold our empirical and philosophical views and our logical judgments.

Note that different people can attain wide reflective equilibrium by means of different

logics. And there is no way to compare the internal logic that allowed one of the subjects to

attain wide reflective equilibrium starting with her initial intuitions and beliefs, with the

different logic that allowed a second person to achieve wide reflective equilibrium beginning

with her own initial beliefs and logical intuitions. Nevertheless, Resnik acknowledges that at

some point we work locally:

“Yet in applying a logical theory to a particular argument, we simply work on

representing the latter in the former. Philosophical debates concerning the formal system, its

semantics or translation rules are set aside. […] More crucially, after we have represented the

argument in question, we do not take the further steps, found in wide reflective equilibrium,

of reforming the practice from which the argument is drawn or correcting the logical theory

itself—steps that may well open philosophical debates” (Resnik, 1985, p.225)

This is what happens when logical theories (understood in his way) are used in order,

for instance, to reconstruct a given argument.

Resnik (Resnik, 1996) describes a logical theory composed of: a formal system, an

appropriate semantics, metatheory, and translation method. It is clear that his notion of

logical theory is more complex than, for instance, that of Corcoran, since he introduces not

only the metatheory that establishes, among other things, that the formal system is adequate

for the semantics and vice versa, but also the formalization method, that is to say, the process

of translating the argument formulated in natural language into the formal language. This

conveys that all the background psychological, historical, linguistic and philosophical theories

we use in order to determine which argument is to be formalized, are, in some sense, part of

the model. In other words, this seems to entail the aspect of the modeled that is being

represented in relation to a given purpose or purposes is part of the model.

His quote above implies that whereas we might work locally in some cases, narrow

methodology is inadequate to determine which logic should be used. For Resnik, separatists—

those who consider statements can be tested independently and thus a category that includes

those defending the view of logic as model—are taking into consideration only part of the

methodology, namely, the one in which we work locally:

“The main difference between holists and separatists arises when a well-established

piece of axiomatic mathematics is part of a model conflicting with experience. Separatists

recognize only the Euclidean rescue, while holists also admit the options of rejecting the

mathematics as inconsistent or altering its underlying logic. They see nothing in the

epistemology of mathematics that excludes using these options in the rare instances when

doing so will benefit science more that the Euclidean option” (Resnik, 1997, p.135).

III. The given

Subscribing to the view of logic as model implies accepting there are data of some sort,

the given, that allow us to test our logical models. Given separatists hold that our hypothesis

can be tested independently of the rest of the theory, if one is a separatist one accepts the

existence of something like ‘logical data’. According to Hartry Field (Field, 2005), making

sense of what those ‘data’ amount to in the case of logic is not possible. He claims

Benacerraf’s dilemma for the logic case can be understood as follows:

“It would be better, then, to put the Benacerraf problem in terms of the lack of an

explanation of how our logical inferences depend on the logical facts. […] So the way to put a

Benacerraf problem for logic is something like the following:

(i) It seems in principle impossible to explain such things as how our acceptance of the

inference from A to A o B depends on the logical fact that necessarily if A then either A or B;r

(ii) Without such an explanation, to believe in a correlation between our accepting the

inferences we do and the logical facts requires belief in a massive coincidence;

(iii) The need to believe in such a massive coincidence undermines the belief in the

correlation, which in turn should undermine our acceptance of the inference.

[…]

The Benacerraf problem in mathematics or logic seems to arise from the thought that

we would have had exactly the same mathematical or logical beliefs, even if the mathematical

or logical facts were different; […] But in the logical case, we have no idea how to determine

what we would have believed had the logical facts been different: reasoning about what our

beliefs would be in alternative circumstances requires logic, and if we contemplate a radically

altered logic we have no idea how to conduct the reasoning. (See, Field, 2005, pp.7-9)

Quite clearly, the view of logic as model can be seen as an attempt to provide a picture

in which mathematical deductive practice constitutes the data.12

On the other hand, holists contend that it is our web of belief that is tested as a whole.

