IndividuationConditions
Let us begin by studying the possible worlds theory of propositions. In the propositions-first setting, this view is characterized by the thesis that propositional individuation is given by necessary equivalence.8 
(ii) we appeal to above.[147] However, given (ii), it straightforwardly follows that every proposition is necessarily equivalent to a precise proposition.[148] But by (i), that means every proposition is a precise proposition.
We can see the problem with the modal account manifest itself in many different ways.
Necessarily equivalent things that are not determinately equivalent cannot always be substituted within the scope of a determinacy operator. Assuming that baldness facts supervene on facts about hair number, the proposition that the cutoff for baldness is 2,024 hairs is either necessarily true or necessarily false, although it's borderline which. Thus, this proposition is either necessarily equivalent to 0 = 0 or 0 = 1: but I cannot substitute the first claim for either 0 = 0 or 0 = 1 in ‘it’s determinate whether 0 = 0’ or ‘it’s determinate whether 0 = 1’ without changing their truth values.It should be noted, by symmetrical reasoning, that determinate equivalence is not a good condition of individuation either, for there are determinately equivalent things that are only contingently equivalent. For example, it’s determinate that Bruce Willis is bald if and only if Patrick Stewart is (since they are both determinately bald); however, this biconditional is contingent.
Less obvious is the fact that determinate necessary equivalence and necessary determinate equivalence will not do either.
The reason is that second-order indeterminacy can prevent the intersubstitutability of necessary determinate equivalents and determinate necessary equivalents within determinacy contexts; similar problems arise for other finite combinations of ‘necessary’ and ‘determinate’ and vagueness at higher orders.Let us now try to reformulate the supervaluationist theory in a propositions-first setting. Again, we shall achieve this by determining what individuation conditions are imposed by the stipulation (in the ordinary setting) that propositions are represented by sets of ordered pairs of worlds and precisifications. If A and B express the same set of world-precisification pairs, then, relative to any world-precisification pair, the sentence π(A B) is true, where π is any finite sequence consisting of □ and Δ symbols. (We will present the supervaluationist semantics in more detail in chapter 12.) This means that A and B are determinately equivalent and necessarily equivalent, so that they are substitutable in the context of one iteration of a □ or a Δ. Butthe abovefactalsomeans that A and B are determinately determinately equivalent, so we can also substitute A and B within two iterations of Δ—indeed, because they are π equivalent for any string π of □s and ∆s it follows that we can substitute A for B in any context regardless of the number of iterations of □ and Δ.
If one had infinite conjunction in the language, one could define an operator (□Δ)*P:= ∕∖ππP where π ranges over arbitrary finite sequences consisting of 
Indeed, with a suitably rich logic of infinite conjunction, determinacy, and necessity, it is possible to show that ≡, when simply defined in terms of (□Δ)* by the above biconditional, satisfies the principles Substitution, Identity, and The Rule of Equivalence.
Although this might serve as a good approximation of the supervaluationist account of propositional individuation, it is not entirely accurate.
There are supervaluational models in which A and B are (□Δ)*-equivalent, but in which they do not correspond to the same set of world-precisification pairs. Such discrepancies arise in models in which there are at least two world-precisification pairs that cannot be reached from each other by following the □ and Δ accessibility relations any number of times: for if A and B are different but agree on the worlds that can be reached from (w, v) by following the accessibility relations, then they will count as (□Δ)*-equivalent at (w, v). We discuss one such model in section 14.3; see also Figure 14.2. (Such models are not just curiosities either: models that do not validate the principle B for determinacy—a principle that plays a role in many of the paradoxes of higher-order vagueness—often have this feature.)11.3