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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

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