Fully Adequate Gentzen Systems and the Deduction Theorem

An infinite sequence ∆ = 〈∆n(x0, . . . , xn−1, y, ū) : n < ω〉 of possibly infinite sets of formulas in n + 1 variables x0, . . . , xn−1, y and a possibly infinite system of parameters ū is a parameterized graded deduction-detachment (PGDD) system for a deductive system S over a S-theory T if, for every n < ω and for all φ0, . . . , φn−1, ψ ∈ FmΛ, T, φ0, . . . , φn−1 `S ψ iff T `S ∆n(φ0, . . . , φn−1, ψ, θ̄) for every possible system of formulas θ̄. 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, θ̄) as θ̄ 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,FiS A〉, where FiS 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-