Identity, Equality, Nameability and Completeness

Authors

  • María Manzano Department of Philosophy, University of Salamanca, Spain
  • Manuel Crescencio Moreno

DOI:

https://doi.org/10.18778/0138-0680.46.3.4.02

Keywords:

first-order logic, type theory, identity, equality, indiscernibility, comprehension, completeness, translations, nameability

Abstract

This article is an extended promenade strolling along the winding roads of identity, equality, nameability and completeness, looking for places where they converge. We have distinguished between identity and equality; the first is a binary relation between objects while the second is a symbolic relation between terms. Owing to the central role the notion of identity plays in logic, you can be interested either in how to define it using other logical concepts or in the opposite scheme. In the first case, one investigates what kind of logic is required. In the second case, one is interested in the definition of the other logical concepts (connectives and quantifiers) in terms of the identity relation, using also abstraction. The present paper investigates whether identity can be introduced by definition arriving to the conclusion that only in full higher-order logic a reliable definition of identity is possible. However, the definition needs the standard semantics and we know that with this semantics completeness is lost. We have also studied the relationship of equality with comprehension and extensionality and pointed out the relevant role played by these two axioms in Henkin’s completeness method. We finish our paper with a section devoted to general semantics, where the role played by the nameable hierarchy of types is the key in Henkin’s completeness method.

References

[1] H. Andreka, J. van Benthem, N. Bezhanishvili and I. Németi, Changing a Semantics: Opprtunism or Courage?, [in:] The Life and Work of Leon Henkin. Essays on His Contributions, Springer Basil, (2014), pp. 307–338.

[2] P. B. Andrews, General models and Extensionality, The Journal of Symbolic Logic 37/2 (1972), pp. 395–397.

[3] A. Church, A formulation of the simple theory of types, The Journal of Symbolic Logic 5/2 (1940), pp. 56–68.

[4] G. Frege, Über Sinn und Bedeutung, Zeitschrift für Philosophie und philosophische Kritik, C, (1892), pp. 25–50. English Translation: On Sense and Meaning, [in:] B. McGuinness (ed.), Frege: collected works, Oxford: Blackwell, pp. 157–177.

[5] L. Henkin, The completeness of the first order functional calculus, The Journal of Symbolic Logic 14 (1949), pp. 159–166.

[6] L. Henkin, Completeness in the theory of types, The Journal of Symbolic Logic 15 (1950), pp. 81–91.

[7] L. Henkin, Banishing the rule of substitution for functional variables, The Journal of Symbolic Logic 18/3 (1953), pp. 201–208.

[8] L. Henkin, Some Notes on Nominalism, The Journal of Symbolic Logic 18 (1953), pp. 19–29.

[9] L. Henkin, Identity as a logical primitive, Philosophia. Philosophical Quarterly of Israel 5/1-2 (1975), pp. 31–45.

[10] L. Henkin, The discovery of my completeness proofs, The Bulletin of Symbolic Logic 2/2 (1996), pp. 127–158.

[11] J. Ketland, Identity and Indiscernibility, The Review of Symbolic Logic 4/2 (2011), pp. 171–185.

[12] M. Manzano, Extensions of First-order Logic, Cambridge University Press (1996).

[13] M. Manzano, Model Theory, Oxford Logic Guides, Oxford University Press, (1999). (Translated by Ruy de Queiroz from Teoría de Modelos, Alianza Editorial, 1989).

[14] M. Manzano, April the 19th, [in:] The Life and Work of Leon Henkin. Essays on His Contributions, Springer Basil, (2014), pp. 265–278.

[15] M. Manzano, A. Kurucz and I. Sain, The little mermaid, [in:] Truth in Perspective, Concepción Martinez et al. (eds.). Ashgate. Aldershot (U.K.), (1998).

[16] M. Manzano, I. Sain and E. Alonso (eds.), The Life and Work of Leon Henkin. Essays on His Contributions, Springer Basil, (2014).

Downloads

Published

2017-12-30

How to Cite

Manzano, M., & Moreno, M. C. (2017). Identity, Equality, Nameability and Completeness. Bulletin of the Section of Logic, 46(3/4), 169–195. https://doi.org/10.18778/0138-0680.46.3.4.02

Issue

Section

Research Article

Most read articles by the same author(s)