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Sorry for a high-school-level question, but something is bugging me about the liar's paradox.

The liar's paradox can be formally written as an axiomatic system with an axiom $P \equiv \lnot P$. Due to the equivalence, we must have

  • $P \iff \lnot P$,
  • which is the same as $(P \implies \lnot P) \land (\lnot P \implies P)$
  • which is the same as $(\lnot P \land \lnot P) \land (P \land P)$
  • which is the same as $\lnot P \land P$

And now, due to the principle of explosion, we can prove any statement. In particular, we can prove $P$, and we can prove $\lnot P$.

What I want to say is that, according to my understanding, the liar's paradox is the paradox in exactly the same sense as an axiomatic system with an axiom False (i.e. a trivially contradictory axiomatic system), where for any statement $P'$, we can prove that $P'$ is true and $P'$ is false.

My understanding of the "philosophical" meaning of the liar's paradox is that it shows an example of a statement $P$ that can't be assigned a "True" or "False" value since it would lead to a contradiction. However, as I show above, the contradiction is already there: it has nothing to do with the value of $P$, in exactly the same sense as in the trivially contradictory axiomatic system, a truth value of any new statement $P'$ doesn't affect the contradiction.

So, the only thing that the liar's paradox achieves is a construction of a contradictory axiomatic system. However, this is already achieved in a less obfuscated manner by the trivially contradictory axiomatic system.

Is the above understanding correct? Or did I do something wrong in the above derivation? Or did I misunderstand the motivation behind the paradox?

Dmitry
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    I don't think you have a correct understanding of the liar paradox. It's not just about a contradiction, it's about a statement like "This statement is false" showing that you can't have both a full blown theory of reference and full blown theory of truth. It is possible to say something about this in a context like mathematical logic but you'll probably have better luck with your question in a philosophy forum. – blargoner Oct 08 '22 at 20:26
  • The paradox is about modalities like "$A$ asserts $\phi$" that don't fit in the framework of classical logic. If you reduce the paradox to classical logic, then your analysis is correct, but that throws out the nub of the paradox. – Rob Arthan Oct 08 '22 at 20:31
  • Related to your question, see this post about the liar paradox and some others. By the way, do note that paraconsistent logic is utterly useless. Don't be surprised that there are many proponents of useless things, just like homeopathy, alkaline water, choprawoo, traditional chinese medicine, and many more. – user21820 Jun 21 '23 at 08:58
  • The sentence "What time is it?" is neither true nor false, There is nothing paradoxical about it. It's truth value can be thought of as somehow indeterminate, thus preserving bivalent logic. A similar classification can be used for "This sentence is false." – Dan Christensen Jul 25 '23 at 22:45

2 Answers2

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As blargoner says in the comments, thinking about the liar's paradox this way misses the real heart of it. The liar's paradox is about a statement $P$ which itself asserts that $P$ is false; in your rendition the assertion $P \equiv \neg P$ is made "outside of $P$." Said another way, you can introduce the assertion $P \equiv \neg P$ into an arbitrary set of axioms, but in order to state the liar's paradoxical sentence you need a set of axioms which is powerful enough to discuss its own propositions and their truth values.

What the liar's paradox implies in broad strokes is that there is no such set of axioms; in other words, we do not have an unlimited capacity for self-reference in logic. Other results in logic can be thought of as precise versions of this, most famously the incompleteness theorem but also Tarski's undefinability theorem. The Berry paradox is a similar paradox about the limits of self-reference and a personal favorite of mine.

Edit: To say everything a bit more explicitly, motivated by the discussion in the SEP article, one way to somewhat formalize the liar's paradox is that it implies a formal logic cannot simultaneously have all of the following ingredients:

  1. The law of excluded middle; every proposition is either true or false (not neither, not both).
  2. The ability to refer to its own propositions; if $P$ is a proposition, write $\ulcorner P \urcorner$ for its "name" in the formal logic.
  3. A truth predicate $\text{true}(\ulcorner P \urcorner)$ which takes as input the name of a proposition and returns as output its truth value.
  4. The ability for propositions to refer to themselves.

