Roughly, a belief about what a speaker meant, but which does not necessarily follow from what they said. That is, it is implicit, not explicit, in their words. Here are some examples:
A: It's rather cold in here.
(Imp.: I want you to shut the window.)

A: Did you get that job you were after?
B: The pay was very low.
(Imp.: I didn't get the job.)

A: Is everything all right?
B: Couldn't be better!
(Imp.: I'm about to cry and I want a cuddle.)

A: Susan's an angel.
(Imp.: Susan is very kind and good-tempered.)

A: I've eaten some of your chocolates.
(Imp.: I haven't eaten all of them.)

Not only is the meaning not explicit, but it does not logically follow. The hearer draws an inference about the speaker's intention, but this is not guaranteed correct. Nevertheless, most of the time we correctly derive most of the speaker's intentions from what they say. Pragmatics is the study of non-encoded meanings like this.

Implicatures were introduced by H.P. Grice in the 1960s. Instead of the idea that all language use was a code merely requiring translation, he proposed that much of what goes on is inference by the hearer of what the speakers intends. There is the sentence meaning, which is coded and fixed, then there is the additional inference needed to decide the speaker meaning: what the speaker meant on this occasion by utterance of the sentence.

So how do we get these non-demonstrative implications out of insufficient words? Grice's explanation for how implicature works is that there is an overarching Cooperative Principle: people in a speech situation are cooperating to get their meaning across. The specific ways of doing this are four subordinate principle now called Grice's Maxims, about being truthful, brief, relevant, and so on. When we violate one of the maxims it can be presumed we are still being cooperative, so the hearer can work out what we meant. Sperber and Wilson's Relevance Theory proposes an alternative that does away with the assumption that people are trying to cooperate, and that there are specific maxims. They have a single cognitive, computational explanation in terms of how much effort it takes to process information, compared to how readily accessible the information is: the combination they call relevance. So we no longer have to assume the speaker's benevolence, nor decide which maxim is being violated how.

Implicatures are not directly said, and they do not logically follow from what is said. This opens up the possibility of a third kind of meaning, ones that are not said but which do explicitly follow. Sperber and Wilson call these explicatures, and claim they are part of 'what is said'. Kent Bach is a neo-Gricean semanticist who prefers to keep 'what is said' rather narrow, so called this sort of thing implicitures (as opposed to implicatures). A lot of the discussion in this area now is about distinguishing explicit vs implicit, what is said vs what is communicated, sentence meaning vs utterance meaning or speaker meaning, and semantics vs pragmatics: variously overlapping concepts.

Semantics is the study of what is said; and pragmatics is the rest of what is meant. But what is explicitly said, or straight-out said rather than merely implied? Everyone agrees some inference must take place in all cases. The sentence 'She is tired' doesn't say who 'she' is. But it probably does mean that some one, explicit person is meant. In contrast, an implicature from that would be 'She can't talk to you now' or 'We should go home soon'. But does it explicitly say how tired she is? Does the word 'tired' here mean 'too tired to talk', or is this an implicature? What about if what they say is 'She's too tired'? This makes no sense at all without some explicit enrichment to a more definite meaning: too tired to do X. This is not an implicature, not something that might or might not follow, but is necessary to make sense of the proposition expressed.

What about 'It's snowing'? It can't just snow: that's impossible, it has to be in some place. Whenever anyone says it's snowing they mean explicitly that it's snowing in some place they're thinking about: usually 'here', where they are. What about 'I've had breakfast'? It looks as if it doesn't give any time restriction, as if it could be true even if you've only had breakfast once in your life, but plainly that's not what it means, and no-one ever means that when they say it. If someone says it they explicitly mean 'I've had breakfast today/recently'; and a possible implicature is 'I don't want to go to the cafeteria with you now'. This process of filling in essential elements has been called saturation.

Utterances have illocutionary force: when you say 'I'm tired' you are therein doing something like affirming, claiming, admitting, or pleading that you're tired; which of these you're doing makes a difference, but it's on top of the ground-level meaning. Sperber and Wilson call these higher-level explicatures in which the basic explicature is embedded.

Relevance Theory sees saturation, enrichment, reference assignment, embedding, and some other processes such as disambiguation as contributing to explicit meaning; and which enter into further computations, together with our knowledge of the context, which is speaker's and hearer's beliefs, memories etc. as well as external or prior context. Implicatures are beliefs that can be derived from this mixture of input and context, and which may contribute to further computations. They are accessed in order of salience or relevance, and the relevance may be adjusted as more information and implicatures come to light. They are revisable, and therefore uncertain.

A scalar implicature is one of degree: that I've eaten some of your sweets implies or at least implicates that I haven't eaten them all.

A conventional implicature is one inherent in the meaning of a word, but which doesn't affect the truth of the statement: 'X but Y' is true just whenever 'X and Y' is, but carries an additional implicature that what you'd expect to follow from X may be cancelled by implications of Y.

A conversational implicature is one that isn't conventional, but occurs by virtue of the particular context of the conversation. An ironic statement has the form of a non-ironic statement, and can only be told apart by knowing the people or circumstances. My 'I want a cuddle' example up the top requires detailed knowledge of the people, such as they themselves would have.

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