. Categories of first-order quantifiers Urszula Wybraniec-Skardowska Abstract. One well known problem regarding quantifiers, in particular the 1storder quantifiers, is connected with their syntactic categories and denotations. The unsatisfactory efforts to establish the syntactic and ontological categories of quantifiers in formalized first-order languages can be solved by means of the so called principle of categorial compatibility formulated by Roman Suszko, referring to some innovative ideas of Gottlob Frege and visible in syntactic and semantic compatibility of language expressions. In the paper the principle is introduced for categorial languages generated by the Ajdukiewicz's classical categorial grammar. The 1st-order quantifiers are typically ambiguous. Every 1st-order quantifier of the type k > 0 is treated as a two-argument functorfunction defined on the variable standing at this quantifier and its scope (the sentential function with exactly k free variables, including the variable bound by this quantifier); a binary function defined on denotations of its two arguments is its denotation. Denotations of sentential functions, and hence also quantifiers, are defined separately in Fregean and in situational semantics. They belong to the ontological categories that correspond to the syntactic categories of these sentential functions and the considered quantifiers. The main result of the paper is a solution of the problem of categories of the 1st-order quantifiers based on the principle of categorial compatibility. Mathematics Subject Classification (2000). Primary 03C07, 03H05; Secondary 03A99. Keywords. 1st-order quantifiers, Categorial languages, Syntactic categories, Denotation, Ontological categories, Denotational Semantics, Compositionality, Categorial compatibility. 1. Introduction Sentential logic beginnings date back to ancient times, times the Stoics (3rd c BC), while the notion of a quantifier arose only in XIX century and of the calculus of quantifiers, predicate logic – only at the beginnings of the XX century. The beginnings of this logic, as we know, were created by Gottlob Frege (Begriffschrift, [11]). The main problems and their solutions connected with the subject of this paper were presented at the Logic Colloquium'15 held in Helsinki on August 3-8, 2015, co-located with the 15th International Congress of Logic, Methodology and Philosophy of Science, CLMPS (see my abstract [35]; see also [36]). 2 Urszula Wybraniec-Skardowska Around 1879 Frege and – independently – Charles Sanders Peirce developed a way to extend sentential logic by introducing symbols representing determiners, such as 'all', 'some', 'no', 'every', 'any', and so on. Frege and Peirce used two symbols: the universal quantifier (which we will write ∀) corresponding roughly to the English 'all', 'every' and 'each' and the existential quantifier (which we will write ∃) corresponding to the English 'some', 'a', 'an'. In this paper we will consider only standard, Fregean quantifiers ∀ and ∃ of the 1st-order as individual variable-binding operators. They are used in formulas of predicate logic of the 1st-order and in formalized languages of elementary theories based on this logic. Their syntactic role and semantic references, i.e. denotation, extension, created some problems that have not been satisfactorily solved yet. In the next part (Section 2) I shall partially explicate the problem of quantifiers. In Section 3, I'll outline some intuitive foundations of my theory of categorial languages which gives the formal direction for justification of my solution of the problem of quantifiers. It corresponds to the principle (CC ) of categorial compatibility based on some Frege's ideas and was formulated by Roman Suszko [27]. The solution of the problem is presented in Section 4. 