1.4.6. Laws for relative exhaustiveness and entailment

Most of the laws of deductive reasoning we will study will be generalizations about specific logical forms that will be introduced chapter by chapter, but some very general laws can be stated at this point. We have already seen a couple of these, the principles of reflexivity and transitivity for implication. And, in fact, more general forms of these principles are at the heart of the basic laws for entailment and relative exhaustiveness.

Both entailment and relative exhaustiveness differ from implication by covering cases of multiple assumptions or no assumptions at all, and relative exhaustiveness similarly flexible with regard to alternatives. This difference is mainly reflected in form taken by the principles analogous to reflexivity and transitivity, and we will turn to them shortly. In addition to those principles we need only say that adding assumptions or alternatives never undermines a claim of entailment or relative exhaustiveness. This idea can be stated formally as follows, where Σ and Θ are the added assumptions and alternatives, respectively:

Although the principle for entailment is stated separately, it is just the special case of the first where the set Δ has φ as its only member and the set Θ is empty.

The idea behind monotonicity is that adding assumptions or alternatives can only make it harder to find a possible world that divides one group from the other, so, if no possible world will divide Γ from Δ, we can be sure that no world will divide the (possibly) larger set of assumptions we get by adding the set Σ from the (possibly) larger set of alternatives we get by adding the set Θ. (The groups are only possibly larger because either set of additions might be empty.)

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 by them and adding alternatives will never lead to a decrease in the sets of assumptions rendering them exhaustive.

It is a distinguishing characteristic of deductive reasoning that such a principle holds. For, when reasoning is not risk free, additional data can show that a initially well-supported conclusion is false and do so without undermining the original premises on which the conclusion was based. If such further data were added to the original premises, the result would no longer support the conclusion. This means that risky inference is, in general, non-monotonic in the sense that additions to the premises can reduce the set of conclusions that are justified. This is true of inductive generalization and of inference to the best explanation of available data, but the term non-monotonic is most often applied to another sort of non-deductive inference, an inference in which features of typical or normal cases are applied when there is no evidence to the contrary. One standard example is the argument from the premise Tweety is a bird to the conclusion Tweety flies. This conclusion is reasonable when the premise exhausts our knowledge of Tweety; but the inference is not free of risk, and the conclusion would no longer be reasonable if we were to add the premise that Tweety is a penguin.

Now let us turn to principles analogous to the principles of reflexivity and transitivity for implication. Whenever we have an implication φ ⊨ ψ, monotonicity tells us that we also have φ, Σ ⊨ ψ, Θ for any assumptions Σ and alternatives Θ. Applying this idea to implications given by reflexivity (and adjusting the notation), we have the following principle:

The principle for entailment here has been given a different name to reflect its use in proofs, but it is again a special case of the principle for relative exhaustiveness. The idea in both cases is just the same as that behind reflexivity: since there is no danger of a sentence φ being false given that it is true, there is no way to divide a set of assumptions containing φ from a set of alternatives in which φ also appears.

Before looking at the analogues to transitivity for relative exhaustiveness and entailment, let us look a little more closely at transitivity itself. Recall, that it says that if φ ⊨ ψ and ψ ⊨ χ, then φ ⊨ χ. This comes to the same thing as saying that if φ ⊭ χ then either φ ⊭ ψ or ψ ⊭ χ. And we can justify it in this latter form by noting, first, that if φ ⊭ χ, there is a possible world that divides φ from χ and this world must also make ψ either true or false. If the world makes ψ false, it will divide φ from it, showing that φ ⊭ ψ, and, if it makes the ψ true, it divide it from χ, showing that ψ ⊭ χ. So, when φ ⊭ χ, we are bound to have either φ ⊭ ψ or ψ ⊭ χ; and if φ ⊨ ψ and ψ ⊨ χ (so we have neither φ ⊭ ψ nor ψ ⊭ χ), we must have φ ⊨ χ.

When we consider division of sets rather than just sentences, the same idea gives us the following principles:

Any world that divides a set Γ from a set Δ must assign some truth value to the sentence φ; and, depending on the value it assigns, it will either divide the set Γ together with φ from Δ or it will divide Γ by itself from φ together with the set Δ. If we take the special case where the set Δ is a single sentence ψ, we get something like the second principle—in particular, if Γ, φ ⊨ ψ and Γ ⊨ φ, ψ, then Γ ⊨ ψ—and the second principle follows by monotonicity since knowing that Γ ⊨ φ is enough to tell us that Γ ⊨ φ, ψ.

The name cut simply reflects the appearance of the principle for relative exhaustiveness: we drop reference to φ in moving from the left side to the right. The name of the principle for entailment again reflects a use of the principle in proofs. The term lemma can be used for a conclusion that is drawn not because it is of interest in its own right but because it helps us to draw further conclusions. This law tells us that if we add to our premises Γ a lemma φ that we can conclude from them, anything ψ we can conclude using the enlarged set of premises can be concluded from the original set Γ. Or, to put it in a way that emphasizes its relation to cut, we can drop from a set of premises any sentence that is entailed by the remaining premises.

Although the cut law and the law for lemmas are based on the same idea as the principle of transitivity, they do not themselves assert transitivity for the more general relations they concern (though transitivity for implication follows from them using monotonicity). We should not expect relative exhaustiveness to obey any principle like transitivity because the significance of sets on the left and right of the turnstile is so different: while Γ ⊨ Δ tells us that the truth of all members of Γ guarantees the truth of at least one member of Δ, something like Δ ⊨ Σ would only guarantee the truth of at least one member of Σ if we were assured of the truth of all members of Δ. And a true transitivity principle would not even make sense for entailment because it is relation between different sorts of things (sets and single sentences) so individual cases of entailment usually cannot be linked in a chain.

However, we can state a principle that amounts to the transitivity of a relation closely tied to entailment.

This principle follows from the same line of reasoning as the cut law and the law for lemmas: if a possible world divides Γ from ψ, it must also either make all members of Δ true, in which case it divides Δ ⊨ ψ, or make some member Δ false, in which case it divides Γ from that member of Δ and Γ would not entail every member of Δ.

The name of this principle reflects its role as a principle of transitivity for a relation that holds between sets Γ and Δ when Γ entails every member of Δ. Let us call this set entailment. The chain law tells us that, if Γ entails each member of Δ and Δ entails each member of Σ, then Γ entails each member of Σ. Set entailment is also reflexive since the law for premises tells us that any set Γ entails each of its members. Since, in fact, the chain law can be combined with the law for premises to yield both the law for lemmas and the monotonicity of entailment, principles stating the reflexivity and transitivity of set entailment might be thought of as the basic principles of entailment.