Notes on Strong Completeness in Lukasiewicz, Product and BL Logics and in Their First-Order Extensions

In this paper we investigate the problem of characterizing infinite consequence relation in standard BL-algebras by the adding of new rules. First of all, we note that finitary rules do not help, therefore we need at least one infinitary rule. In fact we show that one infinitary rule is sufficient to obtain strong standard completeness, also in the first-order case. Similar results are obtained for product logic and for Łukasiewicz logic. Finally, we show some applications of our results to probabilistic logic over many-valued events and to first-order many-valued logic. In particular, we show a tight bound to the complexity of BL first-order formulas which are valid in the standard semantics.

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