Leśniewskian Term Logic

Simons / Lingua e Stile / page 1

Leśniewskian Term Logic Peter Simons*

Published in Lingua e Stile 27 (1992), 23–46.

1 The Preparation Problem Any system of formal logic that can lay claim to be of use in assessing the validity of arguments carried out in natural language—and a system is at best of specialized interest if it cannot—must confront the preparation problem. Just as a specimen organism caught in the wild must be specially prepared for the biology laboratory, so a specimen argument caught in ordinary life must be prepared for logical analysis. Every logical system to date has been markedly simpler in its syntactic forms and semantic variety than any natural language, and would have no use were it not so. Every logic student since logic began has had to learn the knack of taking an argument in natural language and finding its equivalent or near equivalent in the straitened idiom of the logic being taught. This procedure is more like translation than what Quine calls ‘regimentation’, though the two are akin. Like preparation in laboratory work, it belongs to the craft rather than the high art of its discipline. Students of traditional logic, by which I mean the central core of categorical syllogistic with whatever further forms were studied at the time, were drilled in putting the sentences occurring in arguments into “correct logical form”, and present-day students do no different when replacing their natural language sentences by the formulas or semi-formulas of predicate logic. Both procedures involve doing some violence to natural modes of expression. A sentence like Whoever flies saves time1 must be replaced by something like Every flier is a time-saver by traditional logicians and by For all x: if x flies then x saves time by modern logicians. As this makes clear, different logical systems may compete in offering prepared forms proximate to a natural specimen, so there may be a real choice as to which system is preferable for a given purpose. This is familiar to observers of modern logic since there are competing logics of definite descriptions, modality, and so on. Of course, if we confine attention just to the opposition between * I wish to extend my thanks to Eva Picardi for encouraging me to write on this topic and to the Departments of Philosophy and Cognitive Science at the University of Texas at Austin for their hospitality during the writing. 1 Specimens will be marked by italicization rather than quotes.


categorical syllogistic and predicate logic, there seems to be no contest. Predicate logic is expressively much the more powerful system, and as these two are the only two logical systems to have enjoyed widespread acceptance as tools for analysing validity of natural arguments, it might seem that only predicate logic remains as a general vehicle for workaday argument assessment. But the large number of introductory logic textbooks which still contain material on categorical syllogistic bears witness to the fact that, within its more limited sphere, the traditional logic of terms is widely felt to be a more natural and useful alternative to monadic predicate logic. Historical interest alone could not compensate for the inconveniences of introducing two quite different systems, with their different sentential analyses, laws, and terminology, to cover the same ground. It is apparent that one disadvantage of predicate logic for these purposes is its use of bound individual variables, which natural languages do not have, and which they can simulate and match only by rather tortuous use of pronouns and pronominal phrases. Of course this helps to account for the greater perspicuity of predicate logic once we leave the simplest sentences behind, but at the most elementary level it is a hindrance. The singular term/predicate analysis of simple predications compels common noun phrases and adjectives used attributively to appear as syntactically inseparable parts of predicates, which correspond most closely to verb phrases in natural language. Again, this is not a huge sacrifice, but it is pervasive, is felt to be unnatural, and contributes to beginners’ difficulties in learning logic. So it is worth considering from a practical and pedagogical point of view whether, in order to gain the considerable benefits conferred by predicate logic—quantification, multiple generality, relational predicates—it is necessary to put up with the disagreeable features of standard predicate logic. I shall argue that it is not, and that a more natural and flexible medium for which to prepare natural language sentences and arguments is provided by the term logic invented around 1920 by Stanisław Leśniewski (1886–1939) and usually known as Ontology.2

2 The possible confusion of the system of logic with the branch of metaphysics of the same name is not a danger in this context, and in any case I will write the name of the system with a capital letter. Sometimes Ontology is called ‘calculus of names’, but this is misleading, since much more than names are involved. It would be nice to have a better name for Ontology.


2 Distinctive Features of Ontology 2.1 Presentation and Propositional Basis For my purposes it is not necessary to give a formal presentation of a system of Ontology. Leśniewski himself always presented his systems in what is now a very unorthodox way, as collections of concrete inscriptions growing in time by the addition of new theses, rather than as an abstract platonic structure. This concretist approach to logic has philosophical points in its favour, but I shall here just introduce pertinent features of Ontology. Like predicate logic, and like syllogistic as formalized e.g. by Łukasiewicz, Ontology is based on a “propositional” calculus, more appropriately perhaps described as a calculus of two truth-values. The kind of propositional logic favoured by Leśniewski and called ‘protothetic’ by him has several features which distinguish it from standard two-valued systems. It allows quantification of propositional and higher functorial variables, has inference rules for the quantifiers, and is based on material equivalence and the universal quantifier alone. For our purposes it is enough to know that we have all the classical tautologies and valid inferences of classical propositional logic, and more if we need it. 2.2 Kinds of Term The major relevant difference between Ontology and predicate logic lies in the kinds of semantic values its terms may have. However, since terms standardly come in different kinds, constants and different kinds of variables, we need to introduce some terminology to be precise about how Leśniewski’s terms are different. We distinguish as usual between the terms bound by quantifiers or other operators in a proposition, calling these variables, and those terms which are not bound, called free terms. Among free terms we wish to make a further distinction, semantically motivated, but perhaps also syntactically marked, between those free terms which have a fixed meaning in the system as interpreted, called term constants, and those without a fixed meaning, sometimes called free variables, but here called parameters. These distinctions are not novel, but they will be carefully observed. Among simple (non-complex) expressions of any grammatical category we may similarly distinguish constants, variables, and parameters. A complex expression containing a parameter does not itself have a fixed meaning, since the part corresponding to the parameter is indeterminate.


