Logical consequence

Logical consequence (also entailment) is one of the most fundamental concepts in logic. It is the relationship between statements that holds true when one logically "follows from" one or more others. A valid logical argument is one in which the conclusions follow from its premises, and its conclusions are consequences of its premises. The philosophical analysis of logical consequence involves asking, 'in what sense does a conclusion follow from its premises?' and 'what does it mean for a conclusion to be a consequence of premises?' All of philosophical logic can be thought of as providing accounts of the nature of logical consequence, as well as logical truth.

Logical consequence is taken to be both necessary and formal with examples explicated using models and proofs. A sentence is said to be a logical consequence of a set of sentences, for a given language, if and only if, using logic alone (i.e. without regard to any interpretations of the sentences) the sentence must be true if every sentence in the set were to be true.

Entailment (pragmatics)

In pragmatics (linguistics), entailment is the relationship between two sentences where the truth of one (A) requires the truth of the other (B).

For example, the sentence (A) The president was assassinated. entails (B) The president is dead. Notice also that if (B) is false, then (A) must necessarily be false. To show entailment, we must show that (A) being true forces (B) to be true, or, equivalently, that (B) being false forces (A) to be false.

Entailment differs from implicature (in their definitions for pragmatics), where the truth of one (A) suggests the truth of the other (B), but does not require it. For example, the sentence (A) Mary had a baby and (B) got married implicates that (A) she had a baby before (B) the wedding, but this is cancellable by adding – not necessarily in that order. Entailments are not cancellable.

Entailment also differs from presupposition in that in presupposition, the truth of what one is presupposing is taken for granted. A simple test to differentiate presupposition from entailment is negation. For example, both The king of France is ill and The king of France is not ill presuppose that there is a king of France. However The president was not assassinated no longer entails The president is dead (nor its opposite, as the president could have died in another way). In this case, presupposition remains under negation, but entailment does not.