Extensions of paraconsistent weak Kleene logic

Abstract

Paraconsistent weak Kleene ($\textrm{PWK}$) logic is the $3$-valued logic based on the weak Kleene matrices and with two designated values. In this paper, we investigate the poset of prevarieties of generalized involutive bisemilattices, focussing in particular on the order ideal generated by Α$\textrm{lg} (\textrm{PWK})$. Applying to this poset a general result by Alexej Pynko, we prove that, exactly like Priest’s logic of paradox, $\textrm{PWK}$ has only one proper nontrivial extension apart from classical logic: $\textrm{PWK}_{\textrm{E}}\textrm{,}$ PWK logic plus explosion. This $6$-valued logic, unlike $\textrm{PWK} $, fails to be paraconsistent. We describe its consequence relation via a variable inclusion criterion and identify its Suszko-reduced models.

A Simple Logical Matrix and Sequent Calculus for Parry’s Logic of Analytic Implication

“…A structure of this sort is an algebra counting with two partial orders (in our case, where x, y ∈ {t, o, e, f }, the orders ≤ ∧ and ≤ ∨ respectively defined by letting x ≤ ∧ y if and only if x ∧ y = x, and x ≤ ∨ y if and only if x ∨ y = y), with an involution (in our case, the operation ¬), with no constants for the infimum and the supremum elements. Interestingly enough, these authors show that every generalized involutive bisemilattice is decomposable as a P lonka sum over a semilattice direct system of Boolean algebras [32,Theorem 4]. Whence, our comments below oriented at representing FP in this way provide an exemplification of their general result.…”

“…Before moving on, it may as well be noted that FP is a special kind of algebra. It is what F. Paoli and M. Pra Baldi call in [32] a generalized involutive bisemilattice. A structure of this sort is an algebra counting with two partial orders (in our case, where x, y ∈ {t, o, e, f }, the orders ≤ ∧ and ≤ ∨ respectively defined by letting x ≤ ∧ y if and only if x ∧ y = x, and x ≤ ∨ y if and only if x ∨ y = y), with an involution (in our case, the operation ¬), with no constants for the infimum and the supremum elements.…”

“…for every i, j ∈ I such that i ≤ I j, f ij is an homomorphism from A i to A j , and moreover f ii is the identity map for every i ∈ I, and whenever 4,5,6,32]). The P lonka sum Pl(A) of a semilattice direct system of algebras A = {A i } i∈I , I, {f ij | i ≤ I j} is the algebra of language L whose universe is i∈I A i , and for which every basic n-ary operation ¶ of A (with n ≥ 1) is defined as follows, where a 1 , .…”

“…[4,5,6,32]). A semilattice direct system of algebras of language L is a triple A = {A i } i∈I , I, {f ij | i ≤ I j} , where:(i) I = I, ≤ I is a join-semilattice, (ii) {A i } i∈I is a family of similar algebras of language L with parwise disjoint universes,…”

Immune Logics

“…For example, the algebraic structure underlying the two-valued semantics for Classical Logic is a Boolean algebra, as we said before-that is to say, a complemented bounded distributive lattice. In this respect, it is interesting to highlight that the weak Kleene algebra is a (generalized) involutive bisemilattice-see [24] and references therein.…”