On strong standard completeness in some MTL $$_\Delta $$ Δ expansions

Abstract

In this paper, inspired by the previous work of Franco Montagna on infinitary axiomatizations for standard $$\mathsf {BL}$$BL-algebras, we focus on a uniform approach to the following problem: given a left-continuous t-norm $$*$$ź, find an axiomatic system (possibly with infinitary rules) which is strongly complete with respect to the standard algebra [InlineEquation not available: see fulltext.] This system will be an expansion of Monoidal t-norm-based logic. First, we introduce an infinitary axiomatic system $$\mathsf {L}_*^\infty $$Lźź, expanding the language with $$\Delta $$Δ and countably many truth constants, and with only one infinitary inference rule, that is inspired in Takeuti---Titani density rule. Then we show that $$\mathsf {L}_*^\infty $$Lźź is indeed strongly complete with respect to the standard algebra [InlineEquation not available: see fulltext.]. Moreover, the approach is generalized to axiomatize expansions of these logics with additional operators whose intended semantics over [0, 1] satisfy some regularity conditions.