Łukasiewicz' theory of truth, from the quantum logical point of view (Q2702733)

This paper considers Lukasiewicz' 1920 classic 2-pager ``On three-valued logic'', noting his conclusion that the logic may turn out to have applications to physics once ``the consequences of the indeterministic philosophy, which is the metaphysical substratum of the new logic, [is] compared with empirical data.'' Lukasiewicz himself never directly addressed the issues of quantum indeterminacy alluded to in this passage. NEWLINENEWLINENEWLINESection II ``Determinism, bivalence and quantum theory'', compares future conditional statements such as ``John will be at home tomorrow noon'' with the indeterminacy situations encountered in quantum theory. The authors argue that the owing to the uncertainty relations our information on quantum systems is logically incomplete, and furthermore that past and future are similarly indeterminate in quantum theory. Apparently Lukasiewicz claimed that past happenings can only be asserted as possible, not as true. NEWLINENEWLINENEWLINEIn Section III, ``Lukasiewicz and quantum logic'' , the distinction between Excluded Middle and Semantic Bivalence is identified as ``one of Lukasiewicz's celebrated contributions to the history of logic''. From this ``we obtain the typical quantum logical situation: the truth of a disjunction does not generally imply the truth of at least one term''. The authors then distinguish Lukasiewicz from Quantum Logic. The lattice \(V\) representing Lukasiewicz disjunction is distributive but Excluded Middle may fail. However in the ``von Neumann quantum logic QL'', representing the lattice of projection operators in Hilbert Space, \(V\) is not distributive but Excluded Middle holds. There follows a very brief discussion of ``Lukasiewicz Quantum Logic'' (LQL), apparently a sublogic both of Lukasiewicz logic and of QL -- a fuller treatment by Guintini is referenced. LQL can apparently be regarded both as ``an interesting non distributive weakening'' of Lukasiewicz logic, and a ``fuzzy form'' of Quantum Logic, though the details here are only very briefly sketched. NEWLINENEWLINENEWLINEFinally in Section IV ``Is logic an empirical science?'' a long quotation from Lukasiewicz on the nature of a priori truths, argues against the Vienna Circle that logic is empirical, ``that one and only one logic is valid in the real world''. A quote from sixteen years later however seems to indicate a rather different view, with Lukasiewicz concluding that ``The more useful and richer a logical system is the more valuable it is''.NEWLINENEWLINEFor the entire collection see [Zbl 0948.00030].