The Liar's Paradox: A Supertask?
The classical Liar's Paradox is probably the most infamous paradox in philosophy. Self-referential and negating, it creates a contradiction that challenges our basest intuitions of semantics and syntax. I'll quickly go over the classic way of viewing the paradox as an apparently necessary contradiction. Then, I'll go over a very different way of interpreting the statement that looks like an infinite regress or a supertask. I use the classic and supertask interpretations to synthesize the abstract causes of the liar's statement paradoxicality. I finally evaluate notable responses to the liar's paradox and similar semantic problems in light of how they address these abstracted problems.
So let's get into it! This is the liar's statement:
Is that statement true or false? That's the paradox.
The Classic Contradiction Interpretation
The statement is either true or false, so let's just run through each case. If it's true, then we believe the statement, and the statement is false; a contradiction. So it can't be true. But if it's false, then it's not false—it's true. Again, we have a contradiction! Whether the sentence is true or false, we get a contradiction.
Responses to the Liar's paradox are addressing this interpretation of the statement. Answers either accept that contradictions are possible and attempt to rework logic, or try to find an assumption in the argument that they can reject.
But this classical interpretation isn't the only interpretation of the statement that leads to a puzzle, which leads me into...
The Supertask Interpretation
I sometimes see the liar's paradox presented to laypeople as an infinite regress. We take the statement at its word, which means we believe that the statement is false. But if it's false, that means it must actually be true, but if it's true, that means it's false, which means it's true, which means it's false, and so on ad infinitum. Our hypothesis of the statement's truth value flips back and forth between true and false. The same thing would happen if we had presumed it to be false. Notably, no contradiction is reached because the liar's statement is never declared simultaneously both true and false. The truth value of the liar's statement cannot be decided because we never reach the end of this infinite process. However, most people would agree that, even after all of that, it makes sense that the liar’s statement ought to have a definite, objective truth value.
This closely represents another set of paradoxes involving the concept of the supertask. These supertasks deal with an infinite sequence compressed into a finite interval. It's not in our human capacity to fully comprehend infinity, but because we can easily grasp finite lengths of time or space, we can run into something incomprehensible. Here is the most famous paradox related to the supertask, Thomson's Lamp.
Achilles' Race and Thomson's Lamp
Thomson imagines a new take on one of Zeno's paradoxes. Zeno describes Achilles running a race. Before Achilles can reach the finish line, he has to get halfway to the finish line. After he gets halfway, he then needs to get halfway between his new position and the finish line. He has an infinite number of halfway tasks to complete before he reaches the end, and since it's impossible to complete an infinite number of tasks in a finite amount of time, Achilles will never finish the race.
A modern, accepted answer is that we are able to split time into infinite intervals in a corresponding, infinite way. We do have infinite periods of time to complete infinite tasks, and these periods of time, because they become smaller and smaller, add up to a finite amount of time.
It seems like we've solved the problem, right? Well, let's see what Thomson has to say. Let's split up a minute into an infinite number of periods of time, ala Zeno. There's the first 30 seconds, then the 15 seconds after that, then 7.5 after that, ... until after an infinite number of splits we reach the one minute mark. Now imagine a perfect, hypothetical lamp that we can toggle on and off. At the beginning of the minute, it's turned on. After 30 seconds, we turn it off. 15 more seconds, and we turn it back on. After every interval, we toggle the switch. So when the minute is up, will the lamp be on or off? There doesn't seem to be a good answer. In fact, any answer seems to imply that infinity is either even or odd.
Unlike Zeno's Paradox, Thomson's Lamp accepts that infinite tasks can be done in finite time, but then shows that accepting this will still lead us into incomprehensibility.
The Supertask Interpretation and Thomson's Lamp
The classical interpretation of the Liar depends on self-contradiction, while the supertask interpretation depends on examining the end of an infinite sequence. Just as Thomson's lamp turns on and off, the Liar's statement flips our conception of its truth value between truth and falsity. And just as we get a puzzling result when we try to imagine the final state of Thomson's lamp, we reach incomprehensibility when we attempt to determine the definite truth value of the statement.
Because this infinite sequence occurs in logic rather than in space or time, it might be difficult to see why it converges in the same way as an infinite sequence of halving intervals does. It seems obvious that Thomson's lamp will be either on or off after the minute, but why can't the liar's statement's infinite toggling go on forever? To see why we think it must have a definite truth value, we need to look at a common assumption of grammatical, meaningful statements.
Grammatical and Meaningful: The quick brown fox jumped over the lazy dog.
