A NOTE ON COMPOSITIONALITY IN THE FIRST ORDER LANGUAGE

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1 Janusz Maciaszek Luis Villegas-Forero A NOTE ON COMPOSITIONALITY IN THE FIRST ORDER LANGUAGE Abstract The main objective of this paper is to present the syntax and semantic of First Order Language in a way such that all functors (including the quantifiers) are both treated categorematically and considered as (quasi) intensional. It is possible by introdicing only one composition condition. 1. Introduction The logical expressions in First Order Language (FOL) are invariably treated syncategorematically, i.e. they are defined, by context, in the syntax and semantics. This is particulary evident in the case of quantifying expressions 1. Tarski s version of FOL semantics, based on the notion of the satisfaction of a formula by sequences of individuals in a given universe, is a case in point: a quantifying expression has no explicit denotation, but simply shows the way in which the satisfaction of the formula by a particular sequence depends on the satisfaction of the subformula within the quantifier s scope by one or all of the sequences that are accessible to the sequence in question. Let us consider the FOL formulae yp (y) and yp (x), and let us suppose the universe of interpretation to be the set {a, b}. The relevant interpretation sequences are s 1 =< a, a >, s 2 =< a, b >, s 3 =< b, a > and s 4 =< b, b >, where the first position corresponds to x and the second to y. Let f be an interpretation function such that f(p ) = {a} (or the corresponding characteristic function). According to Tarski s definition of satisfaction, yp (x) and P (x) are satisfied by s 1, their denotations for this 1 An important exception is Lewis in [3]:

2 sequence is therefore 1. Hence if y has its own denotation and this denotation is an extensional truth function, the denotation of yp (y) for s 1 must also be 1, since P (y) too is satisfied by s 1. But the denotation of yp (y) is not 1, so either y has no denotation of its own (the conclusion accepted by Tarski in developing his syncategorematic account of quantification and elaborated in a very systematical way by Suszko in [5]), or it has a denotation that is not an extensional truth function. The above example shows that if, unlike Tarski, we insist that quantifying expressions have explicit denotations, then we are bound to accept that these denotations cannot be extensional truth functions. In fact, the well-known analogy between FOL quantifiers and S 5 modal operators (Kuhn, [2]), suggests that it is feasible to endow quantifying expressions with quasi-intensional denotation by using sequences in the same way that possible worlds were used by Montague to construct denotation for modal operators. We shall therefore define the interpretational counterpart of an expression (its quasi-sense) as a quasi-intension, i.e. as a function from sequences to a extension of the kind appropriate to the syntactic category of the expression 2. The quasi-sense of a formula, for example, will be a function from sequences to truth values, and the quasi-sense of a quantificational prefix will assign for each sequence an extension which is a function whose arguments are quasi-intensions of formulae and its values are truth values. So, quantificational prefixes will be treated categorematically as complex monary connectives whose quasi-intensions and extensions depend systematically on the quasi-intensions and extensions of their component parts (quantifiers and variables); and this will involve specification and application of an appropriate composition condition. Let be, in general, α the intension 3 of any expression α (whose extension for any sequence s is therefore α (s)). One can introduce in the beginning two composition conditions: the first (Montague s Condition) establishes that if α is an extensional functor (e.g. a sentential connective), β its argument and s a sequence, then α(β) (s) = α (s)( β (s)). 2 In Montague proper senses are treated as intensions, i.e. as functions from possible worlds to appropriate extensions. 3 In the following, both in the main text and the footnotes, we will supress in order to simplify where there is no risk of confusion the prefix quasi. 207

3 The second (henceforth the (*) Condition) establishes that if α is now an intensional functor (e.g., a quantifying expression), then α[β] (s) = α (s)( β ), where we use square brackets to enclose the argument of the intensional functor. Problem: the new condition works well in intensional contexts, but it does not work in purely extensional contexts where Montague s Condition is still necessary. Montague could deal with the first condition by introducing the operator of intensialization. We do not follow him and we get rid of the need for Montague s Condition by making all functors intensional, and hence amenable to application of the (*) Condition. In consequence we must furnish the well know extensional functors like sentential conections and extensional predicates, with new intensional types. In order to develop the application of the new composition condition for all functors of FOL, let us introduce its syntax and semantics in a systematic way. 2. Syntax The syntax of FOL is a simple application of the general Montagovian scheme (Montague [4]). For simplicity, the version developed here only admits unary and binary predicates, but generalization to predicates of arbitrary order is quite straightforward. is the set of category symbols, containing CONST (individual constants), VAR (individual variables), TER (terms), FOR (formulas), 1 PRED (unary predicates), 2 PRED (binary predicates), 1 CON (unary connectives), 2 CON (binary connectives), QUANT (quantifiers). The lexicon of FOL is (X δ ) δ, where X δ is the set of basic expressions of category symbol δ. In particular X CONST = {c 1, c 2,...} X VAR = {v 1, v 2,...} X TER = X CONST X VAR X 1PRED = {P 1 1, P 2 1,...} X 2PRED = {P 1 2, P 2 2,...} X 1CON = { } X 2CON = {,,, } X QUANT = {, } X FOR = 208

