Abstract
We enrich intuitionistic logic with a lax modal operator ○ and define a corresponding intensional enrichment of Kripke models M = (W, ⊑, V) by a function T giving an effort measure T(w, u) ∈ \({\mathbb{N}} \cup\) {∞} for each ⊑-related pair (w, u). We show that ○ embodies the abstraction involved in passing from “ϕ true up to bounded effort” to “ϕ true outright”. We then introduce a refined notion of intensional validity M |= p : ϕ and present a corresponding intensional calculus iLC-h which gives a natural extension by lax modality of the well-known G: odel/Dummett logic LC of (finite) linear Kripke models. Our main results are that for finite linear intensional models L the intensional theory iTh(L) = {p : ϕ | L |= p : ϕ} characterises L and that iLC-h generates complete information about iTh(L).
Our paper thus shows that the quantitative intensional information contained in the effort measure T can be abstracted away by the use of ○ and completely recovered by a suitable semantic interpretation of proofs.
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References
Curry, H. B., ‘The elimination theorem when modality is present’, Journal of Symbolic Logic, 17:249–265, 1952.
Curry, H. B., A Theory of Formal Deducibility, vol. 6 of Notre Dame Mathematical Lectures. Notre Dame, Indiana, second edition, 1957.
Dragalin, A. G., Mathematical Intuitionism. Introduction to Proof Theory. American Mathematical Society, 1988.
De Swart, H. C. M., ‘Another intuitionistic completeness proof’, Journal of Symbolic Logic, 41:644–662, 1976.
Dummett, M., ‘A propositional calculus with a denumerable matrix’, Journal of Symbolic Logic, 24(2):97–106, June 1959.
Dummett, M., Elements of Intuitionism, Clarendon Press, Oxford, 1977.
Fairtlough, M. and M. V. Mendler, ‘Propositional Lax Logic’, Information and Computation, 137(1):1–33, August 1997.
Fairtlough, M., M. Mendler, and X. Cheng, ‘Abstraction and refinement in higher-order logic’, in R. J. Boulton and P. B. Jackson, editors, 14th International Conference on Theorem Proving in Higher Order Logic (TPHOLs'2001, LNCS 2152, pp. 201–216, Springer, September 2001.
Fairtlough, M., M. Mendler, and M. Walton, First-order lax logic as a framework for constraint logic programming, Technical Report MIP-9714, University of Passau, July 1997.
Friedman, H., ‘Intuitionistic completeness of Heyting's predicate calculus’, Notices of the American Mathematical Society, 22, 1975.
Goldblatt, R. I., ‘Grothendieck topology as geometric modality’, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 27:495–529, 1981.
Goldblatt, R., Topoi: The categorical analysis of logic, North Holland, 2nd edition, 1986.
Johnstone, P. T., Stone spaces, Cambridge University Press, 1982.
Kolmogoroff, A., ‘Zur Deutung der intuitionistischen Logik’, Mathematische Zeitschrift, 35:58–65, 1932.
Kripke, S., ‘Semantical analysis of intuitionistic logic I’, in J. Crossley and M. Dummett, editors, Formal Systems and Recursive Functions, pp. 92–129, North-Holland, 1963.
Lawvere, F. W, ‘Introduction to toposes, algebraic geometry and logic’, in Lecture Notes in Mathematics, no. 274, pp. 1–12, Springer-Verlag, 1972.
Lopez-Escobar, E. G. K. and W. Veldman, ‘Intuitionistic completeness of a restricted second-order logic’, in ISLIC Proof Theory Symposium, LNCS 500, pp. 198–232, Springer, 1974.
Lipton, J., ‘Constructive Kripke semantics and realizability’, in Y. N. Moschovakis, editor, Proc. Logic for Computer Science, pp. 319–357, Springer, 1991.
Mac Lane, S., Categories for the Working Mathematician, Springer-Verlag, 1988.
Macnab, D. S., ‘Modal operators on Heyting algebras’, Algebra Universalis, 12:5–29, 1981.
Medvedev, Ju. T., ‘Interpretation of logical formulas by means of finite problems’, Soviet Math. Dokl., 7(4):857–860, 1966.
Mendler, M., A Modal Logic for Handling Behavioural Constraints in Formal Hardware Verification. PhD thesis, Edinburgh University, Department of Computer Science, ECS-LFCS–93–255, 1993.
Mendler, M., ‘Characterising combinational timing analyses in intuitionistic modal logic’, The Logic Journal of the IGPL, 8(6):821–852, November 2000. Abstract appeared ibid. vol. 6, No. 6, (Nov 1998).
Mendler, M. and M. Fairtlough, ‘Ternary simulation: A refinement of binary functions or an abstraction of real-time behaviour?’, in M. Sheeran and S. Singh, editors, Proc. 3rd Workshop on Designing Correct Circuits (DCC'96), Springer Electronic Workshops in Computing, 1996.
Mitchell, J. and E. Moggi, ‘Kripke-style models for typed lambda calculus’, in Proc. Logic in Computer Science, 1987.
Miglioli, P., U. Moscato, M. Ornaghi, S. Quazza, and G. Usberti, ‘Some results on intermediate constructive logics’, Notre Dame Journal of Formal Logic, 30(4):543–562, 1989.
Moggi, E., ‘Notions of computation and monads’, Information and Computation, 93:55–92, 1991.
Troelstra, A. S., ‘Realizability’, in S. R. Buss, editor, Handbook of Proof Theory, chapter VI, pp. 407–474, Elsevier, 1998.
Troelstra, A. S. and D. Van Dalen, Constructivism in Mathematics, North-Holland, 1988.
Van Dalen, D., ‘Intuitionistic logic’, in D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume III, chapter 4, pp. 225–339, Reidel, 1986.
Veldman, W., ‘An intuitionistic completeness theorem for intuitionistic predicate logic’, Journal of Symbolic Logic, 41:159–166, 1976.
Wadler, P., ‘Comprehending monads’, in Conference on Lisp and Functional Programming, ACM Press, June 1990.
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Fairtlough, M., Mendler, M. Intensional Completeness in an Extension of Gödel/Dummett Logic. Studia Logica 73, 51–80 (2003). https://doi.org/10.1023/A:1022937306253
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DOI: https://doi.org/10.1023/A:1022937306253