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A Categorical Aspect of the Analogy Between Quan- Tifiers and Modalities

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A Categorical Aspect of the Analogy Between Quan- Tifiers and Modalities A categorical aspect of the analogy between quan- tifiers and modalities Brice Halimi Abstract. The formal analogy relating existential quantification to a modal possibil- ity operator, and universal quantification to a modal necessity operator, is well-known. Its obvious limitation is, too: it holds on the condition of confining quantification to a single variable. This paper first presents that analogy in the terms of dynamic logic, which allows one to overcome the limitation just mentioned and to translate fragments of first-order logic into the language of propositional modal logic. It then sets out how the modal semantics for first-order logic which is established by dynamic logic lends itself to a classical algebraic development based on “cylindric algebras.” Finally, the main part of the paper shows how this algebraic semantics can be presented in a cate- gorical setting, and how the algebraic counterparts of first-order structures correspond to “fibered categories” which are shown to be substructures of the “syntactic” fibered category representing first-order logic. Mathematics Subject Classification (2000). Primary 03G15, 03G30; Secondary 03B10, 03B45, 03-03, 01A60. Keywords. Cylindric algebras; mirror cubes; fibered categories; Arthur Prior; dynamic logic; modal translation of fragments of first-order logic. 1. Analogy between quantifiers and modalities The analogy between the universal and the existential quantifiers, on the one hand, the modal operators of necessity and possibility, on the other, is well-known and dates back to Arthur Prior at least. That formal analogy turns on the the parallel double implications 2p ) p ) 3p and 8xfx ) fa ) 9xfx, and on the parallel dualities 8 = :9: and 2 = :3:. Prior harnessed and developed that formal analogy so as to formalize as substantial as possible a fragment of first-order logic within the modal system S5.1 Using ‘p’, ‘q’, ‘r’, . for propositional variables and for predicate schematic letters alike, he wrote ‘Lxpx’ for “for any x, px” as well as ‘Mxpx’ for “there exists x such that px.” Prior also used Polish prefix notations, with ‘C’ as the symbol for implication. The modal 1[7], p. 21. 2 Brice Halimi axiom CLαα (i.e., 2p ! p) of S5 thus translates the axiom scheme CLvαβ, where β differs from α at most by the replacement of all free occurrences of v in α with another variable symbol for which v is free in α. Prior remarked that certain derivations available in first-order logic did not corre- spond to any sound derivation in propositional modal logic. For instance, as soon as y does not contain any free occurrence of x, existential generalization allows one to conclude (9x)(φx) ! y from φx ! y; however, the step leading from φ ! to 3φ ! is permissible only if is fully modalized, i.e., if all the propositional vari- ables contained in are immediately prefixed with a modal operator. As such, the above- mentioned formal analogy does not overcome the major discrepancy due to quantifiers not being operators: it is not possible to (non vacuously) prefix arbitrarily many quantifiers in front of a formula, as it is possible to prefix arbitrarily many modal operators in front of a proposition. Prior concludes: The use of name-variables thus introduces moves that cannot be paralleled in modal logic, however similar the laws of modal operators and quantifiers might be. The basic reason is of course that different quantifiers in a for- mula may bind different name-variables. If, however, we remove this variety, a much closer analogy can be drawn.2 Indeed, the restriction to a fragment of first-order logic —which Prior calls UM1PC— with a single variable symbol and monadic predicates makes it possible to establish a direct correspondence between respective derivations in both systems, which amounts to identifying the modal system S5 with a variant of UM1PC. The core of the formal analogy between quantifiers and modal operators, or a sys- tematic way of presenting it, lies in the strong similarity relating the adjunction about the functors representing the former and the adjunction about the functors representing the latter. Given a binary relation R on a domain Y and V ⊆ Y , let us define: hRi(V ) = fx 2 X : 9y 2 Y (xRy & y 2 V )g (“possibly V ”); [R](V ) = fx 2 X : 8y 2 Y (xRy ) y 2 V )g (“necessarily V ”). Writing R0 for the converse of R (xR0y iff yRx), one gets: hRi a [R0] (1) hR0i a [R] (2). Paul Taylor3 mentions that any adjunction between two powersets }(X) and }(Y ) is naturally isomorphic to a modal adjunction as stated above. The existential and universal quantifiers of first-order logic (underlaid by a given theory of types) can also be characterized through an adjunction, due to William Law- vere.4 Let C be a Cartesian closed category. To any object X of C is attached the Carte- sian closed category of “attributes of type X”, and to any morphism f : X ! Y in C the functor f ∗ : }(Y ) ! }(X) sending each 2 }(Y ) to the attribute of type X obtained 2[7], p. 24. 