Identity, equality, nameability and completeness. Part II
Keywords:identity, equality, completeness, nameability, first-order modal logic, hybrid logic, hybrid type theory, equational hybrid propositional type theory
This article is a continuation of our promenade along the winding roads of identity, equality, nameability and completeness. We continue looking for a place where all these concepts converge. We assume that identity is a binary relation between objects while equality is a symbolic relation between terms. Identity plays a central role in logic and we have looked at it from two different points of view. In one case, identity is a notion which has to be defined and, in the other case, identity is a notion used to define other logical concepts. In our previous paper, , we investigated whether identity can be introduced by definition arriving to the conclusion that only in full higher-order logic with standard semantics a reliable definition of identity is possible. In the present study we have moved to modal logic and realized that here we can distinguish in the formal language between two different equality symbols, the first one shall be interpreted as extensional genuine identity and only applies for objects, the second one applies for non rigid terms and has the characteristic of synonymy. We have also analyzed the hybrid modal logic where we can introduce rigid terms by definition and can express that two worlds are identical by using the nominals and the @ operator. We finish our paper in the kingdom of identity where the only primitives are lambda and equality. Here we show how other logical concepts can be defined in terms of the identity relation. We have found at the end of our walk a possible point of convergence in the logic Equational Hybrid Propositional Type Theory (EHPTT),  and .
P. B. Andrews, A reduction of the axioms for the theory of propositional type theory, Fundamenta Mathematicae 52 (1963), pp. 354–350.
P. B. Andrews, An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof, Academic Press, Orlando, San Diego, New York, Austin, Boston, London, Sydney, Tokyo, Toronto (1986).
P. B. Andrews, A Bit of History Related to Equality as a Logical Primitive, [in:] The Life and Work of Leon Henkin. Essays on His Contributions, Springer Basil, (2014), pp. 67–71.
C. Areces, P. Blackburn, A. Huertas and M. Manzano, Completeness in Hybrid Type Theory, Journal of Philosophical Logic 43/2–3 (2014), pp. 209–238. DOI: 10.1007/s10992-012-9260-4.
P. Blackburn and J. van Benthem, Modal Logic: A Semantic Perspective, [in:] Handbook of Modal Logic, Elsevier, (2007), pp. 1–84.
P. Blackburn, A. Huertas, M. Manzano and F. Jorgensen, Henkin and Hybrid Logic, [in:] The Life and Work of Leon Henkin. Essays on His Contributions, Springer Basil, (2014), pp. 279–306.
M. Fitting and R. L. Mendelsohn, First-Order Modal Logic, Kluwer Academic Publishers, Dordrecht, Boston, London (1998).
M. Fitting, Types, Tableaus, and Gödel’s God, Kluwer Academic Publishers, Dordrecht, Boston, London (2002).
L. Henkin, The completeness of the first order functional calculus, The Journal of Symbolic Logic 14 (1949), pp. 159–166.
L. Henkin, Completeness in the theory of types, The Journal of Symbolic Logic 15 (1950), pp. 81–91.
L. Henkin, A Theory of Propositional Types, Fundamenta Mathematicae 52 (1963), pp. 323–344. Errata, Fundamenta Mathematicae 53 (1964), p. 119.
L. Henkin, Identity as a logical primitive, Philosophia. Philosophical Quarterly of Israel 5/1–2 (1975), pp. 31–45.
G. E. Hughes and M. J. Cresswell, A New Introduction to Modal Logic, Routledge, London, New York (1996).
M.Manzano, M. A.Martins and A. Huertas, A Semantics for Equational Hybrid Propositional Type Theory, Bulletin of the Section of Logic 43:3/4 (2014), pp. 121–138.
M. Manzano, M. A. Martins and A. Huertas, Completeness in Equational Hybrid Propositional Type Theory, Studia Logica, first online. DOI: 10.1007/s11225-018-9833-5
M. Manzano and M. C. Moreno, Identity, Equality, Nameability and Completeness, Bulletin of the Section of Logic 46:3/4 (2017), pp. 169–195.
W. V. O. Quine, Reference and Modality, [in:] From a Logical Point of View, Harvard University Press, Cambridge, MA (1953), pp. 139–159.
F. P. Ramsey, The Foundation of Mathematics, Proceedings of the London Mathematical Society, Ser. 2, Vol. 25, Part 5, (1926), pp. 338–384. DOI: 10.1112/plms/s2-25.1.338
L. Wittgenstein, Tractatus Logico-Philosophicus, Routledge & Kegan Paul, London (1922). Originally published as Logisch-Philosophische Abhandlung, Annalen der Naturphilosophische XIV: 3/4 (1921).
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