Condensed detachment is a method of finding the most general possible conclusion given two formal logical statements. It was developed by the Irish logicianCarew Meredith in the 1950s and inspired by the work of Łukasiewicz.
Informal description
A rule of detachment says: "Given that implies, and given, infer." The condensed detachment goes a step further and says: "Given that implies, and given an, use a unifier of and to make and the same, then use a standard rule of detachment." A substitutionA that when applied to produces, and substitution B that when applied to produces, are called unifiers of and. Various unifiers may produce expressions with varying numbers of free variables. Some possible unifying expressions are substitution instances of others. If one expression is a substitution instance of another, 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. Since any weaker inference you can get is a substitution instance of the most general one, nothing less than the most general unifier is ever used in practice. Some logics, such as classicalpropositional 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, can be generated by condensed detachment alone. For example, if is a theorem of a D-complete system, condensed detachment can prove not only that theorem but also its substitution instance by using a longer proof. Note that "D-completeness" is a property of an axiomatic basis for a system, not an intrinsic property of a logic system itself. J. A. Kalman proved that any conclusion that can be generated by a sequence of uniform substitution and modus ponenssteps can either be generated by condensed detachment alone, or is a substitution instance of something that can be generated by condensed detachment alone. This makes condensed detachment useful for any logic system that has modus ponens and substitution, regardless of whether or not it is D-complete.
D-notation
Since a given major premise and a given minor premise uniquely determine the conclusion, 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 led to the "D-notation" for proofs. This notation uses the "D" operator to mean condensed detachment, and takes 2 arguments, in a standard prefix notation string. For example, if you have four axioms a typical proof in D-notation might look like: DD12D34 which shows a condensed detachment step using the result of two prior condensed detachment steps, the first of which used axioms 1 and 2, and the second of which used axioms 3 and 4. This notation, besides being used in some automated theorem provers, sometimes appears in catalogs of proofs. Condensed detachment's use of unification predates the resolution techniques of automated theorem proving.
Advantages
For automated theorem proving condensed detachment has a number of advantages over raw modus ponens and uniform substitution. At a Modus Ponens and substitution proof you have an infinite number of choices for what you can substitute for variables. This means that you have an infinite number of possible next steps. With condensed detachment there are only a finite number of possible next steps in a proof. 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