Such expressions we shall call schemata of their category. The most important schemata are sentence schemata. Most so-called sentences of classical propositional and predicate logic are in fact sentence schemata, even if they do not contain individual parameters, since they contain propositional and predicate parameters respectively. Leśniewski required all theses of his logics to be sentences rather than sentence schemata, since he thought all theses of logic should be truths in the ordinary sense. A formula which is a schema is often called “true” in a derived sense, namely when any specification of it obtained by uniform substitution of constants for parameters is true. From the standard point of view according to which logic is primarily concerned with formally correct or valid inference rather than logical truth, inference schemata are a natural and uncomplicated device, so provided their exact status is clear, it would be niggardly to reject valid sentence schemata as a special case. We shall allow them. 2.3 Singularity: Syntax vs. Semantics It is usually said that the terms, bound and free, of standard predicate logic are singular terms. We must be clear what this means. Firstly, the distinction between singular and non-singular may be reflected in syntax. In predicate logic, there are no terms in a syntactic category which is non-singular, but out of deference to grammar, where the terms ‘singular’ and ‘plural’ characterize the form of a nominal expression, irrespective of what it denotes, I shall reserve the terminology of singular and plural for syntax. The more important side of the concept of singularity for our purposes is semantic, and has to do with how many individuals a term denotes. We shall describe semantically singular terms as singly denoting.3 What this means depends on the kind of term. A constant term is singly denoting simply when it denotes exactly one individual. A term parameter on the other hand may be called singular when all its admissible meaning specifications are singly denoting. And finally, a bound term variable may be called singly denoting when all its admissible validity-preserving substituends in the standard quantifier inference rules of instantiation and generalization are constants or parameters which are, in the senses defined above, singly denoting. Thus we have three rather different senses of ‘singly denoting’ linked by analogy, of which the basic sense is that of a constant’s being singly denoting. It is useful to maintain this analogy for ease of expression. In standard predicate logic all terms are singly denoting, in whichever sense is applicable for their 3 If we want a single word we might call such terms mononoemic. We could say that singular terms are those which purport to be mononoemic. Then of course plural terms purport to be polynoemic (plurally denoting).


kind. Hence there is no inaccuracy involved in aligning the syntactic name ‘singular’ with the semantic property of denoting (in whichever sense) just one thing. In free logic not all terms are singly denoting. Free simple terms, that is, constants and parameters (if these are distinguished), are allowed to be non-denoting, that is, they may fail to denote any individual. Again, the basic sense of this is that for constants. Syntactically, however, no distinction is made in free logics between free terms which are singly denoting and those which are non-denoting: they are intersubstitutible salva con-gruitate with one another. Variables on the other hand are always singly denoting, in the sense appropriate to variables. This is so despite the existence of empty (non-denoting) singular terms in free logic, since the definition of being singly denoting for variables requires all admissible substituends for which the standard quantifier rules are valid to be singly denoting. This they are, since the empty terms in free logic are precisely those for which the standard quantifier rules fail to hold. If we were to allow non-denoting term variables in a predicate logic, the inferences For all x: Fx, therefore Fa, and Fa, therefore for some x: Fx would be valid for empty a, where x is no longer a singular but now a non-plural (singular-or-empty) variable. Clearly in such a logic the expression For some x: Fx could not mean the same as There exists x such that Fx, it does not carry existential import, and the for some quantifier has to be called the particular rather than the existential quantifier.4 In such a logic one would want to introduce a singular existence predicate, or some other predicate in terms of which the existence of an individual could be expressed. It is no longer what is standardly called a free logic, since in such logics the quantifiers retain their standard meaning. A system of this sort was published in 1967 by Czeslaw Lejewski, and he has an identity predicate which is false when flanked by non-denoting terms, so he can define singular existence.5 2.4 Plural Denotation In Ontology, terms may have a third kind of semantic value: in addition to being singly denoting or empty, they may also denote more than one individual, or be plurally denoting. As before, the primary sense of this is for constants. Strictly, what we have for parameters is that they may be plurally denoting in some of their admissible 4 Peter Geach has suggested calling the quantifiers by the translations of their Polish names, ‘major’ and ‘minor’. But I think ‘universal’ and ‘particular’ are perfectly adequate. 5 See Lejewski 1967.


