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Truth-logics. (English) Zbl 0694.03006
A variety of logical calculi are developed. Their vocabulary is that of traditional propositional logic (PL) enriched with a symbol T for a truth operator, “it is true that”. Formally, the role of T in the calculus is similar to the role of modal operators in a propositional modal logic. The use of the operator allows distinction between ways of negating a proposition, \(\sim Tp\) “it is not true that p”, and \(T\sim p\) “it is true that not p”. The author develops nonclassical truth logics, TL, \(T'L\), and \(T''L\) to admit, respectively, truth-value gaps (propositions neither true nor false), truth-value overlaps (propositions both true and false), and both gaps and overlaps. TL is called paracomplete and \(T'L\) is a paraconsistent logic. TL resembles intuitionist logic in that, if a proposition is true then it is false that it is false, TP\(\to T\sim T\sim p\), but not conversely. In \(T'L\) the reverse implication holds. TLM, \(T'LM\), and \(T''LM\) are corresponding logics extended to handle “mixed” formulas, i.e., formulas composed of T-expressions (of the previous logics) and PL-expressions (of ordinary propositional logic). By way of example, Tp\(\leftrightarrow p\) is a mixed formula, in fact one with a well known role for a general discussion of truth. The use of truth- logics in studying antinomies is discussed. For reasoning with vague concepts the author suggests that a logic like TL which allows truth- value gaps is better suited than classical logic. And for reasoning about processes and the flux of a changing world the use of a paraconsistent type of logic may be more commendable. There are other possibilities still. A good many of them may be systematized within a general theory of what the author calls truth-logic(s).
Reviewer: L.Löfgren

03A05 Philosophical and critical aspects of logic and foundations
03B60 Other nonclassical logic