Implication and Presupposition

Dave Bagwill asks:

To be more clear: Do all propositions imply an ontology? Is 'imply' strong enough to bear the weight of 'assertion'? Or is 'imply' basically an equivalent of 'presuppose'?

Still not clear enough. Dave. Not even the third question is clear since you didn't specify the  sense of 'imply.'  But the third question is clear enough to warrant a brief answer, which is: No. Consider the following which is an intuitively clear example of a proposition resting on a presupposition:

Tom regrets lying to his wife.

Necessarily, if Tom regrets lying to his wife, then Tom has lied to his wife. The antecedent implies (in the sense of 'entails') the consequent. (I have defined 'entails' on many occasions.) But note that it is also true that, necessarily, if Tom does not regret lying to his wife, then Tom has lied to his wife.

This yields a criterion of one type of presupposition. A proposition p presupposes a proposition q just in case both p and its negation ~p entail q. One could also say that an entailment of a proposition p is a presupposition of p if and only if p's presupposition survives the negation of p.  (If the preceding sentence does not make sense to you, forget it, and focus on the one preceding it.) Consider now:

Tom is drunk.

Necessarily, if Tom is drunk, then someone is drunk. But it is not the case that, necessarily, if Tom is not drunk, then someone is drunk.

So by the criterion lately enunciated, the 'survival of negation' criterion to give it a name, 'Tom is drunk,' while it implies (entails) that someone is drunk, does not presuppose that someone is drunk.

Therefore, to answer Dave's question, 'imply' (in the sense of 'entails') is not equivalent to 'presuppose.'

Alles klar?  Vielleicht nicht!

One could conceivably balk, or baulk in the case of the Bad Ostrich, as follows: It is not clear, or it is false, that if Tom does not regret lying to his wife, then Tom has lied to his wife.  The Ostrich could say, "Tom does not regret lying because he didn't lie in the first place."

As you can see, the topic of presupposition is a murky one, and part of the murkiness is due to the fact that presupposition is at the interface of the semantic and pragmatic, and it is not clear how they gear into each other, if you will excuse the mixed metaphors.