Hence, someone who is an objectivist about logic and a holist should provide an explanation

of how logical theories are tested together with the rest of our beliefs. Resnik, a well-known

moderate holist, claims that there is no strong analogue to the Quine-Putnam indispensability

argument for mathematical realism for the case of logical realism:

“A third reason that might come to mind is that much of contemporary philosophy,

especially metaphysics and philosophy of language, contains theories built upon a realist view

of logical possibility, necessity, implication, or equivalence. This reason is thus analogous to

indispensability arguments for mathematical realism. Now, the indispensability arguments for

mathematics are predicated upon our use of mathematics in science, and we dismiss

mathematized science at our peril. But it can hardly be said that we dismiss any contemporary

metaphysics of philosophy of language at our peril, much less possible-world semantics…”

(Resnik, 1997, p.164).

For his part, Corcoran understands that mathematics is “a class of sciences (geometry,

number theory, set theory, group theory, etc.) each having its own subject-matter or universe

of discourse” (Corcoran, 1973, p.23), and “each of which is typically axiomatically organized,

axioms are the initial ‘truths’ of the science and the rest of them are recovered by a

combination of definition and logical deduction” (Idem, p. 25). Besides, he assumes

12 It is also transparent that the picture is extensible to other human argumentative practices. But, if there is a practice that is objective, then it is mathematical practice. In consequence, if it were not possible to argue for its objectivity, the possibility of succeeding in arguing for that of any other practice looks implausible.

mathematical objects exist and that “… their properties are not subject to alteration […] they

are not created by an act of will” (Idem, 24). In other words, he assumes a realist conception of

mathematics, 13 thus taking for granted the objectivity of mathematical discourse. From this

perspective, ‘the given’ is clearly something objective and logical data amount to the

underlying logic of a given mathematical practice.

But, if a realist standpoint is not taken from the start, understanding logic as a science

that develops models of argumentative practices in general and of mathematical practice in

particular could be understood as propounding that logic is an account of the arbitrary

agreements that mathematicians or human beings devoted to this or that activity have come

to achieve. It might appear to amount to changing logic into sociology or into modeling the

inferences some mathematician has performed.

Our purpose in this paper will be to analyze the prospects for arguing for the cognitive

character of logic from the perspective provided by the view of logic as model without a priori

assuming a realistic conception of mathematics. The analysis will be done in terms of Wright’s

criteria for objectivity—for determining the cognitive character—of a given discourse.14 We will

also apply Wright’s criteria from the perspective provided by the wide reflective equilibrium

methodology and Resnik non-cognitivist view of logic. The paper has two main aims. First, to

measure the actual explanatory power of both conceptions in relation to mathematical and

logical practice. Second, to check to what extent Wright’s criteria provide a useful tool

independently of the philosophical position in which we place ourselves as Wright aimed them

to do.

IV The objectivity of mathematical practice

Crispin Wright has advocated several conditions that should allow us to determine

whether a given discourse is or is not cognitive. Following Shapiro (Shapiro, 2000), though

with some differences, we will use those criteria to evaluate the case of logic. We will also take

into consideration Resnik’s arguments for the non-cognitive character of logic and try to

argue against them.

13 He calls his position ‘neutral platonism’. For a more detailed presentation, see Corcoran 1973. 14 See Wright 1992.

The Euthyphro contrast

In the debate between Euthyphro and Socrates, Euthyphro claimed that an act is pious if

and only if it is pleasing to the gods. Euthyphro contended that there is no more to piety than

what the gods want. Socrates argued that at best the gods have the ability to detect piety.

Socrates’ view here is that piety is objective. The debate focussed on which of these comes

first, the chicken or the egg. Euthyphro's view is consistent with a non-cognitivism about

piety, in which case there are no piety facts at all.15 Shapiro extends Wright’s outline of

constraints on a Euthyphro contrast for discourses concerning color, shape, morality,

modality, etc. to the case of logic:

“For the sake of discussion, let us agree on the following (material) biconditional:

An argument is valid if and only if an ideal agent acting under optimal conditions judges

it to be valid.

Does the agent's verdict detect validity, as Socrates might contend? Does the verdict

constitute validity, as a logical subjectivist or relativist might hold? Or does the verdict serve to

manifest a non-cognitive stance towards the argument (e.g. signaling its role in inferential

practice) as Resnik contends? The issue is whether the methodology of wide reflective

equilibrium is a very good (fallible or infallible) method for figuring out whether a given

argument is valid, or whether there is no more to validity than the end result of the process?”

