Fully Adequate Gentzen Systems and the Deduction Theorem

An infinite sequence ∆ = ∆n(x0,. .. , xn−1, y, ¯ u) : n < ω of possibly infinite sets of formulas in n + 1 variables x0,. .. , xn−1, y and a possibly infinite system of parameters ¯ u is a parameterized graded deduction-detachment (PGDD) system for a de-ductive system S over a S-theory T if, for every n < ω and for all ϕ0,. ϑ) for every possible system of formulas ¯ ϑ. A S-theory is Leibniz if it is included in every S-theory with the same Leibniz congruence. A PGDD system ∆ is Leibniz generating if the union of the ∆n(ϕ0,. .. , ϕn−1, ψ, ¯ ϑ) as ¯ ϑ ranges over all systems of formulas generates a Leibniz theory. A Gentzen system G is fully adequate for a deductive system S if (roughly speaking) every reduced generalized matrix model of G is of the form A, Fi S A, where Fi S A is the set of all S-filters on A. Theorem. Let S be a protoalgebraic deductive system over a countable language type. If S has a Leibniz-generating PGDD system over all Leibniz theories, then S has a fully adequate Gentzen system. Theorem. Let S be a protoalgebraic deductive system. If S has a fully adequate Gentzen system, then S has a Leibniz-generating PGDD system over every Leibniz theory. Corollary. If S is a weakly algebraizable deductive system over a countable language type, then S has a fully adequate Gentzen system iff it has the multiterm deduction-detachment theorem. Corollary. If S is a finitely equivalential deductive system over a countable language type, then S has a fully adequate Gentzen system iff there is a finite Leibniz-generating GDD system for S over all Leibniz S-filters.