Why isn't the Liar's Paradox just accepted to be complete nonsense?

Question

I can understand that some self-referential sentences can be sensible and have truth/false values (e.g. "This sentence is written in English." is true, "This sentence has 1,000 words." is false). However, the sentence "This sentence is false." has no meaning. Its as nonsensical as "This sentence is true". What does it mean for this sentence to be true? How would you even prove/disprove this sentence is true? If "This sentence is true." is nonsensical, why can't "This sentence is false." be nonsensical too?

Thanks.

Please explain in simple layman English.

Answer 1

You ask:

Why isn't the Liar's Paradox just accepted to be complete nonsense?

You want an answer in "plain English." The easy explanation is that "complete nonsense" intuitively would be a construction like "Fliba jibba jobby ho funkle" where there is no meaning (as "sense" is a synonym for "meaning"). Clearly, "This sentence is false" is unlike a completely nonsensical sentence composed of random sounds because it is both understandable, and one can have intelligent discourse with and about it.

All of the words have meaning, and the meaning of the entire sentence at first glance is meaningful, and any sentence with those properties isn't nonsensical. Even "Colorless green ideas sleep furiously" isn't nonsensical, because by the Principle of Compositionality, its words and some of its phrases have meaning. Completely nonsensical statements have to be statements that contain no meaning by part or whole and that doesn't describe the Liar's Paradox at all.

In fact, the sentence is so meaningful, that it meets the definition of a paradox which means that not only does one find it meaningful, but it's surprising that's there's a logical problem with the sentence at all! Paradoxes are logical claims like fallacies are logical arguments: they are very persuasive and misleading on the surface, and it is only a careful analysis that reveals they are problematic.

Answer 2

We use the predicates "... is true" and "... is false" in everyday speech and in scientific speech to make comments about propositions. Those comments are themselves also intended as propositions.

Problems arise when those predicates are applied to the sentences themselves. That is, in particular when somebody utters "This sentence is false". Even though the apostle Paul in Titus 1:12,13 writes

One among themselves, even a prophet of their own, said, “The Cretans are always liars, evil beasts, slothful gluttons." This witness is true. Therefore rebuke them sharply, that they may be sound in the faith.

showing that he perhaps didn't have a sharp eye for logical finesse, most people find it paradoxical since the Liar's sentence would be true if and only if false.

If you then ask what makes that sentence into a pathological edge case, some will say that this is because of the involved self-reference (or rather the combination of self-reference with semantic notions, as here "is true" and "is false"). In many other semantic paradoxes, such as Berry's paradox, that use semantic terms like "name", "definition" or "description", self-reference also seems to be the major culprit.

But if self-reference is the major culprit here, then the mirror sentence "This sentence is true" also needs to be looked at with suspicion. It's definitely possible to then argue that, since the Liar sentence and the Truth-Teller sentence have exactly the same form (the same schema), then if one is a pathological edge case, without a clear sense, then the other may also be pathological "nonsense".

On the other hand, it's also possible to argue: It's only because we assign a certain sense or meaning to the Liar sentence that we are able to derive a self-contradiction. Argued like that, the Liar is not simple "nonsense", but just a self-contradiction, while the Truth-Teller may makes sense, but doesn't seem to say anything substantial. (There are ways to formalize this in a formal system, however, and then the equivalent of the Truth-Teller could be vacously true. For more technical details see Self-Reference and Paradox) You could argue that "not saying anything substantial" is also a form of "nonsense", but in this whole line of argument, a counter-argument would be: Well, at least it shows a way of using the predicate "is true" in a vacously true sentence.

What is "nonsense" or not really depends, in the end, on which use you want to make of language.

Twas bryllyg, and þe slythy toves
Did gyre and gymble in þe wabe

Did they really or not? In my imagination they truly did. --- In fact, I now imagine that the "slythy toves" may be the sly logicians (and us) who are "gyring and gymbling" in the "wabe" of this question (it definitely has a "long way be fore it and a long way be hind it" as Humpty Dumpty explained). And "bryllyg" suggests the heat wave that sometimes arises in these discussions and causes us to "gymble" like desperate gimlets trying to bore holes in our thoughts.