2. Problem of quantifiers The problem of quantifiers is connected with the difficulty pertaining to establishing their syntactic and semantic categories. Leśniewski's theory of semantic/syntactic categories [16, 17], which was improved by Ajdukiewicz [1] by introducing categorial indices, does not, obviously, solve this problem, which limits the universal character of the theory. Leśniewski's hierarchy of semantic/syntactic category does not include any variable-binding operators. Leśniewski, in his protothetics and ontology systems, allows only one operator – the universal quantifier, noting it as parentheses, Ajdukiewicz, on the other hand, indicates the difficulty of assigning to quantifiers the index s/s or s/ns. Assigning to them the index s/s, i.e. the category of sentence-forming functors of one-sentence argument, would mean that the quantifiers belong to the same category as one-argument connectives, and assigning to them the index s/ns of sentence-forming functors of one-name and one-sentence arguments would mean that we include them into the same category as some expressions of indirect speech, e.g. 'think that', 'know that', etc. It has been suggested that the categorial grammar, which Bar-Hillel derived from Ajdukiewicz's version of the theory of semantic/syntactic categories, does not satisfactorily account for the role of bound variables and operators binding them. Suszko [25, 27] assigns to them the index s//s/n, and thus the index of sentence-forming functor of the argument, which is a one-argument predicate. In Categories of first-order quantifiers 3 this way, the index, for example in the sentence '∀x(x flows)' pertains to the entire quantifier-variable pattern '∀x(x . . .)' (see Simons [23]) which corresponds to English word 'everything' (see also Cresswell [7], Simons [24]). Suszko and many other researchers of language syntax treat quantifiers as expressions independent of the quantifier variable. Generally, researchers avoid bound variables in attempting to solve the problem, for example by means of combinators (Curry [8, 9], Curry and Feys [10], see e.g. Simons [23]). But earlier, Suszko stated that mounting variable-binding operators into a syntactic scheme requires general principles other than the theory of syntactic/semantic categories. The principle (CC ) of categorial compatibility is one such principle. It allows us to assign to every expression of a formalized 1st-order language, which possesses an index symbolizing a syntactic category, a denotation whose ontological category (relative to the universe U of a given model of the language) is indicated by the same index. Suszko assumes that • the denotation of the entire expression ∀x(e(x)), where e(x) is a sentential function with the free variable x, is ether the logical value 1 (of truth) or the logical value 0 (of falsity) which belong to the ontological category with the index s, and • the denotation of the universal quantifier ∀ is the function of generalization which has the value 1 in only one case, if its argument is the universe U . The function of generalization belongs to the ontological category with the index s//s/n because its arguments are any sets belonging to the family P (U) included into the ontological categories with the index s/n. In this way the principle (CC ) holds although the principle of syntactic connection (SC ) does not hold because no index is assigned to quantifier variable x, and the scope of the quantifier ∀ (here e(x)) is not one-argument predicate of the syntactic category with the index s/n. In the next parts of this paper I explicate both the principle (SC ) of syntactic connection and the principle (CC ) of categorial compatibility on the basis of my theory of categorial languages [30–34] which allows us to give some solutions to the problem of quantifiers. The essence of the approach proposed here is considering them to be typical syntactic notions: functors-functions mapping language expressions into language expressions that correspond to some functions on extralinguistic objects – on denotations of arguments of these functors. Let us note that a standard background for research in the field of mentioned quantifiers assumes treating them as some functions or relations on extralinguistic objects, mostly functions with index t//t/e (cf. Mostowski [21], Lindström [18], Montague [19, 20], Nowaczyk [22], van Benthem [3, 4], van Benthem and Westerståhl [5]). 