implication [Lat. implicatio ‘the action of weaving in; an intertwined system’]
The term ‘implication’ is used in everyday language as well as in logic and semantics in different ways, though with much overlap. (a) Material implication (also conditional implication, logical implication, subjunction): quantifier in propositional logic that connects two elementary propositions p and q in a new single proposition that is false if and only if the first part of the proposition is true and the second part is false (notation: p→q): If London is on the Thames, then 3×3=10 (=f(alse)); but: If 3×3=10, then London is on the Thames (= t(rue)). The following (two-value) truth table represents a definition of this type of implication:
p
q
p→q
t
t
t
t
f
f
f
t
t
f
f
t
The definition of implication in the truth table is based on the fact that implication is logically equivalent with the expression ¬p q which can be paraphrased as ‘first part false or second part true,’ and which are exactly the truth conditions for implication. Another property of material implication is that both the rule of inference and the rule of negative inference hold true for it (in contrast with presupposition). Material implication is the appropriate quantifier for formalizations of conditional existential propositions in predicate logic. This truth-functional interpretation of implication is purely an extensional one, therefore any presupposed semantic relation between the two parts of the proposition does not come into play in everyday language. The intensional relation between the two parts of the proposition that exists in natural language use is covered below in (d). (b) Logical implication (also entailment): metalinguistic relation between two propositions p and q: q logically follows from p (notation: p→q), if every semantic interpretation of the language that makes p true automatically (i.e. based solely on the logical form of p and q) makes q true. For example, p =All humans are mortal and Socrates is a human, q=Socrates is mortal, then it holds true that p→q. (c) Strict implication (also entailment): implicational relation in modal logic: ‘p necessarily implies q or ‘It necessarily follows from q that p’ With the operator of necessitation □ this relation can be expressed as □ (p→ q) (modal logic).
(d) Semantic implication (also (semantic) entailment, conditional): a narrower (intensional) interpretation of implication in regard to natural languages. In contrast with logical implication, the partial propositions of semantic implication are in a semantic relation and their validity is based on appropriate (lexical) meaning postulates. Cf. Austin’s (1962) example: from The cat is lying on the mat it follows semantically that The mat is underneath the cat. In contrast with presupposition, q will remain true if p is negated: from The cat is not on the mat it does not follow that The mat is underneath the cat. This relation of implication can be checked with the but-test: if a speaker maintains that The cat is on the mat, but the mat is not underneath the cat his/her semantic competence is called into question. The concept of semantic implication plays a basic role in structural lexical semantics: (unilateral) implication corresponds largely to the semantic relation of hyponymy, bilateral implication (=equivalence) corresponds largely to synonymy. (e) Contextual implication: expansion of the concept of implication with pragmatic aspects. Contextual implications are conversational conditions that must be fulfilled so that an utterance can be seen as ‘normal’ under the given circumstances of a specific speech situation. Thus by uttering an assertion, one implies ‘contextually’ that this assertion is also really true, and the speaker must similarly be able to defend him-/herself if the hearer is doubtful. Cf. allegation, implicature, invited inference for other types of implication.
References
Austin, J.L. 1962. How to do things with words. Oxford.
q which can be paraphrased as ‘first part false or second part true,’ and which are exactly the truth conditions for implication. Another property of material implication is that both the 