4 Let < A, F > be the free algebra generated by (X δ ) δ, where F is the juxtaposition operation such that for all α, β A, F (α, β) = [α, β]. The syntactic system 4 of FOL is defined as S =< A, F, (X δ ) δ, (R η ) η H, FOR>, where H = {1, 2, 3, 4, 5} and (R η ) η H is the family of syntactic rules, where each rule is of the general form < F, < δ 1, δ 2 >, δ 3 >, readable as: if α 1 is an expresion of the category symbol δ 1 and α 2 is an expresion of the category symbol δ 2, then F (α 1, α 2 ) is an expresion of the categry symbol δ 3. In particular: R1 = < F, < 2P RED, T ER >, 1P RED > R2 = < F, < 1P RED, T ER >, F OR > R3 = < F, < 2CON, F OR >, 1CON > R4 = < F, < 1CON, F OR >, F OR > R5 = < F, < QUANT, V AR >, 1CON >. The family (C δ ) δ of syntactic categories of FOL is the smallest family of sets such that 1. (Cδ ) δ A 2. For all δ, X δ C δ 3. For all α 1, α 2 A and all δ 1, δ 2, δ 3 if α 1 C δ1, α 2 C δ2 and < F, < δ 1, δ 2 >, δ 3 > (R η ) η H, then F (α 1, α 2 ) C δ3. Finally, we define FOL as (C δ ) δ. Thus FOL is the set of all its well-formed expressions, not just its formulae. 3. Abstract semantics Adapting Montague, we develop the semantics of FOL in two stages: ontology and interpretation. We define our ontological scheme in terms of an abstract type structure and what we call a Tarskian frame. We define T, the class of types of things, as the smallest set such that 1. e, t T, where e is the type of individuals and t the type of truth values. 2. If σ, τ T, then < σ, τ > T, where < σ, τ > is the type of functions from things of type σ to things of type τ. 4 Following Dowty [1], we use the term syntactic system for what Montague called a disambiguated language. 209

5 3. If τ T then < q, τ > T. < q, τ > is the type of intensions corresponding to things of type τ 5. A Tarskian frame (TF) F is defined as an ordered triple < E F, SEQ F, Ac F > such that 1. E F is a denumerable set of individuals; 2. SEQ F = E ω is the set of all infinite sequences of members of E F. 3. Ac F, the accessibility function, is a function from SEQ F ω to {0, 1} SEQ F, such that for all s, s SEQ F and i ω, 1 if pr j (s ) = pr j (s) Ac F (s, i)(s ) = for all j ω other than i 0 otherwise. where pr j (s) is just the j-th object of s. The notion of Tarskian frame thus incorporates the relation between sequences that in standard Tarskian semantics is used to apply the concept of satisfaction to quantified formulae. We now relate things types with possible denotation. Given a type system T and a T F F, we define the family of sets of typified things (D τ,f ) τ T, as the smallest family such that 1. D e,f = E F 2. D t,f = {0, 1} 3. For all σ, τ T, D <σ,τ>,f = D D σ,f τ,f 4. For all τ T, D <q,τ>,f = D SEQ F τ,f A possible interpretation for FOL is defined as an ordered triple I =< ϑ, ζ, f > such that 1. ϑ =< B, G, (Y <q,τ>,f ) τ T (S ν ) ν H, < q, t >> is a semantic system, such that: a) B (D <q,τ>,f ) τ T is the domain of intensions of ϑ b) < B, G > is the semantic algebra similar to the syntactic algebra < A, F > c) (Y <q,τ>,f ) τ T, the family of sets of basic intensions of ϑ, is such that: 5 Here q represents the syncategorematic type of sequences. It will play an analogous role to Montague s type s. 210