4 Urszula Wybraniec-Skardowska 3. Some intuitive foundations of the theory of categorial languages 3.1. Main ideas of formalization of categorial language In the paper, formal-logical considerations relate to syntax and extensional semantics of any language L characterized categorially : • in the spirit of some ideas of Husserl [15] and Leśniewski–Ajdukiewicz's theory of syntactic/semantic category (see Leśniewski [16, 17], Ajdukiewicz [1, 2]), • in accordance with Frege's ontological canons [13], • in accordance with Bocheński's motto [6]: syntax mirrors ontology, and • some ideas of Suszko [25–28]: language should be a linguistic scheme of ontological reality and simultaneously a tool of its cognition. The paper includes developing and some explications of these authors' ideas. It also presents, in a synthetic form, some ideas presented in my papers published in [30–34]. Language L is there defined, if the set S of all well-formed expressions (briefly wfes) is determined. These expressions must satisfy requirements of categorial syntax and categorial semantics. 3.2. Categorial syntax The categorial syntax of L is connected with generating the set S by the classical categorial grammar and belonging wfes of S to appropriate syntactic/semantic categories. A characteristic feature of categorial syntax is that each composed wfe of the set S has a functor-argument structure, in this sense that, in accordance with the principle originated by Frege [11], it is possible to distinguish in it its constituent called the main functor, and the other constituents –– called arguments of that functor, yet each constituent of the wfe has a determined syntactic category. If e is a functor-argument wfe of S, f is its main functor and e1, e2, . . . , en its subsequent arguments then e can be written in the functional-argument form: e = f(e1, e2, . . . , en). (e) In categorial approach to the language L, syntactic categories of wfes of L are determined by attributing to them, like their expressions, categorial indices of a certain set I. To every wfe e of the set S is unambiguously assigned a categorial index (type) iS(e) of the set I; wfes belonging to the same syntactic category CATa have the same categorial index a. Categorial indices were introduced by Ajdukiewicz [1] into logical semiotics with the aim to determine the syntactic role of expressions and to examine their syntactic connection, in compliance with the principle of syntactic connection (SC ) discussed below. The set S of all wfes of L is then intuitively defined as the smallest set including the vocabulary of L and closed with respect to the principle (SC ), which in free formulation says that Categories of first-order quantifiers 5 (SC ) The categorial index of the main functor of each functor-argument expression of the language L is formed out of the categorial index of the expression which the functor forms together with its arguments, as well as out of the subsequent indices of arguments of this functor. In the formal definition of the set S it is required that each functor-argument constituent of the given expression should satisfy the principle (SC ). If the functor-argument expression e = f(e1, e2, . . . , en) is a wfe (it belongs to the set S ), then in accordance to the principle of syntactic connection (SC ) the index of its main functor f formed from the index a of e and successive indices a1, a2, . . . , an of successive arguments e1, e2, . . . , en of the functor f , can be written in the following quasi-fractional form: iS(f) = iS(e)/iS(e1)iS(e2) . . . iS(en) = a/a1a2 . . . an. (iS) 3.2.1. An algebraic structure of categorial language. In categorial language L we can distinguish two sets: the set B of all basic wfes of S and the set F of all functors of S such that S = B ∪ F and B ∩ F = ∅, where functors of the set F differ from basic expressions of B that they have indices formed from simpler ones. If the functor f has the functoral index of the form (iS), i.e. the index of the form a/a1a2 . . . an then it belongs to the syntactic category CATa/a1a2 . . . an and so to the category of functors forming expressions with the index a if their arguments are n expressions with successive indices a1, a2, . . . , an. So the functor f can be treated as the following partial function defined on wfes of S: f : CATa1 × CATa2 × . . .× CATan → CATa mapping of wfes from Cartesian product of syntactic categories CATa1, CATa2, . . . , CATan into the category CATa. Then we have f ∈ CATa/a1a2 . . . an = CATaCATa1×CATa2×...