MacFarlane calls them the indeterminacy and determinacy intuitions. Assuming the world is indeterministic, the former is that an assertion (now) of a future contingent such as 'There will be a sea battle tomorrow' lacks a truth value and the latter that, once tomorrow passes, it seems strange to deny that the assertion had a truth value.
It is the assumption that utterance-truth is relative only to a context and does not depend "on who is asking about it, or when". Note that contexts tell us when, who and where the utterance is taking place, so those features of context allow us to determine the referents of indexicals such as the 'I' in 'I'm hungry'.
It allows that future contingents lack a truth-value while at the same time preserving logical truths such as 'There will be a sea battle tomorrow or it is not the case that there will be one'.
TRL respects the determinacy intuition by giving utterances of future contingents a truth-value. MacFarlane's objects to the view on the grounds that it gives up indeterminism. He raises the same objection to Lewis's divergence picture, claiming that the sensen of indeterminacy it provides is merely epistemic, not "objective".
The objection is that philosophers have granted all along that propositional truth is not absolute. The truth of propositions intuitively depends on at least worlds and times. The proposition that Obama is president is true at our world and false and others, and true now but false some times in the past and future. MacFarlane notes, however, that we are concerned with utterance-truth, not propositional truth, and utterances are uttered in contexts which include the world and time at which the utterance is made, and so their truth and falsity depends only on the world and time of utterance, hence only on the context, not some additional parameter.
The truth of sentences is relativized to a moment-history pair m/h. So we can only say that 'It will rain tomorrow' is true or false relative to some moment-history pair. If the moment-history pair is one on which it rains tomorrow, then it's true, and if not, not. Recall that is a "history" in the sense that it is a maximal branch of the temporal tree--a complete past and complete future. (The 'history' terminology might be confusing here.)
An utterance of A is true (false) at a context of utterance u and context of assessment a iff the utterance is true (false) at every point m/h such that m = the moment of u, and h passes through m and the moment m(a) of a if m(a) is later than m. (A history passes through a moment if the moment lies somewhere on the history.) Note that falsity is not defined as the negation of truth (i.e. an utterance isn't false just in case it's not true, just like on supervaluationism.)
It does justice to both intuitions since if one utters at m 'There will be a sea battle tomorrow' relative to u and a where the moment m(a) of a is m and the moment m(u) of u is m, then the sentence is indeterminate (respecting the indeterminacy intuition). And if m(u) is m but m(a) is a moment later than m at which there's a sea battle, then the utterance is true.
He argues that assertion aims at "objective" truth simpliciter. If the truth of an assertion (a special kind of utterance) can be true at one moment but false at another, what is the point of assertion? If I assert A which is true today but false tomorrow, do I have to defend myself tomorrow for having asserted what was yesterday a truth but today a falsehood? Can I assert something that is now false knowing (or at least hoping) that it will be true tomorrow?
MacFarlane defends his account as follows. He claims that assertion does not aim at truth but rather that when one makes an assertion, they are committed to producing a justification. That means one could even lie (assert what they know to be a falsehood with the intention of misleading the hearer) and still make an assertion. On this view of assertion, there is no problem for assessment-sensitive truth. If one asserts 'It will rain tomorrow' relative to a context of utterance and assessment, they are obligated to produce a justification of that utterance. If the moment of assessment is the same as the moment of utterance, clearly the assertion will fail since nobody could justify now that tomorrow there will be a sea battle if it is indeterminate that there will be one. But tomorrow while the battle is taking place, they can justify that yesterday's assertion was true (as assessed from the later time).