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 Δ whenever all members of Γ are true and all members of Δ are false. Whatever gives the sentences in Γ and Δ such values will be said to divide these sets.

Notice that this idea is 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 1.2.6-2 and Tautology at the top.

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.