Kriske, Tupman and Quantum Logic: The Quantum Logician's Conundrum

Sonja Smets

Leopoldo Rom�n, B. Rumbos

Enrico G. Beltrametti, Gianni Cassinelli, Peter Carruthers

Sentences Associated with a Quantum System and Their Truth Values In a sense this chapter is marginal to the content of this volume; the reader can skip it without prejudice to the comprehension of chapters to follow. Quantum logic is a discipline that branched off from the 1936 paper of Birkhoff and von Neumann and has the orthomodular-poset structures encountered in previous chapters as basic mathematical carriers: that is why we think it worthwhile to devote some pages to the subject. But the main interests quantum logic calls into play are in the territory of logicians and philosophers: that is why we (who are neither logicians nor philosophers) shall confine ourselves to a very simplified introduction to the subject. Roughly, the starting question is whether the propositions of a quantum system can be associated with, or can be interpreted as, sentences of a language (or propositional calculus) and which rules this language inherits from the ordered structure of propositions. In raising this question one has in mind the fact that when the physical system is classical its propositions form a Boolean algebra, and Boolean algebras are the algebraic models of the calculus of classical logic. Thus the question above can also be phrased as follows: when a Boolean algebra is relaxed into an orthomodular nondistributive lattice, which logic is it the model of? “Quantum logic” is the name that designates the answer, but there are several views about the content of this name. Here we sketch one approach to a quantum logic.

Boris Zelikovich Moroz

Jennifer Chubb, Eskandarian, Ali, Harizanov, Valentina S

Arising from a special session held at the 2010 North American Annual Meeting of the Association for Symbolic Logic, this volume is an international cross-disciplinary collaboration with contributions from leading experts exploring connections across their respective fields. Themes range from philosophical examination of the foundations of physics and quantum logic, to exploitations of the methods and structures of operator theory, category theory, and knot theory in an effort to gain insight into the fundamental questions in quantum theory and logic. The book will appeal to researchers and students working in related fields, including logicians, mathematicians, computer scientists, and physicists. A brief introduction provides essential background on quantum mechanics and category theory, which, together with a thematic selection of articles, may also serve as the basic material for a graduate course or seminar.

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.