Answer 3

The issue with such statements as The Liar, is that their self-reference is entirely to their own semantic truth value.

The semantic "truth" of a statement arises when the mind compares the claim make by that statement to the "state of the world".

The statement P: "Paris is the capital of France" is true while Paris remains the capital of France.

But, to know the truth value of P, we have to compare P to the real world, and P must make some verifiable claim in order to determine its truth value.

If a statement just refers to its own truth value, it skips over the ONLY means by which its own truth value can be determined -- it provides NO way to determine its semantic truth value.

This is known as un-grounded self-reference, and Joshua Matthews at the University of Reading gave a very accessible account of the matter in "Ungrounded Self-Reference: The solution to the Liar Paradox?", which you can find here: https://digitalworks.union.edu/ephemeris/vol13/iss1/4/.

Answer 4

The liar paradox is not nonsense. We understand what it says. The problem is that while the liar sentence appears to state a proposition, it is problematic to assign a truth value to it.

It is worth noting that we cannot avoid the problem by supposing that a sentence may be true, false or nonsensical. We could just reintroduce the paradox with, "This sentence is either false or nonsense". Also, we can construct combinations of sentences, such that no one of them is self-referential, but the combination of all of them is paradoxical to evaluate.

Students of logic and language are interested in paradoxes such as this because they can tell us interesting things about how logic and language work. An analogy is that neuroscientists who study human vision often investigate optical illusions because those can provide information about how our brains process visual data.

There is an enormous literature on the liar paradox, and many ways of handling it. A common approach is to hold that for some reason or other the liar sentence fails to be a proposition with a truth value. Some claim that the sentence is false, or that it is both true and false. Others have supposed that it tells us something about the relation between the truth of a proposition and some state of affairs that grounds it. Others that it tells us something about the concept of truth: maybe it is not always the case that "P is true" if and only if P. Others maintain that it tells us something about the limitations of natural languages: that they are unable to avoid inconsistency when making statements about truth.

Examining these options is valuable to the logician because they help to bring these questions to the surface. There is a good introduction to the paradox in the SEP article.

Answer 5

If A and B have the same meaning/use in language, including in the limit case where neither means anything, then if A is true, so is B. Let A be, "This sentence is true," and B be, "This sentence is false." But A has the property "if we say it's true, there is no contradiction," but B has the property "if we say it's true, there's a contradiction." So A and B have different uses in language, so they don't have the same meaning.

P.S. "complete nonsense" is a completely nonsensical description, so it might not do to use it here.

Outside reference: I'd highly recommend that you read Żełaniec[??]. A representative quote:

These two sentences: “This sentence is false” (the Liar) and “this sentence is true” (the Truth-Teller)1 give rise to well-known difficulties. The Liar is, or at least appears, paradoxical: supposing it is true it comes out false and the other way ‘round. The Truth-Teller is not obviously paradoxical, because no contradiction seems to follow from the hypothesis that it is true: If it is true, it is true, and if it is not, then it is not, and that is it. Yet, some philosophers have found the Truth-Teller paradoxical, too (see e.g. Woleński 1993; Billon 2014). I must say that I sincerely admire the ingenuity (and the non-ingenuousness) with which the philosophers who have taken either sentence seriously attempted to disarm or explain away the (alleged) paradox. And ingenuity does not always mean complexity: Joseph W. Smith once formulated an astonishingly simple proof that the Truth-Teller was true (Smith 1984), which I, too, find convincing—except for the initial presupposition that “one is prepared to take the self-referential ascriptions of falsity [or truth] of a sentence such as the liar sentence […] [or the Truth-Teller or their ilk] at all seriously” (p. 219). I am not prepared to do so and I can’t imagine how I ever could. And the problem is that I, as distinct from Barry, am rather shortbreathed when it comes to arguing from (what I consider) patently absurd premises, adopted just “for the sake of argument.”

To be sure, the author therefore agrees with you, but through him you might find more about the common view.

Answer 6

Why isn't the Liar's Paradox just accepted to be complete nonsense?

