In addition to ‘only if’, another conditional term in English that causes problems of translation is ‘unless’. This differs from ‘only if’ in that embeds a negation, and so it is akin to the complex ‘neither … nor’ connective we encountered in earlier chapters. Anytime a negation combines with another connective, we have trouble, and this is no different. In what follows, I present the orthodox view championed by the book, along with reasons why that might appeal. The authors suggest that seeing this interpretation takes only “a moment’s thought”, although they are much smarter than me, and their moment is my hour (or more). I then proceed to suggest a different interpretation that might square more with your intuitions. (FWIW, you need to go with the orthodox interpretation in your problem sets and on the exams…)
I. The Orthodox View
According to our authors, “Q unless P” and “Unless P, Q” should be translated as “¬P → Q”. Thus, “Q unless P” should be read, “Q if not P”. They motivate this in 7.1 by considering an example similar to the following:
1. I’ll eat cake, unless I’m full.
Here the idea is that you fully intend to eat cake and you will do so if you are not full; that is, if you are not full, then you’ll eat cake, or “¬Full → Eat Cake”. Represented in a table, we get the following interpretation, along with a telling equivalence for comparison:
On this interpretation, “Q unless P” turns out to be logically equivalent to “Q or P”—either I’ll eat cake or I’m full or both. (We’ll come back to the “or both” part in a moment.)
Let’s examine the logic of this expression. First, on the assumption that we are dealing with a conditional—that is, (at least) one of the conditions is a condition on the other—then it would appear that we must add a negation. Consider: If I eat cake, then I am full; if I am full, then I eat cake. The first of these is clearly out of step with (1), and the second seems equally out of step. (1) suggests that there is something incompatible about eating cake and being full, that we shouldn’t expect to get both at the same time. (Or perhaps not—see the next section.)
Second, we’re not dealing with two negations. If I’m not full, then it certainly seems to follow from the truth of (1) that I eat cake and not that I refrain from cake eating; alternatively, given the truth of (1), if I refrain from cake eating, then that should mean that I’m full and not that I’m not full. Interestingly, this conclusion about two negations follows from our conclusion about no negations, given that “P → Q” is logically equivalent to “¬Q → ¬P” by contraposition, and likewise for “Q → P” and “¬P → ¬Q”, and the fact that there are no other such combinations. (Prove this to yourself using the truth tables.)
So we can conclude that if we are dealing with a conditional, there must be only one negation. Given this assumption, the question is how to arrange P and Q around these two connectives. There appear to be four options:
The book tells us that the first of these is the one we want. What can be said about the others? Let’s take them in turn. First, consider (B). This is the contraposition of (A), and as we see in the above truth table, it is logically equivalent to (A). Thus, unsurprisingly, we find that it works to say that if I don’t eat cake, then I was full. This is equivalent to saying that if I wasn’t full, then I eat cake. In both of these cases, I’m saying that it suffices for my eating cake that I am not full.
Moving to (C) and (D), we note that they are contrapositions of each other, and so are logically equivalent as well. So the question really boils down to this: should we include “Q → ¬P” in our interpretation of “Q unless P”? Our book tells us that “a moment’s thought” should tell us “no”, but I am not so sure. If you agree, read on…
II. An Unorthodox Alternative
The Orthodox View implies that if (1) is true—I will eat cake unless I’m full—then if I’m not full, I’ll eat cake (or what is the same thing, if I don’t eat cake, then I must be full). But it does not imply that if I’m full, I won’t eat cake (or what is the same thing, if I eat cake, then I must not be full). Does this make sense?
To see what the Orthodox View denies, return to the truth table. Note that according to that interpretation, it is possible for the sentence “Q unless P” to be true in row 1, where both Q and P are true—this is implied by the fact that the sentence is logically equivalent to “Q or P”. But that means that on the orthodox interpretation, it is possible for (a) it to be true that I will eat cake unless I am full, (b) I’m full, and (c) I eat cake anyway! Intuitions may conflict here, but there are certainly strong intuitions that (b) and (c) should not be consistent with (a). According to these intuitions, we should translate “Q unless P” as “¬P ↔ Q”, or as the conjunction of (A) and (D) above—this would support the idea that if I’m not full, I eat cake, and if I eat cake, then I am not full.
Our authors acknowledge that these intuitions are strong (pp. 187-9), but they recommend interpreting the piece of meaning represented by (D) above as a conversational implicature. Going back to where they first introduce ‘unless’, we find that they argue that “Q unless P” and “Q if not P” are “false in exactly the same circumstances” (p. 180), namely, when both Q and P are false (i.e., row 4 of our truth table). The contrary intuitions under scrutiny here suggest that this is not the case—that “Q unless P” is also false when both Q and P are true (i.e., row 1 of our truth table).
I am on record arguing that we may choose to tell a semantic story that is out of alignment with our intuitions for broadly theoretical reasons, so long as we have the machinery to adjust the overall significance of the sentence to our intuitions. So it might be required that to get the full story here, we must look up from our six-letter word and look around a bit. Still, if this is going to work, the conversational implicature story should be compelling, or at least plausible. Recall that on this story, it should be possible to cancel the additional piece of meaning without contradiction. In this case, that means asserting “If not P, then Q” while denying “If Q, then not P” in the same breath. Using the sentence, “Max is at home unless Claire is at the library,” our authors tell us that “it makes perfectly good sense” for a speaker to say something like this: “Max is at home unless Claire is at the library … On the other hand, if Claire is at the library, I have no idea where Max is” (p. 189). But is this really a cancellation? After all, having no idea where Max is would be consistent with Max not being home, as required by (D). Thus, it appears that this doesn’t really cancel the additional piece of meaning after all. Instead, we need something stronger—something like this: “Max is at home unless Claire is at the library … On the other hand, if Claire is at the library, then Max is home.” Following the authors’ regimentation, this would be equivalent to “(¬P → Q) ⋀ (P → Q)”, where the second conjunct explicitly contradicts what they take to be an implicature. But whereas their “cancellation” seems fine, this real cancellation is much less compelling. Perhaps this works as an argument for the unorthodox alternative. Thoughts?