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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.

Notice that ⊤ is entailed by any set of premises because it will not add information to any set of sentences; and, for the same reason, its presence among the premises contributes nothing to the validity of an argument.

Law for ⊤ as a conclusion. Γ ⇒ ⊤ (for any set Γ).

Law for ⊤ as a premise. Γ,⊤ ⇒ φ if and only if Γ ⇒ φ (for any set Γ and sentence φ).

Although they are stated for ⊤, these laws will hold for all tautologies since they hold simply in virtue of the proposition expressed by ⊤.

These laws are different in character from the ones we have been considering since they concern the logical properties of a specific sort of sentence rather than the general principles governing logical relations. They are also a first sample of a common pattern in the laws of deductive reasoning that we will consider. Entailment is so central to deductive reasoning that an account of the role of a kind of sentence in entailment as a conclusion and as a premise will usually tell us all we need to know about it.

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. 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. Also, 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 simple law describes the role of absurdities as premises. We state it for the specific absurdity ⊥.

Law for ⊥ as a premise. Γ, ⊥ ⇒ φ (for any set Γ and sentence φ).

An argument with an absurdity among its premises is valid by default. Since its premises cannot all be true, there is no risk of new error no matter what the conclusion is.

We will eventually have a law for ⊥ on the right of the sign ⇒, but that will come only once we have assigned a broader meaning to that arrow. The idea of an entailment with an absurd conclusion is a fundamental one and cannot be restated in any simpler way using entailment. Since ⊥ is bound to be false, we can have Γ ⇒ ⊥ only it is not possible for the premises Γ to all be true. That is, we have the following:

Basic law for inconsistency. A set Γ is inconsistent if and only if Γ ⇒ ⊥.

This characterization of inconsistency will help us to concentrate on entailment. Laws governing inconsistency—and, by way of it, principles governing exclusion and mutual exclusiveness—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 simple example.