specifications, and variables may be replaced by plurally denoting terms in the standard quantifier inferences. In Ontology, no terms are syntactically constrained to be in different substitution classes according to whether they can or cannot be singly denoting, or denoting, etc. They are all, as I shall say, syntactically and semantically neutral. No grammatical mistake is made by substituting in a sentence a non-denoting term for a singly denoting one, for instance, though of course the truth-value of the resulting sentence may be different. The other feature of Ontology that sets it apart from other term logics is that, even with terms that are neutral, the standard quantifier inferences are valid. Hence, again, For some x: Fx cannot mean There exist (one or more) x such that Fx. 2.5 Singular Inclusion In Ontology, quantifiers are allowed to bind the new term variables and those of any nominative category we might introduce. Ontology has its own distinctive logical constants to set it apart from the protothetic upon which it is built. There are numerous logical functors which can serve as the primitive. The one originally used by Leśniewski and still used most widely is the functor of singular inclusion, usually written ‘ε’. A sentence like ‘a ε b’ is understood to be true if and only if the term ‘a’ is singly denoting and the term ‘b’ denotes the same individual as ‘a’, and maybe some more besides. Of all the various English readings of this sentence which can be and have been offered, the one which seems to me to best capture its intended meaning is ‘a is one of b’.6 There are numerous different axiomatizations of Ontology, even with ‘ε’ as the primitive functor.7 One I rather like is based on two conditional axioms: LEFT-REFLEXIVITY: For all a, b: a ε b → a ε a TRANSITIVITY: For all a, b, c: (a ε b and b ε c) → a ε c. The axioms alone do not constitute Ontology: there are inference rules for the quantifiers, for constructing definitions, and generalized extensionality principles for all higher functors. These contribute essentially to the formal power of the system, and account for 6 See Simons 1982. It has often been pointed out that in Leśniewski’s native Polish, a language without articles, the simple ‘a jest b’, ‘a is b’, is a pretty good reading for ‘a ε b’. 7 For some of these, and for a clear exposition of Ontology, see Lejewski 1958.


much of its flexibility. In particular the multitude of fruitful definitions that can be made in Ontology suit it for use in analysing ordinary speech, and I shall give a sample of the things we can define later on. The extensionality principles allow one to intersubstitute coextensional expressions of a given category salva veritate in any sentential context. This thoroughgoing extensionality of Leśniewski’s logic has been questioned, but I do not need to address the issue. The systems with which Ontology competes in this arena are categorical syllogistic and standard predicate logic, both of them extensional. Whether or not that are or can be useful non-extensional enlargements of or alternatives to these systems need not be discussed here. 2.6 Syntactic Categories Leśniewski’s logic was the first to be consciously based on a theory of syntactic categories. Leśniewski derived the idea from Husserl, and liked to call them ‘semantic categories’, to emphasise that the expressions in each category are to be meaningful from the start. The basic idea is now very familiar from treatments of categorial grammar and type theory, so I can be brief. In notation and terminology I shall differ in minor details from other expositions. Ontology has two basic categories, SENTENCE, here symbolized ‘S’, and TERM, here symbolized ‘T’.8 Functor categories are those of expressions which combine in a determinate way with one or more expressions of certain input or argument categories to yield a complex expression of a given output or value category. The con-struction of expressions designating functor categories may be illustrated by examples. If the categories of inputs of a given, say three-place functor category are, in an order fixed by the way the functor expressions work, β, γ, δ, and that of the output category is α, then the functor category is symbolized by the functional expression ‘α⟨β γ δ⟩’. Thus the category of the singular inclusion functor ‘ε’ is S⟨TT⟩. Functors and their categories having names with ‘S’ as the leftmost symbol are called propositive, those having a ‘T’ as leftmost symbol are called nominative. Propositive categories are needed to form complex expressions which are sentences, and nominative functors are needed to form complex terms. In Ontology there is no limit to the categories which may occur, so it resembles type theory in this respect. Most modern systems of type theory nevertheless manage with only one-place functors, getting the effect of a functor with more than one place, e.g. one of category α⟨βγ⟩, by the use of a functor-forming functor, say of 8 It is usual to call the second category NAME and symbolize it ‘N’, but I am not using the term ‘name’ as widely as Leśniewski. Relics of the terminology linger in the word ‘nominative’, below.


