1.2.3. Contrasting content

We arrived at the relation of implication by considering entailment by a single premise. If we do the same with exclusion, we arrive at another relation between sentences. If φ excludes ψ, then the set {φ, ψ} formed of the two is inconsistent. Since the members of a set have no order, it will be equally true that ψ excludes φ; and this reversability is reflected in the usual terminology for this relation. When there is no possible world in which φ and ψ are together true, φ and ψ are mutually exclusive. There is no standard notation for this relation; but, when it is convenient to have a symbol for it, we will write φ × ψ to say that φ and ψ are mutually exclusive. This use of the multiplication sign is intended to suggest crossing out of the possibility that sentences are both true.

Mutually exclusiveness is neither reflexive nor transitive like implication, but it does obey the following two laws:

The first of these notes the reversability of ×. The term contravariance in the second alludes to the fact that it tells us that exclusion is transferable by implication but in a direction opposite to the direction of the implication arrow. That is, if ψ excludes χ then anything φ is implied by also excludes χ. This law indicates that strength in sentences is important for mutual exclusiveness: indeed, to say that sentences are mutually exclusive is to say any case of weakness in one—any possibility left open—is made up for by the other.

Mutually exclusive sentences are opposed to one another, and they can be thought of as opposites. But there are different sorts of opposites. Some, like The glass is full and The glass is empty are extremes that may both fail in intermediate cases. Others, like The glass is full and The glass is not full cover all the ground between them and do not leave room for a third alternative.

The difference between these sorts of opposition is tied to another relation between sentences that we haven’t discussed yet. Sentences φ and ψ are jointly exhaustive when there is no possible world in which both are false, when there is no possible world that both rule out. If we put together the possibilities left open by such sentences, the result will include all possibilities because any possibility ruled out by one must be left open by the other; and, in this sense, these sentences jointly exhaust all possibilities. Such sentences certainly differ in meaning—as we will see later, they can be said to have no common content—but they are not opposites since they need not be incompatible. They might be thought of instead as complementary since, in regard to possibilities left open, each picks up where the other leaves off. We will use a simple large circle ◯ as our notation for this relation, with φ ◯ ψ intended to suggest that φ and ψ between them leave open the full range of possibilities.

Like exclusiveness, exhaustiveness is neither reflexive nor transitive but is symmetric. However, its difference from exclusiveness is reflected in the fact that it is not contravariant but is transferable by implication in a different way.

The second law claims a transferability of ◯ by implication that, unlike that for ×, follows the direction of the implication arrow (which is what the term covariance points to). That is, weakness in sentences is what is important for joint exhaustiveness: any point of strength in one of a pair of jointly exhaustive sentences—any possibility ruled out—is matched by a corresponding weakness in the other.

Although neither × nor ◯ is transitive, linking sentences by the two relations in either order does tell us something about the logical relations about the sentences at each end.

Notice that, because of the symmetry of × and ◯, saying that φ × ψ and χ ◯ ψ comes to the same thing as saying either that φ × ψ and ψ ◯ χ or that χ ◯ ψ and ψ × φ. The reason for the law is that, if φ is true and χ is false, then either φ and ψ are both true or χ and ψ are both false. In other words, if φ ⇒ χ fails to be true, then so must one of φ × ψ and χ ◯ ψ.

When sentences are not only mutually exclusive but also jointly exhaustive, they are opposed in the second way described above: since they cannot both be false, one or the other is bound to hold and there is no room for a third alternative. We will say that two sentences for which this is so are contradictory. Contradictory sentences—like The glass is full and The glass is not full—are bound to have opposite truth values. We will combine the notation for the two relations that make up this idea and write φ ⊗ ψ to say that φ and ψ are contradictory (using the symbol circled times). Although our use of the term contradictory is the standard one in discussions of deductive logic, in ordinary speech, it is often applied to sentences that are only mutually exclusive. In particular, when a claim is said to be self-contradictory, what is meant is that part of what it says excludes something else it says. Such a sentence will not contradict itself in the sense in which we will use the term because that would require that it be both true and false in each possible world. (Being true and false in each possible world is a problem only if there are possible worlds, but that’s an assumption we will make.)

Contradictoriness inherits the symmetry of exhaustivness and exclusion. We have just seen that reflexivity fails for it in an even stronger way than for them, and ⊗ also has a special property that implies a similarly strong failure of transitivity.

The second law follows from the alternation law for × and ◯. It is alo easy to understand directly: contradictory sentences have opposite truth values and taking the opposite (in this sense) twice over returns you to where you started. Although this sort of property can be found for other relations (mirror image of is one), there is no standard name for the particular form it takes here. But an operation which is undone when repeated a second time (like the operation of reversing course) is known as an involution (in one sense of the term), and the doubling law tells us that the operation of moving from the content of a sentence to the content of a sentence contradictory to it is an involution.

The four basic deductive relations between two sentences that we have considered are shown in the following table:

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. Ruling out more than one pattern does not give us many more relations. If we rule out the first two patterns, we say φ ⇔ ψ, and if we rule out the last two patterns, we say φ ⊗ ψ; but, if we were to rule out any other pair of patterns, we would simply rule out a truth value for one of the sentences in all possible worlds, and ruling out three patterns would leave just one pattern and would specify the truth values of both sentences. While the idea of a sentence that cannot be false or cannot be true is an important one, it is a property of the single sentence rather than a relation between two. So, in one sense, the six relations for which we have terminology are the only ones possible.

When none of these relations hold 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 tied by any deductive relation.