Time Travel Paradoxes

Καὶ ὁ ἄγγελος... ὤμοσεν... ὅτι χρόνος οὐκέτι ἔσται.
And the angel... swore... that time will be no more.

Revelation 10:5-6

Robert Heinlein's 1941 story "By His Bootstraps" begins with the narrator writing in a philosophy thesis that time travel is impossible because time, in Immanuel Kant's terms, is only empirically real and does not exist independently among things in themselves. The narrator is then suddenly surprised to find two different versions of himself arriving from the future, with conflicting warnings and promises about what he can do. Traveling to the future, he meets an older man who repeats the promises, but whom he ends up distrusting. After some confusion, back in the present, he obtains some supplies and returns to the future to a period significantly earlier than when he would met the older man, intending to contest the future with him. Eventually, however, it turns out that he himself is the older man and his future is in fact, pace Immanuel Kant, secured.

A paradox of time travel arises in relation to this story. The narrator does indeed set himself up "by his bootstraps" -- his present and future selves all interact with each other to produce the events. The paradoxical nature of this comes down to the case of a notebook that was provided to the narrator by the older man in the future. It contained a vocabulary of the language that was spoken by people in the future. The narrator learns the language and, as the book wears out over the years, copies it over into a notebook he had fetched from the present. This notebook, as it happens, is the very one he, as the older man, then provides to his other self. He is therefore the same person who both learns the knowledge from the notebook and put the knowledge into the notebook in the first place. The vocabulary as a certain list of items arranged in a certain way was thus complied by no one whatsoever. The knowledge exists in a closed temporal loop and is in an important sense uncaused or uncreated. The narrator himself notes that there is something peculiar about this.

Peculiar indeed. A very similar paradox, allowed by the possibility of the same kind of temporal loop, can become a reductio ad absurdum for time travel. We see just such a paradox in the 1980 movie Somewhere in Time, staring Christopher Reeve and Jane Seymour. As a young man, Reeve encounters an old woman who gives him a watch. Later he becomes obsessed with the painting of a woman in an old 19th century hotel (actually filmed on the beautiful Mackinac Island, Michigan). He decides that he must meet that woman, and he thinks it is possible because of the theory of a professor he had for physics. The professor thinks that it is possible to will one's self back in time, as long as what one carries along is not anachronistic for that time.

Reeve outfits himself for the 19th century and actually succeeds in willing himself back into it. He meets the woman in the picture, played by Jane Seymour, and he is able to win her heart, so that she returns the love he felt ever since seeing her painting. He gives her the watch that he had acquired many years before from the old woman. Then, as their mutual happiness seems assured, Reeve discovers a penny from the 20th century in his suit, and the anachronism vaults him back into the present. He is unable to endure separation from his beloved, starves himself to death in his hotel room, and, apparently, is reunited with her in the Hereafter.

The old woman who gave him the watch in his youth was, of course, Jane Seymour's character, lived to a ripe old age just to see him again. The watch, therefore, was obtained by Reeve from Seymour and was obtained by Seymour from Reeve. In a closed temporal loop, like the knowledge in the notebook in Heinlein's story, the watch is uncreated. But this is impossible. The watch is an impossible object. It violates the Second Law of Thermodynamics, the Law of Entropy. If time travel makes that watch possible, then time travel itself is impossible.

The watch, indeed, must be absolutely identical to itself in the 19th and 20th centuries, since Reeve carries it with him from the future instantaneously into the past and bestows it on Seymour. The watch, however, cannot be identical to itself, since all the years in which it is in the possession of Seymour and then Reeve it will wear in the normal manner. It's entropy will increase. The watch carried back by Reeve will be more worn that the watch that would have been acquired by Seymour.

The reductio ad absurdum created by the watch can be fixed up in a couple of ways. First, we might think that entropy could be reversed by time travel, so that forms of matter would be restored to that state they would have been at the earlier period. But this will not do, since Reeve himself would then be restored to the state his matter was in in the 19th century, which, whatever it was, would not be in the form of Christopher Reeve.

