One of the end-of-chapter exercises in IFL2 is simply this:
The negation of a proposition can usually be unambiguously expressed by prefixing ‘It is not the case that’. Can you think of any exceptions?
I tweeted the question too, and got some minor variations on familiar examples. Here are my draft notes for the online suggested answers to exercises, slightly tweaked in the light of a couple of those variations. These are notes for beginners, remember: but any comments?
There are, trivially, examples which use a different sense of ‘case’ (as in luggage, or box of wine): thus compare the pair
- I usually pack for a weekend.
- It is not the case that I usually pack for a weekend.
and likewise
- I ordered from the wine shop yesterday.
- It is not the case that I ordered from the wine shop yesterday.
There’s a natural reading of (2) in which it isn’t the negation of (1), and a natural reading of (4) in which it isn’t the negation of (3).
Still, it might be said that those examples are cheats! – the intended question was surely: does ‘It is not the case that’ still used as meaning ‘it is not true that’, when prefixed to a sentence, always unambiguously express the negation of what the original sentence says? (To be sure, that wasn’t originally spelt out. But equally, to adapt an example of Wittgenstein’s, if I say “show the children a game” and you teach them strip-poker, I can reasonably enough complain “I didn’t mean that sort of game” even if I didn’t actually have that exclusion in mind when I made my request!)
So let’s set aside those examples. What about
- This sentence has less than ten words.
- It is not the case that this sentence has less than ten words.
At least in (6) ‘it is not the case that’ is being used with the intended meaning! However, this time, prefixing those words changes the topic of the whole claim by changing the reference of ‘this sentence’. Not so much of a cheat, perhaps, but still an anomalous case, we might reasonably remark: we were thinking of prefixing ‘it is not the case that’ in its usual sense, while also not changing the actual content of what follows.
So what about examples where ‘it is not the case that’ retains its equivalence to ‘it is not true that’, and the topic of the sentence it is applied to stays the same? How can prefixing ‘it is not the case that’ then fail to unambiguously negate what it is applied to?
One way is by it not being clear how much of what follows the prefix does apply to. Thus in IFL2, p. 65, we consider the pair:
- Jack loves Jill and it is not the case that Jill loves Jack.
- It is not the case that Jack loves Jill and it is not the case that Jill loves Jack.
In (8), on by far the more natural reading, the first ‘it is not the case that’ applies to just the clause ‘Jack loves Jill’. So both (7) and (8) are false if Jill does love Jack.
What is happening here is that ‘it is not the case’ hasn’t changed its meaning, isn’t changing the topic of what follows, but its scope isn’t (or at least, isn’t unambiguously) the whole of the sentence it is being applied to.
What about this sort of pair?
- Oedipus, who killed his father at the crossroads, was guilty of murder.
- It is not the case that Oedipus, who killed his father at the crossroads, was guilty of murder.
It might be said that both imply that Oedipus killed his father at the crossroads: so if Oedipus hadn’t killed his father, neither claim would have been true (but is this right? I don’t want to put any weight on this example!). However, if (9) and (10) can both fail to be true together, one is not the outright negation of the other. Again, this is like a scope phenomenon – the clause ‘who killed his father at the crossroads’ is insulated from the negation in (10).
We’ve mentioned reductio arguments. Sometimes we show that some assumption A has to be false by first arguing that if A then C, and then arguing that we also have if A then it isn’t the case that C. And since C and its negation can’t be true together, we conclude that A is false. (For example, the [IFL] Exercises 4 proof in effect shows that if √2 = m/n, a fraction in lowest terms, then m is even, and if √2 = m/n then m is not even, and concludes that√2 isn’t a fraction.)
Only slightly re-arranging, we have propositions of the form
- C, if A
- It is not that case that C, if A
proved true together. So one isn’t the negation of the other. Again, this is a scope phenomenon – on the natural reading, the initial ‘it is not the case’ doesn’t apply past the comma.
Are there examples where prefixing ‘it is not the case that’ (with the usual meaning) doesn’t negate what follows which aren’t to be explained as arising from scope phenomena? Well, what about this pair?
- Anyone can run a mile in four minutes.
- It is not the case that anyone can run in a mile in four minutes.
(13) is false: most of us are far too slow! But on one natural reading, (14) is false too – any elite middle distance runner can run a sub-four-minute mile.
Note though that, like (8), (14) is ambiguous; especially if the ‘anyone’ is stressed, it can also be construed as the simple negation of (13). And as we will see later in IFL2, the ambiguity here can also be thought of as of scope-ambiguity. We can represent the two readings of (14) like this:
- (Anyone is such that)(it is not the case that) they can run a mile in four minutes.
- (It is not the case that)(anyone is such that) they can run a mile in four minutes.
(15) represents the reading of (14) which is not the negation of (13). And thought of like that, we can regard the prefixed ‘it is not the case that’ in (14) as not really governing everything that follows it — i.e. we can treat this as another scope phenomenon.