For many months now this column has been examining logical/mathematical paradoxes. Strictly speaking, a paradox is a kind of argument – for example, in some of my academic work I define paradoxes as follows:
A paradox is an argument that:
- Begins with premises that seem uncontroversially true.
- Proceeds via reasoning that seems uncontroversially valid.
- Arrives at a conclusion that is a contradiction, is false, or is otherwise absurd, inappropriate, or unacceptable.
Often, however, such as in the case of the Liar Sentence:
“This sentence is not true.”
there is a central claim that seems to be the root of the paradox, and in such cases we often talk as if the sentence itself is the paradox, rather than the argument. Let’s adopt this informal usage here. Thus, on this looser way of speaking, sentences that cannot be true and cannot be false are paradoxical. We’ll call the kind of sentences just described “philosophically paradoxical”, or paradoxicalP.
In literary theory, some sentences are also called paradoxes, but the meaning of the term is significantly different. If one is in an English department, rather than a philosophy department, and one claims that George Orwell’s claim from Animal Farm that:
“All animals are equal, but some animals are more equal than others.”
is a paradox, then one is not claiming that this sentence is neither true nor false, or that one can derive a contradiction or absurdity from this claim. Rather, the claim being made is something like this: the sentence in question involves a misleading juxtaposition of concepts and ideas that leads to an unexpected truth. Although this will obscure some of the subtleties, for our purposes we can simplify this as: A sentence is paradoxical (in the literary sense) if and only if it appears to be false, nonsensical, or otherwise problematic, but in fact hides a deeper truth – in short, if it is surprisingly true. Let’s call the kind of sentences just described literarily paradoxical, or paradoxicalL.
It is worth making the following observations, which underlie most of what follows, explicit: If a sentence is paradoxicalP, then it cannot be either true or false. Any sentence that is paradoxicalL, however, must be true, even if it initially appears to be false.
Now, at first glance we might think that this second, literary notion of paradoxicality has little to offer the logician – after all, it is a literary notion, not a logical on. But the opposite is in fact true. We can see this by considering the Literary Liar:
“This sentence is not a paradoxL”
The Literary Liar is either uncontroversially true, and hence is neither kind of paradox, or it is a paradoxP (and hence neither true nor false), but not a paradoxL.
First, the Literary Liar cannot be false. If it were false, then it would not be a paradoxL, since any sentence that is a paradoxL must be true. But it says that it is not a paradoxL, so this would mean that what it says is the case, and hence it would be true. Contradiction.
So the Literary Liar must be true. But now we have two options: either the Literary Liar appears false, nonsensical, or otherwise problematic, or it does not.
Why might we suspect that the Literary Liar is false, nonsensical, or otherwise problematic? Well, paraphrasing loosely, the Literary Liar says something like:
“This sentence is not surprisingly true.”
One might think that this appears, at first glance, to be close enough in content to the Liar Sentence to strongly suggest that similar self-reference induced problems will arise here as well.
If (for whatever reasons) we expect the Literary Liar to be false, nonsensical, or otherwise problematic, then, since it is in fact true, it is a paradoxL. But it says that it is not a paradoxL. So if it is true then it is not a paradoxL. Contradiction, and we have our paradox (of the philosophical variety).
If, however, the Literary Liar does not appear to be false, nonsensical, or otherwise problematic at first glance, then there is no paradox. The Literary Liar, in this situation, would be true, but not surprisingly so, and hence neither a paradoxP nor a paradoxL.
To sum up: If the Literary Liar is not a philosophical paradox, then it is true, but not surprisingly so. Surprising, no?
This example highlights another important fact: Whether or not a sentence is a paradoxL depends on our expectations – that is, on whether or not we expect it to be false, nonsensical, or otherwise problematic. ParadoxicalityP does not depend on our expectations in this manner.
In addition to examining constructions involving the literary notion of paradoxicality, we can combine these two notions to obtain some interesting puzzles. For example, consider the sentence:
“This sentence is a paradox, but is not a paradox.”
On the face of it, this sentence looks plainly false. In fact, however, given the two readings of “paradox” we have to hand, the sentence is ambiguous, and on one reading could be true!
There are four ways that we can disambiguate the sentence in question, depending on how we label the two occurrences of “paradox” with our subscripts “P” and “L”:
- This sentence is a paradoxP, but is not a paradoxP.
- This sentence is a paradoxL, but is not a paradoxL.
- This sentence is a paradoxP, but is not a paradoxL.
- This sentence is a paradoxL, but is not a paradoxP.
Readings (i) and (ii) seem, on the face of it, to be plainly false: a sentence cannot be both a paradoxP and not a paradoxP, nor can it be both a paradoxL and not a paradoxL (dialetheists: please forgive my brushing over some subtleties in case (i) – doing so allows us to get to some less subtle but much more fun issues in the other cases).
Reading (iii) also seems false: It can’t be true, of course, because then by the first conjunct it would also have to be a paradoxP, but a paradoxP is a sentence that can’t be true and can’t be false. But on reading (iii) the sentence in question can be false, since in such a case it wouldn’t be a paradoxP and hence what it says would not be the case. Since it can consistently be false, it isn’t a paradoxP, and so is in fact false.
Reading (iv) is the most interesting, however, since it seems to work a bit like the sentence known as the Truth Teller:
“This sentence is true”
The Truth Teller, intuitively, is indeterminate: If it is true, then what it says is the case, which is exactly what is required of a true sentence. If it is false, then what it says is not the case, which is exactly what is required of a false sentence. Thus, it could be true and it could be false, and there seems to be no additional information that can determine which it is.
Similarly, on reading (iv) our sentence might be true and it might be false. First, notice that the sentence, without our disambiguating subscripts, certainly appears to be false – it has the form “P and not P”. So, if it is true, then the fact that it is true is surprising. So if it is true, then it is a paradoxL. And if it is true then it is certainly not a paradoxP. Thus, if it is true, then what it says is the case, which is exactly what is required of true sentence.
Second, if the sentence is false, then it is not a paradoxL, so it is certainly not the case that it is a paradoxL but not a paradoxP. So what it says is not the case, which is exactly what is required of a false sentence.
The reader is encouraged to consider the following slight variation of this sentence to see what difference inclusion of “unsurprising” makes:
“Unsurprisingly, this sentence is a paradoxL, but is not a paradoxP.
Some interesting issues with regard to the logic of “surprising” arise, consideration of which I will leave to the reader.
Featured image: Abstract by geralt. Public domain via Pixabay.