4.2.2. Proving disjunctions

Now let us look at disjunctions as conclusions. An entailment Γ ⊨ φ ∨ ψ will hold if and only if φ ∨ ψ is true in every possible world in which all members of Γ are true. But this is to say that at least one of φ and ψ is true in every such world, and that is a way of saying that Γ renders φ and ψ jointly exhaustive. So we can state the following principle:

Since the right-hand side has two alternatives, this is not a law concerning entailment alone, and we will not take the principle in this form as our account of the role of disjunctions as conclusions. However, we can use the basic law for conditional exhaustiveness to restate the right-hand side as claim of entailment. Indeed we have two ways of doing that. If φ and φ′ are contradictory, we can say

and if ψ and ψ′ are contradictory, we can say

In short, a disjunction is a valid conclusion from premises Γ if and only if adding to our premises a sentence contradictory to one disjunct enables us to validly conclude the other disjunct.

In stating a principle for disjunction we will limit ourselves to cases where a sentence and its negation are the pair of contradictory sentences. But, when the disjuncts are already negative, that leaves us with two choices for each of the pairs φ and φ′ and ψ and ψ′ since each of φ′ and ψ′ might be the result of either adding or dropping a negation. To avoid stating four principles to cover each of these possibilities, we will introduce some notation to capture the general idea of obtaining a contradictory sentence by either adding or dropping a negation. Let the sentence ¬^± φ be the result of negating φ with an optional added step of deleting a double negation if φ was already negative. Then ¬^± φ will stand for ¬ φ when φ is not a negation and, when φ is the negation ¬ χ, it will stand for either ¬ ¬ χ or χ. That is, ¬^± φ is the result of either negating or, perhaps, de-negating φ, which means that ¬^± φ will either be the negation of φ or have φ as its negation.

Then ¬^± φ and φ form a contradictory pair consisting of a sentence and its negation in one order or the other, so we may formulate a principle to account for conclusions that are disjunctions with only two statements:

When these are implemented as derivation rules, they give us two ways of planning for a disjunctive goal.

The two rules are shown as alternative developments in Figure 4.2.2-1. We will refer to both forms of the rule as Proof of Exhaustion (PE) since it is a way of showing that φ and ψ, taken together, exhaust all possibilities left open by the premises.

In each way of developing a gap, we set one of the components of the disjunction as a new goal and add the negation or de-negation of the other component as a supposition. In each way of developing a gap, we set one of the components of the disjunction as a new goal.

Both forms of planning will lead to the same answer in the end, but one or the other may be more efficient in a particular case. There is no simple way of predicting which choice is best but the following rules of thumb may help:

In many cases none of these suggestions will apply; but, in most such cases, neither one of the two forms of the rule is better than the other.

As an example of this rule, consider the argument below, understanding X was out to be the denial of X was home. The validity of this argument can be established by the English derivation whose first stage is shown at the right.

The overall form is that of a argument that we will call hypothetical (for reasons discussed below) in which we suppose that Ann was at home (a supposition that is one of the two possibilities for ¬^± ¬ A) and establish under this supposition that Bill was out. This shows the connection between Ann being out and Bill being out that we claim when we state, outside the scope of the supposition, that at least one was out.

Notice that if we continue the derivation

we plan for the goal ¬ B by supposing B for reductio. And this example illustrates the different functions of the two sorts of supposition. We suppose that Ann is home in order to show that ¬ B (i.e., Bill is out) is true in all possible worlds in which ¬ A (i.e., Ann is out) is false. We go on to show that ¬ B is true in these cases by showing that to suppose further that B would rule out all possibilities—i.e., that this supposition would be absurd when added to our premises and the supposition A. From one point of view, both suppositions are merely added assumptions. But we add the first in order to show that to accept the second would be to go too far. That is, we add the second in order to show that we cannot accept it given the first, and we add the first to show that the second is related to it in this way.

To complete the derivation, we might exploit the first premise by CR, and this is the only way to proceed using basic rules. Doing this would make the conjunction (A ∧ B) ∧ ¬ C our goal; and, since its components are all resources, it is clear that the gap would close. But, seeing this, we might choose instead to derive that conjunction by Adj.

Either way we are completing the reductio, in one case under the guidance of the rules and in the other under our own direction.

As noted above, the supposition in PE may be described as hypothetical, and this indicates the role it plays, a fourth role on top of those we have seen in Lem and LFR, in RAA and IP, and in PC. In RAA and IP, we make suppositions with the aim of showing that they are false. In Lem and LFR, we make a supposition to investigate the consequences of a claim that we plan to separately show to be true. In PC, we make a pair of suppositions, having already shown that at least one is true. In PE on the other hand, a supposition is made with no expectation of either truth or falsity. It is made instead simply to establish a connection between it and some other claim. As we argue within the scope of the supposition, we are making a hypothetical argument, an argument made under a hypothesis. The conclusion we draw when we discharge the supposition states a connection between the hypothesis and the conclusion of the hypothetical argument. This statement no longer falls under the supposition, and that can be indicated by saying that it is stated categorically.

In each of the two forms of PE, shown here as a rule for tree-form proofs,

the conclusion after the hypothetical argument says that at least one of two sentences is true. This is to say that, if one is false, the other is true. And this is the connection established between the supposition and conclusion of the hypothetical argument in each form of the rule.

There is some danger of getting tangled in the terminology here, so let’s pause and look at it more closely. The terms hypothetical and categorical derive from an ancient classification of sentences into the categorical, the disjunctive, and the hypothetical. Since disjunctions and hypothetical sentences (the conditionals to be studied in the next chapter) are ways of hedging claims, the term categorical has acquired the meaning ‘unhedged’. Now the disjunctive goal to which we applied this term above certainly hedges each of its components, so it does not state them categorically. But, while sentences along the scope line of the hypothetical argument are stated only under a hypothesis—that is, under the supposition of the hypothetical argument—the disjunction following the argument is no longer hedged in this way. That means it is stated categorically with respect to that supposition (though it may still fall in the scope of earlier ones). In short, when the scope line of a hypothetical argument ends, we move from hedged assertion of some claim (in the example above, the assertion of ¬ B under the hypothesis A) to unhedged assertion of a claim that incorporates a hedge (i.e., ¬ A ∨ ¬ B in the example).