A Sound and Complete Tableaux Calculus for Reichenbach's Quantum Mechanics Logic

In 1944 Hans Reichenbach developed a three-valued propositional logic (RQML) in order to account for certain causal anomalies in quantum mechanics. In this logic, the truth-value indeterminate is assigned to those statements describing physical phenomena that cannot be understood in causal terms. However, Reichenbach did not develop a deductive calculus for this logic. The aim of this paper is to develop such a calculus by means of First Degree Entailment logic (FDE) and to prove it sound and complete with respect to RQML semantics. In Section 1 we explain the main physical and philosophical motivations of RQML. Next, in Sections 2 and 3, respectively, we present RQML and FDE syntax and semantics and explain the relation between both logics. Section 4 introduces Q calculus, an FDE-based tableaux calculus for RQML. In Section 5 we prove that Q calculus is sound and complete with respect to RQML three-valued semantics. Finally, in Section 6 we consider some of themain advantages of Q calculus and we apply it to Reichenbach's analysis of causal anomalies.

Automated summary

This paper introduces Hans Reichenbach's three-valued propositional logic RQML developed in 1944 and develops a corresponding calculus Q calculus using First Degree Entailment logic (FDE). By proving the soundness and completeness of Q calculus with respect to RQML semantics, we are able to apply it to analyze causal anomalies.