phi_270_text_1_4

Most of our concern with deductive reasoning will not be with particular examples, but instead with general laws. Most of these will be generalizations about specific logical forms that will be introduced chapter by chapter, but some very general ones can be stated now. We will look first at relative exhaustiveness, the content of whose laws is the clearest, and then turn to entailment.

We will consider three basic principles for relative exhaustiveness, two of which are related to the laws of reflexivity and transitivity for implication that we considered in 1.2.2. For any sentence φ and any sets Γ, Δ, Σ, and Θ of sentences:

Repetition. Γ, φ ⇒ φ, Δ (for any sentence φ and any sets Γ and Δ).

Cut. If Γ ⇒ φ, Δ and Γ, φ ⇒ Δ, then Γ ⇒ Δ (for any sentence φ and any sets Γ and Δ).

Monotonicity. If Γ ⇒ Δ, then Γ, Σ ⇒ Δ, Θ (for any sets Γ, Δ, Σ, and Θ).

The repetition law tells that relative exhaustiveness holds whenever an assumption appears also among the alternatives. When this is so, the truth of the assumptions certainly guarantees the truth of at least one alternative. The reflexivity law for implication is the special case of this where the sets Γ and Δ are both empty, where the only alternative is also the sole assumption. Relative exhaustiveness itself is not reflexive in general, but there is only one counterexample. The empty set does not render itself exhaustive, but the cases of the repetition law where Γ and Δ are the same set tell us that all non-empty sets render themselves exhaustive.

The name of the cut law reflects the disappearance of the φ in the conclusion that is drawn. This is a very fundamental law, and instances and consequences of it (the transitivity of implication is one) are clearer in their import than the law itself. But, to see the import of this law in its full generality, notice that the relation Γ ⇒ φ, Δ implies that Γ guarantees that either φ or a member of Δ is true. But if φ is true, we know that a member of Δ will be true also (because Γ, φ ⇒ Δ). And this means that, given Γ, at least one member of Δ is bound to be true, which is what Γ ⇒ Δ says.

The idea behind monotonicity is that the truth of an instance of relative exhaustiveness can never be damaged by adding assumptions or alternatives. (The law mentions added sets of both assumptions Σ and alternatives Θ, but either of these might be the empty set.) If we add assumptions, we are narrowing the range of possibilities left open for the alternatives to exhaust; and, if we add alternatives, we are adding further ways of covering these possibilities. Either way, we are making it harder to find a counterxample to the claim of relative exhaustiveness. The term monotonic is applied to trends that never change direction. More specifically, it is applied to a quantity that does not both increase and decrease in response to changes in another quantity. In this case, it reflects the fact that adding assumptions will never lead to a decrease in the sets of alternatives rendered exhaustive and adding alternatives will never lead to a decrease in the sets of assumptions rendering them exhaustive.

The cut law and monotonicity combine to yield the transitivity of implication. For, if φ ⇒ ψ and ψ ⇒ χ, then both φ ⇒ ψ, χ and φ, ψ ⇒ χ by monotonicity, and we can cut ψ from these two to get φ ⇒ χ. However, relative exhaustiveness itself is not transitive. It is true that, if Γ ⇒ ψ and ψ ⇒ Δ, then Γ ⇒ Δ. But knowing that Γ ⇒ Σ and Σ ⇒ Δ for a larger set Σ is not enough to insure that Γ ⇒ Δ. With Γ ⇒ Σ we have a guarantee, given Γ, only that at least one member of Σ is true while Σ ⇒ Δ guarantees the truth of at least one member of Δ only given the truth of all members of Σ.

There is a sense in which cut and monotonicity are inverse principles since cut allows us to eliminate assumptions and alternatives while monotonicity allows us to add them. The nature of the inversion can be seen more clearly by considering a generalization of cut:

Generalized cut. If Γ, Σ ⇒ Δ, Θ and moreover Γ, Σ⁠′ ⇒ Δ, Θ⁠′ for all other non-overlapping sets Σ⁠′ and Θ⁠′ such that Σ⁠′ ∪ Θ⁠′ = Σ ∪ Θ, then Γ ⇒ Δ (for any sets Γ, Δ, Σ, and Θ).

This principle says that we can drop a group of premises and alternatives provided the relation holds no matter how they are distributed between assumptions and alternatives. The intuitive idea is that, if it does not matter what specific role these sentences play, they need not appear at all. The basic cut law applies this idea to a single sentence, and its application to any finite set follows from that law. For example, consider the case of two sentences. Putting together the following instances of cut (which cut ψ in the first two cases and φ in the third)

if Γ ⇒ ψ, φ, Δ and Γ, ψ ⇒ φ, Δ, then Γ ⇒ φ, Δ
if Γ, φ ⇒ ψ, Δ and Γ, φ, ψ ⇒ Δ, then Γ, φ ⇒ Δ
if Γ ⇒ φ, Δ and Γ, φ ⇒ Δ, then Γ ⇒ Δ

if Γ ⇒ ψ, φ, Δ and Γ, ψ ⇒ φ, Δ and Γ, φ ⇒ ψ, Δ and Γ, φ, ψ ⇒ Δ, then Γ ⇒ Δ.

That is, if relative exhaustiveness holds no matter how φ and ψ are added to assumptions Γ and alternatives Δ, then it holds without any addition. One reason for considering the generalized principle is that any relation that satisfies it together with repetition and monotonicity will be the relation of relative exhaustiveness corresponding to some set of possibilities, so these three principles encompass all there is to be said in general about relative exhaustiveness.