Routledge Dictionary of Language and Linguistics.pdf

Dictionary of language and linguistics 544The definition of implication in the truth table is based on the fact that implication islogically equivalent with the expression ¬p q which can be paraphrased as ‘first partfalse 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 ofnegative inference hold true for it (in contrast with presupposition). Materialimplication is the appropriate quantifier for formalizations of conditional existentialpropositions in predicate logic. This truth-functional interpretation of implication ispurely an extensional one, therefore any presupposed semantic relation between the twoparts of the proposition does not come into play in everyday language. The intensionalrelation between the two parts of the proposition that exists in natural language use iscovered below in (d). (b) Logical implication (also entailment): metalinguistic relationbetween two propositions p and q: q logically follows from p (notation: p→q), if everysemantic interpretation of the language that makes p true automatically (i.e. based solelyon the logical form of p and q) makes q true. For example, p =All humans are mortal andSocrates is a human, q=Socrates is mortal, then it holds true that p→q. (c) Strictimplication (also entailment): implicational relation in modal logic: ‘p necessarily impliesq or ‘It necessarily follows from q that p’ With the operator of necessitation □ thisrelation can be expressed as □ (p→ q) ( modal logic). (d) Semantic implication (also(semantic) entailment, conditional): a narrower (intensional) interpretation of implicationin regard to natural languages. In contrast with logical implication, the partialpropositions of semantic implication are in a semantic relation and their validity is basedon appropriate (lexical) meaning postulates. Cf. Austin’s (1962) example: from The catis lying on the mat it follows semantically that The mat is underneath the cat. In contrastwith presupposition, q will remain true if p is negated: from The cat is not on the mat itdoes not follow that The mat is underneath the cat. This relation of implication can bechecked with the but-test: if a speaker maintains that The cat is on the mat, but the mat isnot underneath the cat his/her semantic competence is called into question. The conceptof semantic implication plays a basic role in structural lexical semantics: (unilateral)implication corresponds largely to the semantic relation of hyponymy, bilateralimplication (=equivalence) corresponds largely to synonymy. (e) Contextualimplication: expansion of the concept of implication with pragmatic aspects. Contextualimplications are conversational conditions that must be fulfilled so that an utterance canbe seen as ‘normal’ under the given circumstances of a specific speech situation. Thus byuttering an assertion, one implies ‘contextually’ that this assertion is also really true, andthe speaker must similarly be able to defend him-/herself if the hearer is doubtful. Cf.allegation, implicature, invited inference for other types of implication.ReferencesAustin, J.L. 1962. How to do things with words. Oxford.formal logic