1.4.2. Separation

The relation of implication can be characterized briefly by saying that φ ⊨ ψ when φ cannot be separated from ψ. We can do the same for entailment if we extend the idea of separation to say that a set Γ is separated from a sentence φ when the members of Γ are all true but φ is false. Then we can say that Γ ⊨ φ is true in just the cases where Γ cannot be separated from φ, and this way of thinking about entailment will reappear frequently throughout the course.

When we move from thinking of implication by a single premise to thinking of entailment by a set, we broaden our perspective to include inference from multiple premises. But reference to a set also admits the case that there are no premises or assumptions in question, for the set may be the empty set, which has no members. An argument that offers a conclusion without basing it on premises will be valid only if we have an unconditional guarantee that the conclusion is true—only if the conclusion is a tautology. We are provided no premises that could involve some falsehood, so any error in the conclusion is a new error, and a guarantee against new error must be a guarantee that it is true. And, if we look at validity in terms of content, we notice that there are no premises from which the content of the conclusion can be extracted, so it better have no content at all.

The same follows from our more formal definitions of entailment, but the idea may take a little while to get used to. First, let’s adopt some notation for referring to the empty set. We will often use the sign ∅, but we may also denote it by giving an empty list {} of members. The latter notation fits with a way we will often write claims of entailment by the empty set: we might say that ∅ ⊨ φ, by saying ⊨ φ—i.e., by writing an entailment sign with an empty list of assumptions to its left. Now the claim ∅ ⊨ φ or ⊨ φ will hold just in case there is no possible world that separates ∅ from φ. But what is to separate the empty set from a sentence? Well, the possible world would need to be one in which φ was false and every member of ∅ was true. So we need to know what it takes for every member of ∅ to be true. The short answer is it takes nothing at all because ∅ has no members; that is, in every possible world it will be the case that every member of ∅ is true. For, to assert that every member of a set is true is to assert a generalization, so this assertion will be true unless there is a counterexample to the generalization. We can make that more explicit by restating the clause every member of Γ is true as no member of Γ is false, and the latter is clearly true when Γ has no members at all. We will run into generalizations about empty collections of things in other contexts, so it may help to have a label for them: they are often described as vacuous generalizations. The allusion to a vacuum is based on the idea that such generalizations are empty of content because there are no counterexamples available for them to rule out.

Since the condition for separation that concerns ∅ is vacuous, all that is necessary for ∅ to be separated from a sentence φ is for φ to be false. So, to say that ∅ cannot be separated from φ (i.e., that ∅ ⊨ φ or ⊨ φ) is to say that φ cannot be false, that it is a tautology. This gives us a simple notation for tautologousness: we can say that φ is a tautology by saying that ⊨ φ. More importantly, it shows that we can study tautologousness by studying entailment because tautologousness is just a special case of entailment, namely, entailment by an empty set of assumptions.

The rest of this section will be devoted to showing how to study other deductive properties and relations by studying entailment, and a first step in doing that will be to extend the idea of separation further still. The properties and relations that I have been labeling deductive are ones that rule out certain patterns of truth values for the sentences they concern. An individual pattern of this sort will be a specification of truth values for certain sentences. It will make some (perhaps empty) set Γ of sentences all true and some set Σ all false, and we will say that in doing this it separates Γ from Σ. When Σ has a single member, this is just separation as we have been speaking of it, so we are now consideration that idea not only as a relation between a pair of sentences or between a set and a sentence but as one between any set and any other set. It can help, as it did in the case of entailment, to restate the requirement all members of Γ are true as no member of Γ is false and restate the requirement for Σ analogously. That is, separation in its full generality can be defined in either of the following ways, with the second one the clearest:

The requirement this places on a set is automatically satisfied when that set is the empty set ∅. That means that we can say:

Either way, we can see that the empty set is bound to be separated from itself. That particular consequence is only a curiosity, but more interesting, and a taste of things to come, is the use of this broader notion of separation to describe the relations between pairs of sentences discussed in 1.2. In particular, φ ▵ ψ if and only if {φ, ψ}, the set consisting of φ and ψ, cannot be separated from the empty set; for to say that {φ, ψ} cannot be separated from the empty set is simply to say that φ and ψ cannot be made both true. Similarly, φ ▿ ψ if and only if the empty set cannot be separated from {φ, ψ}.