category α⟨γ⟩⟨β⟩. Leśniewski does not allow these two kinds of functor to be collapsed in this way, distinguishing many-place functors from many-link functors. This makes his system more variegated than standard ones. One motivation for not making the distinction is that natural languages seem not to do so, at least not as rigidly as Leśniewski. However Leśniewski allows a kind of flexibility which most logical systems do not, which is to use the same symbol on different occasions for expressions out of different categories, especially if a logical analogy exists between them. This is a little like the Whitehead–Russell device of typical ambiguity, and may be made use of in analysing arguments from ordinary language. 2.7 Some Definitions There are two kinds of definition in Ontology. Propositive definitions define propositions (sentences) or propositive functors and have the form of a formal equivalence: PROPDEF: For all ... : P(...)...(...) ↔ --- where P is the sentence or propositive functor being defined, the definiens is a sentence to the right of the equivalence, and the blanks in the parentheses are filled by suitable variables, all bound to the left by the first universal quantifier. (Certain other requirements have to be fulfilled which will not be reviewed in detail here.9) Nominative definitions define terms or nominative functors and have the standard form NOMDEF: For all a... : a ε N(...)...(...) ↔ (a ε a and ---a---). Here the N is the term or nominative functor being defined, and ---a--- is a sentential context in which the name ‘a’ occurs. Again, all variables are bound to the left and certain conditions are required to exclude circularity and other vices. The first conjunct in the definiens is there to prevent paradoxical definitions. As can be seen, Leśniewski takes definitions not as metalinguistic abbreviations but as object-language formal equivalences. Here are some of the propositive functors which can be defined in Ontology. I have not given the definitions, and the English expression is at best a near approximation to the formula of Ontology, but it will be seen that many useful ideas are expressible. The

9 For details of how such definitions are to be made, see Lejewski 1958, 172-4.


letters ‘a’ and ‘b’ here are place-holders for terms, which in the appropriate definitions would be variables free in the definiendum and definiens, but bound at the beginning of the formula by a universal quantifier. The definiens is in effect in each case the remainder of the expression considered as a whole, apart from these variables. There is at least one a There is at most one a There is exactly one a There is no a There are at least two a All a are b (no existential import of the subject term) Every a is b (with existential import) Some a is b Some a is not b No a is b (no existential import) None of the a is b (with existential import) Many a are b Few a are b Most a are b Three a are b Finitely many a are b Infinitely many a are b Only a are b All and only a are b (general identity or nominal coextensionality, without existential import) All and only the a are the b (general identity with existential import) a is the same object as b (singular identity with existential import) The two a are the same as the two b (dual identity) and so on. Here are some of the names and nominative functors which can easily be defined. Thing (Object, Entity) Nonentity (Nothing) non-a (term negation) a and b (term conjunction) a or b (term disjunction) a which is not b (term difference) Further definable functors will be mentioned below.


2.8 Comments on These Definitions As this list may have made clear, many of the propositive functors we can define bear on the one hand a close relation to what in natural language we think of as quantifier expressions, and the nominative functors show terms to be susceptible to Boolean operators. Both of these features are important. The Boolean character of this calculus of terms was something of which Leśniewski was well aware. He describes his system as embodying an algebra of logic of the sort developed in the nineteenth century, and brought to culmination by Ernst Schröder, and including the theory of atoms or individuals. The algebra of such a theory is now generally known as an atomic Boolean algebra.10 The idea of recognizing Boolean structure, i.e. in effect conjunction, disjunction and negation (complementation) among not just propositions but many other kinds of expression has only recently re-emerged in the logico-linguistic literature,11 but Leśniewski had it, and knew he had it, in the 1920s. Note that because terms in Leśniewski can be multiply denoting, there is a point in having Boolean functors for them, whereas in standard predicate logic term conjunction is trivial, and disjunction and complementation take us outside singly denoting terms, so lack the closure property. The ability to define in a simple way the standard quantifier-functors of traditional term logic was also known and appreciated by Leśniewski. The unbounded variety of quantifier-functors definable allows us to regard Leśniewski as the true precursor of a discipline which developed only much later, the theory of generalized quantifiers. What is more, as can be shown, the treatment of generalized quantifiers which can be given in Ontology is one which can be given largely within the object-language, whereas standard modern treatments use a metalanguage and a metatheory of sets. In these respects Leśniewskian ideas were not only well ahead of their time, but perhaps more theoretically neutral than later, platonizing developments. This will now be taken up. 3 Further Developments 3.1 Generalized Quantifiers The quantifier-type functors, like singular inclusion, are of category S⟨TT⟩. If we were predicate-logicians, we would say this is the category of relations or binary predicates. 10 For an investigation of the interpretation of atomic Boolean algebra in Ontology, see Lejewski 1960–1. 11 See Keenan and Faltz 1985.


