1.4.2. Division

The pattern of truth values for premises and conclusion that is ruled out by entailment (i.e., true premises with a false conclusion) will recur often enough that it will be convenient to have special vocabulary for it. Let us say that a set Γ is divided from a set Σ whenever all members of Γ are true and all members of Σ are false. Whatever gives the sentences in Γ and Σ such values will be said to divide these sets. The source of the truth values will differ from context to context though, for the time being, it will be a possible world. When there is something of the appropriate sort that divides a set Γ from a set Σ, we will say that Γ and Σ are divisible; otherwise we will say they are indivisible.

Notice that these ideas are asymmetric. When one set is divided from another it is the members of the first set that true and the members of the second that are false. You might think of sets being divided vertically, with the first set above the second. In this spatial metaphor, truth is thought of as higher than falsehood; and, although this is only a metaphor, it is a broadly useful one and is consistent with the appearance of Absurdity at the bottom of Figure 1.2.5-2 and Tautology at the top. The asymmetry of division is especially important to remember in the case of the terms divisible and indivisible since this way of expressing the idea could suggest a symmetric relation between the results of a division.

As with talk of sets of sentences as premises, it is really only the list of members of a set that we care about here, and we speak of sets only because the order of the list and the occurrence of repetitions in it do not matter. In particular, we will not distinguish between a sentence and a set that has only it as a member. So we can restate the negative definition of entailment as follows:

We will also say that an argument is divided when its premises are divided from its conclusion, so we can say that an argument is valid when no possible world divides it. So to say that a possible world divides an argument is to say that the world is a counterexample to the argument’s valdity. The divisibility or indivisibility of an argument thus amounts to the existence or non-existence of such a counterexample.

It can help when thinking about cases of division where one or both of the sets Γ and Σ is empty 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, the most generally useful form of definition of division is this:

Notice that 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 in particular that the empty set is bound to be divided from itself. This consequence is no more than a curiosity, but it serves to emphasize that we are using the term divides in a rather special sense.