Logics based on linear orders of contaminating values

Ciuni, Roberto; Ferguson, Thomas Macaulay; Szmuc, Damián Enrique

doi:10.1093/logcom/exz009

Cited by 25 publications

(29 citation statements)

References 20 publications

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“…[13]). These connections have been clarified in a number of recent papers [2,6,7]; here, we underscore the role played in the algebraic analysis of DAI by involutive bisemilattices, a subquasivariety of which was already recognised in [1] as the "algebraic counterpart" (in a sense to be specified below) of Paraconsistent Weak Kleene (PWK), a typical logic of nonsense.…”

Section: Introductionmentioning

confidence: 66%

Algebraic Analysis of Demodalised Analytic Implication

2019

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The logic DAI of demodalised analytic implication has been introduced by J.M. Dunn (and independently investigated by R.D. Epstein) as a variation on a timehonoured logical system by C.I. Lewis' student W.T. Parry. The main tenet underlying this logic is that no implication can be valid unless its consequent is "analytically contained" in its antecedent. DAI has been investigated both proof-theoretically and model-theoretically, but no study so far has focussed on DAI from the viewpoint of abstract algebraic logic. We provide several different algebraic semantics for DAI, showing their equivalence with the known semantics by Dunn and Epstein. We also show that DAI is algebraisable and we identify its equivalent quasivariety semantics. This class turns out to be a linguistic and axiomatic expansion of involutive bisemilattices, a subquasivariety of which forms the algebraic counterpart of Paraconsistent Weak Kleene logic (PWK). This fact sheds further light on the relationship between containment logics and logics of nonsense.

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Section: Introductionmentioning

confidence: 66%

Algebraic Analysis of Demodalised Analytic Implication

2019

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“…This element can be seen as a universal zero element, as well as an element working in a value-in-value-out fashion. In fact, these two different approximations can be seen to be equivalent as noted in the definitions below, taken from [31] and [10].…”

Section: Infectious Logicsmentioning

confidence: 99%

“…these logics already quite well-known and have received a lot of attention at least in the past years. Respectively, they are called the paracomplete weak Kleene logic K w 3 and the paraconsistent weak Kleene logic PWK-see, e.g., [22], [20], [8], [10].…”

Section: Restricting the Sets Of Designated Valuesmentioning

confidence: 99%

Immune Logics

SZMUC

2021

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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.

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“…Thus it will be enough to prove that Mod Su ( l ) ⊆ P ł (Mod * ( )). We have that Mod Su ( l ) = P sd Mod * ( l ) (18) ⊆ SPMod * ( l ) (19) ⊆ SPP ł (Mod * ( )) (20) ⊆ P ł (SPMod * ( )) (21) = P ł (Mod * ( )).…”

Section: Corollary 30 If Is An Equivalential Finitary Logic With a Pmentioning

confidence: 99%

Logics of left variable inclusion and Płonka sums of matrices

2020

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The paper aims at studying, in full generality, logics defined by imposing a variable inclusion condition on a given logic . We prove that the description of the algebraic counterpart of the left variable inclusion companion of a given logic is related to the construction of Płonka sums of the matrix models of . This observation allows to obtain a Hilbert-style axiomatization of the logics of left variable inclusion, to describe the structure of their reduced models, and to locate them in the Leibniz hierarchy.

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Logics based on linear orders of contaminating values

Cited by 25 publications

References 20 publications

Algebraic Analysis of Demodalised Analytic Implication

Algebraic Analysis of Demodalised Analytic Implication

Immune Logics

Logics of left variable inclusion and Płonka sums of matrices

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