5.2.3. Unless

Yet another sort of conditional appears in this example:

Here the main clause is hedged, but in a different way than if the subordinate clause were introduced by if. The speaker’s intent is to leave open some cases where the main clause fails (where they still have food) but to limit this failure to the sort of situation described in the subordinate clause. We can compare the function of unless to the function of only if by paraphrasing the sentence above as

The second sentence limits the truth of They still have food to cases where The received new supplies is true. So it asserts the truth of They have run out of food with the possible exception of such cases. Similarly, the first sentence asserts They have run out of food but hedges this by allowing the exception expressed by the subordinate clause.

φ	ψ	ψ	unless	φ
T	T		T
T	F		T
F	T		T
F	F		F

So, like an only-if-conditional, an unless-conditional is automatically true in cases where the subordinate clause is true; but unlike an only-if-conditional its truth value is the same as the main clause in cases where the subordinate clause is false. That is, the form ψ unless φ has the table at the right. This account of truth conditions appears also in Figure 5.2.3-1. Continuing the example of these diagrams, 5.2.3-1B represents the proposition expressed by The number shown by the die is less than 4 unless it is odd.

There are two ways of describing the proposition on the right. First of all, it fences in the failure of the main clause. That is, it rules out some of the possibilities in which the main clause ψ fails, those that are ruled out by the subordinate clause φ. This is to see the conditional as the proposition expressed by ¬ ψ only if φ. But the possibilities left open by the denial of the main clause are those ruled out by the main clause itself. So the conditional can be seen also to whittle down the possibilities ruled out by the main clause to those left open by the denial of the subordinate clause. And this is to see the conditional as the hedge of the main clause expressed by ψ if ¬ φ.

The same restatements appear if we trace our way back to if-conditionals in order to get a way of expressing this conditional symbolically. The form ψ unless φ amounts to ¬ ψ only if φ and we are treating the latter as ¬ ¬ ψ ← ¬ φ. If we use the principle of double negation to simplify this last expression, we get ψ ← ¬ φ as a rendering of ψ unless φ. The corresponding English paraphrase of ψ unless φ as ψ if it is not the case that φ is usually pretty good (good enough that if not is a common dictionary definition of unless).

The negation used to analyze the subordinate clause of only-if-conditionals means that they are covariant with both their clauses. That will be no surprise if you have noticed that they have the same truth conditions as disjunctions, but it is also to be expected if it is regarded as an assertion of the main clause with the subordinate clause as a possible exception. Such a claim will say more as the main clause says more, and it will say more also as the subordinate clause says more because a narrower exception will apply in fewer cases.

There are enough steps in the path from unless to ← ¬ to justify a fear that the implicatures are not all in order when we arrive, but this account of unless works better than using or. How far the synonymy of unless and or extends beyond truth conditions can be seen by considering a few examples. We might paraphrase the example above as

and we would do so with reasonable success. But things do not work out as well in other cases, particularly with unless-conditionals concerning the future. The following two sentences have quite different implicatures:

Disjunction is not symmetric when it comes to an implicated connection between its two components, and we could paraphrase the first sentence better by We’ll either get to a town soon or run out of gas, but the need to change the order of the clauses reduces the advantages of or over if not as a paraphrase of unless.

The remaining issues regarding unless pretty well parallel those concerning if and only if. It is possible to find an unless-clause at the front of a sentence (e.g., Unless we get to town soon, we’ll run out of gas). And the form ψ unless φ has, in addition to its core implicature that the truth of φ is necessary for the falsity of ψ, a secondary and easily canceled implicature of sufficiency. In our initial example (They have run out of food unless they received new supplies), this secondary implicature is rather weak if it is present at all, so there might be no need to add the canceling clause and they might have run out even if they got them. But, in other cases, the implicature is stronger. For example, in We’ll go unless it rains, we would have to add and we might go even if it does if we did not want to suggest that rain would be enough to keep us from going.