Josep Moragrega Font

We establish some relations between the class of truth-equational logics, the class of assertional logics, other classes in the Leibniz hierarchy, and the classes in the Frege hierarchy. We argue that the class of assertional logics belongs properly in the Leibniz hierarchy. We give two new characterizations of truth-equational logics in terms of their full generalized models, and use them to obtain further results on the internal structure of the Frege hierarchy and on the relations between the two hierarchies. Some of these results and several counterexamples contribute to answer a few open problems in abstract algebraic logic, and open a new one.

An infinite sequence $\bgD=\ $ of possibly infinite sets of formulas in $n+1$ variables $\seq x0{n-1},y$ and a possibly infinite system of parameters $\vu$ is a \emph{parameterized graded deduction-detachment} \emph{system} for a deductive system $\bcS$ over a $\bcS$-theory $T$ if, for every $n $, where $\Fi_\bcS\sbA$ is the set of all $\bcS$-filters on $\sbA$.Theorem.Let $\bcS$ be a protoalgebraic deductive system over a countable language type. If $\bcS$ has a Leibniz-generating PGDD system over all Leibniz theories, then $\bcS$ has a fully adequate Gentzen system.Theorem.Let $\bcS$ be a protoalgebraic deductive system. If $\bcS$ has a fully adequate Gentzen system, then $\bcS$ has a Leibniz-generating PGDD system over every Leibniz theory.Corollary.If $\bcS$ is a weakly algebraizable deductive system over a countable language type, then $\bcS$ has a fully adequate Gentzen system iff it has the multiterm deduction-detachment theorem.Corollary.If $\bcS$ is a finitely equivalential deductive system over a countable language type, then $\bcS$ has a fully adequate Gentzen system iff there is a finite Leibniz-generating \textup{GDD} system for $\bcS$ over all Leibniz $\bcS$-filters.