## 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.

### MSC:

 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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### References:

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