Entailments and presuppositions versus conversational implicatures
The most prominent difference between entailments and presuppositions on the one hand, and conversational implicatures on the other, is that the latter tend to be defeasible, while the former tend not to be. Conversational implicatures are defined as arising from the maxims, after all—so you can always just claim that you weren’t actually following them!
The non-defeasability of entailments
#Jo has at least three siblings, {and, but} she doesn’t have at least two siblings.
#Jo is Bo’s sibling, {and, but} Jo and Bo aren’t siblings.
#No one came to class, {and, but} it’s not the case that Julian didn’t come to class.
Each of these examples seems to be weird and contradictory. That’s because an entailment of the first clause in each example is negated in the second clause. That is, Jo has at least three siblings entails Jo has at least two siblings; Jo is Bo’s sibling entails Jo and Bo are siblings; and no one came to class entails Julian didn’t come to class.
The non-defeasibility of presuppositions
Like entailments, presuppositions also appear to be non-defeasible.
#Jo stopped smoking, {and, but} she didn’t used to smoke.
#Jo loves that it’s raining, {and, but} it’s not raining.
#Bo failed the assignment, {and, but} there isn’t an assignment.
#Bo failed the assignment again, {and, but} he hasn’t failed it before.
The defeasibility of conversational implicatures
Jo did some of the readings. In fact, she did all of them.
I want something with lots of caffeine. In fact, I want a matcha latte.
Context: Jo is writing a letter of recommendation for Bo for a linguistics instructor position.
Jo: Bo is very punctual. He has great handwriting. He kicked my ass in Hungry Hungry Hippos. Last, but not least, he’d be a great instructor in your linguistics program!
These examples illustrate that conversational implicatures, unlike entailments, may be canceled. In (8), the implicature that Jo did not do all of the readings is canceled; in (9), the ignorance implicature is canceled; and in (10), the implicature that Bo should not be hired as a linguistics instructor that arises when Jo flouts relation is canceled.