Algebraic semantics for propositional superposition logic

Abstract

We provide a new semantics and a slightly different formalisation for the propositional logic with superposition (PLS) introduced and studied in Tzouvaras [(2018). Propositional superposition logic. Logic Journal of the IGPL, 26(1), 149–190]. PLS results from Propositional Logic (PL) by adding a new binary connective construed as the ‘superposition operation’ and a few axioms about it. The original semantics used in the above paper was the so-called sentence choice semantics (SCS), based on choice functions for all pairs of classical formulas of PL. In contrast, the algebraic or Boolean-value choice semantics (BCS) developed in this paper is based on choice functions for pairs of elements of a Boolean algebra in which the classical sentences take truth values. The Boolean-value choice functions can be subject to similar constraints as those imposed on sentence choice functions. The new axiomatisation is based on the same set of axioms as the previous one but uses a new inference rule, called Rule of Analogy (RA), in place of the rule Salva Veritate (SV) of the previous systems. The Deduction Theorem fails in the systems containing the new rule. As a consequence the completeness theorems for them hold conditionally again, namely the systems are complete with respect to BCS if and only if every consistent set of sentences is extended to a consistent and complete set. Finally connections are established between tautologies of the semantics SCS and those of BCS.