1.4.8. Duality
In the context of relative exhaustiveness all that need be said about the logical properties of Tautology ⊤ and Absurdity ⊥ is that Tautology is a tautology (i.e., ⊨ ⊤) and that Absurdity is absurd (i.e., ⊥ ⊨). The first of these makes sense for entailment and, together with the basic laws of entailment, provides the basis for the sort of laws for ⊥ we will consider shortly. However, it is the latter laws that we will focus on since they state the role of ⊤ in entailment. And, in the case of ⊥, saying merely that it is absurd tells us nothing from the point of view of entailment since that is to say only that ⊥ ⊨ ⊥.
Tautology ⊤ is entailed by any set of premises (the empty set included) because it cannot go beyond the information contained in any set of sentences; and, for the same reason, the presence of ⊤ among the premises of an argument contributes nothing to the argument’s validity. These two ideas can be expressed more formally in the following laws.
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 consider in the last subsection because 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.
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. There is no law restating the significance of having ⊥ as a conclusion because that is simplest way we have of using entailment to say that a set of assumptions is inconsistent.
Although entailment will be our focus, it is enlightening to consider analogues for relative exhaustiveness of the laws just stated. In particular, we can state a law for ⊥ as an alternative in the context of relative exhaustiveness, and all the properties of ⊤ and ⊥ take a particularly symmetric form when stated in terms of that relation.
| as a premise | as an alternative | |
| Tautology | if Γ, ⊤ ⊨ Σ, then Γ ⊨ Σ | Γ ⊨ ⊤, Σ |
| Absurdity | Γ, ⊥ ⊨ Σ | if Γ ⊨ ⊥, Σ, then Γ ⊨ Σ |
That is, while ⊤ contributes nothing as a premise and may be dropped, it is enough for a claim of relative exhaustiveness to hold that it be an alternative (no matter how small the set Γ of premises or the set Σ of other alternatives). And while it is enough to have ⊥ as a premise (no matter how small the set of alternatives is), it contributes nothing as an alternative and may be dropped.
Notice that the converses of the principles at the upper left and lower right hold by monotonicity because they are just the addition of a premise in one case and an alternative in the other. If we take the if and only if principle that results from adding the converse to the lower right and consider a case where Σ is empty, we get
This is the principle for relative exhaustiveness that lies behind the law providing inconsistency via Absurdity of 1.4.6. The moral is that our use of ⊥ as a conclusion to define inconsistency in terms of entailment really involves the same idea as the principle for ⊥ as an alternative that may be stated for relative exhaustiveness.
The symmetry exhibited by the set of principles in the table above might be traced to the fact that ⊤ and ⊥ are contradictory since then having one as an assumption comes to the same thing as having the other as an alternative according to the law of 1.4.6 providing alternatives via assumptions. However, there is a more general idea behind this symmetry that will apply also to cases where sentences are not contradictory.
The essential difference between the lower left and upper right in the table above lies in interchanging ⊥ and ⊤ and, at the same time, interchanging premises and alternatives. And the same is true of the upper left and lower right. That is, if we apply this transition to the lower left, we get
and that differs from the upper right only in the order of the alternatives and the exchange of Σ for Γ. And neither of these differences is important. Alternatives function only as a set, so the order in which they are listed does not matter. And, since each of Γ and Σ could be any set, exchanging these labels does not alter the content of the principle. Either way, we say that it is enough to have ⊤ as an alternative no matter what premises and what further alternatives we have. The possibility of the sort of transformation used to get from the lower left to the upper right can be expressed by saying that ⊤ and ⊥ on the one hand and premise (or assumption) and alternative on the other constitute pairs of dual terms. We will run into other pairs of terms later that combine with these pairs in an even broader sort of duality.