(Shapiro, 2000, p.362)

In order to try to answer these questions we will apply Wright’s Cognitive Command

test, according to which we need to establish:

• whether agreement among mathematicians is commanded of mathematicians; 16

• whether there is some characteristic of arguments that accounts for their validity and

the fact that a given move has or lacks this characteristic is something that is commanded of

the mathematician / logician.

The criterion of cognitive command has been set by Wright in order to articulate

Socrates’ side in the Euthyphro contrast. Wright’s proposal maintains that ‘being commanded

of the mathematician’ at least evaluates whether there is some ‘resistance’ on the side of

mathematical practice that corresponds to the sort of ‘resistance’ natural data offer to our

15 I follow Shapiro 2000 and Wright 1992. 16 Where ‘commanded of’ is understood in the sense formulated by Wright in his 1992 book.

scientific theories that is independent of the mathematician / logician. Wright has formulated

and articulated the intuition underlying the words ‘being commanded of’ and has made it

operative in terms of the cognitive command principle:

“… we may say that a discourse exerts Cognitive Command if and only if it is a priori

that any differences of opinion formulated within the discourse, unless excusable as a result

of vagueness in a disputed statement, or (vaguenes )s

17 in the standards of acceptability, or

variation in personal evidence thresholds, so to speak, will involve a cognitive shortcoming.”

(Wright, 1992, p. 144)

Shapiro (Shapiro, 2000) provides a very clear elucidation of the underlying idea we now

paraphrase: in the case of a given area of discourse functions to describe some mind-

independent reality, whenever two speakers diverge about something in that area of

discourse, then at least one of them has misrepresented the reality, and so is guilty of some

cognitive shortcoming. On the other hand, if a given area of discourse does not serve to

describe a mind-independent realm, then disagreements need not involve cognitive

shortcoming on the side of either party. So, what of logic?

Is agreement among mathematicians commanded of them?

It seems obvious that whenever one works with a given mathematical theory and makes

moves in order to obtain proofs of different mathematical statements, a given move does not

turn into a correct one simply because a given mathematician or a given community of

mathematicians arbitrarily decides it to be so. One can arbitrarily decide to use the terms of a

given mathematical theory in a certain way, but whether that method will result in an

interesting theory or in one that complies with our expectations is clearly beyond us. History

provides us with many different examples: Frege’s project for the foundations of mathematics

failed because it proved to be inconsistent, Hilbert’s project falls short because non-

constructive methods cannot be reduced to constructive ones, and so forth. So, in a sense

logical and mathematical systems clearly offer some sort of ‘resistance’ to the logician and

mathematician that can be interpreted as analogous to that offered by the data to the

scientist. Therefore, it seems agreement among mathematicians is not the result of arbitrary

conventions. It looks as if there exists some condition a proof must satisfy in order to be

17 The text inside brackets and in italics, ‘(vagueness)’, has been added by me because Wright explicitly clarifies he means this in the text on the same page as the quotation.

considered as such and that such a condition is independent of the mathematician. Resnik

(and Wright 1986) agrees that if a sentence ϕ follows from Γ according to logic L, then the

conditional, ‘if Γ then ϕ (according to logic L)’ is objectively true. It is clear then, that in order

to see whether a given logic (formal or informal) complies to certain standards it is not

necessary to take a compromise with those principles, that is to say, in order to see whether a

given proof is intuitionistically acceptable one does not have to be an intuitionist, but only to

use the rules of the corresponding logic. Thus, it is a priori that any difference of opinion

formulated within the logical di cour e, unless excusable as a re ult of vagueness in a

disputed statement, or (vagueness) in the standards of acceptability, or variation in personal

evidence threshold , so to speak, will involve a cognitive shortcoming.

s

s s s

s

Consequently, it seems there is some objectivity at this level. Nevertheless, Michael

Resnik has argued that non realists can also account for agreement in these cases in terms of

shared conventions, shared psychologies, conceptual frameworks or inferential practices:

“On these views, you and I agree that the Russell paradox leads to an absurdity because

all humans are inclined to react this way or because we share the same language or inferential

practices.” (Resnik, 2000, p. 185)

If such an agreement is the result of a shared logic to be explained in psychological

terms, then some objectivity obtains, since the fact that we as a species share a given

psychological behavior is ‘independent of us’, in the sense that we have not decided what our

psychological structure should be.