6 (i) (Y <q,τ>,f ) τ T B (ii) Y <q,e>,f E SEQ F F (iii) Y <q,t>,f {0, 1} SEQ F (iv) Y <q,τ>,f = if τ e or τ t d) (S η ) η H is the family of semantic rules of ϑ (1 1 corresponding to syntactic rules) such that for each η H, S η is of the form < G, << q, σ 1 >, < q, σ 2 >>, < q, τ >>, where σ 1, σ 2, τ T. e) < q, t > is the distinguished intension type. 2. ζ is any function from to T such that ζ(for) = t (ζ assigns to each syntactic category symbol the type of extensions of the members of that category). 3. f is a function from (X δ ) δ to B such that if α X δ, then f(α) K <q,ζ(δ)>,ϑ, where in general (K <q,τ>,ϑ )t T is the family of intensions generated by ϑ, defined as the smallest family of sets such that a) (K <q,τ>,ϑ ) τ T B b) for all τ T, Y <q,τ>,f K <q,τ>,ϑ c) for all b 1, b 2 B, σ 1, σ 2, τ T if b 1 K <q,σ1>,ϑ, and b 2 K <q,σk >,ϑ and there exists η H such that if S η =< G, << q, σ 1 >, < q, σ 2 >>, < q, τ >>, then G(b 1, b 2 ) K <q,τ>,ϑ. (f assigns to each basic expression α of FOL an intension of the type corresponding to the type of extension assigned by ζ to the appropriate category symbol) 6. By the freedom of the syntactic algebra, f induces a generalized meaning function. I, the only homomorphism from the syntactic algebra to the semantic algebra such that f. I. This homomorphism permits to connect syntactic and semantic rules in the following way: For each η H, δ 1, δ 2, δ 3 and α 1, α 2 A, if R η =< F, < δ 1, δ 2 >, δ 3 >, α 1 C δ1 and α 2 C δ2, then a) S η =< G, << q, ζ(δ 1 ) >, < q, ζ(δ 2 ) >>, < q, ζ(δ 3 ) >> b) α 1 I K <q,ζ(δ1)>,ϑ and α 2 I K <q,ζ(δ2)>,ϑ c) G( α 1 I, α 2 I ) K 7 <q,ζ(δ3)>,ϑ 6 It is perhaps opportune at this point to stress that the property of being extensional or intensional is purely ontological, being only determined by the formal ontology adopted. 7 These notions allow us to define a model of FOL as an ordered pair M =< I, s > where I is an interpretation for FOL and s SEQ F, where F is the TF underlying the semantic system on which I is based. This definition deviates obviously from Tarski and 211

7 4. Logical constraints The semantic notions introduced so far have merely ensured algebraic compatibility between the syntax of FOL and its possible interpretations. We now impose logical constraints, i.e. we constrain possible interpretations to conform to the traditional use of both logical expressions (connectives and quantifying expressions) and extralogical expressions (variables, constants and predicates). A possible interpretation I =< ϑ, ζ, f > based on a semantic system ϑ is a logically admissible interpretation iff ζ complies with the conditions 1 and. I complies with the conditions a) ζ(var) = ζ(const) = ζ(ter) = e b) ζ(1pred) = << q, e >, t > c) ζ(2pred) = << q, e >, << q, e >, t >> d) ζ(1con) = << q, t >, t > e) ζ(2con) = << q, t >, << q, t >, t >> f) ζ(quant) = << q, e >, << q, t >, t >> Thus the extension type assigned to each functorial category symbol is a type of function defined on a set of intensions. The type of sense corresponding to each category symbol is the intension type corresponding to its denotation type. 2. a) General composition constraint ((*) Condition) For all δ 1, δ 2, δ 3, α 1, α 2 A and η H, if α 1 C δ1, α 2 C δ2 and R η =< F 1, < δ 1, δ 2 >, δ 3 >, then for all s SEQ F F (α 1, α 2 ) I (s) = [α 1, α 2 ] I (s) = G( α 1 I, α 2 I )(s) = α 1 I (s)( α 2 I ) b) Constraints on extralogical expressions (i) For all v i V VAR and s SEQ F, v i I (s) = pr i (s) (ii) For all c i C CONST and s, s SEQ F, c i I (s) = c i I (s ) even somewhat from Montague s [6], but allows us to give specific definitions of some well-known semantic concepts of FOL, for example, truth relative to a model (which is the counterpart of the Tarskian notion of satisfaction by a sequence), truth in an interpretation, (which is the counterpart of the Tarskian concept of truth in a model), and validity. So the logic that can be expressed by our version of FOL is just First Order Logic. 212

