Quine, Heinlein and the Terminator: A Self-Referential Journey Through Spacetime
SPOILERS! This post spoils stuff from the movie Predestination as well as multiple Terminator movies
This well-known paradox, known as the Liar paradox, is where our journey starts. If This sentence is false is a true sentence, then it is false: it says so itself. If it is false, it’s false that it is false and thus must be true. A nice paradox of self-reference. Or consider
This sentence — a conditional — seems to logically prove that 2 + 2 = 5. Conditionals are true if the consequent (in this case 2 + 2 = 5) is true given that the antecedent (in this case if this sentence is true) is true. For example, If it’s raining, then the streets get wet is true if the streets in fact get wet given that it rains. And if that sentence is true and it is also true that it’s raining, we can conclude that the street get wet. So the streets get wet is true if two things are true:
- If it’s raining, then the streets get wet
- It’s raining
Back to
In order to decide whether 2 + 2 = 5, two things must be true:
- If this sentence is true, then 2 + 2 = 5
- This sentence is true
(1) is, in fact, true: 2 + 2 = 5 follows logically from this sentence is true, because if it is true that this sentence is true, then, since this sentence refers to If this sentence is true, then 2 + 2 = 5, it must be that If this sentence is true, then 2 + 2 = 5 is true as well — so if it is true that this sentence is true, then it is also true that If this sentence is true, then 2 + 2 = 5 and therefore 2 + 2 = 5. 2 + 2 = 5 thus follows logically from this sentence is true in this situation. So (1) is true.
But then (2) is also true, because (2) simply states that (1) is true! So both (1) and (2) are true, which means 2 + 2 = 5?! This is known as Curry’s paradox, and it’s not an easy one to resolve — but if I ever have the knowledge to resolve it, I hope I find the time to share it here.
So self-reference is pretty cool, huh? Certainly, but in these problems, it’s not just the self-reference that’s the problem. Self-reference itself doesn’t necessarily lead to a paradox. Consider:
This one is just as self-referential as This sentence is false, yet there is no paradox: the sentence is true, as it has indeed five words. Likewise,
isn’t a paradox either: it’s just false.
So self-reference doesn’t necessarily lead to a paradox. And it’s even true that the same paradox of This sentence is true — or at least a very similar one — can be constructed without self-reference:
If the first sentence is true, then so is the second one (as the first one says so). But since the second sentence says the first one is false, the first one must be false. So if the first sentence is true, it is false… Similarly, if it’s false, it must be true. The paradox is back, without any sentence referencing itself! The sentences only reference each other in this two sentence version of the Liar paradox.
Or consider Quine’s paradox:
This one can be a bit more difficult to grasp, but when you do, it’s very cool. yields a falsehood when appended to its own quotation is, in fact, appended to its own quotation. So the sentence
“yields a falsehood when appended to its own quotation” yields a falsehood when appended to its own quotation.
yields a falsehood — in other words, is false — according to itself. But then it is, in fact, true, as it says that it is false (and is thus correct in saying that it is false). A nice contradiction, even though the sentence doesn’t refer to itself — though it does refer to a part of that sentence. More specifically, it (the word it in its quotation) refers to yields a falsehood when appended to its own quotation.
So if it isn’t just self-reference that creates these problems, what does? It seems that combining self-reference with negation in the right way is a good recipe, as is the case in This sentence is false: it refers to itself with a negation (false), and that causes a paradox. Of course, the negation has to be added to the mix in the right way. Consider:
This one refers to itself and has a negation (doesn’t), but there’s no paradox: it’s just true that this sentence doesn’t have ten words. In this case, the negation only applies to part of the sentence — the ten words thing — instead of the whole sentence, as is the case in This sentence is false.
Then again, self-reference and negation — while definitely a paradoxical mix — aren’t the only way to create these paradoxes, as we saw with Quine’s paradox and the two-sentence Liar paradox.