On the other hand, concurrence about these issues according to the other possible

explanations Resnik points at—shared conventions, shared conceptual frameworks or shared

inferential practices— is trivially due to obvious non-objective reasons. In those cases,

agreement is ascribable to the fact that we all share a norm— for instance, that logical rules

should be truth-preserving— therefore we concur as to which applications of the norm are

correct.

Does this observation by Resnik show that Wright’s cognitive command test does not

apply? Not quite since, though there is agreement in the cases depicted above, there is a level

at which there is no agreement. Problems arise when we ask ourselves, for instance, which

logic is the right logic:

“So far we have no argument that a pronouncement either for classical logic or relevant

logic would count as detecting the facts, constituting them, or manifesting a stance. Two

logicians can agree that the irrelevant inferences preserve truth, …, without agreeing on the

status of the inference. […] A logical realist would say that the logicians are attempting to

detect the correct logic, while the subjectivist, non-cognitivist and relativist would say

otherwise.” (Shapiro, 2000, p. 363)

For Resnik logic is non-cognitive and there is only one logic, but this logic is not a

priori; it is a posteriori, as follows from the description of the wide reflective equilibrium

methodology. Also, he is a (moderate) holist,18 that is to say, he rejects the separatists’

distinction between internal and external matters. As a result:

“The main difference between holists and separatists arises when a well established

piece of axiomatic mathematics is part of a model conflicting with experience. Separatists

recognize only the Euclidean rescue, while holists also admit the options of rejecting the

mathematics as inconsistent or altering its underlying logic. They see nothing in the

epistemology of mathematics that excludes using these options in the rare instances when

doing so will benefit science more than the Euclidean option” (See Resnik, 1997, p.135).

On the other hand, the thought underlying the view of logic as model is that different

logics can be adequate models for dissimilar practices or even acceptable models for different

aspects of a given practice. Thus different logics can be taken as prescriptive depending on

the practice that is to be modeled and the goals to be achieved. From this stance, the point of

coincidence between logical models and scientific models is that the former, though truth-

preserving, can be unsuitable to characterize a given mathematical practice. But, in the case of

logical models, having certain internal properties seems to be enough to consider a given

mathematical model interesting. This should allow us to understand why logicians develop

putative logical systems without any intention whatsoever of accounting for human

argumentative practices. Resnik’s explanation of this feature of mathematical practice goes as

follows:

“In proving a mathematical theorem we do not come to know a logical fact (for instance,

that the axioms logically necessitate the theorem) which holds independently of our beliefs,

proofs and practices. However, we do come to know the conditional truth that axioms hold

18 He is a moderate holist since he takes confirmational holism to be the correct methodology for science; but against what he calls ‘strict’ holists (Resnik, 1997), he acknowledges that scientists and mathematicians often work locally. He assumes the justification for their working locally follows from confirmational holism for pragmatic reasons.

only if the theorem does and that the axioms and theorem are connected via deduction. This

permits us to insulate the truth from empirical refutation.” (Resnik, 2000, p.193)

This isolation from refutation can be revised, if necessary to attain wide reflective

equilibrium. Resnik also claims we can apply Euclidean rescues to scientific theories by

reinterpreting them, though this is not usually the case. Whether we apply it or not depends

on what our overall scientific activity establishes to be best.

In relation to the relevantist disagreeing with the classical logician about which logic is

the correct one, Shapiro’s conclusion is that if Resnik is right (among other things about his

defense of monism), then either the cognitive command test shows that logic is non-cognitive

or the condition will be useless since logic will be part of our cognitive abilities (quite clearly if

we apply the criterion assuming the same logical frame there will trivially be agreement). That

seems correct since, on one hand, Resnik does not seem to have an answer to Shapiro’s toy

example: if both the logician and the mathematician are in wide reflective equilibrium, there is

no possibility of agreement. For the logical realist one can be in reflective equilibrium and still

be wrong. According to Shapiro (Shapiro, 2000), since Resnik is a realist about mathematics

and science, he needs to explain why one can be wrong about a given belief (or set of

scientific or mathematics beliefs) even if one applies the methodology of wide reflective

equilibrium, while in the case of logic, there is no more to validity than the end result of the

process.

