1.2.4. Truth conditions and propositions

In making the various comparisons we have been considering, what have needed to know about a sentence in order to compare it to others is its truth values in various possible worlds. We will describe this aspect of a sentence’s meaning as its truth conditions. That is, when we know, for any given possible world, whether or not a sentence is true, we know the conditions under which the sentence is true; and, when we know those conditions, we can tell whether or not it is true in a given possible world.

It will also be convenient to be able to speak of this kind of meaning or aspect of meaning as an entity in its own right. We will do this by speaking of the truth conditions of a sentence as encapsulated in the proposition expressed by the sentence. This proposition can be thought of as a way of dividing the full range of possible worlds into those in which the sentence is true and those in which it is false—i.e., into the possibilities it rules out and the ones it leaves open. And a proposition can be pictured as a division of an area representing the full range of possibilities into two regions.

Fig. 1.2.4-1. A proposition dividing the full range of possible worlds into possibilities ruled out and possibilities left open.

Since a sentence that rules out more possibilities makes a stronger claim, the size of the region occupied by the possibilities it rules out can be thought to correspond to the strength of the claim it makes.

Relations between the propositions expressed by a pair of sentences can also be depicted in this way. The regions ruled out are shown shaded in the top row in Figure 1.2.4-2, and the regions left open are shown hatched in the bottom row.

a	b	c
d	e	f

Fig. 1.2.4-2. Three relations between sentences φ and ψ. (a, d) φ implies ψ. (b, e) φ and ψ are mutually exclusive. (c, f) φ and ψ are jointly exhaustive. Regions ruled out by sentences are shaded in the top row—in green for φ and in blue for ψ. The regions left open are hatched in the bottom row—hatched horizontally for φ and vertically for ψ.

When φ ⇒ ψ (see a and d above), the implied sentence ψ cannot rule out any possibility not already ruled out by the implying sentence φ, so the region ruled out by φ must include the region ruled out by ψ (and the region left open by φ must therefore be included in the region left open by ψ). If φ and ψ are mutually exclusive (see b and e above), there can be no overlap in the regions they leave open so the regions ruled out by the two must together cover the full range of possibilities. Here φ rules out all worlds at the left of the rectangle and ψ rules out all worlds at the right, with both ruling out a swath of worlds in the middle. Finally, when φ and ψ are jointly exhaustive, the situation is reversed (see c and f above): the regions left open by the two must together cover all possibilities so the regions they rule out cannot overlap. In the diagram a swath of worlds through the middle is left open by both.