It is, however, one thing for a proposition or set of propositions to be false, another altogether for it to be necessarily false.
Against completeness, one might say that there are circumstances in which the level of disagreement on a particular proposition (or set of propositions) is so great that forming a collective view on it is undesirable or counterproductive.
'Internal' relations are often of the 'nucleus-satellite' type: they are deployed to increase the success of a nuclear proposition (or set of propositions) or proposal (or set of proposals); but they can also be 'multi-nuclear', as in internal sequence used to link steps in an argument (firstly, secondly … furthermore).
But this is not to commit to the claim that propositional quantification is to be reduced to objectual quantification over sets of propositions or sets of possible worlds or what have you.
These representations are often described as measures on events, or states, or propositions, or sets of possibilities.
Just as deductive logic consists of the study of a certain consequence relation between propositions or sets of propositions (the relation of valid implication), so defeasible (or nonmonotonic) logic consists of the study of a different kind of consequence relation.
Note also that Classical Semantics is compatible with most traditional views on semantics; in particular, it is compatible with all the standard views on the meanings of sentences, namely that they are truth-values, propositions, or sets of possible worlds.
It will be convenient to rephrase AC as the assertion where \(\langle M_{ij}: i \in I, j \in J\rangle\) is any doubly indexed family of propositions (or sets).
According to ersatzist views, which identify possible worlds with abstract entities like sentence-types, properties, propositions, or sets, the truth of World Indiscernibility will turn, not on whether there are distinctively haecceitistic possibilities, but on whether such entities are qualitatively indiscernible from one another.
We have seen that there are various sorts of thing a narrow content could be: a description, a conceptual role, a function from contexts to broad contents, a diagonal proposition, or a set of maximal epistemic possibilities.
Then we could coherently talk of truth-preservation relations on (sets of) sentences, on (sets of) propositions, or on (sets of) characters (as in Kaplan 1989) and ultimately on any truth-bearer whatsoever.