But because of plural denotation in Ontology we must distinguish the syntactic property of being a binary sentence-forming functor in two terms from standing for a binary relation. We can indeed define what it is for such a functor to be relational: For all F: Rel(F) ↔ for all a, b: F(ab) → (a ε a and b ε b), but the ability to have first-order functor expressions for quantifier-like inclusion functors allows us to treat on the first-order object language level what modern generalized quantifier theorists treat as at the second-order level, and interpret in terms of relations between sets which are the denotations of predicates. Leśniewski was very antithetical to set theory in the form due to Zermelo, regarding it as unintuitive, so any version of generalized quantifier type phenomena which can be accomplished without appeal to sets would have met his approval. The question of whether Ontology is somehow indirectly committed to sets through its semantics will come up for discussion later. Of course the standard laws and inferences of traditional syllogistic and Boolean algebra do come out correctly in Ontology, so Leśniewski could rightly view his logic as subsuming previous term logics. We might add that it does so in a way which stays much closer to these, and to natural language, than predicate logic. It will have been noticed that the inclusion functors do not isolate the quantificative role from the copulative, whereas in syllogistic and natural language we do this. We have noun phrases like ‘all men’, ‘some mad dogs’, which are syntactic units. In some versions of generalized quantifier theory this is expressly recognized.12 In Ontology we in fact have the means to do this too. If f is a variable functor of the category S⟨T⟩, we have the definitions of multi-link functors ‘All’, ‘Every’, ‘Some’, ‘No’, as follows

For all a, f : All(a)(f) ↔ for all b: b ε a → f (b) For all a, f : Every(a)(f) ↔ for some b: b ε a, and for all c: c ε a → f (c) For all a, f : No(a)(f) ↔ for all b: b ε a → ~ f (b) For all a, f : Some(a)(f) ↔ for some b: b ε a and f (b).

These definitions are not of course startlingly original, but the point is that the use of definitions for multi-link functors, by employing the variables f and a as parameters, enable the quantifier words to be given meanings of their own. Using higher-order

12 See Barwise and Cooper 1981.


definitions (quantifying over relations) we can further define such interesting quantifiers as the equinumerative quantifier with four parameters: As many a f as b g (e.g. As many men smoke as women drive). This can be used to derive a whole family of related quantifiers with fewer parameters, but we consider here instead the similar family definable in terms of the majoritative quantifier More a f than b g (e.g. More men drive than women smoke), which is again a 4-parameter quantifier with two names and two intransitive verbs, and we can use this to define two three-parameter ones More a than b f (e.g. More men than women smoke) and More a f than g (e.g. More men drive than smoke) and finally, via More a f than do not f the two-parameter Most a f. Likewise we may define proportion quantifiers, such as Three times as many a f as b g A quarter of a f and so on. Bounds of the definable seem to be set more by human ingenuity than inherent limitations of the system. An advanced feature of such quantifiers in Ontology is the ability to define com-pounds, especially using easily definable Boolean operators, so we can employ quantifiers like Some but (i.e. and) not all Between seven and twenty (i.e. not fewer than seven and not more than twenty) Either twelve or seventeen and many more, without having to define each one specially, by using the ‘and’, ‘not’ and ‘or’ definable for the quantifier categories, in particular the ‘and’ and ‘or’ etc. of category

S⟨S⟨T⟩⟩⟨T⟩ ⟨S⟨S⟨T⟩⟩⟨T⟩ S⟨S⟨T⟩⟩⟨T⟩⟩. 3.2 Copulae The original quantifier-functors mentioned in Section 2.7 are also as it were quantifi-cational copulae, joining two terms in a sentence. What might a “mere” or “pure” copula without a quantificative aspect look like? The simplest kind of predication in most natural


languages consists of a term and an intransitive verb, such as Socrates runs. A second term does not itself have the form of an intransitive verb phrase, so it will not cohere with another name to give a sentence unless we provide some syntactic “glue” which enables it to do so. So if we want to put two names together in a predication, as in Socrates is a Greek, we need something like the copula ‘is’ or ‘is a’. In Ontology the singular inclusion functor ε does this already pretty well, but it does it in a particular way, namely by taking both terms at once to form a sentence. If we prefer the idea of first taking the predicate term and using a copula to adapt it for use as an intransitive verb phrase, then in Ontology we have to go about this slightly differently. Many people will think it is petty to distinguish the two ways of forming the predication, and indeed it may be, but that judgement can first be made when we have separated the two and compared them. Suppose then we define a new copula ι which forms a singular transitive verb phrase from a term, like the ‘is a’ of English or the ‘jest’ of Polish can be taken to do. We can of course define this in terms of ε. In doing so I shall put the intransitive verb phrase in front of its term argument, the standard functional notation, which is not only a more consistent use of this notation but also brings out clearly that we are defining a quite new functor. The definition is the obvious one For all a, b: ι(b)(a) ↔ a ε b. (Note that if ε were written first, as it often is in Leśniewski, we should have ‘ε(ab)’ on the right-hand side.) So ι is another multi-link functor. Obviously, since we have an equi-valence here, it would be perfectly possible to take ι rather than ε as the sole distinctive primitive functor of Ontology. Now we are in a position to see how an English sentence like No man is an island can be represented in Ontology in a way which very closely fits its actual syntactic structure in English: No(man)(ι(island)) and as we should naturally expect, this turns out, via the definitions of the quantifier No and the copula ι, to be equivalent to the No ... is --- exclusion functor mentioned in Section 2.7. In the particular negative sentence Some man is not an island we might contemplate defining a negative copula isn’t, and indeed we could do so, but it stays closer to English structure if we recognize that a copula, of category S⟨T⟩⟨T⟩, can also have its own kind of Boolean negation, and insert that into our syntactic structure just as it stands. It is tiresome but elementary to check that the standard immediate inferences, like obversion


