On Jaśkowski's discussive logics
`Discussive logics' are apparently extensions of classical propositional or predicate systems which are `inconsistent' in the sense that \(\alpha\) or \(\neg \alpha\) may be included in a theory, yet are not `trivial' in the sense that the theory coincides with the set of all logical formulae. This paper seeks to show that such logics provide a proper foundation for the axiomatization of physics. The authors claim `it is the purpose of this paper to lay those foundations; we leave their consequences to physics to a future paper'. `Laying the foundations' consists in presenting the well-known modal calculus S5 along with Kripke semantics, axiomatization and some completeness results, then developing the extension S5Q\(^ =\) which is `S5 with quantification and necessary equality'. Next `discussive' analogues of these systems are presented, called \(J\) and \(J^*\) with corresponding axiomatizations, `Kripke type' semantics and model- theoretic results. At the end of the paper a formal theory of `pragmatic truth' is sketched founded on the notion of a `simple pragmatic structure', or sps, \(\langle A_ 1, A_ 2, R_ i, R_ j, P\rangle\) where \(A_ 1\) is a set of `real objects (e.g. vapour trajectories,\dots spectral lines\dots)', \(A_ 2\) is a set of ideal objects `(e.g. quarks or wave functions\dots)', \(R_ i\) are relations over \(A_ 1\) and \(R_ j\) relations over \(A_ 1\cup A_ 2\) and \(P\) are `true decidable propositions' and `general sentences that express laws or theories already accepted as true'. A kind of Kripke semantics founded on these structures is supposed to furnish an appropriate sense of `pragmatic truth' for the axiomatization of physics. After the rather laboured `laying out' of the logic, this comes at breathtaking speed, lacking detail (e.g. of the conditions defining \(P\)) and with no results yet to suggest a fruitful application to quantum mechanics. The paper ends with discussion of Dalla Chiara's similar formalism.
S5 with quantification and necessary equality
foundation for the axiomatization of physics
scientific article; zbMATH DE number 724128
scientific article