Cancellability is the ability of cancelling a conclusion, logically/inferentially derived from a [set of] premise[s] without causing a logical error.

A presupposition necessarily holds (its truth is taken for granted) independently of the truth value of a proposition. ex.1. "the king of France is bald" (the existence of "the king of France" is presupposed here - in this particular case, the presupposition is called "existential presupposition") ex 2. "I am glad that you came" --> the presupposition is "you came"

If we try cancelling these presuppositions we see that we end up with phrases that sound irrational and are necessarily unacceptable:

*The king of France is bald, but there is no King of France

*I'm glad that you came, but you didn't come

An implicature, on the other hand, is the conclusion of a reasoning, which can be called "nonmonotonic" i.e. you can add further conclusions, cancelling the previous ones, without causing any logical problems. Paul Grice was the one who first noted that "cancellability" is one of the 5 main characteristics of implicatures. An implicature is cancellable, because some of its premises are inferences which have to do with pragmatic knowledge (knowledge about the world or the specific context of discussion) and not with purely logical operations.

Here's an example:

John says: "It's a bit chilly in here"

Mary knows that the window is open in that room and she also knows that John knows that the window is open, she also knows that she shares the common assumption with John that if they close the window, it will stop being chilly [all these are premises depending on the general knowledge of Mary]

So Mary infers that what John implied by the phrase "it's a bit chilly in here" is "I would like you to close the window" (this is the implicature)

However, if John decides to cancel that implicature, although it would be a little bit unexpected, it wouldn't be unacceptable.

So John can say "It's a bit chilly in here, but I don't want you to close the window" (he can add further explanations such as "I like the fresh air that's coming in" or "I 'd rather wear one more sweater") and the phrase will be perfectly fine.

So there you have the basic difference between presupposition and implicature: presuppositions are NOT cancellable, while implicatures are.

Moving on to entailments, they are propositions, logically following the initial proposition, so, like presuppositions, they cannot be cancelled either. The rule for entailment is that if p entails q, then if p is True q is necessarily true as well [symbolised p--> q].

CAVEAT: Applying the cancellability test to entailments, is a bit different than applying it for presuppositions. [as we saw in the examples given above for the cancellability of presuppositions] If r presupposes s the phrase that you want to test for its acceptability, is of the form "r but not s" (which, as we saw, is always unacceptable). However, when you test cancellability of entailments you need a more complex phrase: if p entails q, then you have to test a phrase such as "p --> q AND p is true but q is false" (which is also going to be unacceptable)

Example: if someone is a man, then he is mortal.

In order to apply the cancellability test, we need to form the phrase: "If someone is a man, then he is mortal and Socrates is a man but he is immortal", which is obviously unacceptable.

To sum up: Implicature--> cancellable, Presupposition and entailment--> non cancellable.

Additionally, if we need to distinguish between Presupposition and entailment (which are both noncancellable), there is one more test we can apply and this is negation test:

Presuppositions

I am glad that you came (presupposition : you came)

+Negation: I am NOT glad that you came (presupposition still holds: "you came")

Entailment

If someone is human then he is mortal and Socrates is human (entailment: Socrates is mortal)

+Negation: If someone is NOT human then he is mortal and Socrates is human (it is NOT entailed that Socrates is mortal)

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