A new conditional for naive truth theory. (English) Zbl 1273.03073

The paper presents a logical system \(\mathbf{TJK^+}\) which is claimed to be suitable for securing a disquotational reasoning about truth. The intended formalism is constructed in the first-order language of Peano arithmetic enriched with a truth predicate Tr. The system \(\mathbf{TJK^+}\) is defined as an extension of the negation-free fragment of the relevant logic \(\mathbf{TJ}\) by the axiom schema \(\varphi\rightarrow(\psi\rightarrow\varphi)\) known as \(\mathbf{K}\). This system is supplied with a possible-worlds semantics dealing with both binary and ternary accessibility relations. A general form of a fixed-point theorem grasping the conditional of \(\mathbf{TJK^+}\) is proved, thus allowing to accommodate any possible self-referential reasoning with this operator. The key Theorem 5.5 of the paper states the existence of a standard model for \(\mathbf{TJK^+}\) plus intersubstitutivity rule: from \(\varphi\) to infer \(\varphi^\prime\) and vice versa (\(\varphi^\prime\) is any sentence obtained from \(\varphi\) by substituting some occurrences of \(\psi\) for Tr(\(\ulcorner\psi\urcorner\)), where \(\ulcorner\psi\urcorner\) represents the numeral for the Gödel number of \(\psi\)). In the last section the possibility of adding to \(\mathbf{TJK^+}\) some kind of an involutive negation operator is briefly discussed.


03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03A05 Philosophical and critical aspects of logic and foundations
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