and contraposition, and syllogisms with negative terms, are valid inferences in Ontology when we adopt this syntactic analysis of categorical sentences. (Care has only to be taken in ensuring suitable denotata when existential import is required: the additional premisses need to be stated.) 3.3 Syntactic Flexibility It may seem, as we said, rather excessive to have the two copulae ε and ι as alternatives in a syntactic analysis. Of the two, the more flexible is indeed the newly defined ι, since the intransitive verb phrases it builds can go wandering on their own without requiring a first term all the time. This thought is behind those versions of categorial grammar and type theory which dispense with multi-place functors in favour of one-place multi-link ones. In Ontology we can simply defer or avoid the question by employing just one-place functors, that is, using only a fragment of the available language. However, in practice even those logicians who say they only use multi-link functors get the effect of using multi-place ones, especially binary ones, by a notational dodge: an “abbreviation” is introduced which gives us the effect of a multi-place functor, while it is said that the meaning is otherwise. The clearest example of this is Alonzo Church.13 I think this is cheating, and that Leśniewski is more honest. It is because in Leśniewski everything which for other logicians would be an abbreviatory definition or a convenient notational variant is directly in the object language of Ontology that this diversity exists in his language. Nevertheless there is a feature of the rigidity of Leśniewskian syntactic categories which makes them less than a perfect match to natural language. In natural language sentences like John admires Marilyn, Some man admires Marilyn, John admires some woman, and Some man admires some woman, the selfsame transitive verb admires, assumed to be of category S⟨TT⟩, happily pairs up with either a name left and right, or a name left and a quantifier phrase right, or a name right and a quantifier phrase left, or a quantifier phrase on either side. In Ontology this will not work syntactically. This is where categorial grammars score which allow words to have more than one category without being ambiguous, according to what are known as type-change rules. The most frequently invoked type-change rule is Geach’s Rule: an expression of category α⟨β⟩ may also, as occasion demands, have category α⟨γ⟩⟨β⟨γ⟩⟩.14 So some woman, 13 See Church 1940. The square brackets in Church signify that this syntactic switch is taking place. 14 See Geach 1972.


construed as having category S⟨S⟨T⟩⟩, has this category when combining with just a simple verb like flies, but in the phrase admires some woman it must have the category S⟨T⟩⟨S⟨T⟩⟨T⟩⟩ in order to combine with admires. There is a lot to be said for allowing this kind of flexibility into the syntax of Ontology. However there is another possibility. The flexibility of type-changes may be regarded as the outcome of the operation of a number of structure-modifying formal operators, of the type generally known as combinators. The transition from ε to ι can be seen as the result of a general type of combinator which lifts the first argument of a two-place functor out to the right on its own. We can capture the force of such a lifting combinator by the definition schema For all f, a, b: L(f)(b)(a) = f (ab). This is a schema rather than a definition, because a, b and f may be expressions of any categories α, β and γ⟨ab⟩ respectively, and the functor written ‘=‘ is the coextensionality functor of category γ⟨γγ⟩. So we could in fact view ι as just the functor L(ε). Conversely, were we to take ι rather than ε as primitive, we could take ε as the product of ι and a combinator converse in its effect to L . There seems to be some evidence that the kinds of transformation which modify syntactic structures in natural language can be plausibly viewed as natural language combinators.15 If that is so, it is a gain in analytic clarity to know how the various versions are related by combinators, even if for everyday purposes we simply get on with making the transformations and adjustments as usual. I have suggested that structure- modifying combinators in fact explain the type-change rules,16 so it is at least helpful to know we can have them in Ontology. We could also take advantage of Leśniewski’s device of allowing different tokens of the same symbol type to stand for closely analogous functors and simply write ε in place of ι. The disadvantage of such a practice, once started, is that there are no strict rules for controlling it, so it could be abused, and its limitations are not circumscribed. It would seem that for pedagogical purposes one would do better, as now, to rely on students’ knowledge of their own language, and introduce them a rather more straitened idiom such as a fragment of Ontology for their argument preparation.

