Identity, Equality, Nameability and Completeness

María Manzano; Manuel Crescencio Moreno

Bulletin of the Section of Logic (2017)

  • Volume: 46, Issue: 3/4
  • ISSN: 0138-0680

Abstract

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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.

How to cite

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María Manzano, and Manuel Crescencio Moreno. "Identity, Equality, Nameability and Completeness." Bulletin of the Section of Logic 46.3/4 (2017): null. <http://eudml.org/doc/295560>.

@article{MaríaManzano2017,
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.},
author = {María Manzano, Manuel Crescencio Moreno},
journal = {Bulletin of the Section of Logic},
keywords = {first-order logic; type theory; identity; equality; indiscernibility; comprehension; completeness; translations; nameability},
language = {eng},
number = {3/4},
pages = {null},
title = {Identity, Equality, Nameability and Completeness},
url = {http://eudml.org/doc/295560},
volume = {46},
year = {2017},
}

TY - JOUR
AU - María Manzano
AU - Manuel Crescencio Moreno
TI - Identity, Equality, Nameability and Completeness
JO - Bulletin of the Section of Logic
PY - 2017
VL - 46
IS - 3/4
SP - null
AB - 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.
LA - eng
KW - first-order logic; type theory; identity; equality; indiscernibility; comprehension; completeness; translations; nameability
UR - http://eudml.org/doc/295560
ER -

References

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  10. [10] L. Henkin, The discovery of my completeness proofs, The Bulletin of Symbolic Logic 2/2 (1996), pp. 127–158. 
  11. [11] J. Ketland, Identity and Indiscernibility, The Review of Symbolic Logic 4/2 (2011), pp. 171–185. 
  12. [12] M. Manzano, Extensions of First-order Logic, Cambridge University Press (1996). 
  13. [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. [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. [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. [16] M. Manzano, I. Sain and E. Alonso (eds.), The Life and Work of Leon Henkin. Essays on His Contributions, Springer Basil, (2014). 

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