Abstract
We propose syntax and semantics for systems of intuitionistic and classical first order dependently sorted logic, with and without equality, retaining type dependency, but otherwise abstracting, from systems for dependent type theory, and which can be seen as generalised systems of multisorted logic. These are presented as extensions to Gentzen systems for first order logic in which the logic is developed relative to a context of variable declarations over a theory of dependent sorts. A generalised notion of Kripke structure provides the semantics for the intuitionistic systems.
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References
Gambino, N., Aczel, P.: The generalised type-theoretic intepretation of constructive set theory (preprint 2005)
Jacobs, B.: Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics, vol. 141. North Holland, Amsterdam (1999)
Aczel, P.: Predicate logic with dependent sorts or types. Unpublished (2004)
Belo, J.F.: Dependently typed predicate logic. Master’s thesis, University of Manchester (2004)
Makkai, M.: First order logic with dependent sorts, with applications to category theory. Unpublished (1995)
Rabe, F.: First-order logic with dependent types. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 377–391. Springer, Heidelberg (2006)
Mosses, P.D. (ed.): CASL Reference Manual. LNCS, vol. 2960. Springer, Heidelberg (2004)
Benke, M., Dybjer, P., Jansson, P.: Universes for generic programs and proofs in dependent type theory. Nordic J. of Computing 10(4), 265–289 (2003)
Pfeifer, H., Rueß, H.: Polytypic proof construction. In: Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C., Théry, L. (eds.) TPHOLs 1999. LNCS, vol. 1690, pp. 55–72. Springer, Heidelberg (1999)
Cartmell, J.: Generalized algebraic theories and contextual categories. PhD thesis, Univ. Oxford (1978)
Cartmell, J.: Generalized algebraic theories and contextual categories. Ann. Pure Appl. Logic 32, 209–243 (1986)
Pitts, A.M.: Categorical logic. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science. Algebraic and Logical Structures, vol. 5, ch.2, Oxford University Press, Oxford (2000)
Hofmann, M.: Syntax and semantics of dependent types. In: Pitts, A.M., Dybjer, P. (eds.) Semantics and Logics of Computation, vol. 14, pp. 79–130. Cambridge University Press, Cambridge (1997)
Troelstra, A.S., Schwichtenberg, H.: Basic proof theory. Cambridge University Press, Cambridge (2000)
Johnstone, P.T.: Sketches of an Elephant: A Topos Theory Compendium, vol. 2. Oxford University Press, Oxford (2002)
Johnstone, P.T.: Notes on Logic and Set Theory. Cambridge University Press, Cambridge (1987)
Shoenfield, J.R.: Mathematical Logic. Association for Symbolic Logic (1967)
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Belo, J.F. (2008). Dependently Sorted Logic. In: Miculan, M., Scagnetto, I., Honsell, F. (eds) Types for Proofs and Programs. TYPES 2007. Lecture Notes in Computer Science, vol 4941. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68103-8_3
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DOI: https://doi.org/10.1007/978-3-540-68103-8_3
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