15 See Steedman 1988. 16 Simons 1989.


4 Disadvantages of Ontology and How to Overcome Them 4.1 Lack of Singular Terms Ontology lacks terms constrained to be singular. In his expositions, Leśniewski used the device of putting a capital letter variable where we should expect from the context that the term would be singly denoting, especially in a context like ‘A ε α’. But this was just an informal aide lire and not a syntactic restriction. For those raised on post-Frege predicate logic, the lack of a special syntactic category of singular terms may seem a retrograde step. Of course in Ontology we can always simulate the effect of singular terms in any context by prefacing a formula containing terms required to be singly denoting by a conditional antecedent of the form (a ε a and b ε β and ...). But this can be tiresome. In any case, do not facts like that Socrates is a man is a grammatical sentence but *Man is a Socrates is not show that there is a genuinely syntactic distinction between singular terms on the one hand and general terms on the other, which is precisely what the traditional term logic covered up for so long? Here I think we have to concede that the fit between Ontology and natural language is not exact. There is, at least in a large class of languages, a genuine and fairly clear distinction between definite terms and common nouns, though how thoroughgoing it is may be debated. To remain within Ontology is to suppress or play down this distinction. On the other hand, the considerable advantages of naturalness that Ontology enjoys over predicate logic seem to me to amply compensate for this. It is simply that, as elsewhere, we need to do our preparation. Could we somehow overcome this limitation and compensate for Ontology’s lack of singular terms? There are several ways to do so. One way is to extend Ontology with one or more classes of term, constrained to be non-plurally denoting or perhaps even singly denoting. These would not belong to a different grammatical substitution class from the usual terms, but their semantic constraints would allow them to be employed without having to express the tiresome uniqueness conditions as antecedents. The variety of possibilities here have been explored by me (for other reasons than this) in an earlier paper.17 It is clear however that this is a largely pragmatic and cosmetic expedient which does not address the claim that syntactically singular and general terms belong to distinct substitution classes. The addition of these extra constrained terms does not enhance the expressive power of the system, so they are also logically dispensable.

17 Simons 1985.


A more radical departure from Ontology would be to have two syntactically distinct classes. Instead of the single category TERM would could have the category NAME (N), restricted to singular or non-plural terms, and a category COMMON NOUN (C) for which ordinary proper names, definite descriptions and the like are ineligible. The first to have suggested this seems to have been Ajdukiewicz.18 To date the only fully worked-out proposal is due to Lejewski.19 He takes his logic of non-reflexive identity for the non-plural terms and adds common nouns and principles for them. The resulting system is a closer fit than Ontology to ordinary language. On the other hand, everything that is expressible in this system has a logical equivalent in the part of the theory (the S–C part) having only common nouns: the non-plural terms prove again to be logically dispensable, and the pure theory of common nouns is just like Ontology with only a change of names to protect the innocent. So nothing is gained logically and the resulting system is, for obvious reasons, more complicated than Ontology. So while naturalness speaks for it and against Ontology, simplicity without loss of power speak for Ontology. I think this is a trade-off which cannot be decided once and for all, but depends on the purposes for which one uses a system. For pedagogic purposes, teaching elementary logic, Ontology seems to me to be preferable by virtue of its greater simplicity. 4.2 Complicated Proof Theory There is no denying that the proof theory available in publications for Ontology is not as adapted to modern needs as those we can find for predicate logic. Ontology was developed in the 1920s when the only known method for giving a rigorous account of a logical system was axiomatic. If we add to this Leśniewski’s concretist approach to logic and his uncompromising insistence on absolute rigour at all stages, we can see why the systems of Ontology preserved in publications are without exception moderately to extremely forbidding. An additional factor is the particularly Polish sense of formal elegance, which enjoins one to look for a single axiom, as short as possible, in a single primitive. Fortunately, Ontology is in this regard by far the least forbidding of Leśniewski’s systems, having in the form based on ε the unsurpassably short axiom AXONT For all a, b: a ε b ↔ for some c: a ε c and c ε b.

18 Ajdukiewicz 1934. 19 Lejewski 1979.


Nevertheless, the use of axiom systems requires more ingenuity than other methods of proof, and they are hardly used in elementary logic any more. Leśniewski’s own recipe for adding new theses (definitions and theorems) to his systems are contained in his famous Terminological Explanations, which have daunted all but the most intrepid explorer. Clearly they are for formal rigour, not for daily use. In conversation, Leśniewski used to compare the system so described with evening dress with a stiff collar, for formal occasions only, to be contrasted with the lounge suit or jacket of everyday attire. Following the sartorial metaphor, we might say that what we need for today’s logic students is a jeans and teeshirt version of Ontology. His own everyday practice for himself and in his lectures was to use a kind of natural deduction system, perhaps the first, which can be seen in the writings of Lejewski, Sobociński and others of his students. Unfortunately, however, neither he nor his followers produced a canon of this system, since it was regarded as a tool for sketching how to make a proper i.e. axiomatic proof. Obviously one way to make Leśniewski’s ideas more accessible would be to combine distinctive principles of Ontology with more well-known systems of natural deduction such as those of Fitch, Lemmon, Quine or Copi.20 This would put the derivation of theses of Ontology within the reach of the average logic student accustomed to such methods, and enable Ontology to be used instead of the disjoint combination of syllogistic and predicate logic often favoured today. Seemingly the easiest kind of proof theory for beginners to master is the tree or tableau method. This virtually eliminates the uncertainties of strategy searching required in natural deduction, and enables students to advance quickly. A proof theory for Ontology using the tree method makes it much more accessible. It builds on propositional trees, and adds quantifier rules analogous to those used in standard predicate logic. There is an added pedagogical advantage to presenting Ontology in this way. As numerous logic teachers from Łukasiewicz to Copi have recognized, the system of categorical syllogistic forms a stage of usefulness and complexity intermediate between propositional logic and full predicate logic with its multiple generality and its undecidability. But syllogistic as it stands cannot be built in between the two since it relies, as we have seen, on quite different methods from either propositional or predicate logic. A valiant and virtuoso attempt to bridge the gap is Quine’s theory of Boolean schemata in his Methods of Logic.21 But the plural in the title tells the story. Each increase in power is approached with new methods. Quine takes pains to optimize his methods at each stage, but there is

