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.

This article builds on a recent paper coauthored by the present author, H. Hosni and F. Montagna. It is meant to contribute to the logical foundations of probability theory on many-valued events and, specifically, to a deeper understanding of the notion of strict coherence. In particular, we will make use of geometrical, measure-theoretical and logical methods to provide three characterizations of strict coherence on formulas of infinite-valued Łukasiewicz logic.

We investigate the property of strict coherence in the setting of many-valued logics. Our main results read as follows: (i) a map from an MV-algebra to [0,1] is strictly coherent if and only if it satisfies Carnap’s regularity condition, and (ii) a [0,1]-valued book on a finite set of many-valued events is strictly coherent if and only if it extends to a faithful state of an MV-algebra that contains them. Remarkably this latter result allows us to relax the rather demanding conditions for the Shimony-Kemeny characterisation of strict coherence put forward in the mid 1950s in this Journal.

In this paper we try to apply universal algebraic geometry to MV algebras, that is, we study “MV algebraic sets” given by zeros of MV polynomials, and their “coordinate MV algebras”. We also relate algebraic and geometric objects with theories and models taken in Łukasiewicz many valued logic with constants. In particular we focus on the structure of MV polynomials and MV polynomial functions on a given MV algebra.