1.2.8. Deductive relations in general
The six basic deductive relations between two sentences that we have considered are shown in the following table:
| relation | pattern ruled out | relation | ||
| φ ⊨ ψ | φ is T | ψ is F | ⎱ ⎰ |
φ ≃ ψ |
| φ ⫤ ψ | φ is F | ψ is T | ||
| φ ▵ ψ | φ is T | ψ is T | ⎱ ⎰ |
φ ⋈ ψ |
| φ ▿ ψ | φ is F | ψ is F | ||
Each says that one or more patterns of truth values occurs in no possible worlds. And there are no other ways of doing this that yield genuine relations between a pair of sentences. If we rule any pair of patterns other the pairs ruled out by equivalence and contradictoriness, we end up specifying the truth value of one of the two sentences—i.e., we say of either φ or ψ that is a tautology or that it is absurd. And any way of ruling out three patterns must do this for both φ and ψ.
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 (so each one is logically contingent). 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 1.2.6) are logically independent since it is possible for the package to arrive next week but not on Wednesday, for it to arrive on a Wednesday but not next week, for it to arrive next Wednesday, and for it to arrive neither next week nor on a Wednesday.
There are a number of connections among the six deductive 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 1.2.8-1. 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 it also shows something more important: 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 1.4.6 that something analogous continues to be true.