Why is the intension of a sentence the set of all possible worlds in which it is true?

Question

In Chapter 1 Section 1.3.3 in Kearns (2011), as for the extension and intension for sentence, Midge is grinning, the extension is "truth value (true or false) in the actual world", and the intension is "the set of all possible worlds in which Midge is grinning is true". But for intension, I'm just confused that why we cannot say "the set of all possible worlds in which Midge is grinning is true or false"? Should the sentence be definitely true in the possible world?

Answer 1

How not to do it

If one were to apply Kearn's treatment of intensions for predicates to sentences, then in analogy to "the set of all dogs in all worlds" one would end up with "the set of all truth values in all worlds". Under this account, every contingent sentence (= a sentence that is not tautological or self-contradictory, and can thus be true in some worlds and false in others) would have as its intension the set {0,1}.
This is not very useful: It doesn't say much about the meaning of the sentence other that it can be true or false, and worse, all contingent sentences would have this same set as their intension -- that is, all these sentences would be intensionally equivalent. This is a rather unintuitive result, given that the intension should be something like the conceptual meaning, and non-synonymous sentences should have different intensions. Hence why I find Kearn's description of intensions for predicates somewhat strange: Applying the same pattern for sentences as for predicates makes for a rather useless definition of intension, and assuming two entirely different approaches for different kinds of expressions seems unelegant.

If one were to adopt your approach, "the set of all possible worlds in which the sentence is true or false", then, since a sentence is either true or false in every given world by the very definition of a sentence, this would just have one end up with the set of all worlds: {w0, w1, w2, ...}.
This is again not a very useful definition. The intension of any sentence would simply be the entire logical space, which is neither very informative nor does it allow for a distinction between the intensions of different sentences.

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How to do it

Following the standard definition of intension, the intension of an expression is a function from possible worlds into the extension of the expression in that world; hence the intension of a sentence is a function from possible worlds into the truth value of the sentence in that world.

Consider the sentence "European robins (a species of bird) have an orange breast". In the actual world (typically named w0), this sentence is true; robins have an orange breast, white belly and brown upper side. We can imagine a different possible world, w1, in which robins are colored like ours but are overall larger, another world, w2, in which robins look almost like ours except that they have a green breast, and yet another world, w3, in which they are green all over. We could of course think of infinitely many other variaties of robins, but for simplicity let's assume that in all other worlds, w4 and upwards, robins look exactly like in our world.

Then the extension of "Robins have an orange breast" in the actual world is true, and the intension is the function w0 ↦ true, w1 ↦ true, w2 ↦ false, w3 ↦ false, w4 ↦ true, w5 ↦ true, ....
Every sentence has a definitive truth value (an extension) in the actual world and in each other possible world; the intension is the function that collects these truth values across all possible worlds.

Now instead of this way of writing it down, one could simply kick out the falsy worlds and collect those worlds in which the sentence is true into a set: {w0, w1, w4, w5, ...} That is, instead of a function which assigns to every world the truth value the sentence has in this world, we can take the intension as the set of possible worlds in which it is true.

There are several reasons why this definition as a set rather than as a function would be preferable: It may be more intuitive conceptually -- the intension (~= concept) of a sentence then amounts to the set of possible worlds (~= conditions) under which it is true --, and one can perform set operations such as the cardinality, union and intersection, subsethood and equality between the intensions of different sentences; for instance, the more specific a sentence is in its content, the fewer worlds there will be that manage to satisfy it, so we can take the size of the intension set as a measure of the degree of informativeness of the sentence. And if one wants, one can always convert the set back into the function in order to have uniformity with other types of expressions.
Note that compared to the first two approaches, this formulation of intension is actually meaningful (or at least non-trivial): Since all possible worlds differ from one another in terms of the propositions that hold in them, every sentence will have a different distribution of truth values across possible worlds, and thus a different set of possible worlds as its intension, such that we now arrived at a formulation of intensions for sentences from which we can read certain properties of the sentence and which effectively distinguishes between sentences with different meanings.