3[8], p. 163. 4[4]. A categorical aspect of the analogy between quantifiers and modalities 3 from by pullback along f. Since (f ◦ g)∗ = f ∗ ◦ g∗ can easily be seen to hold, a functor Cop ! Cat has thus be defined, in other words an indexed category. Moreover, given f : X ! Y in C and ' ⊆ X, the two following functors can be introduced: Σf (') = fy 2 Y : 9x 2 ' : f(x) = yg; Πf (') = fy 2 Y : 8x 2 X(f(x) = y ) x 2 'g: The functor Σf (') (resp. Πf (')) is called the existential quantification (resp. the uni- versal quantification) of ' along f. The two functors Σf ; Πf : }(X) ! }(Y ) satisfy a ∗ double adjunction Σf a f a Πf : ! Πf (') Σf (') ! f ∗( ) ! ' ' ! f ∗( ) which can be rewritten as follows, as soon as f is expressed as a relation (xfy iff y = f(x)):5 ∗ −1 hfi = f a Πf = [f ] (1’) −1 ∗ hf i = Σf a f = [f] (2’). The straightforward correspondence between (1) and (10), on the one hand, between (2) and (20), on the other, makes fully explicit the formal analogy at the bottom of Prior’s remarks. Whereas the modal semantics in the Carnap-Kripke tradition has interpreted modal clauses in quantificational terms (possibility of φ as the existence of some appropriate possible world in which φ is true, necessity of φ as the truth of φ in all appropriate pos- sible worlds), the modal translation of first-order logic works in the opposite direction, toward interpreting quantification logic in modal terms —an idea clearly implemented by dynamic logic. The technical motivation here is to delineate a fragment of first-order logic obtained by syntactic interpretation into a certain modal calculus, so as to export decid- ability properties of the latter into the realm of first-order logic. That perspective ushers in a modal analysis of quantification from the perspective of algebraic logic. Before turning to the latter framework, let us consider first the case of dynamic logic. 2. Dynamic logic The seminal tenet of dynamic logic consists in looking at an assignment of values to the variable symbols of a first-order language, as being a possible world; accordingly, for any given variable symbol x, in looking at two assignments differing at most on the value assigned to x, as being related by an accessible relation Rx associated to x. As a result, the usual semantic clause for existential quantification over x can be rephrased as the usual clause for the possibility operator 3x in Kripkean modal semantics: M 9xφx [σ] , there exists an assignment θ : σRxθ and M φ [θ] (1) , M; σ 3xφ ; σRxθ meaning that σ(z) = θ(z) for all z 6= x. 5[8], p. 166. 4 Brice Halimi The principle of that rephrasing can even to extended to generalized quantifiers, as shown by [2], with the prospect of establishing a “modal logic for quantifiers.” The start- ing point of that paper is the interpretation of generalized quantifiers as subsets of }(D), where D is the domain of the interpretation under consideration. In order to describe a generalized quantifier Q with inference rules similar to those of a sequent calculus, the ap- plication of Q can be associated with the following first-order rule Q: (Qx)'(x; x1; : : : ; xn) becomes 8x(R(x; x1; : : : ; xn) ! '(x; x1; : : : ; xn)). The possible values of a variable bound by Q can be restricted to a specific domain, that restriction being expressed through an accessibility relation: the sequence x1; : : : ; xn corresponds to the actual world, and the variable x to a possible world which has to be accessible from the actual world. It turns out that axioms φ governing Q can be translated into first-order conditions about a relation RQ, in the sense that any set of formulae about Q is consistent with φ(Q) iff the set of their translations is consistent with (RQ). For instance, (Qx)'^(Qx) ! (Qx)('^ ) translates into R(x; y¯z¯) ! R(x; y¯), hence the following elimination rule for Q: (Qx)'(x; y¯) ; '(xz¯; y¯) where xz¯ 2 fx : R(x; z¯)g and y¯ ⊆ z¯. Alechina and van Benthem then put forward the following modalized clause for the existential quantifier: ¯ ¯ ¯ M; [d=y¯] 3x'(x; y¯) , 9e 2 jMj : R(e; d)& M; [e=x; d=y¯] '(x; y¯): The classical interpretation of existential quantification thus corresponds to taking R to be the universal relation on the domain jMj of M, in other words: Ordinary predicate logic then becomes the special case of flat individual do- mains admitting of “random access”, whose R is the universal relation.6 The modalization of first-order quantifiers initiated by clause (1) above can be pur- sued by a similar treatment of substitution: M φ[y=x][σ] , 8θ(σRx;yθ ) M φ [θ]) (2), 7 where σRx;yθ means that t(z) = s(z) for all z 6= x and t(x) = s(y).
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