On strong standard completeness of MTLQ∗ expansions

Within the mathematical logic field, much effort has been devoted to prove completeness of different axiomatizations with respect to classes of algebras defined on the real unit interval [0, 1] (see for instance [1] and [2]), but in general, what has been mainly achieved are axiomatizations and results concerning fini-tary completeness, that is, for deductions from a finite number of premises. In this work we are concerned with the problem of strong completeness, i.e., completeness for deductions from an arbitrary number of premises. In particular, we will focus on showing strong completeness for logics of a left-continuous t-norm. These will be extensions of the monoidal t-norm based logic, MTL, the logic of prelinear, bounded, commutative and integral residuated lattices [3], expanded with rational truth-constants and with an arbitrary set of connectives under some constraints. It is known that MTL is strongly complete with respect to the class of all standard algebras based of left-continuous t-norms [2]. Some particular extensions of MTL enjoy more concrete completeness results: BL, Gödel, Product or Lukasiewicz logics are complete wrt single particular standard algebras. However , these completeness results are for finitary deductions, only in a few cases (e.g. in Gödel logic) they hold in general. Regarding the issue of enforcing strong completeness in MTL logics expanded with rational constants, the main references are [4] and [5]. While the former focuses on the strong standard completeness for Product logic extended with rational constants following the usual algebraic approach, the latter is framed in the context of Pavelka-style completeness, a different (infinitary) notion of completeness originally introduced by Pavelka in the context of Lukasiewicz logic [6]. We will not deal here with this kind of completeness, we only notice that it is a weaker notion than that of strong standard completeness. The paper by Cintula [5] explores different notions of rational expansions of MTL, and shows that adding a pair of infinitary deduction rules for each discontinuity point in the connectives' truth-functions on the unit real interval [0, 1] makes these logics Pavelka-style complete. Cintula also makes an observation that will partially orient our work: for a rational standard algebra (i.e., over [0, 1] with rational constants) with a non-continuous operation, there is no finitary axiomatic system that is strongly complete with respect to it. In this abstract we present an alternative way (with respect to the Pavelka-style approach) to enforce strong standard completeness of rational …