Proof Theory of Paraconsistent Weak Kleene Logic

Paoli, Francesco; Baldi, Michele Pra

doi:10.1007/s11225-019-09876-z

Cited by 20 publications

(11 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Given these results, one may conjecture that the logics induced by applying these standards for logical consequence over there weak Kleene algebra and, respectively, the Sobociński algebra will be similarly related. To this extent, we may confirm that such a connection between wST and PWK has been recently established in [25]. Whether it is possible to produce similar proofs for the case of the remaining systems is as of today, yet to be determined.…”

Section: ϕ ⇒ ϕmentioning

confidence: 51%

Immune Logics

SZMUC

2021

AJL

View full text Add to dashboard Cite

This article is concerned with an exploration of a family of systems—called immune logics—that arise from certain dualizations of the well-known family of infectious logics. The distinctive feature of the semantic of infectious logics is the presence of a certain “infectious” semantic value, by which two different though equivalent things are meant. On the one hand, it is meant that these values are zero elements for all the operations in the underlying algebraic structure. On the other hand, it is meant that these values behave in a value-in-value-out fashion for all the operations in the underlying algebraic structure. Thus, in a rather informal manner, we will refer to immune logics as those systems whose underlying semantics count with a certain “immune” semantic value behaving in a way that is somewhat dual to that of the infectious values. In a more formal manner, carrying out this dualization will prove to be not as straightforward as one could imagine, since the two characterizations of infectiousness discussed above lead to two different outcomes when one tries to conduct them. We explore these alternatives and provide technical results regarding them, and the various logical systems defined using such semantics.

show abstract

Section: ϕ ⇒ ϕmentioning

confidence: 51%

Immune Logics

SZMUC

2021

AJL

View full text Add to dashboard Cite

show abstract

“…In this respect, a crucial thing to highlight is that the operations of this algebra induce two different semilattice orders which cannot be identified. Indeed, WK is an instance of a generalised involutive bisemilattice as defined in [32,33], whose meet-induced semilattice order ≤ ∧ differs from the join-induced semilattice order ≤ ∨ , as it can be seen in Figure 3.…”

Section: Semanticsmentioning

confidence: 99%

Pure Variable Inclusion Logics

Paoli

Baldi

Szmuc

2021

LLP

Self Cite

View full text Add to dashboard Cite

The aim of this article is to discuss pure variable inclusion logics, that is, logical systems where valid entailments require that the propositional variables occurring in the conclusion are included among those appearing in the premises, or vice versa. We study the subsystems of Classical Logic satisfying these requirements and assess the extent to which it is possible to characterise them by means of a single logical matrix. In addition, we semantically describe both of these companions to Classical Logic in terms of appropriate matrix bundles and as semilattice-based logics, showing that the notion of consequence in these logics can be interpreted in terms of truth (or non-falsity) and meaningfulness (or meaninglessness) preservation. Finally, we use Płonka sums of matrices to investigate the pure variable inclusion companions of an arbitrary finitary logic.

show abstract

“…To wit, if the matrix in question is a proper p-matrix but not a regular matrix, one may wonder whether the resulting logic will be identical to CL, or if it will differ with this system in some respect. Lengthy debates have been had in the past few years in this regard, mostly revolving around the logic ST defended by Cobreros, Egré, Ripley and van Rooij in many works-some of which include [9], [8], [10], [11], [34] and [35]. For future reference, the logic ST is induced by the p-matrix SK, {t}, {t, n} built on top of the 3-element strong Kleene algebra SK from S. Kleene's [26], whose carrier set is {t, n, f } and whose characteristic operations can be presented in the form of the "truth-tables" appearing in Figure 1.…”

Section: Semanticsmentioning

confidence: 99%

“…This alternative option is built on top of the 3-element weak Kleene algebra WK-instead of the aforementioned strong Kleene algebra. Thus, either implicitly or explicitly it is possible to find semantics along these lines in F. Correia's [13], F. Paoli and M. Pra Baldi's [32] and in our own [42]. In order for things to be clear in what follows, let us state here that the WK algebra is the structure whose carrier set is {t, e, f } and whose characteristic operations can be presented in the form of the "truth-tables" appearing in Figure 2.…”

Section: Semanticsmentioning

confidence: 99%

The (Greatest) Fragment of Classical Logic that Respects the Variable-Sharing Principle (in the FMLA-FMLA Framework)

Szmuc

2021

B Sect Log

View full text Add to dashboard Cite

We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics is defined in terms of a \(p\)-matrix built on top of a 5-valued extension of the 3-element weak Kleene algebra, whereas the calculus is defined in terms of a Gentzen-style sequent system where the left and right negation rules are subject to linguistic constraints.

show abstract

Proof Theory of Paraconsistent Weak Kleene Logic

Cited by 20 publications

References 22 publications

Immune Logics

Immune Logics

Pure Variable Inclusion Logics

The (Greatest) Fragment of Classical Logic that Respects the Variable-Sharing Principle (in the FMLA-FMLA Framework)

Contact Info

Product

Resources

About