I understand there is still another possible way of considering the dilemma posed by

Shapiro. Maybe the cognitive command test does not apply because reports of logicality are

theory-laden. In other words, in order to apply the cognitive command test we need to

determine that evaluation of logical models actually complies with the requisites posited by

Wright (Wright, 1992) on theory-ladenness:19 To argue that one faces a problematic case of

theory-ladenness one has at least to make a case that reports of correct mathematical practice

do not count as acceptable, or unacceptable, independently of one's logical beliefs; that what

is proper to report oneself as having observed as logically sound is a function of, inter alia,

elements of logical theory which one carries into the observational situation. As a result, if we

succeed in showing that evaluations of the satisfactoriness of a logical model to account for a

given mathematical practice are not theory-laden, we will have found some sign of the

objective character of logic.

19 Wright distinguishes four cases of theory-ladenness and points out that only the one depicted below is problematic. (See Wright, 1992, pp. 160-161)

On the other hand, if reports of which logic is the right logic are theory-laden, that is to

say, are not independent of the logic one takes into the situation, then the non-objective

character of logic follows.

Different logical models are frequently compared in order to determine: which provides

the best account of the way mathematicians reason; which deductive mechanism is easier to

use for a human / artificial logician; whether they are extensionally adequate (in the sense that

they allow us to characterize the set of valid arguments); whether or not the language used

has enough expressive capacity to formalize mathematical structures, and so forth. As a

result, one logical system or another may be used depending on its adequacy to attain a

certain goal.

For example, intuitionists reject classical logic on philosophical grounds, not because it

does not succeed in explaining the assumptions and methodology a mathematician employs.

They take that classical logic cannot be an adequate model of mathematical practice because

it does not provide an adequate epistemology. Classical logicians argue that intuitionist logic

is unacceptable because it mutilates classical mathematics, but constructivists in general and

intuitionists in particular contend that given Bishop's claim (Bishop, 1967) intuitionistic logic

succeeds in accounting for the interesting part of classical mathematics; again classical

mathematicians reject this because that part of mathematics that can be explained in

intuitionistic terms is not adequate for certain foundational purposes,…

If we analyze these arguments and counterarguments it can be seen that: i) the dispute

between intuitionists and classical logicians was initially a dispute between a descriptive

account of classical mathematics (a prescriptive practice) and a prescriptive account of a

prescriptive practice; ii) the best argument classical logicians waged against intuitionists was

that the intuitionist account mutilated classical mathematics, in other words that it forced us

to reduce significantly our body of mathematical knowledge; iii) moreover part of that body of

mathematical knowledge has proved very useful to other disciplines such as physics; iv)

constructivists, such as Bishop, reconstructed part of classical mathematics trying to solve

mathematical problems in a way that allowed us to compute the result; v) it is now claimed

that that part of classical mathematics Bishop succeeded in reconstructing is enough to

account for that part of mathematics that is relevant to science.

The view of logic as model sheds a new light on traditional questions such as 'which

logic is the underlying logic of classical mathematics?', or 'which logic is the right logic?', or ‘is

second order logic?’, etc. because, according to it, what we have is different logical systems

that succeed in accounting for different aspects involved in mathematical practice. And though

it makes sense to ask which model provides the best balance between simplicity and closeness

of fit, given a certain feature of the practice, typically there is no question of ‘getting it exactly

right’ (Shapiro, 1998, pp.138-139).

In the case of the dispute about which logic is the right logic our concern is whether

determining, for instance, which logic is the underlying logic of classical mathematics, is free

of theory-ladenness in the sense depicted above.

In order to see this, let us consider a possible argument to choose between classical and

intuitionistic logic. To start with let us consider it as a particular case of the general problem

of theory choice. Burgess and Rossen20 have formulated a list of criteria they take to be

"generally accepted criteria of theory-choice". If we apply (a slightly modified version of) these

criteria, the view of logic as model allows us to supply an argument to conclude that classical

logic is the best model we count on to explain classical mathematical practice. That argument

would consist in surveying the classical model and the intuitionistic model in the light of the

criteria pointed out by them:

1) Correctness and accuracy of the model;

2) Precision, range and breadth of the model;

3) Internal rigor and consistency;

4) Minimality or economy of assumptions in various respects;

5) Consistency and coherence with familiar, established theories (or failing this,

minimality of change);

6) Perspicuity of the basic notions and assumptions; and fruitfulness, or capacity for

extension.

