AN OPEN PROBLEM IN TARSKI'S CALCULUS OF DEDUCTIVE SYSTEMS
D. W. Miller · 2005
The notation and terminology of this paper follow [2], and are dual to those of [6] and [7]. If L is a language in the narrow sense, Cn may be any consequence operation on sets of sentences of L that includes classical sentential logic. Henceforth when we talk of the language L we intend to include reference to some fixed, though unspecified, operation Cn. X is a deductive system ifX = Cn(X). Sentences x, z that are logically equivalent with respect to Cn – that is x ∈ Cn({z}) and z ∈ Cn({x}) – are identified. If X and Z are systems we often write X ` x instead of x ∈ X and Z ` X instead of X ⊆ Z. If X = Cn({x}) for some sentence x, X is (finitely) axiomatizable. The set theoretical intersection of X and Z has the logical force of disjunction, and is written X∨Z; Cn(X∪Z), the smallest system to include both X and Z, is written XZ. If K is a family of systems,