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

Summary: In this study, we investigate the lattice of extensions of four-valued Belnap-Dunn logic, known as super-Belnap logics. By utilizing the antiaxiomatic (or explosive) part operator, we describe the overall structure of this lattice and prove new completeness theorems for super-Belnap logics. We discover the intriguing connection between the antiaxiomatic extensions of Belnap-Dunn logic and graph theory, and show that the lattice of finitary antiaxiomatic extensions is isomorphic to the lattice of upsets in the homomorphism order on finite graphs. Additionally, we find a continuum of finitary and non-finitary super-Belnap logics, and derive some algebraic consequences related to De Morgan algebras.

Summary: This paper demonstrates that Reichenbach's three-valued logic exhibits many interesting features when analyzed with contemporary tools in logic, and even offers new responses to old objections. Additionally, connections between Reichenbach's three-valued logic and some contra-classical logics are established.