Grammatical but Meaningless: Colorless green ideas sleep furiously.
(Note that individual words still have meaning, but as a whole, the statement does not because the concepts cannot meet the semantic requirements of the grammar that connects them. For example, colorless cannot be used to describe green, although that is what the grammar of the sentence demands.)
Ungrammatical: Can't between yesterday could shall is.
It really seems as though, This statement is false, is both grammatical and meaningful. It's commonly accepted that all grammatically correct, meaningful statements have an objective, unaltering truth value associated with them, so this may be interpreted as the need for a final convergence of truth values after the infinite toggling. And just as we cannot figure out if Thomson's lamp is on or off after a minute, in the supertask interpretation, we cannot figure out whether the liar's statement is true or false when thinking about its final, definite answer.
A Synthesis of the Two Interpretations' Problems
The two interpretations are quite different from each other, but they both have strange outcomes. And their shared puzzling nature shouldn't be so surprising – they both are analyzing the same statement. What is it about the statement that makes both interpretations fall into incoherence? Do they fail for unrelated reasons? Or, as I suspect, do the problems of the two interpretations both stem from the same problem?
If we find a single problem causing both puzzles, I think this would provide reason to seriously consider that a solution to the Liar's Paradox should try to solve that specific problem. Fixing it in another way would be like treating the symptom of the paradox instead of the disease in our notions of language and logic.
So what do the two interpretations share?
They both seemingly lead to contradictory beliefs. I tend to think that this is just a result of the problem, and not the problem in itself.
Before we start evaluating each interpretation, we are unable to justify whether we should first assume that it is true or false, and go into a proof by cases. Is there a contradiction if we assume the statement to be true? What about if we assume it to be false? Both assumptions lead to the same incoherency in both interpretations, but nonetheless, I think this ambiguity is relevant.
The statement's holistic meaning is in reference to itself. With other cases such as, My shoe is red, we determine the truth value of each statement by examining the corresponding real experience of my shoe and being red. Statements are confirmed or denied depending on whether the real concepts relate to each other as the propositions state. The objects and connectives in the statements defer their meaning to things examinable outside of the sentence. In the liar's statement, meaning is also deferred, but it is deferred to itself. This semantic self-dependence causes the meaning not to be well-founded, which is a primary cause of both interpretations’ problems.
Another primary cause is that whenever it defers meaning to itself, it inverts what its truth value must be. Not only does it defer meaning, it also affects the meaning in the process. This is why the classical interpretation ends in contradiction, and why the supertask’s infinite sequence never converges to a single truth value.
These are the four main causes for our distaste with the liar’s statement, appearing in both of the interpretations of the statement. An answer to our problems would hopefully change as little as possible about our outlook of language and logic while still fixing all four of the stated problems above. However, Never use contradictory sentences like the liar’s statement, is a fix, but are not a pleasing one because it seems arbitrary and without justification; a more general answer with justification would seemingly be better.
Notice that (1) is the broadest problem, and stems from (4). If we fix (4), we get rid of contradictions and antinomies.
Similarly, (4) depends on (3); without semantic self-dependence, problems with a statement changing its own meaning cannot occur. This will be more precisely argued for in my post on the Truth-Teller’s Paradox.
(2) is a nuanced problem, but I argue that it also is a result of (3), albeit epiphenomenally. In statements like, My shoe is red, we similarly cannot say that we should presume either truth or falsity from the statement alone. Luckily, we don’t have to presume anything! We simply assume that there exists an objective answer, and withhold judgement until the statement is either confirmed or shown to be false. We don’t need any proof by cases. But in the case of the liar’s statement, as (3) states, we have no real-world counterpart that we can use to confirm or deny it. All we have is the statement, and the only tool we have at our disposal to get a truth value is thus a proof by cases.
It seems like (3), the liar’s statement’s self-deferring, ungrounded property, is the root of all our problems in the liar’s paradox, in both of the interpretations I have presented. A decent solution to this problem will thus address ungroundedness (in this case caused by self-deferrence). If it does not, it will be like a band-aid on a gaping wound – the problem has technically been addressed, but not in a way that fixes our main problem. These band-aid solutions will likely also be unnecessarily more complicated than a solution that simply recognizes (3) as the core problem.
Notable Responses to the Liar
Keeping in mind the criterion for a good answer that was established in the previous section, we can evaluate different responses to the paradox. Here are extremely brief overviews of five notable answers to the liar’s paradox and similar semantic problems.
