D-complete Axioms for the Classical Equivalential Calculus

Virtually all previously known axiom sets for the classical equivalential calculus, EQ, are D-incomplete: not all theorems derivable by substitution and detachment can be derived using the rule D of condensed detachment alone. The only exception known to the author is Wajsberg’s base {EEpEqrErEqp, EEEpppp}. This axiom set, albeit inorganic since one of its members contains a theorem of EQ as a proper subformula, is here shown to be D-complete. D-complete single axioms for EQ are then constructed, culminating with EEpqEErqEsEsEsEsEpr which has the distinction of being both D-complete and organic. 1. D-completeness and D-incompleteness The well-formed formulas of the classical equivalential calculus are built in the usual way from a binary connective E and denumerably many sentence letters p, q, r, . . . , p1, p2, . . . : each sentence letter is well-formed, and if α and β are well-formed, so is Eαβ. Let EQ be the set of such formulas in which each sentence letter occurring occurs an even number of times. The members of EQ are, as first noted by Leśniewski [7], exactly the formulas that are tautologies of the standard two-valued truth-table for material equivalence. For each set A of formulas, let the A-theorems be the formulas derivable from members of A using the rules of substitution and detachment (that is, modus ponens). A subset A of EQ is a complete axiom set, or base, for EQ, if and only if the set of A-theorems is exactly EQ.