Let us agree, for the sake of the argument, that Bishop's developments allow for an

intuitionistic reconstruction of classical analysis and that this is enough for the mathematician

working in those aspects of classical mathematics that do not involve logic-related fields.

Take it that we acknowledge that both classical logic and intuitionistic logic add similarly in

relation to the three first criteria above. Assume we agree that it is disputable whether

classical or intuitionistic logic should score higher in respect to the fourth criteria—one

implies a stronger ontology and a deficient epistemology but allows simpler proofs, while the

other supplies a constructive epistemology but at the cost of making proofs more complicate).

20 Taken from Shapiro, 2000b, p. 248.

Surely, classical logic punctuates higher in the remaining two. But this of course begs the

question, because what we are doing is modeling an already evaluative practice. And in order

to accept that classical logic punctuates higher in relation to features 5 and 6 one has to apply

classical standards. In other words, this argument will not convince the intuitionist who

considers that the fact that classical mathematics implies assuming a transcendent notion of

truth makes classical mathematics un-perspicuous and unacceptable.

This appears to have to do with the fact that determining whether a given mathematical

practice is right depends to a certain extent on the philosophical thesis we assume. That is so

because our intuitions or judgments about the correctness of a given argument will differ

depending on our philosophical views. Hence, there seems to be a problem with theory-

ladenness here. Note that this is clearly the case if the methodology of wide reflective

equilibrium is applied, since this establishes that in order to achieve wide reflective

equilibrium our philosophical judgments should cohere with our logical ones. Our issue now is

that if the same is achieved when we adopt the methodology underlying the view of logic as

model, then the non-objective character of logic follows. Does our assessment of which logic

is the right one heavily depend on the intuitions we assume in the evaluations?

If we apply the cognitive command test in this case, the result is two different

standards, and disagreement cannot be explained in terms of a cognitive shortcoming on one

on the sides. If, for the sake of the argument, Bishop’s thesis is accepted, we count on two

extensionally coincident accounts of 99 per cent of classical mathematics, that part of

mathematics that is relevant to the working mathematician not in logic-related fields

(Feferman, S., 2000, p. 402). That 99 per cent of classical mathematics that is of interest to

the working mathematician can easily be formalized in ZFC, and that is all mathematics that is

currently being applied in science. Of course, divergent positions arise in the case of

mathematicians working in logic-related fields and philosophy of mathematics. They reject

Feferman’s claim because only classical non-constructive mathematics is enough for

foundational issues. And the same goes for classical logic.

But, if both parties agreed that the fact that both intuitionistic and classical logic can be

seen as accounting for 99 per cent of classical mathematics is all there is to this story, we

would have a case of two extensionally equivalent though conceptually different accounts of a

given practice. This is perfectly compatible with the view of logic as model. We would

determine which model to use depending on, maybe, specific goals.

But Bishop’s claim is not accepted by many. Moreover, even if it comes to be accepted,

classical mathematicians working in logic-related fields would consider that intuitionistic logic

leaves out a lot of classical mathematics. So what we have is that the extensional adequacy of

the intuitionistic / classical approach is evaluated differently depending not only on the

position in philosophy of mathematics one defends, but also on whether it is maintained that

only that part of mathematics that is indispensable for science is to be accepted or whether

one sustains stronger realist views on mathematics.21 This seems to be a clear case of theory-

ladenness.

Yet, we can analyze the issue in different terms. Imagine we, again for the sake of the

argument, use another criterion of objectivity proposed by Wright, wide cosmological role.

Then we would be substituting the Wrightean frame according to which the issue is evaluated

in terms of the criteria of wide cosmological role, for the usual theory-choice criteria that aim

at determining which model provides the best explanation. Our concern would be once more

the question of the objectivity of mathematical practice.

