Awodey, S., Butz, C., Topological Completeness for Higher Order Logic, Journal of Symbolic Logic 65(3) (2000), p. 1168-1182.  Home/Search   Document Details and Download   Summary   Related Articles   Check

No context found.

Awodey, S., Butz, C., Topological Completeness for Higher Order Logic, Journal of Symbolic Logic 65(3) (2000), p. 1168-1182.

....for a theory i Sets is. We also have a corollary similar to Cor 5. 1, that a theory T M = N i for all models [ T DOF (P) M ] N ] 8 Conclusions Since this paper just unravels the results of [2] into elementary and noncategory theoretic terms, one could do the same for [1]. It would be more dicult however since the constructions and semantics are deeply category theoretic. An easier application would be to consider the retract theory T, and build an appropriate P and universal model of T in Sets . By [13] this would give an elementary model of the untyped ....

Awodey, S., Butz, C., Topological Completeness for Higher Order Logic, Journal of Symbolic Logic 65(3) (2000), p. 1168-1182.

....at Heyting valued models to get completeness. The last section contains some remarks about the infinitary variants L . We assume familiarity with basic notions of categorical logic, see for example [14] or [9] The results presented here are closely related to the joint paper with S. Awodey, [2]. In fact, they give a detailed exposition of one of the completeness result presented there. In case of pure typed calculus, a more detailed exposition can be found in [1] Our overall presentation is in the line of categorical model theory, as was done for geometric logic in [15] and for first ....

....one shows that our logic L (but in fact, full intuitionistic second order logic) is complete with respect to Heyting valued Henkin models. Fixing a set of enough Heytingvalued Henkin models S T , points of X are pairs (M; ff) where M is in S T and ff is an enumeration of M , similar as in [2], appendix. The enumerations are used to define the topology. Before we end this section let us mention that the model U in B (T ) is minimal in the following sense: Proposition 4.4 For any model M of T in a Grothendieck topos F there is a unique (up to isomorphism) geometric morphism M : F ....

S. Awodey and C. Butz. Topological completeness for higher order logic. Preprint, June 1997, submitted.

.... preserves finite coproducts and the internal first order logic of W; iii) for any objects Y and Z of W there is a canonical inclusion Gamma(Z Y ) ae GammaZ GammaY : Proof: The proof of statement (i) is straightforward, and (ii) results from 14 (i) and the fact that W is boolean (cf. [3] for details) For (iii) we have Gamma(Z Y ) W(1;Z Y ) W(Y;Z) ae Sets( GammaY; GammaZ ) Gamma is faithful; GammaZ GammaY : Note that by (ii) and (iii) one also has a canonical inclusion Gamma(P X) Gamma(2 X ) ae Gamma2 GammaY = 2 GammaY = P ( GammaX ) 26) ....

S. Awodey and C. Butz. Topological completeness for higher-order logic. Technical report, BRICS, Aahrus University, Denmark, 1997.

....in figure 3 (which also indicates how even fewer would still suffice) The adjoint rules of figure 1 can then be proven. In categorical logic we extend the treatment of propositional logic as a 1 The quantifier rules require the variable x not occur freely in #. For a full statement see [4, 2] 2 Figure 1: Adjoint rules for FOL # and # # and # # # # # ) x) # # (x) 9x: x) # # 8x: x) Figure 2: Algebraic formulation of HOL Types: X Theta Y; Y X ; P(X) P Terms: hs; ti; 1 t; 2 t; x:t; t(s) fx j g ....

.... general than are true of constant ones (think of the difference between the field of real numbers and the ring of real valued functions) What is the logic of continuously varying sets That is, which formulas of HOL are true in all sheaf models The answer is given by the following theorem from [2]: Theorem. Logic of sheaves = classical deductive HOL. The proof of this fact uses recent, non trivial results in topos theory. 2 The sheaf theory on which it rests [3] is rooted in geometry, not logic. It is worth emphasizing that, unlike the preceding theorem, there is no obvious reason why ....

S. Awodey and C. Butz (2000) "Topological completeness for higherorder logic" Journal of Symbolic Logic 65(3), pp. 1168--82.

....ffl [ Gamma j 2 (P ) p 2 ffi [ Gamma j P : oe Theta ] where p 2 : oe] Theta [ is the second canonical projection. 12 Note that a context Gamma is always interpreted by: Gamma] x 1 : 1 ; x n : n ] 1 Theta : Theta n ] [ 1 ]] Theta : Theta [ n ] A closed term M : in the empty context is therefore interpreted as an arrow of the form 1 [ Finally, an interpretation is, of course, a model of T if it satisfies the equations of T, in the sense that one has identity of arrows [ M ] N ] in C for each ....

....in [5] to the presheaf topos Set (C T ) op , taking as a sufficient set of points the evaluation functors eval oe : Set (C T ) op Gamma Set for each object (type) oe 2 CT , and then unpacking the result in terms of the theory T. A similar method was applied to higher order logic in [1], the appendix of which gives a different perspective on the construction, in terms of so called Henkin models . To simplify the description, we shall assume that the theory Thas countably many basic types and terms, and we add a terminal type 1, together with a basic term : 1 and the ....

[Article contains additional citation context not shown here]

S. Awodey and C. Butz, Topological completeness for higher-order logic, Tech. report, BRICS, Aahrus University, Denmark, 1997, (submitted).

No context found.

S. Awodey and C. Butz. Topological completeness for higher-order logic. Journal of Symbolic Logic, 65(3):1168-1182, 2000.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC