In this paper we study the definition and axiomatisation of different (finitary) threshold preserving companions of several extensions of the Monoidal t-norm based fuzzy logic MTL. More in detail, we first focus on the non-falsity preserving logics, where a conclusion follows from a set of premises if, whenever the premises are non-false (i.e., have a truth degree greater than 0), the conclusion is also non-false. We then introduce a new type of companions, namely logics that preserve the notion of acceptability , where a formula is called acceptable whenever it is deemed to be more true than false, or in other words it is more true than its negation. Finally, we also consider a more general stance and consider logics that preserve some intermediate truth-value 0 < a < 1, so that a can be understood as a (strict or non-strict) threshold above which a formula is considered as valid. All these types of threshold-preserving companions of a given MTL logic can be seen as particular cases of matrix logics defined by lattice filters.

Abstract An algebraically expandable (AE) class is a class of algebraic structures axiomatizable by sentences of the form $\forall \exists ! \mathop{\boldsymbol {\bigwedge }}\limits p = q$ . For a logic L algebraized by a quasivariety $\mathcal {Q}$ we show that the AE-subclasses of $\mathcal {Q}$ correspond to certain natural expansions of L, which we call algebraic expansions. These turn out to be a special case of the expansions by implicit connectives studied by X. Caicedo. We proceed to characterize all the AE-subclasses of abelian $\ell $ -groups and perfect MV-algebras, thus fully describing the algebraic expansions of their associated logics.

In this paper we consider the logics |$\mathsf{L}_n^i$| obtained from the |$(n+1)$|-valued Łukasiewicz logics Ł|$_{n+1}$| by taking the order filter generated by |$i/n$| as the set of designated elements. In particular, the conditions of maximality and strong maximality among them are analyzed. We present a very general theorem that provides sufficient conditions for maximality between logics. As a consequence of this theorem, it is shown that |$\mathsf{L}_n^i$| is maximal w.r.t. CPL whenever |$n$| is prime. Concerning strong maximality (i.e. maximality w.r.t. rules instead of only axioms), we provide algebraic arguments in order to show that the logics |$\mathsf{L}_n^i$| are not strongly maximal w.r.t. CPL, even for |$n$| prime. Indeed, in such case, we show that there is just one extension between |$\mathsf{L}_n^i$| and CPL obtained by adding to |$\mathsf{L}_n^i$| a kind of graded explosion rule. Finally, using these results, we show that the logics |$\mathsf{L}_n^i$| with |$n$| prime and |$i/n < 1/2$| are ideal paraconsistent logics.