×CATan . (CATf ) In this way we simultaneously can regard the categorial language L as an algebraic structure L, partial algebra with the carrier S and the set Fo ⊆ F of partial functions on S (simple functors of L): L = 〈S,Fo〉 . 3.3. Categorial semantics Categorial extensional semantics is connected with denotations of wfes of S and with their belonging to an appropriate semantic extensional category. Each constituent of the composed wfe has determined a semantic extensional category and also a denotation, and thus – an ontological category (the category of ontological objects). Denotations (extensions) of wfes of L are sets of object references (references) of wfes of L, objects of the cognized reality, e.g.: individuals, sets of individuals, states of affairs, operation on the indicated objects, and the like. 6 Urszula Wybraniec-Skardowska We will concentrate only on referential relationships between expressions of L and reality to which they refer. We enrich the categorial grammar generating L by the denotation operation δ regarded as its semantic component. The denotation operation δ assigns to every wfe of the set S an object of ontological reality ONT describing by the language L – its denotation belonging to an ontological category. So δ : S → ONT, (δ) where ONT is the sum of all ontological categories corresponding to wfes of S. According to some innovative ideas of Frege [12, 13], Bocheński's (his famous motto: syntax mirrors ontology) and Suszko [25–27] who anticipated the research in categorial semantics and was the first to use categorial indices as a tool for coordination of expressions and their references, extralinguistic objects, the mutual dependence of syntactic and semantic formal description of L should be considered by keeping the principle (CC ) of categorial compatibility, based on the compatibility of the syntactic category of each language expression of L with the ontological category assigned to its denotation. The principle (CC ) of syntactic and semantic, i.e. also ontological categorial compatibility in Suszko's formulation can be given by keeping for any wfe e of categorial language L the relationship: e ∈ CATι iff δ(e) ∈ ONTι, (CC ) where CATι and ONTι are: the syntactic category and the ontological category, respectively, with the same categorial index ι, and δ is the operation of denotation. From the principle (CC ) it follows that for any e = f(e1, e2, . . . , en) ∈ S with the main functor-function f ∈ CATa/a1a2 . . . an satisfying the condition (CATf ) the following conditions are satisfied: δ(f) ∈ ONTa/a1a2 . . . an = ONTaONTa1×ONTa2×...×ONTan (ONTf ) and δ(f(e1, e2, . . . , en)) = δ(f)(δ(e1), δ(e2), . . . , δ(en)). (PCD) The condition (ONTf ) states that the denotation (object reference) of the main functor of the composed wfe e of the set S is the set-theoretical function mapping the Cartesian product of ontological categories ONTa1 ×ONTa2 × . . .× ONTan into the ontological category ONTa and it is defined by means of the condition (PCD) connected with some Frege's ideas and called the principle of compositionality of denotation. 3.3.1. An algebraic ontological structure corresponding to the partial algebra L. The operation δ assigns the following ontological structure RL of a reality corresponding to language L to the algebraic structure L: RL = 〈ONT,ONTFo〉 , where ONTFo is the sum of all ontological categories corresponding to all functors of the set Fo. The structure RL is a partial algebra similar to the algebra L and the Categories of first-order quantifiers 7 principle (PCD) is simultaneously the condition of homomorphism of the algebra L into the algebra RL, i.e. δ : 〈S,Fo〉 −−−→ hom 〈ONT,ONTFo〉 . Amodel of language L is the structure of homomorphic images of components of L, i.e. the substructure ML = 〈δ(S), δ(Fo)〉 of the structure RL. If we distinguish in the set B of basic wfes of S the category CATs of all sentences of language L, then the notion of truthfulness of any sentence e ∈ CATs in the model ML is defined as follows: e is a true sentence in the model ML iff δ(e) ∈ T, (T ) where T is primitive notion of the considered theory intuitively understood either as the singleton with the true value (in Freegan semantics) or as the set of all states of affairs that take place (in situational semantics). 