phi_270_text_1_2

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. When sentences φ and ψ are related in this way, it is equally true that ψ excludes φ. This reversability of this relation is reflected in the usual terminology for it: when there is no possible world in which φ and ψ are together true, φ and ψ are said to be mutually exclusive. There is no standard notation for the relation, and we will shortly have a way of expressing it in terms of entailment; but, when it is convenient to have special notation, we will write φ ▵ ψ to say that φ and ψ are mutually exclusive. This use of the up-pointing triangle is intended simply to reflect the shape of signs for some related ideas. One of these related ideas is Absurdity. In particular, notice that sentences φ and ψ are mutually exclusive if and only if they form an inconsistent set—that is, if they together entail ⊥.

Mutually exclusive sentences provide one example of the differences in propositions that made for the horizontal spread of the logical space of Figure 1.2.5-2. Indeed, one of the examples cited there, the sentences The package will arrive next Wednesday morning and The package will arrive next Wednesday afternoon was a pair of mutually exclusive sentences. Mutually exclusive sentences differ to the extent that there is no overlap in the possibilities they leave open. From one point of view, that is a pretty considerable difference; but, as this example illustrates, such sentences can still have a lot in common. And, in general, sentences that rule out many possibilities may express propositions that divide the space of possibilities in very similar ways even though they have no overlap in the ones they leave open.

The diagrams below depict mutually exclusive sentences φ and ψ. Notice that in none of the three regions are both true. The diagram on the left shows that the two sentences taken together rule out all possibilities. In this sense, the relation of mutual exlcusivity is an indication of the strength of the two taken together: any possibility not ruled out by one is ruled out by the other. In the diagram on the right, we see the lack of overlap in the possibilities left open.

Fig. 1.2.6-1. The relation betwen mutually exclusive sentences φ and ψ, depicted on the left in terms of the possibilities each rules out and, on the right, in terms of the possibilities each leaves opens. (The region in the middle, where both φ and ψ are false, is a feature of this example and is not required for sentences to be mutually exlucsive.)

The region in the middle of the diagram could be contracted to a line and the sentences would still be mutually exclusive, for then the sentences would still combine to rule out all possibilites and would still show no overlap in the possibilities left open.

This suggests a distinction that may be made among pairs of mutually exclusive sentences. All 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, and the example depicted above is like this. 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. Opposites of the latter sort might be described as exactly opposite.

The difference between these sorts of opposition is tied to another way in which sentences can differ. 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—since there is no overlap in the possibilities they rule out, they can be said to have no common content—but they are not opposites in the sense of being 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 down-pointing triangle ▿ as our notation for this relation, as in the case of ▵ because of the similarity in shape between ▿ and some ideas related to joint exhaustiveness. (Tautology is one of these ideas but we will not consider the relation between it and joint exhaustiveness until 1.4.)

In the diagrams below, the absence of common content is depicted on the left. The regions consisting of possibilities ruled out, which indicate the contents of the two sentences, show no overlap. On the right, we see how the regions of possibilities left open by the two sentences combine to exhaust all possibilities whatsoever.

Fig. 1.2.6-2. The relation betwen jointly exhaustive sentences φ and ψ, depicted on the left in terms of the possibilities each rules out and, on the right, in terms of the possibilities each leaves opens. (The region in the middle, where both φ and ψ are true, is a feature of this example and is not required for sentences to be jointly exhaustive.)

As with the depiction of mutually exclusive sentences, the region in the middle could be contracted to a line. Notice that, if it was, the two sentences would not be not only jointly exhaustive but also mutually exclusive since every possibility would be ruled out by one or the other and there would be no overlap in possibilities left open.

Fig. 1.2.6-3. The relation betwen sentences φ and ψ which are contradictory—i.e., both mutually exclusive and jointly exhaustive—depicted on the left in terms of the possibilities each rules out and, on the right, in terms of the possibilities each leaves opens.

When sentences are not only mutually exclusive but also jointly exhaustive, one or the other is bound to hold—i.e., there is no space between the possibilities left open by the two—and there is no room for a third alternative between the possibilities they rule out. 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 write φ ⋈ ψ to say that φ and ψ are contradictory (using the symbol bowtie). (You might think of the symbol as indicating that things get turned upside down when moving from one sentence to the other.)

Although our use of the term contradictory is the standard one in discussions of deductive logic, in ordinary speech this term 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, and that cannot happen if there are any possible worlds at all (an assumption we can feel safe in making).

Just as the propositions expressed by logically strong sentences need not be far different even when they are mutually exclusive, the propositions expressed by logically weak sentences need not be far different even when they are jointly exhaustive. It is contradictory sentences that provide the true extreme examples of difference. When logical space in Figure 1.2.5-2 is thought of in three dimensions, the contradictory sentences appear in diametically opposite positions. Notice that mutually exclusive sentences cannot both appear above the middle level (for such sentences leave open more than half the possibilities), and jointly exhaustive sentences cannot appear both below the middle. Contradictory sentences fall under both restrictions. A pair of contradictory sentences might both appear on the middle level, but it is also possible for one to be of more than average logical strength if the other is relatively weak. The extreme case of this is provided by ⊥ and ⊤, which are contradictory. They also constitute the only example of a contradictory pair one of which implies the other (because ψ can be both true in any case where φ is and also false in any such case only if φ can never be true at all).