On product logic with truth-constants.

*(English)*Zbl 1102.03030Summary: Product logic \(\Pi\) is an axiomatic extension of Hajek’s Basic Fuzzy Logic BL coping with the 1-tautologies when the strong conjunction & and implication \(\to\) are interpreted by the product of reals in \([0,1]\) and its residuum respectively. In this paper we investigate expansions of product logic by adding into the language a countable set of truth-constants (one truth-constant \(\overline r\) for each \(r\) in a countable \(\Pi\)-subalgebra \({\mathcal C}\) of \([0,1])\) and by adding the corresponding book-keeping axioms for the truth-constants. We first show that the corresponding logics \(\Pi({\mathcal C})\) are algebraizable, and hence complete with respect to the variety of \(\Pi({\mathcal C})\)-algebras. The main result of the paper is the canonical standard completeness of these logics, that is, theorems of \(\Pi({\mathcal C})\) are exactly the 1-tautologies of the algebra defined over the real unit interval where the truth-constants are interpreted as their own values. It is also shown that they do not enjoy the canonical strong standard completeness, but they enjoy it for finite theories when restricted to evaluated \(\Pi\)-formulas of the kind \(\overline r\to\varphi\), where \(\overline r\) is a truth-constant and \(\varphi\) a formula not containing truth-constants. Finally we consider the logics \(\Pi_\Delta ({\mathcal C})\), the expansion of \(\Pi({\mathcal C})\) with Baaz’s well-known projection connective \(\Delta\), and we show canonical finite strong standard completeness for them.