Identity, equality, nameability and completeness. Part II

María Manzano; Manuel Crescencio Moreno

Bulletin of the Section of Logic (2018)

  • Volume: 47, Issue: 3
  • ISSN: 0138-0680

Abstract

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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, [16], 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), [14] and [15].

How to cite

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María Manzano, and Manuel Crescencio Moreno. "Identity, equality, nameability and completeness. Part II." Bulletin of the Section of Logic 47.3 (2018): null. <http://eudml.org/doc/295513>.

@article{MaríaManzano2018,
abstract = {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, [16], 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), [14] and [15].},
author = {María Manzano, Manuel Crescencio Moreno},
journal = {Bulletin of the Section of Logic},
keywords = {identity; equality; completeness; nameability; first-order modal logic; hybrid logic; hybrid type theory; equational hybrid propositional type theory},
language = {eng},
number = {3},
pages = {null},
title = {Identity, equality, nameability and completeness. Part II},
url = {http://eudml.org/doc/295513},
volume = {47},
year = {2018},
}

TY - JOUR
AU - María Manzano
AU - Manuel Crescencio Moreno
TI - Identity, equality, nameability and completeness. Part II
JO - Bulletin of the Section of Logic
PY - 2018
VL - 47
IS - 3
SP - null
AB - 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, [16], 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), [14] and [15].
LA - eng
KW - identity; equality; completeness; nameability; first-order modal logic; hybrid logic; hybrid type theory; equational hybrid propositional type theory
UR - http://eudml.org/doc/295513
ER -

References

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  16. M. Manzano and M. C. Moreno, Identity, Equality, Nameability and Completeness, Bulletin of the Section of Logic 46:3/4 (2017), pp. 169–195. 
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