1.2.7. Deductive relations
The four basic deductive relations between two sentences that we have considered are shown in the following table:
| Relation | pattern ruled out | |
| φ implies ψ (φ ⊨ ψ) | φ is T | ψ is F |
| φ is implied by ψ (ψ ⊨ φ) | φ is F | ψ is T |
| φ and ψ are mutually exclusive (φ ▵ ψ) | φ is T | ψ is T |
| φ and ψ are jointly exhaustive (φ ▿ ψ) | φ is F | ψ is F |
These are the only relations that can be defined by ruling out a specific pattern of truth values for two sentences because there are only four such patterns.
And while ruling out more than one pattern, does produce relations beyond these four, it does not give us any that do not already have means to express. If we rule out the first two patterns, we are saying that the sentences entail each other—i.e., that they are equivalent. And if we rule out the last two patterns, we are saying that they are contradictory. If we were to rule out any other pair of patterns, we would simply rule out a truth value for at least one of the sentences in all possible worlds. The claim we would then make about the two sentences could be expressed by saying about one of them that it is tautologous or that it is absurd. For example, ruling out the first and last patterns amounts to saying that ψ cannot be false (since it would be to say that ψ cannot be false when φ is true and also that it cannot be false when φ is false), and this to say that ψ is a tautology. If we were to rule out three patterns, we would have specified the truth value of each sentence (by ruling out all other alternatives), so we would be of each that is was tautologous or that is was absurd. For example, ruling out the last three patterns is to say that the first pattern occurs in all possibile worlds—i.e., that φ is a tautology and that ψ is absurd. It follows that any deductive relation between pairs of sentences can be understood in terms of ordering by implication and the sorts of opposition discussed in the last subsection.
There are a number of connections between these relations that can be depicted in a traditional form of diagram known as a square of opposition. In the case of the examples that were used to illustrate various sorts of opposites, the square can be arranged as shown in Figure 1.2.7-1. The vertical structure of the diagram displays ordering by implication in the way we have before: each of the sentences in the bottom row implies the sentence show above it. The horizontal structure of the diagram displays the sorts of opposition. The sentences along the bottom are mutually exclusive, those along the top are jointly exhaustive, and the sentences along the diagonals are contradictory.
Given one side of the square, the other side can be reconstructed by taking contradictories. For example if φ ⊨ ψ, then φ will be mutually exclusive with any sentence contradictory to ψ and any sentence contradictory to φ will be jointly exhaustive with ψ. This provides a way of generating squares of opposition; but, more importantly, it shows that implication and contradictoriness can be seen as the fundamental deductive relations between pairs of sentences. There is more to be said about deductive relations when we consider larger collections of sentences, but we will see in 1.4.5 that something analogous continues to be true.
When no deductive relation holds between a pair of sentences φ and ψ—that is, when each of four patterns of truth values for the two appears in some possible world—we will say that φ and ψ are logically independent. Not only are logically independent sentences unordered by implication, they are not mutually exclusive or jointly exhaustive. And it follows from this, of course, that they are not equivalent or contradictory and also that neither is a tautology or absurd. This sort of thing is true for most pairs of sentences. Although sentences on different topics almost always provide examples, logically independent sentences do not need to differ in subject matter. For example, the sentences The package will arrive next week and The package will arrive on a Wednesday (a pair of sentences mentioned in 1.2.5) are logically independent since it is possible for the package to arrive next week but not on Wednesday (so the first doesn’t imply the second), for it to arrive on a Wednesday but not next week (so the first isn’t implied by the second), for it to arrive next Wednesday (so they aren’t mutually exclusive), and for it to arrive neither next week nor on a Wednesday (so they aren’t jointly exhaustive).