1.2.4. Tautologies and absurdities

There are two extreme examples of truth conditions or propositions. A sentence that is true in all possible worlds says nothing. It has no informational content because it leaves open all possibilities and rules nothing out. For example, the weather forecast Either it will rain or it won’t has no chance of being wrong and is, therefore, completely worthless as a prediction. We will say that such a sentence is a tautology. Although there are many (indeed, infinitely many) tautologies, all express the same proposition; and the words that they use to express it are beside the point since they all say nothing in the end. In short, any two tautologies are logically equivalent. It will be convenient to establish a particular tautology and mark it by special notation. We will call this sentence Tautology and use the sign ⊤ (down tack) as our notation for it. Since the logical properties and relations we will consider depend only on the propositions expressed by sentences, any logical property or relation of ⊤ will hold for all tautologies, and we will often simply speak of ⊤ in order to say things about tautologies generally.

At the other extreme of truth conditions from tautologies are sentences that rule out all possibilities. The fact that such a sentence is the opposite of a tautology might suggest that it is maximally informative, but it sets an upper bound on informativeness in a different way: any genuinely informative sentence must say less than it does. The ultimate aim of providing information is to narrow down possibilities until a single one remains, for this would provide a complete description of the history of the universe. To go beyond this would leave us with nothing because there is no way to distinguish among possibilities if all are ruled out. For example, the forecast It will rain, but it won’t is far from non-committal since it stands no chance of being right, but it is no more helpful than a tautologous one.

Sentences that rule out all possibilities make logically impossible claims, and we will refer to them as absurd. As was the case with tautologies, any two absurd sentences are logically equivalent. So, as with tautologies, we will introduce a particular example of an absurdity, named Absurdity, and we use the special notation ⊥ (the perpendicular sign, or up tack) for it.

A tautology is implied by any sentence φ since, as it rules out no possibilities, it cannot rule out any possibility that is left open by φ. The sentence ⊤ is thus the weakest sentence there could be and it can stand at the top of any ordering by logical strength like that depicted in 1.2.3. Analogously, an absurd sentence implies all sentences, and the sentence ⊥ can stand at the bottom of any ordering by logical strength.

Any sentence implying ⊥ is thus equivalent to it and is itself absurd. More generally, the idea of entailing ⊥ provides way characterizing inconsistency. That is, we can have Γ ⊨ ⊥ only when it is not possible for the premises Γ to all be true, and premises that cannot all be true will entail any conclusion, including ⊥. This characterization of inconsistency in terms of entailment will help us to keep our focus on entailment. Laws governing inconsistency—and, by way of it, principles governing related ideas like exclusion—will appear as principles governing valid arguments with the conclusion ⊥. In fact, we are not really dispensing with the idea of inconsistency since an absurdity amounts to a sentence that forms an inconsistent set all by itself. The role of entailment will be to enable us to study the full range of inconsistent sets by way of this one very special example.