Entailment

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In logic, entailment (or logical implication) is a relation between sets of sentences and a sentence. Typically entailment is defined in terms of necessary truth preservation: some set T of sentences entails a sentence A if and only if it is necessary that A be true whenever each member of T is true.

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Introduction - entailment in logic

Entailment relates, by logical implication, two sentences, A and B, such that the truth of B follows from the truth of A.

This linguistic concept can be generalized logically and mathematically. For a set of A's — call this set T — each member of the set T is a proposition such that every valuation will model the proposition B. In other words entailment holds that the class models of T is a subset of the class model of B.

Without using the language of models, entailment states that the material conditional formed from the conjunction of all the elements of T and B (i.e. the corresponding conditional) is valid. That is, it is valid that

where the Ai are the elements of T. (If T has infinite cardinality then, provided the language of T has the compactness property, some finite subset of T implies B.) The statement in terms of the material conditional holds only in logics that have the semantic equivalent of the deduction theorem (and, as noted earlier, if T is infinite, then the compactness property is also required if the language disallows conjunctions over infinite sets of formulas). Thus, the original statement in terms of models is more general. The weaker truth function material implication, denoted by '→', should not be confused with the stronger logical implication.

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