Peter Mittelstaedt, The Modal Logic Of Quantum Logic

Peter Mittelstaedt, The Modal Logic Of Quantum Logic

Author: Peter Mittelstaedt

Year: 1910

Peter Mittelstaedt develops a modal extension of quantum logic that captures the notions of “necessity” and “possibility” for quantum‐mechanical propositions. He uses a dialogical proof framework to define both the object language (quantum propositions) and a meta‐language (propositions about proofs). Formally true meta‐propositions coincide with the effective (intuitionistic) logic, extendable to classical logic. Quantum logical modalities arise when one views necessity and possibility as statements about the material truth of quantum propositions. The resulting modal calculi M(Q_eff) and M(Q) differ fundamentally from standard modal logics (e.g., S4). 1. The Language of Quantum Physics 1.1 Object Language Elementary propositions A,B,… about a quantum system S are assumed value‐definite. Logical connectives (∧, ∨, →, ¬) are defined via a material dialog game D_m involving a proponent (P) and opponent (O), where “attacks” and “defences” correspond to proof obligations. Availability (commensurability) k(A,B) and its negation track whether proving B disturbs A. Propositions defended successfully in every D_m‐dialog (regardless of system state) are formally true and form the effective