A Kalmár-style completeness proof for the logics of the hierarchy 𝕀 n ℙ k

Víctor Fernández

Commentationes Mathematicae Universitatis Carolinae (2023)

  • Volume: 64, Issue: 4, page 485-509
  • ISSN: 0010-2628

Abstract

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The logics of the family 𝕀 n ℙ k := { I n P k } ( n , k ) ∈ ω 2 are formally defined by means of finite matrices, as a simultaneous generalization of the weakly-intuitionistic logic I 1 and of the paraconsistent logic P 1 . It is proved that this family can be naturally ordered, and it is shown a sound and complete axiomatics for each logic of the form I n P k . The involved completeness proof showed here is obtained by means of a generalization of the well-known Kalmár’s method, usually applied for many-valued logics.

How to cite

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Fernández, Víctor. "A Kalmár-style completeness proof for the logics of the hierarchy ${\mathbb {I}}^n {\mathbb {P}}^k$." Commentationes Mathematicae Universitatis Carolinae 64.4 (2023): 485-509. <http://eudml.org/doc/299327>.

@article{Fernández2023,
abstract = {The logics of the family $\{\mathbb \{I\}\}^n \{\mathbb \{P\}\}^k$:=$\lbrace \{ I^n P^k\}\rbrace _\{(n,k) \in \omega ^2\}$ are formally defined by means of finite matrices, as a simultaneous generalization of the weakly-intuitionistic logic $I^1$ and of the paraconsistent logic $P^1$. It is proved that this family can be naturally ordered, and it is shown a sound and complete axiomatics for each logic of the form $I^n P^k$. The involved completeness proof showed here is obtained by means of a generalization of the well-known Kalmár’s method, usually applied for many-valued logics.},
author = {Fernández, Víctor},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {mathematical logic; Kalmár's completeness proof; many-valued logic},
language = {eng},
number = {4},
pages = {485-509},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A Kalmár-style completeness proof for the logics of the hierarchy $\{\mathbb \{I\}\}^n \{\mathbb \{P\}\}^k$},
url = {http://eudml.org/doc/299327},
volume = {64},
year = {2023},
}

TY - JOUR
AU - Fernández, Víctor
TI - A Kalmár-style completeness proof for the logics of the hierarchy ${\mathbb {I}}^n {\mathbb {P}}^k$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2023
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 64
IS - 4
SP - 485
EP - 509
AB - The logics of the family ${\mathbb {I}}^n {\mathbb {P}}^k$:=$\lbrace { I^n P^k}\rbrace _{(n,k) \in \omega ^2}$ are formally defined by means of finite matrices, as a simultaneous generalization of the weakly-intuitionistic logic $I^1$ and of the paraconsistent logic $P^1$. It is proved that this family can be naturally ordered, and it is shown a sound and complete axiomatics for each logic of the form $I^n P^k$. The involved completeness proof showed here is obtained by means of a generalization of the well-known Kalmár’s method, usually applied for many-valued logics.
LA - eng
KW - mathematical logic; Kalmár's completeness proof; many-valued logic
UR - http://eudml.org/doc/299327
ER -

References

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