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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.01274 |
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| _version_ | 1866910538606313472 |
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| author | Bonzio, Stefano Zamperlin, Nicolò |
| author_facet | Bonzio, Stefano Zamperlin, Nicolò |
| contents | Weak Kleene logics are three-valued logics characterized by the presence of an infectious truth-value. In their external versions, as they were originally introduced by Bochvar and Hallden, these systems are equipped with an additional connective capable of expressing whether a formula is classically true. In this paper we further expand the expressive power of external weak Kleen logics by modalizing them with a unary operator. The addition of an alethic modality gives rise to the two systems $\B^{\square}$ and $MPWK$, which have two different readings of the modal operator. We provide these logics with a complete and decidable Hilbert-style axiomatization w.r.t. a three-valued possible worlds semantics. The starting point of these calculi are new axiomatizations for the non-modal bases B and PWKe, which we provide using the recent algebraization results about these two logics. In particular, we prove the algebraizability of $\PWKe$. Finally some standard extensions of the basic modal systems are provided with their completeness results w.r.t. special classes of frames. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01274 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Modal weak Kleene logics: axiomatizations and relational semantics Bonzio, Stefano Zamperlin, Nicolò Logic 03B45 Weak Kleene logics are three-valued logics characterized by the presence of an infectious truth-value. In their external versions, as they were originally introduced by Bochvar and Hallden, these systems are equipped with an additional connective capable of expressing whether a formula is classically true. In this paper we further expand the expressive power of external weak Kleen logics by modalizing them with a unary operator. The addition of an alethic modality gives rise to the two systems $\B^{\square}$ and $MPWK$, which have two different readings of the modal operator. We provide these logics with a complete and decidable Hilbert-style axiomatization w.r.t. a three-valued possible worlds semantics. The starting point of these calculi are new axiomatizations for the non-modal bases B and PWKe, which we provide using the recent algebraization results about these two logics. In particular, we prove the algebraizability of $\PWKe$. Finally some standard extensions of the basic modal systems are provided with their completeness results w.r.t. special classes of frames. |
| title | Modal weak Kleene logics: axiomatizations and relational semantics |
| topic | Logic 03B45 |
| url | https://arxiv.org/abs/2403.01274 |