The external version of a subclassical logic
Massimiliano Carrara, Michele Pra Baldi
TL;DR
This work develops the external version $L^{e}$ of a three-valued subclassical logic $L$ by adjoining external unary operators, notably $\\Delta_{1}$, and analyzes the semantic conditions under which an $L$-model can be expanded to an $L^{e}$-model. A central result shows that such an expansion is possible precisely when the underlying algebra $\\mathbf{B}$ admits a complete Boolean conucleus, with $\\Delta_{1}$ computable as the join of Boolean elements of a power extension. The paper proves that $L^{e}$ is algebraizable with equivalent semantics $\\mathop{\\mathrm{Alg}}L^{e}=\\mathbb{I}\\mathbb{S}\\mathbb{P}(\\mathbf{A}^{e})$, and that external operators enable recapture of classical features like the Deduction Theorem, often via a natural substitution. It further introduces $\\Delta_{0}$ and $\\Delta_{1/2}$ to define an implication and shows that $L^{e}$ can recover many classical inferences, with the external versions $L^{e}_{1}$ and $L^{e}_{1/2}$ being deductively equivalent, yielding a robust framework for unifying external versions across subclasses of three-valued logics.
Abstract
A three-valued logic L is subclassical when it is defined by a single matrix having the classical two-element matrix as a subreduct. In this case, the language of L can be expanded with special unary connectives, called external operators. The resulting logic L^e is the external version of L, a notion originally introduced by D. Bochvar in 1938 with respect to his weak Kleene logic. In this paper we study the semantic properties of the external version of a three-valued subclassical logic L. We determine sufficient and necessary conditions to turn a model of L into a model of L^e . Moreover, we establish some distinctive semantic properties of L^e.