Second, we might think that time travel puts one in an alternative universe. In some universe, the watch is manufactured and bought in the ordinary way, and then the older Jane Seymour, for whatever reason, gives it to the young Christopher Reeve. He goes back in time, to an alternative universe where Seymour did not acquire a manufactured watch, and gives her his. Then she gives it to him later; and he returns to a different universe, where Seymour does not buy a watch but acquires a somewhat more worn watch from him. The temporal loop thus generates a spiral of alternative universes. Unfortunately, it would require a spiral of an infinite number of alternative universes, as each watch in a particular universe is returned to a new universe where it can exist in its increasingly worn state. In some universe, the watch would disintegrate while in Seymour's or Reeve's keeping and need to be discarded; but Reeve would keep returning to the past, unless the watch turned out to be some causal factor in his falling in love with the picture.

Every instance of time travel generating an infinite number of alternative universes might be thought to violate Ockham's Razor, especially since the idea that an alternative universe could be generated in the first place has disturbing consequences for the metaphysics of identity. What does it mean if there are an infinite number of each of the characters, all facing a universe slightly different? Simplicity and common sense rebel against such principles -- although serious versions of such metaphysics have been produced to deal with quantum mechanics, and multiple real universes were proposed by the philosopher David Lewis to explain possibility and necessity (after Saul Kripke used Leibniz's idea of "possible universes" to produce a quantified version of modal logic) [note]. But without them, time travel, that would allow for the sort of temporal loop in which the paradoxical and impossible watch of Somewhere in Time becomes possible, is itself impossible.

While the notebook of "By His Bootstraps" and the watch of Somewhere in Time produce paradoxes for the concept of time travel, nothing is as simple or stark as the "Grandfather" paradox. Thus, you go back in time and murder your grandfather (take your choice) before he has any children. Thus, one of your parents never would have been born. And you never would have been born. But if you never had been born, then you would not exist to go back in time and murder your grandfather. The possible murderer of your grandfather is removed from existence by the very act of murder. An act that now cannot happen. But then, if it doesn't happen, you do exist; and you can go back in time and murder your grandfather.
The Liar Paradox
"This sentence is false."
If that sentence is false, then what it says is true. Therefore, it is true.If that sentence is true, then what it says is false. Therefore, it is false.
Introduced and debated in Hellenistic philosophy, resulting in at least one suicide [note]
This begins to sound like the Liar paradox, where, if a sentence is true, it's false, and, if it is false, it's true. You neither can nor cannot murder your grandfather.

This could be fixed up, again, with multiple universes. You don't just travel in time. You are leaving the universe in which you exist and enter a universe where your actions mean that you will not exist. No problem. The original universe is not altered. However, this does mean that there actually is not true time travel. There is only travel between alternative universes, where you can enter the alternative universe at a random point in time. Does this already falsify the premise? I expect so. But the simpler move is then to forget the alternative universes and just deny that there can be time travel. The paradox is resolved, without all the paradoxes inherent in the theory of multiple universes itself.

Each of these paradoxes involves a kind of loop in time. The knowledge in "By His Bootstraps" circulates from past to future and back without having an actual causal origin external to the loop. There is something "peculiar" about that, but it doesn't seem to violate any logic or laws of nature. The watch in Somewhere in Time similarly is uncreated, but it does violate the laws of nature, for its existence through time on the loop involves wear that will not be undone by its transport from the future to the past. The watch at the point of transfer in the future cannot be the same watch at the point of arrival in the past, since the watch in the past becomes the watch in the future through many years of the operation of entropy.

In the Grandfather paradox, the problem with the loop in time becomes more acute. The violation moves from being of one of the laws of nature into the realm of actual logic, albeit the logic of the basic nature of causality. Thus, the Third Noble Truth of Buddhism is that you remove the effect by removing the cause. Removing the grandfather thus removes the effect, namely the murderer himself. Here we have, not a continuity along the loop -- of the linguistic knowledge or the watch -- but a discontinuity -- the causal chain is broken rather than continued. This is why we get a phenomenon similar to the Liar paradox. The existence of the murderer makes the murder possible; but the murder makes the murderer, and the murder, impossible. With Ockham's Razor, the simplest fix is the denial of time travel.

Kant's theory of time may go unrefuted after all.