Given all four ingredients we can construct the liar's paradoxical sentence $P$ which asserts $\neg \text{true}(\ulcorner P \urcorner)$ but which cannot be either true or false. So there are (at least) four different ways to resolve the liar's paradox: throw out LEM, throw out the ability to refer to propositions, throw out the truth predicate, or throw out self-reference specifically. (Although, as the SEP article explains, there are variants of the liar's paradox showing that self-reference specifically is not necessary.)

First-order Peano arithmetic satisfies LEM, is capable of referring to its own propositions and is capable of constructing self-referential propositions (via the diagonal lemma); this is used crucially in the proof of the incompleteness theorem, for example. It resolves the liar's paradox by containing no truth predicate (this is Tarski's undefinability theorem).

Other ways of resolving the liar's paradox throw something else out; for example you could try to throw out LEM. See the SEP article, specifically the section on paracomplete and paraconsistent logics for more. It seems that a lot of work here has been motivated by the desire to not throw out the truth predicate:

Working in classical logic, Tarski (1935) famously concluded from the Liar paradox that a language cannot define its own truth predicate. More generally, he took the lesson of the Liar to be that languages cannot express the full range of semantic concepts that describe their own workings. One of the main goals of the non-classical approaches to the Liar we have surveyed here is to avoid this conclusion, which many have seen as far too drastic. However, how successful these approaches have been in this regard remains a highly contentious issue.

Qiaochu Yuan
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  • I'm not sure I understand. I guess I should ask the following: what is the "correct" formal statement of the paradox? Does your phrase "there is no such set of axioms" mean that we can't state it formally? – Dmitry Oct 08 '22 at 20:46
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    @Dmitry: the liar's paradox is not a formal statement. It's an informal idea that can be formalized in multiple ways, in the same way that infinity is not a formal concept but an informal concept that can be formalized in multiple ways. Tarski's undefinability theorem is a good formalization if you really want one: it says that it's not possible for $P$ to state "$P$ is false" in first-order logic because it's not possible to say "false" at all. – Qiaochu Yuan Oct 08 '22 at 20:50
  • Thank you. I guess it was a mistake to look too deeply into the liear's paradox instead of the actual proven formal statements? It roughly outlines an idea, and the formal statements are quite different? – Dmitry Oct 08 '22 at 20:55
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    I haven't said anything of the sort! I think this is a completely natural question. Maybe reading the SEP article on the liar's paradox would be helpful: https://plato.stanford.edu/entries/liar-paradox/ The point I am trying to make, which maybe the SEP article will make better, is that the liar's paradox is something that happens prior to the whole concept of formal logic; it places constraints on what any formal system of logic can look like, rather than being a specific formal statement in or about a fixed such system. – Qiaochu Yuan Oct 08 '22 at 20:56
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For what it is worth, here is my proposed resolution of the Liar Paradox. It has nothing to do with the principle of explosion, or any notion of "meaninglessness," or a third truth value. It requires only some elementary set theory.

Given any set of sentences $S$, a trichotomy of 3 disjoint subsets exists within it:

  • The subset of true sentences $T\subset S$
  • The subset of false sentences $F\subset S$
  • The subset of sentences with indeterminate truth value $I\subset S~$ (e.g. "What time is it?" and "Wash your hands.")

$S=T\cup F\cup I$

Side note: Given ANY set $A$, there exists a 3 disjoint subsets $B, C, D \subset A$ that will satisfy the trichotomy rule. (Hint: Construct $B = A, ~C = D = \emptyset$)

Now, consider any sentence that is an element of $T$ if and only if it is an element of $F$. By the trichotomy rule, it can be neither an element of $T$, nor an element of the $F$. By default, it must be an element of $I$. If I understand correctly, "The sentence is false" is just such a sentence. It is one of indeterminate truth value.

Follow-up

For what it is worth, I presented my proposed resolution of LP to ChatGPT. It concluded:

"This approach is a valid way to deal with the Liar Paradox …[Y]our particular articulation and application of this approach to the Liar Paradox are unique and add to the ongoing discourse around paradoxes and truth value."

Full text: http://www.dcproof.com/ChatGPT.htm

Dan Christensen
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