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Abstract: Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces---so-called "topological semantics". The first is classical higherorder logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic. (Update)
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.... ) 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...
...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...
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7: Introduction to Higher Order Categorical Logic (context) - Lambek, Scott - 1986
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BibTeX entry: (Update)
S. Awodey and C. Butz, Topological completeness for higher-order logic, Tech. report, BRICS, Aahrus University, Denmark, 1997, (submitted). http://citeseer.ist.psu.edu/article/awodey98topological.html More
@misc{ awodey97topological,
author = "S. Awodey and C. Butz",
title = "Topological completeness for higher-order logic",
text = "S. Awodey and C. Butz, Topological completeness for higher-order logic,
Tech. report, BRICS, Aahrus University, Denmark, 1997, (submitted).",
year = "1997",
url = "citeseer.ist.psu.edu/article/awodey98topological.html" }
Citations (may not include all citations):359 Introduction to higher-order categorical logic (context) - Lambek, Scott - 1986
86 Completeness in the theory of types (context) - Henkin - 1950
65 Sheaves in geometry and logic: A first introduction to topos.. (context) - Lane, Moerdijk - 1992
30 Sheaves and logic (context) - Fourman, Scott - 1977
15 La logique des topos (context) - Boileau, Joyal - 1981
8 Topological representation of sheaf cohomology of sites - Butz, Moerdijk - 1996
6 Toposes as homotopy groupoids (context) - Joyal, Moerdijk - 1990
6 Logical and cohomological aspects of the space of points of .. (context) - Butz - 1996
6 A foundation for the simple theory of types (context) - Church - 1940
3 Topological representation of the -calculus (context) - Awodey - 1998
3 Journal of Pure and Applied Algebra (context) - Barr, Par'e et al. - 1980
2 The University of Chicago (context) - topoi, for et al. - 1997
The graph only includes citing articles where the year of publication is known.
Documents on the same site (http://www.andrew.cmu.edu/user/awodey/): More
Continuity and Logical Completeness: An application of sheaf.. - Awodey (2000) (Correct)
Sheaf Representation for Topoi - Awodey (1997) (Correct)
Axiom of Choice and Excluded Middle in Categorical Logic - Awodey (1995) (Correct)
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