Logics of Variable Inclusion

Abstract

Bochvar algebras consist of the quasivariety BCA playing the role of equivalent algebraic semantics for Bochvar (external) logic, a logical formalism introduced by Bochvar [3] in the realm of (weak) Kleene logics. In this paper, we provide an algebraic investigation of the structure of Bochvar algebras. In particular, we prove a representation theorem based on Płonka sums and investigate the lattice of subquasivarieties, showing that Bochvar (external) logic has only one proper extension (apart from classical logic), algebraized by the subquasivariety N BCA of BCA. Finally, we prove that both BCA and N BCA enjoy the Amalgamation Property (AP).

The Structure of Locally Integral Involutive Po-monoids and Semirings

“…All these results point toward the idea that locally integral ipo-semigroups are built up from integral ipo-monoids, or at least their semigroup reducts are, by means of a Płonka sum. This construction was first introduced and studied in [10,11,12]; for more recent expositions see [13] and [14]. Given a compatible family of homomorphisms between algebras of the same type {ϕ ij : A i → A j : i j}, indexed by the order of a join-semilattice (I, ∨), its Płonka sum is the algebra S defined on the disjoint union of their universes S = i∈I A i , so that for every nonconstant n-ary operation symbol σ and elements…”

A Logical Modeling of Severe Ignorance

“…Examples of algebraizable logics include, among many others, classical, intuitionistic logic, all substructural logics and global modal logics. Not all logics though are algebraizable: examples of non-algebraizable logics can be found in the realm of Kleene logics, such as the Logic of Paradox (see [37]), Paraconsistent weak Kleene (see [6]) and Bochvar internal logic (see [7][8][9]). The above definition of algebraizable logic can be drastically simplified: is algebraizable with equivalent algebraic semantics K if and only if it satisfies either ALG1 and ALG4 (or, else ALG2 and ALG3).…”

“…The usefulness of the above result will be explored in a fore-coming paper, focused on a deeper understanding of the properties of Bochvar algebras [10].…”

Paraconsistent Weak Kleene Logic

“…This tendency has only marginally involved weak Kleene logics. These (three-valued) logics, originally introduced by Bochvar [4], and subsequently investigated by Kleene [27], to deal with mathematical paradoxes and formulas possibly referring to non-existing objects and/or incorrectly written computer programs, have attracted much attentions in the recent past from several points of view: semantical [15], algebraic [7], [9], epistemic [35], [5], computer-theoretic [14], [16], topic-theoretic [1], and belief revision [13]. The unique proposal of a modal logic whose propositional basis is a logic in the weak Kleene family is due to a work of K. Segerberg [32], who studied several modal logic based on the external version of Paraconsistent Weak Kleene [25].…”

“…This is part of our motivation to study modal Kleene logics in the external language: despite the present work does not take in consideration the global version of modal logics over a certain relational structure, but only their local versions, we still expect to provide a modal basis on which algebraizability can be carried over from the propositional level (if one considers the global consequence relations instead of the local ones). Moreover, as weak Kleene logics turned out to be a particular case of a more general phenomenon, that of the "logics of variable inclusion" [9] -then "internal" modal Kleene logics shall turn out to be an example of logics of variable inclusion. As described in details in [9], these logics could be approached, syntactically, by imposing certain constraints on the inclusion of variables to standard modal logics and, semantically, by considering the construction of the Płonka sum of modal algebras or of its subvarieties "corresponding" to the extension of the modal logic K.…”