Abstract

BB 0 IW logic (or T! ) is known to be D-complete. This paper shows that there are infinitely many weaker D-complete logics and it also examines how certain D-incomplete logics can be made complete by altering their axioms using simple substitutions. Keywords: condensed detachment 1 Introduction The condensed detachment rule, first proposed by C. A. Meredith in Lemmon et al [5], is a form of modus ponens preceded by `just enough' substitution to make the modus ponens possible. The substitution mechanism, for implicational formulas, was a precursor to Robinson's unification algorithm [8]. Roughly, a system of implicational logic is D-complete if the system with the same axioms, but with condensed detachment (D) instead of modus ponens and substitution, has the same theorems. To show that a logic is D-complete it is sufficient to show that all the substitution instances of its axioms are deducible in the corresponding condensed logic (i.e. the logic with rule D only). It is well know...

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