4. The solution of the problem of quantifiers of 1st-order The unsatisfactory efforts to establish, in the sense of the principle (CC ) of categorial compatibility, the category of quantifiers in formalized 1st-order languages can be solved by means of notions and statements of the above outlined theory of categorial languages. Let L1 be any 1st-order formalized language. Let us treat any standard quantifier of L1 as a context-dependent functor of two arguments: 1. a quantifier variable (the variable accompanying this quantifier) and 2. its scope, i.e. a sentential function including as a free variable the same variable as the quantifier variable. 4.1. Different types of the 1st-order quantifiers and their syntactic categories A standard, the 1st-order quantifier is a functor forming a new sentential function (in particular a sentence of L1) in which there occur one free variable less than in the scope of this quantifier (the variable bound by the quantifier). As such a functor, a quantifier can be treated as a set-theoretical function relative to the number of free individual variables occurring in its scope. So, we should not speak of one existential ∃ or one universal quantifier ∀ but about different types of such quantifiers depending of the number of free variables in their scope. We will use numerical superscripts in order to point out these different types of quantifiers. Let • Var be the set of all individual variables for L1, with categorial index n1; • S = S0 – the set of all its sentences, with the categorial index s; • Sk(k ≥ 1) – the set of all sentential functions in which exactly k free variables occur, with the index sk. 8 Urszula Wybraniec-Skardowska For example, if α(x1, x2, x3) ∈ S3, where x1, x2, x3 ∈ Var, then the expressions: ∀3x2α(x1, x2, x3) ∈ S2, ∃2x3∀3x2α(x1, x2, x3) ∈ S1, ∀1x1∃2x3∀3x2α(x1, x2, x3) ∈ S0 and quantifiers ∀3, ∃2, ∀1 belong to different syntactic categories with indices s2/n1s3, s1/n1s2, s/n1s1, respectively. More generally, the quantifiers ∀k and ∃k (k ≥ 1) are treated as the functorsfunctions: ∀k, ∃k : Var× Sk → Sk−1 (S0 = S). Thus, in accordance to (CATf ), for k > 0 we have (CAT∀k,∃k) ∀k,∃k ∈ CATsk−1/n1sk (s0 = s), and the principle of syntactic connection (SC ) for them is satisfied. Their denotations and ontological categories should be defined in such a way as to satisfied the principle (CC ) of categorial compatibility (their denotations should belong to the ontological category ONTsk−1/n1sk) and the principle (PCD) of compositionality of denotation. Let the denotation operation for the language L1 be the function d in Fregean, standard semantics and the function d in the situational, non-standard semantics: d, d : S(L1) → ONT(L1) mapping the set S(L1) of all wfes of L1 into the set ONT(L1) which is the sum of all ontological categories in the ontological structure RL1 . We will give here two possible solutions of denotations of quantifiers of the 1st-order taking into account two different ways of understanding of the denotation of sentences and sentential functions presented below. 4.2. Denotations of 1st-order quantifiers and their ontological categories 4.2.1. Fregean semantics. We assume that if U is the universe of individuals in an established model ML1 of L1, 1 is the value of truth, 0 – the value of falsity then d(x) ∈ {U} = ONTn1 for any x ∈ CATn1 = Var; d(p) ∈ {0, 1} = ONTs for any p ∈ CATs = S; d(sf) ∈ 2Uk = ONTsk for any sf ∈ CATsk = Sk(k ≥ 1) and for any x1, x2, . . . , xk ∈ Var and for any sf = α(x1, x2, . . . , xk) ∈ Sk d(α(x1, x2, . . . , xk)) = {(u1, u2, . . . , uk) ∈ Uk | d(αo(x1/u1, x2/u2, . . . , xk/uk)) = 1}, where αo(x1/u1, x2/u2, . . . , xk/uk) is a sentence which we get from sentential function sf by replacement of its all free variables x1, x2, . . . , xk of Var by suitable individual names of individuals u1, u2, . . . , uk of the universe U , i.e. the denotation of sf is the set of all k-tuples from Uk which satisfy this sentential function. Categories of first-order quantifiers 9 Denotation for the quantifier ∀k of the type