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 of the same form, then use a standard rule of detachment."
A substitution A that when applied to produces, and substitution B that when applied to produces, are called the 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 (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. (And 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.)
In some logics (such as standard PC) have a set of defining axioms with the "D-completeness" property. If a set of axioms is D-Complete, then any vaild theorems of the system can be generated by condensed detachment alone. 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 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. This makes condensed detachment useful for any logic system that has modus ponens and substitution, regardless of whether or not it is D-complete.
Read more about this topic: Condensed Detachment
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