I had an instructor who thought hard about formal systems that might underwrite analysis of speech acts like supposition and asserting. He argued as a matter of logic, asserting entailed supposing, though supposing did not entail asserting. For example, the following would be counted as valid:

1. John AST(The store is open)

2. Hence, John SUP(The store is open)

While the following would not count as valid:

1. John SUP(The store is open)

2. Hence, John AST(The store is open)

I agree with the latter not counting as valid. Asserting seems clearly associated with a norm of truth in every case, while supposing does not. John might claim to know p while supposing not p, without intuitive conflict. In contrast, John claiming to know p while asserting not p, seems a misuse of asserting, i.e. a lie. He's surely asserted, but he's violated a norm.

Does this make trouble for the first argument too? Not obviously. John asserting p comes with a norm of assertion, and if it's entailed John supposes p as well, we might think John's supposing in this case comes with a norm of assertion as well. That doesn't mean John can't suppose without the norm, and indeed, in many cases he will do just that.

That said, I do think there's trouble holding the first entailment. Supposing as an attitude seems to me to involve - in every case - direction towards some further goal. John doesn't simply suppose the store is open. Rather, John supposes the store is open for a reason. This is clearest, I think, in situations where one might suppose something for the sake of contradiction, i.e. reductio proofs. John might suppose p with the intention of drawing out some inconsistency in a premise set combined with background logical constraints. This strikes me as how supposing works in natural language as well. When John supposes the store is open, it's natural to ask - if you aren't already party to reason for the supposition - why John is supposing such a thing, e.g. do you need milk? do you have a shift today?

This feature of supposing in mind, return to the first argument. If John asserts the store is open, then if this argument is valid, it follows John supposes the store is open. But if John supposes the store is open, then there is some goal X John has in mind which motivates this supposition. Hence, in every case of assertion, there is some goal X agents have in mind which motivates the assertion. I find this implausible.

I'll detail why in another post. In the meantime, what do you think?