k(k ≥ 1) is defined by induction as follows: a) for k = 1 and any α(x) ∈ S1 d(∀1xα(x)) = d(∀1)(d(x)), d(α(x))) = { 1, d(x) = U = d(α(x)) 0, d(x) = U 6= d(α(x)); . According to a) the quantifier sentence obtained from any sentential function α(x) by preceding it with the universal quantifier ∀1 is a true sentence in the established model ML1 of L1 with the universe of individuals U iff every object of the universe U satisfies the α(x) which is the scope of ∀1. b) for k = j + 1(j > 0) and any α(x1, x2, . . . , x, . . . , xj+1) ∈ Sj+1 d(∀j+1xα(x1, x2, . . . , x, . . . , xj+1)) = = d(∀j+1)(d(x), d(α(x1, x2, . . . , x, . . . , xj+1))) = = {(u1, u2, . . . , uj+1) ∈ U j | d(αo(x1/u1, x2/u2, . . . , x/u, . . . , xj+1/uj+1)) = 1 for each u ∈ U}. According to b) the denotation of the sentential function sfk−1 ∈ Sk−1 obtained from the sentential function α(x1, x2, . . . , x, . . . , xj+1) ∈ Sk(k > 1) by binding the variable x by the universal quantifier ∀k(k = j + 1 > 1) is the set of all j = (k − 1)-tuples (u1, u2, . . . , uk−1) of individuals of U such that all sentences obtained by the substitution of all j free variables in sfk−1, respectively, by names of individuals of these tuples and names of any individuals of U representing x are true; in other words the denotation of sfk−1 is the set of all such (k − 1)-tuples (u1, u2, . . . , uk−1) of individuals of U that for any individual u of U k-tuples (u1, u2, . . . , u, . . . , uk−1) build from them satisfy the scope α(x1, x2, . . . , x, . . . , xj+1) of the quantifier ∀k. Thus for any k ≥ 1 d(∀k) ∈ ONTsk−1/n1sk = ONTsONTn1×ONTskk−1 . Similarly for d(∃k): a) for k = 1 and any α(x) ∈ S1 d(∃1xα(x)) = d(∃1)(d(x)), d(α(x))) = { 1, d(x) ∩ d(α(x)) 6= ∅ 0, d(x) ∩ d(α(x)) = ∅; . b) for k = j + 1(j > 0) and any α(x1, x2, . . . , x, . . . , xj+1) ∈ Sj+1 d(∃j+1xα(x1, x2, . . . , x, . . . , xj+1)) = = d(∃j+1)(d(x), d(α(x1, x2, . . . , x, . . . , xj+1))) = = {(u1, u2, . . . , uj+1) ∈ U j | d(αo(x1/u1, x2/u2, . . . , x/u, . . . , xj+1/uj+1)) = 1 for some u ∈ U}. 10 Urszula Wybraniec-Skardowska According to a) the quantifier sentence obtained from any sentential function α(x) by preceding it with the existential quantifier ∃1 is true sentence in the established model ML1 of L1 with the universe of individuals U iff at least one object of the universe U satisfies the α(x) which is the scope of ∃1. According to b) the denotation of the sentential function sfk−1 ∈ Sk−1 obtained from the sentential function α(x1, x2, . . . , x, . . . , xj+1) ∈ Sk(k > 1) by binding the variable x by the existential quantifier ∃k(k = j + 1 > 1) is the set of all j = (k − 1)-tuples (u1, u2, . . . , uk−1) of individuals of U such that all sentences obtained by the substitution of all j free variables in sfk−1, respectively, by names of individuals of these tuples and the substitution some individual name of u for x are true; in other words the denotation of sfk−1 is the set of all such (k − 1)-tuples (u1, u2, . . . , uk−1) of individuals of U that for some individual u of U k-tuples (u1, u2, . . . , u, . . . , uk−1) build from them satisfy the scope α(x1, x2, . . . , x, . . . , xj+1) of the quantifier ∃k. Thus, for any k ≥ 1 d(∃k) ∈ ONTsk−1/n1sk = ONTsONTn1×ONTskk−1 . Moreover, the principle (CC ) is also valid for ∀k and ∃k in situational semantics. 4.2.2. Situational semantics. In situational semantic we assume that d(x) ∈ {U} = ONTn1 for any x ∈ CATn1 = Var; d(p) ∈ {St} = ONTs for any p ∈ CATs = S, where St is the set of all states of affairs, St = T ∪ F, T ∩ F = ∅ and T is the nonempty set of all states of affairs that take place and F – the nonempty set of remaining states of affairs. Stk ⊂ St is the set of states of affairs with k individuals. d(sf) ∈ 2Stk = ONTsk for any sf ∈ CATsk = Sk and for any x1, x2, . . . , xk ∈ Var and for any sf = α(x1, x2, . . . , xk) ∈ Sk d(α(x1, x2, . . . , xk)) = {s ∈ Stk | s = d(α0(x1/u1, x2/u2, . . . , xk/uk)) for any (u1, u2, . . . , uk) ∈ Uk}. So, if the denotation operation is understood here as the operation d then the denotations of sentences are states of affairs and the denotation of any sentential function is the set of all states of affairs that are denotations all sentences represented by the sentential function. Denotation