Dialetheism and Paraconsistent Logics
Some people take the liar’s paradox to mean that we should accept some contradictions as true (Graham Priest). Usually, we take contradictions to mean that one of premises is false, or that our line of reasoning went wrong, but dialetheists say that contradictions aren’t the problems we make them out to be. This has led to the creation of paraconsistent logics (meaning logics which have transcended the need for consistency). They reject Aristotle’s Law of Non-Contradiction and the explosion principle.
This answer works, sure, but it is much more complicated than others, and only addresses problem (1), writing off the other three as non-problematic. By accepting contradictions, dialetheists might only address the symptom of a more nuanced problem in our logic.
Contradictions are how we find problems in our reasoning, and I’m not prepared to give that up just yet, especially when there are other answers to the liar’s (which I find more palatable).
Tarski's Undefinability Theorem
The classic solution comes from Alfred Tarski, who proposes that there’s a rule about semantics and references in language that normally isn’t that important, but is revealed by self-referential statements like the liar. This has the effect that the liar’s statement isn’t as grammatical or meaningful as it seems to be. This answer directly addresses (3).
When we want to define truth, or any other semantic notion, like redness or English, Tarski claims that we must to use a higher-level language, a metalanguage. Semantics is essentially the meaning, or conceptual notion of something, rather than the word or symbol being used to represent it.
Essentially, “My shoe is red” is a true statement if, and only if, my shoe is red. Here, the statement “My shoe is red” has just been talked about in a higher order language.
Formally, these hierarchies of language would have subscripts that label what number in the hierarchy the language/metalanguage is. This rigid hierarchy is a rule that prevents self-reference in any semantic notion, so the problem of deferring meaning to oneself can never occur. Self-reference in such a way would be ungrammatical and meaningless. This prevents the Liar’s Paradox and variations like the Strengthened Liar and the Truth-Teller. Quine argues that natural language can easily be interpreted to already contain this hierarchy.
Kripke's Value Gap
Saul Kripke’s Outline of a Theory of Truth is a highly influential paper addressing certain pitfalls in Tarski’s method. In certain cases, sentences are grounded or ungrounded, paradoxical or not, because they are based on empirical evidence or contingent on other statements and matters of fact. Kripke builds on the idea of a truth-value gap, where some statements are meaningful but cannot be given a truth value. Although only an outline, the paper is quite comprehensive, providing a mathematical formalization of truth involving the gaps, rather than claiming the gaps are meaningless and ungrammatical (as Tarski’s hierarchy would recommend).
Van Fraassen's Supervaluation
Bas Van Fraassen also accepts a truth value gap, except specifically with the notion of a supervaluation. Kripke’s work supports most truth gap interpretations, including Van Fraassen’s. Essentially, this means that although the liar’s statement falls into the truth gap, the conjunct of the liar’s statement (or any other statement that falls into the gap) and its negation must come out true. So if L represents the liar’s statement, L & ~L is true, even though L is undefined and ~ L is undefined.
Both Kripke and Van Fraassen address (3), but allow ungroundedness in some cases that they define to be a truth-value gap. Their work then goes on to try to figure out how we might live with some ungroundedness in our formal languages.
Gupta's Revision Theory of Truth
A relatively new formulation of truth, Anil Gupta formulates how truth can be self-referential. His interpretation of the liar is extremely similar to the supertask approach, except he denies that we must converge to a single truth value. Instead, a hypothesis is created about the statement’s truth, and is revised in each self-deferrence of meaning. This revision of hypotheses continues infinitely.
Although technically accomplished, this method seems similar to the dialethist’s approach of saying that there is nothing problematic about (3). I haven’t read up enough on this method, but it doesn’t seem like it gets to the problem presented by the liar’s statement. Instead, it finds a new way of interpreting the problem and uses that to say that the statement is no longer problematic. The Revision Theory therefore seems to ignore the classical interpretation by simply saying that that is the incorrect way of interpreting the statement. This is how it gets around (1).
Conclusion
In this post we’ve seen a second interpretation of the infamous liar’s statement (something I realized independently before stumbling upon Gupta’s work) and used that to synthesize the core problem of the statement: that self-deferrence causes a certain ungroundedness in meaning. We then evaluated five notable responses in light of those four abstracted problems. I lean towards favoring Tarski, Kripke, and Van Fraassen’s work; they seem to be the most direct in addressing the core problem without massive overhauls of the way language works.
References and Nice Links
Internet Encyclopedia of Philosophy on the Liar Paradox
Stanford Encyclopedia of Philosophy on Self-Reference
Stanford Encyclopedia of Philosophy on Tarski's Truth Definitions
Stanford Encyclopedia of Philosophy on the Revision Theory of Truth