Fregean logics with the multiterm deduction theorem and their algebraization
Studia Logica: An International Journal for Symbolic Logic
A deductive system S (in the sense of Tarski) is Fregean if the relation of interderivability, relative to any given theory T , i.e., the binary relation between formulas f h ; i : T; 'S and T; 'S g; is a congruence relation on the formula algebra. The multiterm deduction-detachment theorem is a natural generalization of the deductive theorem of the classical and intuitionistic propositional calculi (IPC) in which a nite system of possibly compound formulas collectively plays the role of the
... lication connective of IPC. We investigate the deductive structure of Fregean deductive systems with the multiterm deduction-detachment theorem within the framework of abstract algebraic logic. It is shown that each deductive system of this kind has a deductive structure very close to that of the implicational fragment of IPC. Moreover, it is algebraizable and the algebraic structure of its equivalent quasivariety is very close to that of the variety of Hilbert algebras. The equivalent quasivariety is however not in general a variety. This gives an example of a relatively point-regular, congruence-orderable, and congruence-distributive quasivariety that fails to be a variety, and provides what apparently is the rst evidence of a signi cant di erence between the multiterm deduction-detachment theorem and the more familiar form of the theorem where there is a single implication connective.