Quantum Logic and Quantum Reconstruction

Miklós Rédei

By quoting extensively from unpublished letters written by John von Neumann to Garret Birkhoff during the preparatory phase (in 1935) of their ground-breaking 1936 paper that established quantum logic, the main steps in the thought process leading to the 1936 Birkhoff–von Neumann paper are reconstructed. The reconstruction makes it clear why Birkhoff and von Neumann rejected the notion of quantum logic as the projection lattice of an infinite dimensional complex Hilbert space and why they postulated in their 1936 paper that the quantum propositional system should be isomorphic to an abstract projective geometry. Looking at the paper now I see, that I forgot to say this, which should be said somewhere in the first §: That while common logics did apply to quantum mechanics, if the notion of simultaneous measurability is introduced as an auxiliary notion, we wished to construct a logical system, which applies directly to quantum mechanics – without any extraneous secondary notions like simultaneous measurability. And in order to have such a consequent, one-piece system of logics, we must change the classical class calculus of logics. (J. von Neumann to G. Birkhoff, November 21, 1935)

George Svetlichny

Roberto Giuntini

Abstract The relative Lindenbaum property is considered in orthomodular quantum logic and in partial class ical logic. The properties of these models are connected with the possibility of hidden‐variable reconstruction of particular (1/2‐spin particle e.g.) quantum physical systems.

Mladen Pavičić, Norman D. Megill

Martin Strauss

Elżbieta Drozdowska

This paper reconstructs a forgotten episode in the early history of quantum logic by examining the work of Polish philosopher Zygmunt Zawirski. In the early 1930s, Zawirski developed an original system of many-valued logic explicitly designed to interpret quantum mechanics. Anticipating later views by Reichenbach and Putnam, Zawirski argued that the indeterminacy and probabilistic structure of quantum theory required a departure from classical two-valued logic. Drawing on Łukasiewicz's many-valued logic and Post's formal constructions, he built a system of truth-functional operators capable of representing probabilistic conjunction and disjunction. This article offers a technical and conceptual reconstruction of Zawirski's logic, based on his 1934 monograph and recent interpretations by Garbacz. While the system's practical limitations have been noted, its philosophical ambition and originality remain significant. Zawirski's logic not only constitutes the first formal system of quantum logic but also anticipates contemporary debates on the empirical foundations of logic and the relation between logic and probability. By recovering his overlooked contribution, this paper broadens our understanding of early responses to quantum theory and highlights the role of Polish logicians in shaping 20th-century philosophy of science.