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On Jaśkowski’s discussive logics. (English) Zbl 0818.03012
‘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,…spectral lines…)’, \(A_ 2\) is a set of ideal objects ‘(e.g. quarks or wave functions…)’, \(R_ i\) 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.

MSC:
03B53 Paraconsistent logics
03B45 Modal logic (including the logic of norms)
03B80 Other applications of logic
03B30 Foundations of classical theories (including reverse mathematics)
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