Product Logic Π is an axiomatic extension of Hájek's Basic Fuzzy Logic BL coping with the 1-tautologies when the strong conjunction & and implication → 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 for each r in a countable Π-subalgebra of [0, 1]) and by adding the corresponding book-keeping axioms for the truthconstants. We first show that the corresponding logics Π() are algebraizable, and hence complete with respect to the variety of Π()-algebras. The main result of the paper is the canonical standard completeness of these logics, that is, theorems of Π() 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 Π-formulas of the kind → φ, where is a truth-constant and φ a formula not containing truth-constants. Finally we consider the logics Π(Δ)(), the expansion of Π() with the well-known Baaz's projection connective Δ, and we show canonical finite strong standard completeness for them.

The authors thanks the anonymous reviewers for their helpful comments and suggestions. They are also indebted to Rostislav Horcˇík for his helpful comments and for pointing out the reference [14]. P. Savicky´ acknowledges the support from the Czech Ministry of Education, Youth and Sports by the project 1M0021620808, Institute for Theoretical Computer Science - ITI. R. Cignoli, F. Esteva and L. Godo also acknowledge a bilateral Argentinian-Spanish project CONICET-CSIC, and F. Esteva and L. Godo a bilateral Czech-Spanish project CSAV-CSIC. Finally, F. Esteva, L. Godo and C. Noguera also acknowledge partial support from the Spanish project TIN2004-07933-C0301

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