Finite limitations on Dummet's LC.
Ivo Thomas · Notre Dame Journal of Formal Logic · 1962
The propositional system LC of [l] can be based on axioms for 3 (implication), Λ (conjunction), a constant f, and definitions for v (alternation) and Ί (negation), as hereunder.In primitive notation, elementary variables and f are wffs, and if α, β are wffs so are ( a D β), (a A β).To restore primitive notation in the sequel, replace dots by left parentheses with right terminal mates; in a sequence of wffs separated only by implications, restore parentheses by left association; enclose the whole in parentheses.If S is a system, S c is its implicational fragment, containing only variables and implications.If a is provable (not provable) in S, we write | -Ot (-| a); if 0! is uniformly valued 0 (is not uniformly valued 0) by the matrix ϊί, we write |-0? (-| α).As a basis for LC we take, with detachment and substitution, the axioms and definitions: