Defining LFIs and LFUs in extensions of infectious logics

Abstract

The aim of this paper is to explore the peculiar case of infectious logics, a group of systems obtained generalizing the semantic behavior characteristic of the {¬, ∧, ∨}-fragment of the logics of nonsense, such as the ones due to Bochvar and Halldén, among others. Here, we extend these logics with classical negations, and we furthermore show that some of these extended systems can be properly regarded as Logics of Formal Inconsistency (LFIs) and Logics of Formal Undeterminedness (LFUs).

Track-Down Operations on Bilattices

“…It can be proved (as we do in Observation 1) that the logic defined below, semantics for which were given for the first time in [13], is indeed a Dual Parry logic. 3 Definition 11.…”

“…As highlighted in [8], the following is a Parry logic. The logic S w fde , semantics for which were first given in [13], is also a Parry logic. This can be established by noting, as we do below, that S w fde is a fragment of S fde .…”

“…Moreover, the logic dS w fde , presented first in [13], is also a Dual Parry logic. This can be established-once again-by noting, as we do below, that dS w fde is a fragment of dS fde .…”

Logics of left variable inclusion and Płonka sums of matrices

“…This logic is equivalently defined, syntactically, by imposing the variable inclusion constrain, as in Definition 10, to classical logic or, semantically via the so-called weak Kleene tables with two of the three truth values as designated (see [10,20]). [54]) have been introduced and discussed in [61]. They are semantically defined, by adding a nonsensical truth value to the (single) matrix inducing Strong Kleene and the logic of Paradox, respectively.…”

Dualities for Płonka Sums

“…The models of the logic ⊢ l are obtained out of matrix models of ⊢ via the construction of the P lonka sum (see [4] for details). As a consequence, logics of variables inclusion embrace the class of logics often referred as infectious (see [32,11]), as they are semantically defined by a matrix containing a value that infects every operation in which it takes part (the logic PWK is a prototypical example): P lonka sums is indeed to most appropriate algebraic tool to express, algebraically, the notion of contamination. Examples of logics of variables inclusion are also introduced in [7,6].…”