The ‘Logical’ Approach to Axiomatic Quantum Theory

Quantum mechanics, ontology, and non-reflexive logics

“…Yuri Manin has recalled that quantum mechanics has no its own language, making use of a fragment of the language of standard functional analysis (the Hilbert space formalism) [Man10, p.80]. But, inspired by Dalla Chiara and Toraldo di Francia [DalTor93] and by G. Cattaneo [Catt93], we can suppose a suitable language incorporating the standard logical vocabulary, plus the following nonlogical symbols: (i) A collection of monadic predicates P i (1 = 1, . .…”

“…(i) The parameters a 1 , a 2 etc. are interpreted either as elements of S or as subsets of S. Let me observe that in assuming the structure above, we are introducing explicitly the quantum systems in the semantic considerations, something that is omitted in the usual approaches (see [DalTor93], [Catt93], [vanF75], [Ish95,pp.203ff]).…”