for the quantifier ∀k is defined by induction as follows: a) for k = 1 and any α(x) ∈ S1 d(∀1xα(x)) = d(∀1)(d(x)), d(α(x))) ∈ T iff d(αo(x/u)) ∈ T for each u ∈ U ; Categories of first-order quantifiers 11 b) for k = j + 1(j > 0) and any α(x1, x2, . . . , x, . . . , xj+1) ∈ Sj+1 d(∀j+1xα(x1, x2, . . . , x, . . . , xj+1)) = = d(∀j+1)(d(x), d(α(x1, x2, . . . , x, . . . , xj+1))) = = {s ∈ St | s = d(αo(x1/u1, x2/u2, . . . , x/u, . . . , xk/uk)) for each u ∈ U, any (u1, u2, . . . , uj+1) ∈ U j}. According to a) the quantifier sentence obtained from any sentential function α(x) by preceding it with the universal quantifier ∀1 is a true sentence in an established model ML1 of the language L1 with the universe of individuals U iff every sentence representing this sentential function is true (because their denotations are states of affairs that take place). According to b) the denotation of sentential function sfk−1 ∈ Sk−1 obtained from the sentential function α(x1, x2, . . . , x, . . . , xj+1) ∈ Sk(k > 1) by binding the variable x by the universal quantifier ∀k is the set of all denotations of sentences (intuitively – the set of all states of affairs describing by these sentences) which can be obtained from sfk−1 by replacing all free variables in it with individual names of any individuals of U ; in other words, it is the set of all denotations of sentences (all states of affairs) which can be obtained from α(x1, x2, . . . , x, . . . , xj+1) by replacement for the variable x binding by ∀k individual names of any individual of U (of the denotation of this variable) and for remaining variables in it also individual names of any individuals of U . Thus, for any k ≥ 1 d(∀k) ∈ ONTsk−1/n1sk = ONTsONTn1×ONTskk−1 . Similarly for d(∃k): a) for k = 1 and any α(x) ∈ S1 d(∃1xα(x)) = d(∃1)(d(x)), d(α(x))) ∈ T iff T ∩ d(α(x)) 6= ∅ b) for k = j + 1(j > 0) and any α(x1, x2, . . . , x, . . . , xj+1) ∈ Sj+1 d(∃j+1xα(x1, x2, . . . , x, . . . , xj+1)) = = d(∃j+1)(d(x), d(α(x1, x2, . . . , x, . . . , xj+1))) = = {s ∈ St | s = d(αo(x1/u1, x2/u2, . . . , x/u, . . . , xj+1/uj+1)) for some u ∈ U, any (u1, u2, . . . , uj+1) ∈ U j}. Thus, for any k ≥ 1 d(∃k) ∈ ONTsk−1/n1sk = ONTsONTn1×ONTskk−1 . 4.3. The syntactic and semantic compatibility of quantifiers In our categorial approach to syntax and semantics of the 1st-order formalized language L1 its quantifiers have been treated as context-dependent two-argument functors-functions of different categorial types k > 0 (defined on the set Var of all its individual variables and the set of all its sentential functions Sk with exactly 12 Urszula Wybraniec-Skardowska k free variables) and with values in the set of sentential functions Sk−1 possessing one free variable less or, in particular, in the set of sentences S: ∀k, ∃k : Var× Sk → Sk−1 (S0 = S). Thus, according to the condition (CATf ), quantifiers ∀k, ∃k belong to syntactic categories: (CAT∀k,∃k) ∀k, ∃k ∈ CATsk−1/n1sk = CATsCATn1×CATskk−1 (s0 = s), and it means that they satisfy the principle (SC ) of syntactic connection. It was also shown that for the denotation operations: d, d : S(L1) → ONT(L1) their denotations, according to the condition (ONTf ), belong to ontological categories: d(∀k), d(∀k), d(∃k), d(∃k) ∈ ONTsk−1/n1sk = ONTsONTn1×ONTskk−1 . (ONT∀k, ∃k) 5. Conclusions From the conditions (CAT∀k, ∃k) and (ONT∀k, ∃k) follow the following conclusions: 1. the 1st-order quantifiers ∀k, ∃k(k > 0) satisfy the principle of syntactic connection (SC ) and the principle of categorial compatibility (CC ) and 2. the problem of standard quantifiers is solved by employing the conceptual apparatus and statements of the outlined theory of categorial languages. It should also be noted that 3. in languages with other operators biding variables the problem of their denotations can be solved in an analogous way, but 4. for branching quantifiers used in Independence-Friendly logic (see Hintikka [14]) the outlined here denotational (compositional) semantics does not work. 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Theory and Applications 8, 134–140 (2016) Urszula Wybraniec-Skardowska Cardinal Stefan Wyszyński University in Warsaw Department of Philosophy Warsaw, Poland e-mail: skardowska@gmail.com