Josep M. Font1
This paper studies, with techniques of Abstract Algebraic Logic, the effects of
putting a bound on the cardinality of the set of side formulas in the Deduction
Theorem, viewed as a Gentzen-style rule, and of adding additional assumptions
inside the formulas present in Modus Ponens, viewed as a Hilbert-style rule. As
a result, a denumerable collection of new Gentzen systems and two new
sentential logics have been identified. These logics are weaker than the
positive implicative logic. We have determined their algebraic models and the
relationships between them, and have classified them according to several
standard criteria of Abstract Algebraic Logic. One of the logics is
protoalgebraic but neither equivalential nor weakly algebraizable, a rare
situation where very few natural examples were hitherto known. In passing we
have found new, alternative presentations of positive implicative logic, both
in Hilbert style and in Gentzen style, and have characterized it in terms of
the restricted Deduction Theorem: it is the weakest logic satisfying Modus
Ponens and the Deduction Theorem restricted to at most 2 side formulas. The
algebraic part of the work has lead to the class of quasi-Hilbert algebras, a
quasi-variety of implicative algebras introduced by Pla and Verdú in 1980,
which is larger than the variety of Hilbert algebras. Its algebraic properties
reflect those of the corresponding logics and Gentzen systems.