Other answers here are giving a narrow definition of "nonsense". But yes it is. Or at least not true/false.

A statement "Please buy milk at the store." is not true/false. "This statement is true" is even more so.

My understanding is: A true/false statement must either match/conflict with fact or be provable/disprovable from given axioms. I don't see how the phrase "this statement" is sufficiently specific to ever be true or false.

Answer 7

The concise way I've heard it is to simply ask the question:

"What part of 'This sentence is false' don't you understand?"

To a lot of people, this comes across as a defensive dismissal. But, to a philosopher, this is a prompt to start a reflective investigation! There are four words in the bit in single quotation marks, and each of them suggests a slightly different track to start going down:

• "Sentence". What is the substance of language? Is a sentence a particular individual utterance in time, a composition of words, of concepts, of sound waves, of meanings, of a logical or mathematical relationship of things to ideas?
• "This". What is reference? How do we connect to things in the world, how is it that humans apprehend, form ideas about what we perceive, and come to relate what we say to those things in everyday speech? Can sentences or abstract things like meanings and concepts be the subjects of reference, and if so, how does that even work?
• "Is". What is the logical structure between things and their properties? How do you identify and distinguish individuals, determine that certain features apply or do not apply to individuals. Are there individuals, are there properties, is there any "thing" at all?
• "False". What does it mean for a speech act to correctly or incorrectly connect to the things in the world? If a person speaks falsely, is this just a nonsense statement? Is it possible to speak falsely, or counterfactually, but still be referring to things that have an underlying truth to them? How can we tell the difference, if at all, or use that difference strategically in our actions?

It is by no means clear that this entire realm of philosophical consideration is ultimately nonsensical, but neither does that mean that we should presume that everything is fine and well, because yes, this does give us some really bizarre conclusions about the possibility of relating truth to assertion. In fact, each element seems, on the face of it, to talk about something really quite primal in our shared, linguistic experience of the world, but if we can't clip them together in this way, what does that say about our system of language as a whole, and is there anything we can do about that?

Answer 8

The Liar Paradox, as many others of its kind, is considered "interesting" or non-nonsensical because it gets people to think long and hard about what is going on.

Naively, i.e. if you tell it to 5-year-old, the receiver will probably not be able to get to any conclusion about how to get out of the conundrum. They will think and think and think and eventually give up with much frustration; or depending on their character, they will just hear it and immediately be uninterested.

On the other spectrum, you end up with one of the deepest results we know about computation from the 20th century, i.e. Gödels incompleteness theorem with has incredibly far-fetching impact on the whole of maths, computer sciences and philosophy of both.

In between these two arguable extremes, such a paradoxon could spawn many other fields within logic, semantics etc. (check out that Wikipedia page for many instances where there have been rather deep explanations of how to resolve the Liar's Paradoxon).

A separate famous example is Achilles and the tortoise - seemingly nonsensical, but it took over 2000 years to figure out, and could only be explained when we invented infinite limits in calculus in the 19th century.

The same can be said for many if not all paradoxons; their worth is in honing our wits, figuring out ways to explain them, end old misconceptions, avoid naive worldviews and so on and forth.

Answer 9

Paradoxes are effectively linguistic anomalies. They are statements that should make sense according to the normal rules and usages of language, but somehow aren't entirely sensible. They're important in the same way that scientific anomalies are: they point out that we don't quite understand something we think we understand, and open the door for advancement.

One can certainly write off the Liar's Paradox as nonsense and it won't affect daily life one whit. By that same token, one can write off other anomalies as nonsense and assert that the earth is a flat, 6000 year old disc; that won't change how one buys groceries or gets to work either. "Who cares?" is a common enough response to things one doesn't understand, but it's not something I recommend as a lifestyle.

Answer 10

It is nonsense, for some definition of "nonsense", but what can make it interesting is why it is nonsense, what makes it nonsense, how a sentence like that might not be nonsense, how language functions, the nature of self-reference, etc. Analogous problems occur in mathematics, like Russell's paradox, which have motivated new and some would say better formalizations and theories like type theory.