Geodesic
A Kalmár-style completeness proof for the logics of the hierarchy ${\mathbb{I}}^n {\mathbb{P}}^k$
Commentationes Mathematicae Universitatis Carolinae, Tome 64 (2023) no. 4, pp. 485-509
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The logics of the family ${\mathbb{I}}^n {\mathbb{P}}^k$:=$\{{ I^n P^k}\}_{(n,k) \in \omega^2}$ 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.
The logics of the family ${\mathbb{I}}^n {\mathbb{P}}^k$:=$\{{ I^n P^k}\}_{(n,k) \in \omega^2}$ 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.
DOI : 10.14712/1213-7243.2024.009
Classification : 03B50, 03B53
Keywords: mathematical logic; Kalmár's completeness proof; many-valued logic 
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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, Tome 64 (2023) no. 4, pp. 485-509. doi: 10.14712/1213-7243.2024.009

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