A. Pitts. On an interpretation of second order quantification in first order intuitionistic propositional logic. Journal of Symbolic Logic, 57:33--52, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....uniform presentation of Craig interpolants becomes available, has been illustrated in this paper. The uniform interpolants are instances of the same syntactic pattern, and the usually hard process of computing interpolants is thus trivialised. The latter is consistent with the insight of [30] and the remark of [9] that uniform interpolants have the same implicit (meta)form. In our case, this common form is revealed in the formal syntax. In particular, Pitts in [30] uses second order propositional quantifiers implying that his analogues to our 8 , 9 should range over ....

....and the usually hard process of computing interpolants is thus trivialised. The latter is consistent with the insight of [30] and the remark of [9] that uniform interpolants have the same implicit (meta)form. In our case, this common form is revealed in the formal syntax. In particular, Pitts in [30] uses second order propositional quantifiers implying that his analogues to our 8 , 9 should range over propositions; we have have shown that similar quantifiers can range over atoms, as well. This is somewhat closer to how D Agostino and Hollenberg explain the existence uniform ....

A. Pitts. On an interpretation of second order quantification in first order intuitionistic propositional logic. Journal of Symbolic Logic, 57:33--52, 1992.

....with respect to any L 0 L(OE) Thus, classical Propositional Logic enjoys uniform interpolation. Intuitionistic Propositional Logic IL is another example of a logic in which uniform interpolation holds, albeit in this context the proof becomes nontrivial. It was first proved by Pitts in [56], by means of proof theoretical methods. Using a contraction free variant of the Gentzen sequent calculus for IL, it is possible to define a wellfounded relation on sequents in such a way that the hypotheses of a rule of the calculus are always smaller than the conclusion. The proof of the ....

....2 ffi(l(w) LA (w) f(a; q 0 ) a; q 1 )g. 2 3.3 Uniform interpolation In this section we prove the Uniform Interpolation Theorem for the Modal Calculus. This is done via an interpretation of second order quantifiers within the Modal Calculus itself. This approach is inspired by [66] and [56]. The results of this section were proved in collaboration with Marco Hollenberg in [21] We start by proving that the Modal Calculus allows existential quantification on proposition constants, modulo bisimulation. 3.3.1. Theorem. Let OE be a sentence and L its language. Let p be a proposition ....

A. Pitts. On an interpretation of second-order quantification in first-order intuitionistic propositional logic. Journal of Symbolic Logic, 57:33--52, 1992. Bibliography 151

....the method did not give any positive result for incomplete logics. In this paper we shall combine Thomason s use of the 0 categoricity of atomless boolean algebras with some techniques introduced in [10] to prove that decidability, interpolation and uniform interpolation in the sense of Pitts [13] transfer in general. The paper does not use Kripke semantics but only algebraic methods. Acknowledgements. I thank M. Kracht, W. Rautenberg and M. Zakharyaschev for helpful discussions. 1 Syntax A modal similarity type S = hF; aei consists of a set F of modal operators and a map ae : F ....

....Kripkecomplete logics. a We say that a modal logic has uniform interpolation if for any formula and variables q = fq 1 ; q k g there exists a uniform interpolant 9 q for , i.e. ffl 9 q 2 , ffl var9 q var Gamma q, ffl 9 q 2 whenever 2 and var q = Pitts [13] proved that intuitionistic propositional logic has uniform interpolation. 5] and [16] prove that K, provability logic GL and Grzegorzcyk s system Grz have uniform interpolation but that S4 lacks it. It is easily proved that a normal modal logic has uniform interpolation whenever it has ....

A. Pitts. On an interpretation of second order quantification in first order intuitionistic propositional logic, Journal of Symbolic logic, 57: 33 -- 52, 1992

....for the calculus extends to polymodal logic. Finally, section 5 contains negative results for infinitary modal logic, Propositional Dynamic Logic (PDL, 11] first order logic and monadic second order logic. First, let us present some motivation for studying uniform interpolation. Pitts [19] proves uniform interpolation for intuitionistic propositional logic (just another modal logic, as far as we are concerned) and demonstrates the connection with implicit definability of second order like quantifiers. To see this connection, let OE be a formula and consider the language L that ....