8 (iii) For all Pi 1 C 1PRED, t 1, t 2 C TER and s, s SEQ F, if t 1 I (s) = t 2 I (s ), then Pi 1 I (s)( t 1 I ) = Pi 1 I (s )( t 2 I ) (iv) For all Pi 2 C 2PRED, t 1, t 2, t 1, t 2 C TER and s, s SEQ F, if t 1 I (s) = t 1 I (s ) and t 2 I (s) = t 2 I (s ) then Pi 2 I (s)( t 1 I )( t 2 I ) = Pi 2 I (s )( t 1 I )( t 2 I ) both (iii) and (iv) can be regarded as extensional constraints on predicates. c) Constraints on logical expressions (i) For all φ C FOR, s SEQ F, [ φ] I (s) = 1 iff φ I (s) = 0 what implies the extensionality of negation: For all φ, φ C FOR and s, s SEQ F, [ φ] I (s) = [ φ ] I (s ) iff φ I (s) = φ I (s ). (ii) For all φ, φ C FOR, s SEQ F [[ φ]φ ] I (s) = 1 iff φ I (s) = φ I (s) = 1 [[ φ]φ ] I (s) = 1 iff φ I (s) = 1 or φ I (s) = 1 [[ φ]φ ] I (s) = 1 iff φ I (s) = 0 or φ I (s) = 1 [[ φ]φ ] I (s) = 1 iff φ I (s) = φ I (s) what implies condition of extensionality For all ξ C 2CON, φ 1, φ 2, φ 1, φ 2 C FOR, s, s SEQ F, [[ξφ 1 ]φ 2 ] I (s) = [[ξφ 1]φ 2 ] I (s ) iff φ 1 I (s) = φ 1 I (s ) and φ 2 I (s) = φ 2 I (s ). (iii) By successive applications of the (*) Condition, the following constraints establish the intensions and extensions of the universal and existential quantifiers and the quantifying prefixes, in a way that is equivalent to the Tarskian definition of satisfaction for quantified formulae. The difference is that whereas Tarski s definition is contextual, ours is explicit. So, the result of successively applying the intension of the universal (existential) quantifier to a sequence s, applying the resulting function to the intension corresponding to the i-th variable, and applying the function finally resulting to the sense corresponding to the formula quantified, is 1 iff the function determining accessibility to s at i is a subfunction of (has non-empty intersection with) the sense corresponding to the formula quantified. It is clear that the final truth value depends only on the sequences that are accessible in each case, not on the whole of SEQ F. 213

9 For all i ω, v i C VAR, s SEQ F, φ C FOR [[ v i ]φ] I (s) = [ v i ] I (s)( φ I ) = I (s)( v i I )( φ I ) = 1 iff {s : ac(s, i)(s ) = 1} {s : φ I (s ) = 1} [[ v i ]φ] I (s) = [ v i ] I (s)( φ I ) = I (s)( v i I )( φ I ) = 1 iff {s : ac(s, i)(s ) = 1} {s : φ I (s ) = 1} = Conclusion. Suszko in [5] was the first in proposing an elaborated account of the syntactic structure, underlying ontology, and semantic of a restricted version of FOL. The approach presented in our paper is wider. It introduces (in the context of Montague s program) a unique and novel composition condition, which permits a systematic, simple and uniform computation of the denotation (semantic value) of all formulas, including e.g., xp y or x xp x, which were not well formed in Suszko s account. References [1] D. D. Dowty et all, Introduction to Montague Semantics, Reidel, Dordrecht, [2] S. Kuhn, Quantifiers as Modal Operators, Studia Logica 39 (1980), pp [3] D. K. Lewis, General Semantics Sythese 22 (1970), pp [4] R. Montague, Universal Grammar, Formal Philosophy. Selected Papers of R. Montague, edited by R. Thomason, Yale University Press, New Haven, 1976, pp [5] R. Suszko, Syntactic Structure and Semantical Reference, Studia Logica 8 (1958), pp , 9 (1960), pp Department of Logic University of Lódź Matejki 34a Lódź Poland janmac@krysia.uni.lodz.pl Department of Logic University of Santiago de Compostela Campus Universitario Sur Santiago de Compostela Spain lflpvill@.usc.es 214