Many sentences can be evaluated as true or false relative to a (possible) context. For example, 'it is raining' is true (in English) at all and only those possible contexts at which it is raining.
This relation between sentences (of a language) and contexts is arguably central to a theory of communication. At a first pass, what is communicated by an utterance of 'it is raining' is that the utterance context is among those at which the uttered sentence is true. (You can understand what is communicated without knowing where the utterance takes place.)
Truth-at-a-context is also arguably central to a theory of validity, entailment, and other logical concepts. At a first pass, 'it is raining' entails 'it is raining or snowing' because the latter is true in every possible context in which the former is true.
A context (unlike an "index") is not an arbitrary list of parameters to which truth might be relativized. An ordinary utterance context, for example, does not contain or determine an "assignment function" that maps infinitely many variables to a referent.
But sometimes it would be useful to expand contexts by an extra parameter.
Take conditionals. Some have argued that subjunctive conditionals validate the principle of "Conditional Excluded Middle":
(CEM) (If A then B) or (if A then not B).
This seems to require that for all sentences A and B, and for all contexts c, either 'if A then B' is true at c or 'if A then not B' is true at c. But for some instances, it is hard to see how an ordinary, concrete utterance situation could render either of these true.
If I had tossed one of the coins in my pocket, how would it have landed? We can't say. It depends on which coin I would have chosen, how I would have tossed it, and possibly on how various superpositions of particle interactions would have collapsed. And there's no fact of the matter about which coin I would have chosen, how I would have tossed it, and how the superposition states would have collapsed. Or so it seems.
Philosophers like crazy views. Some have argued that there is indeed a fact of the matter about which coin I would have chosen, etc.: we just don't know, and can't find out, which of these irreducible "counterfacts" obtains. Let's set this view aside. Let's assume real utterance situations do not contain primitive counterfacts.
But maybe this doesn't matter. Couldn't we define validity and entailment relative to an expanded notion of context that does involve primitive counterfacts?
Formally, this is trivial. The question is why we should be interested in the so-defined concepts of validity and entailment.
An obvious answer (for friends of CEM) is that this gives us a "model theory" that (supposedly) matches our intuitive judgements about entailment, validity, equivalence, etc.
OK. But I don't want to give up the idea that truth-at-a-context is central for a theory of entailment and validity. And by a "context" I don't mean an arbitrary list of parameters. As I tell my logic students, the model-theoretic concept of validity is meant to formalize the intuitive concept of truth in all conceivable scenarios (under every interpretation of the non-logical vocabulary). A conceivable scenario is not an arbitrary list.
On the other hand, it is also not obvious that we must restrict contexts (or scenarios) to metaphysically real features of possible utterance situations.
Imagine a community in which everyone believes in primitive counterfacts. When they imagine possible scenarios, they imagine scenarios with counterfacts; they distinguish scenarios that only differ with respect to counterfacts; they judge any scenario to be incomplete unless it fixes all the counterfacts; they often profess their ignorance of various counterfacts; and so on.
Members of this community may well judge that CEM is valid. Moreover, they will insist that in every conceivable scenario, either 'if A then B' is true or 'if A then not B' is true (on any interpretation of A and B).
Now, metaphysical reality does not contain primitive counterfacts, nor could it. But why should we define validity and entailment in terms of what is metaphysically real and metaphysically possible, if users of the language are systematically mistaken about what is real and possible?
Similar considerations apply to the theory of communication. If we want to systematize the information that is communicated with utterances of different sentences, should we cash out this information in terms of genuine possible worlds, or in terms of "worlds" speakers and hearers take to be possible?
We can imagine other communities, for which we might want to give different answers.
Imagine a community in which, on reflection, people don't really believe in primitive counterfacts, but some of their cognitive processes effectively assume such facts. Perhaps their "semantics module" assumes primitive counterfacts, insofar as it creates judgements that seem to require primitive counterfacts and that clash with judgements coming from their "metaphysics module".
It is not obvious to me what we should say about this kind of case. It might depend on what exactly we want from semantics.
Personally, I am not convinced by the data that supposedly shows the validity of CEM. But it's worth noting that we could accept the data without postulating counterfacts, and without resorting to supervaluationism to get rid off extra parameters.
There are many other possible applications. For example, in the case of vagueness, we could accept classical logic, bivalence, and the T-schema without postulating unknown sharp boundaries.