Past, Present, and Future, A Philosophical Essay about Time, by Irwin C. Lieb

Philosophy of Science


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Time Travel Paradoxes, Note 1

Since this page was originally posted, correspondence has been received from some people enthusiastic for multiple universes. Their enthusiasm did not seem to add much to the case. On the other hand, the science writer Martin Gardner has just published a brief but cogent essay on multiple universes, "Multiverses and Blackberries," in Are Universes Thicker Than Blackberries? [W.W. Norton & Company, 2003, pp.3-11]. In addressing the use of "many worlds" in interpreting quantum mechanics, Gardner says:

It is hard to imagine a more radical violation of Occam's razor, the law of parsimony which urges scientists to keep entities to a minimum. [p.4 -- notice that there are various spellings of William of Ockham's name]

In the end he concludes:

The stark truth is that there is not the slightest shred of reliable evidence that there is any universe other than the one we are in. No multiverse theory has so far provided a prediction that can be tested. In my layman's opinion they are all frivolous fantasies. [p.9]

Well, "frivolous fantasies" is a little strong when there is some serious philosophy and science that uses multiple universes. But such theories are, at best, a reach, and a very questionable one.

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Time Travel Paradoxes, Note 2;
The Liar and the Anti-Liar Paradoxes

The Anti-Liar Paradox
"This sentence is true."
If that sentence is false, then what it says is false. Therefore, it is false.If that sentence is true, then what it says is true. Therefore, it is true.
Note the peculiarity of the Liar paradox if we switch its assertion from false to true. The contradictions generated by the paradox disappear, but we are left with a further question. OK, we might think, if it's true, it's true; and if it's false, it's false. That is the case with any non-paradoxical proposition. If it is raining today, it's raining. If it's not raining today, then it's not raining. Big deal. But then we are left to wonder, Is the Anti-Liar true? Or, Is it false? How do we determine that?

Well, ordinarily we look to the cognitive ground of the proposition. To know if it is raining, we look out the window. But both the Liar and the Anti-Liar statements only refer to themselves. Therefore, if they are either true or false, there must be something about them that makes them true or false. We find lots of things like that in logic, where the sentences that cannot be false are tautologies, and where the sentences that cannot be true are contradictions.

The Liar Paradox
"This sentence is false."
If that sentence is false, then what it says is true. Therefore, it is true.If that sentence is true, then what it says is false. Therefore, it is false.
Now, with the Liar, we have the problem that L L, i.e. the Liar itself implies its own negation. Or L L. What actually is a tautology of logic is that (L L) L. This is the Law of
Clavius, which would mean that the Liar is false. However, it is also true that (L L) L. So the Liar is true. So the paradox persists.

This bothered Hellenistic logicans, especially the Stoics, because of the Principle of the Excluded Middle, which is that propositions are either true or false. It doesn't look like the Liar can be either true or false, since each one even implies the other, which just drove them bananas. However, notice that, without generating any contradictions, we have the same problem with the Anti-Liar. How can it be either true or false? It is self-referential, which means that its truth can only be determined by its own form or meaning. But the form of a proposition can determine its own truth or falsehood only if it is either a tautology or a contradiction. Neither the Liar nor the Anti-Liar is either a tautology or a contradiction. And the content of the proposition is simply to refer to itself. The Anti-Liar, as much as the Liar, violates the Principle of the Excluded Middle. So it is actually the Anti-Liar that exposes the issue, without the distractions of the contradictions generated by the Liar. So the problem is with that.

Indeed, there is nothing, internally or externally, form or meaning, that provides a ground for the truth of the Anti-Liar. It is as though we are given a proposition, P, and are asked whether it is true or false. Well, we can't say; and our objection will be that we cannot determine truth or falsehood without knowing its meaning (logicians say an "intepretation"). But its meaning is merely self-reference, with nothing else. In a sense, this means that P consist of nothing but p with a predication of truth, i.e. Tp -- just as the Liar is a predication of falsehood, i.e. Fp or p. That is not much help. But a predication of truth can be added or removed at will, i.e. p Tp -- or (p Tp) & (Tp p). So, strictly speaking, P remains without meaning -- it has no semantic content beyond a redundant logical form -- and it is no more paradoxical than any other uninterpreted p, such as is used in logical symbolism all the time -- in fact since Aristotle. So our answer to both paradoxes may be that they actually have no meaning and so cannot have a truth value -- unless we have a three value logic that adds "unknown" or "undetermined" to truth and false. That does indeed violate the Excluded Middle, but then it may be necessary, for instance, in Quantum Mechanics. It is also in fact the practice of logic were uninterpreted symbols, like P, are used. Unless they are tautologies or contradictions, their truth value is undetermined until interpreted.

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