Wright establishes that a subject has a wide cosmological role, just in those cases where

mention of its state of affairs can feature in at least some kinds of explanation of

contingencies which are not of that sort, explanations whose possibility is not guaranteed

merely by the minimal truth aptitude of the associated discourse. The use of the term

'cosmological' in the title does not imply that the explanations have to be causal. "Explain" is

understood as to make plain, clear or intelligible.22

Hence, at this point the difficulty is not whether a given logical model (that is consistent,

truth-preserving, and so on) is adequate to account for a given mathematical practice, but

whether a given logical theory plays a part in some kinds of explanation of contingencies

which are not logical. As a pertinent illustration, we could talk about mathematics: at least that

part of mathematics that is indispensable for science plays a wide cosmological role. But our

subject here is logic and we have quoted above Field’s reconstruction of Beneacerraf’s

dilemma for logic as questioning the possibility of establishing a relation between our

accepting an inference and accepting ‘logical facts’. Yet, our discussion above has illustrated

how the choice of classical or intuitionistic logic has implications for the set of mathematical

truths that obtains. In fact, mathematics shares with logic the special status of being allowed

proof as its standard methodology; thus, in order to see whether a given logical theory plays a

wide cosmological role, we have to see whether that part of mathematics that only obtains if

21 Quite obviously, if the consistency of Set Theory could be proved, we would not be discussing this issue either. 22 Again I follow both Wright 1992 and Shapiro’s discussion in his 2000.

classical logic is taken to be the underlying logic of classical mathematics plays a wide

cosmological role. The only part of classical mathematical practice that is nowadays necessary

for science is ZFC. The wide cosmological role of that part of classical mathematics that is

used in physics is clear, in that it is indispensable for science, according to the famous Quine-

Putnam argument. But that part of (classical) mathematical practice that features in scientific

explanations is the one both classical logic and intuitionistic (or constructive) logic can

account for (assuming Bishop’s result). So, it is precisely that part of mathematics that

satisfies the wide cosmological role requisite. Therefore, as things stand at the moment, that

part of classical logic that is not constructive cannot be claimed to satisfy the wide

cosmological role requisite. Nonetheless,

1) It is not clear that "non-constructive mathematics does not satisfy the wide

cosmological role requisite" is not temporally indexed. The issue is whether we do know that

no more mathematics than that 99 per cent that can easily be formalized in ZFC will be ever

applied, for instance to the development of more powerful models for physics. Imagine it

were. In that case, classical logic would feature in some kind of explanation of contingencies

whose possibility is not guaranteed merely by the minimal truth aptitude of the classical (non-

constructive) mathematics discourse. In other words, classical non-constructive mathematics

would be said to have a wide cosmological role. And again, the traditional contention against

intuitionistic logic would be in place. For the time being, applying the wide cosmological role

criteria in these terms leaves us unable to decide.

2) However, note that even if Bishop’s claim is right, classical logic is still indispensable

to account for full classical mathematical practice. Thus, in as far as one considers that there

is some way to isolate conditional truths as ‘if Γ then ϕ according to L’, full classical logic

features significantly in facts that are not logical. Note Wright adds that, in order for the

objectivity of the discourse to obtain, those facts have to be contingent. As a result, only those

philosophers who reject the necessary character of mathematical truths can claim that

classical logic has a wide cosmological role.

3) Moreover, even if Bishop is right, that part of classical mathematics that is not

indispensable for science in as much as it is not directly used to formulate scientific theories is

still useful:

“It is hard to see how investigations of the consistency of various systems of analysis

could be directly useful in constructing scientific models or in deriving empirical

consequences from physical hypotheses formulated in the vocabulary of analysis. Despite this,

the information these proofs provide about the reliability of those models is relevant to

evaluating the sciences using them. Similarly, it is often useful for scientists to know of

mathematically equivalent ways of formulating scientific theories or of probabilistic

approximation methods that might substitute for rigorous calculations. Yet in deriving

empirical results they can often use alternative formulations directly without having to cite

metamathematical equivalence proofs.” (Resnik, 1997, p. 131)

Acknowledgements

Earlier drafts of this paper were read during 2003 at the Colloquium of Logic and

Analytic Philosophy at the Universidad de Santiago de Compostela, at the Seminar of the Logic

Department of the Universidad Complutense de Madrid and at the Congress “Ontology and

Mathematics” organized by the Dipartamento di Filosofia of the Università degli Studi di

Padova. Agustin Rayo and specially Stewart Shapiro provided great help with their comments.

In 2005 a version very similar to this one was presented at the Instituto de Filosofia da

Linguagem of the Universidade Nova de Lisboa. I am very grateful to all those who helped with

comments on earlier versions of this paper; however, the responsibility for any remaining

unfortunate assertions is mine.

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