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Propositions in possible-world semantics

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In possible-world semantics, the same sentence can have different truth values at different worlds. "We can thus define a function that maps each (sentence, world) pair onto the truth value (0 for false and 1 for true) of the sentence of that world"(Steinhart, p. 100). Keeping in mind what we noted before about an intension of a word being a function which associates each world with its referent at a specific world, we

"can let the intension of a sentence be a function which associates every world with the truth-value of the sentence at that world. We have a set of worlds W. The set of characteristic functions over W is { f | f : W ---> {0, 1}}. Each of those characteristic functions is an intension"(Steinhart, p. 101).

As we saw in our article on the distinction between intension and extension, and intension is usually understood by philosophers as the formal definition of a world, as opposed to extension, which refers to the set of objects included in that definition. For our purposes, the semantic value or meaning of a sentence is referred to by philosophers as a "proposition." "The intension of a sentence is...the proposition expressed by that sentence"(Steinhart, p. 101). Thus, for our purposes, "proposition" is not identical with "sentence." Sentences express propositions, which are the meaning of the sentences. A proposition "is a function that associates each world with a truth-value"(Steinhart, p. 101). As in formal logic, 0 will represent falsity and 1 will represent truth. Let's use Steinhart's example to see how we can use this technical equipment to express the truth or falsity of sentences at different worlds.

[Charlie is sad] = {(w1, 1), (w2, 1), (w3, 1), (w4, 1)}

[Charlie is happy] = {(w1, 0), (w2, 0), (w3, 0), (w4, 0)}

[Allan loves Betty] = ({w1, 1), (w2, 1), (w3, 1), (w4, 0)}

Keeping in mind that each number associated with a w is a subscript, we see that it is true, for example, that Allan loves Betty in world 1, but it is false that Allan loves Betty in world 4.

Keeping in mind that Charlie is sad in every world and that he is not "happy" in any of the worlds, Steinhart gives us an example of many philosophers "prefer to identify the proposition expressed by a sentence with the set of worlds at which a sentence is true"(Steinhart, p. 101):

[Charlie is sad] = {w1, w2, w3, w4};

[Charlie is happy] = {}

[Allan loves Betty] = {w1, w2, w3}.

Note that Charlie is sad at all the worlds under consideration and the set of all worlds in which Charlie is happy is an empty set. Likewise, Allan loves Betty in worlds 1, 2, and 3 but not in world 4. The latter, therefore, is missing from the set.

We see the importance of possible-world semantics for a rich modal logic. A proposition is necessarily true if it is true at all possible worlds, and it is necessarily false if it is false at all possible worlds"(Steinhart, p. 101).

"A proposition that is true at every world is necessarily true; it is a necessary truth. A proposition that is false at every world is a necessary falsehood. For example, "Charlie is sad" is a necessary truth while "Charlie is happy" is a necessary falsehood. Charlie isn't happy at any world. Poor Charlie"(Steinhart, p. 101)

Steinhart, Eric. "More Precisely: The Mathematics You Need To Do Philosophy." Broadview Press, 2009