A NOTE ON IMPLICATIONAL INTERMEDIATE CONSEQUENCES
Wolfgang Rautenberg · 2007
complete) if C ′(∅) = C(∅) implies C ′ = C for each (structural) strengthen-ing C ′ ≥ C. C is s.f.a. (strongly finite approximable) if G is determined by a set K of finite matrices; if K itself can chosen to be finite, C is said to be tabular (alias strongly finite). In the sequel the letters C,D range exclusively over →-intermediate consequences, i.e. consistent structural C ≥ CH where CH is the intu-itionistic consequence restricted to the propositional language F (→). CH is simply the smallest structural consequence C in F (→) such that Modus Ponens holds in C and (dt) Q ∈ C(X,P) ⇒ P → Q ∈ C(X) (deduction theorem) where X,Y range over formula sets, P,Q over formulas. In Section 2 we will prove the following Theorem. C is s.s.c. iff C is s.f.a. Moreover, each s.s.c. C satisfies (dt). Let 4 denote the set of all C satisfying (dt) and 4ω = {C ∈ 4|C