Condensed detachment (Rule D) is a method of finding the most general possible conclusion given two formal logical statements.
It was developed by the Irish logician Carew Meredith in the 1950s and inspired by the work of Łukasiewicz.
[1] A rule of detachment (often referred to as modus ponens) says: "Given that
Various unifiers may produce expressions with varying numbers of free variables.
If one expression is a substitution instance of another (and not just a variable renaming), then that other is called "more general".
If the most general unifier is used in condensed detachment, then the logical result is the most general conclusion that can be made in the given inference with the given second expression.
Some logics, such as classical propositional calculus, have a set of defining axioms with the "D-completeness" property.
If a set of axioms is D-Complete, then any valid theorem of the system, including all of its substitution instances (up to variable renaming), can be generated by condensed detachment alone.
A. Kalman proved that any conclusion that can be generated by a sequence of uniform substitution (all instances of a variable are replaced with the same content) and modus ponens steps can either be generated by condensed detachment alone, or is a substitution instance of something that can be generated by condensed detachment alone.
[1] This makes condensed detachment useful for any logic system that has modus ponens and substitution, regardless of whether or not it is D-complete.
Since a given major premise and a given minor premise uniquely determine the conclusion (up to variable renaming), Meredith observed that it was only necessary to note which two statements were involved and that the condensed detachment can be used without any other notation required.
This notation, besides being used in some automated theorem provers, sometimes appears in catalogs of proofs.
[3] Condensed detachment's use of unification predates the resolution technique of automated theorem proving which was introduced in 1965.
[4][5] For automated theorem proving condensed detachment has a number of advantages over raw modus ponens and uniform substitution.
With condensed detachment there are only a finite number of possible next steps in a proof.
[clarification needed] The D-notation for complete condensed detachment proofs allows easy description of proofs for cataloging and search.
A typical complete 30 step proof is less than 60 characters long in D-notation (excluding the statement of the axioms.)