20 There are some steps in this direction in Miéville 1984. 21 Quine 1982.


an inevitable discontinuity of development. If the tree method can be harnessed for all stages from propositional logic through simple term logic to full Ontology the pedagogical benefits would be considerable, and the learning process can be straightforwardly cumulative. For a simple term logic without variable-binding quantifiers a tree proof procedure can be given.22 This is complete and decidable. So syllogistic can be mastered by the same general methods as propositional logic: a basis of term conjunction, term negation and the existence predicate There is a --- will suffice with propositional logic to cope with anything Aristotle, Boole or Carroll may throw at us. This does not yet give us even the basic level of Ontology, since it does not give us a way to talk about individuals (it is Boolean, not atomic Boolean). The easiest way to get this is to add quantifiers binding term variables and define There is at most one a as For all a: There is at most one a ↔ for all b: if there is an a b there is no a non-b. Singular inclusion can then be defined as For all a, b: a ε b ↔ there is an a and there is at most one a and there is no a non-b. Still we may stay before unleashing the full power of a system equivalent to predicate calculus, by restricting variables and parameters to those for terms, giving the decidable system known as elementary Ontology. With relational parameters but no relational variables we enter a system as strong as the complete but undecidable monadic second-order predicate logic. Thereafter, the rungs of type theory beckon. 4.3 Lack of Formal Semantics Formal semantics was only beginning to be developed late in Leśniewski’s life by his former student Tarski. Because Leśniewski regarded his systems as vested with meaning from the start, he saw no need to develop a semantics for them in the sense of a theory intended to breathe meaning into a system of meaningless marks, and his example was followed by his students. There are isolated starts on semantics and model theory for Ontology, of which the most concerted is due to Stachniak.23 The main restraint on this development, apart from a widespread lack of knowledge about or interest in Ontology,

22 Simons 1989a. 23 Stachniak 1981. See also Rickey 1985, Simons 1985a.


has been reconciling Leśniewski’s known views, which were antithetical to abstract objects in general and sets in particular, with the standard machinery for doing semantics. I am not convinced that a nominalistically acceptable semantics is impossible,24 but in any case we are free to interpret Leśniewski’s logical system as we like, and there is nothing in his system incompatible with a semantics involving sets or hierarchies of functions. So while it is true that Ontology lacks a formal semantics, this is a contingent matter and can be remedied by relatively straightforward modification of existing techniques. 5 Prospects It will be apparent that I am an admirer, though not an unreserved one, of Leśniewski and his Ontology. It has been a pleasing experience to see how often those apprised of the features of Ontology that I have mentioned have reacted positively and even enthus-iastically that the system is natural and appealing, while at the same time regretting that it is so little known. It has been then a more frequent and disheartening experience to find how few philosophers and logicians, especially outside Poland, know much about his ideas or are prepared to take the trouble to find out. Nearly everyone reacts to Leśniewski’s heterodoxy by trying to interpret his views in terms of their orthodoxy, such as set theory or type theory, and ends up doing slight to immoderate violence to his views. It should be possible to present the attractive aspects of the theory, especially its ontologically innocent use of plural denotation, and its definitional fecundity, in a palatable, user-friendly and teachable way. Then the gap between traditional and modern logic which Leśniewski saw himself as bridging would be well and truly bridged.

University of Texas at Austin Postscript (2015) This piece should have mentioned Leśniewski’s ‘Definitions and Theses of Leśniewski’s Ontology’, published in 1988, based on lecture notes from 1929–30 and comprising a condensed outline of his system, its axiom and rules, its subsumption of syllogistic and Boolean algebra, some condensed proofs and indications of how to derive more, totalling 59 definitions and 633 theorems. For more on generalized quantifiers in Ontology, see my 1994. Both items have been added to the references. 24 In particular the use of plurally denoting terms does not compel one to use sets as denotata. The ontological innocence of plural terms has been stressed by Boolos: see his 1984.


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Leśniewskian Term Logic

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