Canonical Propositional Gentzen-Type Systems

Cut-free sequent calculi for C-systems with generalized finite-valued semantics

“…(k) is anyway a theorem of almost every important C-system ever studied. This is due to the fact that it is derivable in B from each of the three most important axioms concerning • which have been studied in the literature: the axiom denoted below by (i), as well as axioms (l) ¬(ϕ ∧ ¬ϕ) ⊃ •ϕ, and (d) ¬(¬ϕ ∧ ϕ) ⊃ •ϕ, which are not handled in this paper 10 . Those dependencies are easily established (see Example 42 for the case of (l)).…”

“…Our main semantic tool in what follows will be the following generalization of the concept of a many-valued matrix introduced in [10,11] (for a comprehensive survey on non-deterministic matrices, see also [13]):…”

Strict Canonical Constructive Systems

“…Proposition 2 ensures that ⊢ G is a structural and finitary tcr. That the coherence of G implies the consistency of the multiple conclusion consequence relation which is naturally induced by G was shown in [5,6]. That consequence relation extends ⊢ G , and therefore also the latter is consistent.…”

“…, ψ n ) ∈ T and there does not exist T ′ ∈ W , T ⊆ T ′ , and an elimination rule for ⋄ whose set of premises is satisfied in T ′ by a substitution σ such that σ(p i ) = ψ i (1 ≤ i ≤ n). 5 First we prove that W is a generalized L-frame:…”

“…For the multiple-conclusion framework the above-mentioned facts about the classical connectives were considerably generalized in [5,6] by defining "multipleconclusion canonical propositional Gentzen-type rules and systems" in precise terms. A constructive necessary and sufficient coherence criterion for the nontriviality of such systems was then provided, and it was shown that a system of this kind admits cut-elimination iff it is coherent.…”