....to one all of whose states are nonempty. 3 Quantifiers and uniform interpolation In this section we prove the uniform interpolation theorem for the modal calculus. It is done via an interpretation of second order quantifiers within the calculus itself. This approach is inspired by [22] and [19]. Theorem 3.1 Let OE be a sentence and L its language. Let p be an arbitrary proposition constant. Then there is a sentence in the language L n fpg such that: M fl iff there is a model N with M (Lnfpg) N fl OE. Proof. Let A = Q; Sigma p ; Sigma r ; q 0 ; ffi; Omega Gamma be an ....

A. Pitts. On an Interpretation of Second Order Quantification in First Order Intuitionistic Propositional Logic. Journal of Symbolic Logic, 57:33--52, 1992.

....in the previous sections has been constructed with a computer program building a diagram and calculating for each class the so called carriers of interpolation. The notion of a carrier of interpolation may have an interest of its own. It is related to the notion of uniform interpolation (see [Pitts 92] the left uniform interpolant of a formula OE can be regarded as the singleton left carrier of OE 1 . The partial order on equivalence classes can be used to define a left and a right carrier of interpolants of a formula. Definition 4 Let OE be a formula in the fragment F ( p; q) The ....

A. Pitts, 'On an interpretation of second order quantification in first order propositional logic', JSL) 57, 33--52 (1992).

....the ordinary Interpolation Theorem employing characters (see: 6] The methods of Gleit Goldfarb and later of Shavrukov can be viewed as model theoretical. For IPC, A. Pitts proved Uniform Interpolation by proof theoretical methods, using proof systems allowing efficient cut elimination (see: [14]) developed, independently, by J. Hudelmaier (see: 10] and R. Dyckhoff (see: 2] Later S. Ghilardi and M. Zawadowski (see: 4] and, independently but later, A. Visser, found a model theoretical proof for Pitt s result using bounded bisimulations. We prove an amalgamation lemma. Note that ....

A. Pitts. On an interpretation of second order quantification in first order intuitionistic propositional logic. Journal of Symbolic Logic, 57:33--52, 1992.

....again by the Induction Hypothesis, m = b fl oe( The converse is similar. 2 Consider oe 2 sub p;IPC(P) Dick de Jongh and Albert Visser prove the following theorem. See [5] Theorem 2.9 p;IPC(P) oe) is finitely axiomatizable. The proof uses Pitts Uniform Interpolation Theorem. See [17], 9] 40] Par abus de langage, we call an axiom of p;IPC(P) oe) oe . Note that oe is only determined up to provable equivalence. We call a formula axiomatizing some p;IPC(P) a p; P exact formula. The set of p; P exact formulas is exact p;P . Silvio Ghilardi proved that ....

A. Pitts. On an interpretation of second order quantification in first order intuitionistic propositional logic. Journal of Symbolic Logic, 57:33--52, 1992.

....algebra. We have just seen that every Boolean algebra can arise as Sub E (1) for some topos E. However, it is not known whether every Heyting algebra can arise in this way. Probably the free Heyting algebra on countably many generators cannot be the Heyting algebra of truth values of a topos; see Pitts 1992, Section 1 for more on this topic. 5. Conclusion The notion of tripos was motivated by the desire to explain in what sense Higg s description of sheaf toposes as H valued sets and Hyland s realizability toposes are instances of the same construction. The construction itself involves building ....

Pitts, A. M. (1992). On an interpretation of second order quantification in first order intuitionistic propositional logic. Jour. Symbolic Logic 57, 33--52.

....algebra. We have just seen that every Boolean algebra can arise as Sub E (1) for some E. However, it is not known whether every Heyting algebra can arise in this way. Probably the free Heyting algebra on countably many generators cannot be the Heyting algebra of truth values of a topos; see Pitts 1992, Section 1 for more on this topic. 5 Conclusion The notion of tripos was motivated by the desire to explain in what sense Higg s description of sheaf toposes as H valued sets and Hyland s realizability toposes are instances of the same construction. The construction itself involves building ....

Pitts, A. M. (1992). On an interpretation of second order quantification in first order intuitionistic propositional logic. Jour. Symbolic Logic 57, 33--52.

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