Let’s explore that last one a bit more. To recap:
The underlying pattern here is that of two things pointing towards each other — one of them in a positive manner (the following sentence is true) and one of them in a negative manner (the previous sentence is false), thereby creating a paradox. And this pattern can be expressed in different ways. Consider the Grandfather paradox: what happens if you travel back in time and kill your own grandfather before he had children? If you do so, you can never be born. But if you’re never born, you don’t go back in time and thus never kill your grandfather! In this paradox, your grandfather has the same role as the following sentence is true, whereas you are analogous to the previous sentence is false: your grandfather ensures your existence, and you negate his, creating a paradox.
Or consider the movie Terminator 2: Judgment Day. In this movie’s universe, there is a future in which John Connor leads humanity in a fight against Skynet, a very advanced Artificial Intelligence. John Connor wins this fight, and as a last resort, Skynet sends terminators — human-like robots — back in time to wipe out John Connor’s existence. Specifically, in Terminator 2, Skynet sends back a T-1000 to kill John Connor. A less advanced terminator — a T-800, also built by Skynet — is captured by the future John Connor, who reprograms it and sends it back in time to protect his younger self. Together with John Connor, Sarah Connor (John’s mother) and Miles Dyson (more on him later), this T-800 then destroys a project of Cyberdyne Systems — a project that would have led to the creation of Skynet. So future Skynet ensures the existence of the T-800 (the following sentence is true), who then acts to prevent Skynet from ever existing: the previous sentence is false.
But Terminator’s party of referential awesomeness doesn’t stop here. Note that John Connor helped in preventing the creation of Skynet — a system that tried to kill him. John and Skynet therefore point to each other in a negative way, just like
This problem is known as the No-no paradox. It can’t be the case that both sentences are true: because if so, they are also both false (as they claim this about each other). Similarly, they can’t be both false. We can say the first sentence is true and the second one is false — or vice versa — without it leading to such a paradox, but this is a bit random: the sentences are perfectly symmetrical, so assigning them opposing truth values seems unjustified (and perhaps even paradoxical in its own right). Terminator 2’s sequel Terminator: Dark Fate does resolve the problem this way, as it turns out John Connor succeeded in preventing the creation of Skynet. Of course, this doesn’t bring up the symmetry problem of the No-no paradox, as John Connor and Skynet are not at all the same — but it does once again bring up the two sentence Liar paradox, as John Connor only acted to prevent Skynet from ever existing as a direct result of Skynet sending terminators back in time.
By the way, the earlier discussed Miles Dyson, who worked on a project leading to the creation of Skynet, also did so as a result of Skynet sending terminators back in time: some of his colleagues found a chip and an arm of a destroyed T-800 that was sent back in time to kill Sarah Connor in the first movie. It’s clearly implied that Dyson’s project would not have been possible without the chip. So by sending a T-800 back in time, Skynet causes its own existence (if it wasn’t for John, Sarah, Miles and the other T-800’s work), which is known as a Bootstrap paradox. It’s also analogous to
which doesn’t seem to have a definitive truth value: the sentence says of itself that it is true, so if it’s true, it’s, well, true. If it’s false, what is says about itself is false — so it’s false that it is true, and thus false. Neither truth nor falsehood lead to a paradox, but the fact that both are equally justifiable seems to me to be paradox in itself.
An even stranger example of the Bootstrap paradox can be found in the movie Predestination — after the short story All You Zombies by Robert A. Heinlein — which tells the story of Jane. Jane, an intersex person who was born female, grows up in an orphanage. At some point in her life, she meets John, and together, they have a baby — but due to birth complications, Jane has to undergo sex reassignment surgery. Continuing live as a man, she adopts the name John — and goes back in time, falling in love with Jane. The baby they have is also transported back in time, and brought to an orphanage… In other words, Jane gives birth to herself, and Jane, John and the baby are all the same person.
And with that awesome instantiation of the Bootstrap paradox, this post is finished. As always, thanks for reading!
