Broadly speaking, these propositions assert that certain conditions are sufficient to ensure that a partially ordered set contains at least one maximal element, that is, an element such that, with respect to the given partial ordering, no element strictly exceeds it.
If our aim were merely to utter truths, we should be willing to assert the proposition.
Speakers will assert the proposition after seeing the result of the lottery drawing or hearing about the winning ticket from a newscaster or remembering what the winning ticket was.
When the speakers make a practice of asserting the proposition under these conditions, they become the proposition's truth conditions.
If we concede that these sentences assert relations to propositions, then we are conceding that there are propositions.
In effect, Brouwer is saying that we can assert the weak negations of the propositions expressed in these questions; hence, these propositions are so-called "Brouwerian counterexamples" or "weak counterexamples" to PEM.
If we construe assertively uttering a sentence exemplifying 'S is P' as asserting the proposition that A is A, then we fail to capture the point of the assertion.
Finally, one asserts the proposition.
Then, plausibly, John asserts the proposition that (16) semantically expresses in his context.
The one might assert many true propositions, while the other fails to assert any true proposition.
(Compare again with assertion: John can assert the singular proposition that he is shopping in at least two ways, by uttering either 'John is shopping' or 'I am shopping'.) Moreover, a rational agent can rationally believe both a proposition and its negation, as long as he does so in suitably different ways.
