Some sentences are true. Some are false. However, sentences like "What time is it?" or "Wash your hands" can be said to be of indeterminate truth value. This suggests a trichotomy on any set of sentences. Every sentence in that set is precisely one of: a true sentence, a false sentence or a sentence of indeterminate truth value.

Now, "This sentence is false" is said to be a true sentence if and only if it is false sentence. This contradiction eliminates both possibilities, leaving only the possibility that it is a sentence of indeterminate truth value.

Formal proof: https://dcproof.com/LiarParadox2.htm (only 44 lines in DC Proof format)

Trichotomy Lemma: For every set, there exists 3 disjoint subsets on which a trichotomy rule holds. https://www.dcproof.com/LiarParadoxLemma.htm (84 lines)


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    – Philip Klöcking
    Aug 21, 2023 at 6:08
  • Lumping the Liar sentence with interrogatives and indexicals is not very helpful. The latter are syntactically identifiable, and have transparent semantics explaining why they do not have truth values. The former is a declarative, and there is no apparent semantic reason for it to lack a truth value. What we need is a syntactically identifiable feature that would tell us which declaratives lack truth values. Saying "those that lead to a paradox" is circular, as it has to explain why they do that.
    – Conifold
    Aug 22, 2023 at 0:34
  • The only characteristic of "This sentence is false" that we need here is that it is an element of the subset of true sentences if and only if it is an element of the subset of false sentences. From this contradiction, we can infer that it is not an element of either of those subsets. We know that there are sentences that are elements of the subset of sentences of indeterminate truth values, e.g. "Wash your hands." "This sentence is false" must then be an element of that set. Aug 22, 2023 at 3:40

4 Answers 4


Statements in logic are propositions: statements that assert (propose) a truth-value. Not every natural language utterance is a proposition. That doesn't mean they are 'undecided'; that means they are outside the purview of logic. Further, the nature of a proposition is that its truth-value is naturally indeterminate (unless it's an axiom) until its truth-value is derived from other 'known' propositions according to systematic rules.

The point is that adding an 'indeterminate' category to make this trifecta doesn't add anything to logic, unless you mean it to be an 'absolute' category - the way that 'true' and 'false' are absolute — in which case it just seems like a different sort of paradox. I mean, We can prove something is true, we can prove something is false, but how can we prove that some given proposition cannot be proven to be true or false?

The Liar's Paradox (and similar things like Russell's Paradox), are best viewed as a kind of category error: the moment of self-referential negation that lies at the heart of them merely highlights that the self-reference has gone wrong: the referring statement is not actually identical to the referred statement, despite the linguistic implication. Recognizing the limitations of language here seems more reasonable than trying to revise the entirety of logic.


Quine and others have an "indeterminacy principle"[1] where there are different, equally correct translations of language, which seems to completely allow for your interpretation.

Opponents like Katz and Keenan respectively have maximal and weakened effibility principles where any thought/proposition can be communicated, contra indeterminacy.

We get two relatively opposed modes of attack: what should we interpret (Quine) vs. what is being communicated (K&K).

I think a plausible answer is: if following Quine, you simply can take an indeterminate position regarding the liar paradox and language writ large. But opening this Pandora's box will come at cost if you try to argue anywhere else for objectivity through language, which seems to exist in some cases like math say. If you follow K&K, the liar paradox is objectively communicating something, but what? You might find the liar paradox is simply too short/underdetermined to make the case it's communicating indeterminism.

There's many other ways, and more fine-grained ones, to approach this question, but I hope it shows some of the broad approaches philosophers deal with to harness language.

[1] https://plato.stanford.edu/entries/quine/#IndeTran


Firstly, I don't think the trichotomy rule is correct. Maybe I'm misunderstanding your notation, but:

Apply Subset Axiom

 2     EXIST(sub):[Set(sub) & ALL(a):[a e sub <=> a e s & a e s]]
 3     Set(t) & ALL(a):[a e t <=> a e s & a e s]

What is this? I assume "a e s" means "a is an element of s" but there's not really a "mere subset" axiom in set theory, or at least not like you've written down. In fact, if I remember correctly, it is somewhat notoriously difficult to pin down a good formula for arbitrary subsets,E but the special axiom is the one asserting that all subsets of a base set can be composed into another, "higher," set. The definitions that do occur are not (to my knowledge) equivalent to (3) in the above, they're more like (this is from Wikipedia; oddly, the formulas on Wolfram are not altogether perspicuous):

enter image description here

More importantly, we can falsify the trichotomy rule ab initio because it is already false for every set with less than three subsets at all. I.e. the empty set does not have three subsets, but one, itself; the unit set has two subsets, but only two (itself and the empty set). Or that's how it seems to me; the notation used in the program is unfamiliar to me and so are the claims about partition relations (from the material I've looked into when trying to understand Erdős cardinals and amorphous sets, it doesn't seem necessary, much less trivial, that every set whatsoever can be partitioned in such specific ways, but there seem to always be possible sets that are exceptions to various sorts of (obscure) partition principles).

Re: "It's all ordinary true-or-false logic with a bit of set theory": however, you have the set of sentences S as having subsets T, F, and M, where each subset is determined by a predicate. But to have such subsets and their identifying predicates is to have truth values, so if M is a subset of S then M is ipso facto a third truth value (as Bumble notes, using more of the language of extensionality). Now it is typically said that the truth-value powerset of {T, F} is {0, T, F, {T&F}}, then, so contra the desired trichotomy, S would have a subset whose elements are all the elements of {T&F}, it would seem. And this is why your formulation of your argument confuses me:

  1. You seem to be saying that if x ∈ {T&F}, then x ∈ 0.
  2. But for no x is x ∈ 0.
  3. Even assuming otherwise (somehow: maybe not impossibly, I will confess), if x ∈ 0 only on condition that it be first an element of {T&F}, then it seems like it's actually an element of {0, {T&F}}, which seems to allow for a revenge paradox.

On the other hand, I learned somewhat recently that intuitionistic logic's intended interpretation is not as three-valued, but I don't think it's meant to be strictly two-valued, either (or if it is two-valued on one level, it's something else on another, or a rather different framework is in play anyway and the question of "how many truth values are there" isn't very well-formed in intuitionistic metalogic).

ETo be sure, there is no substantive/specific formula for all subsets of some infinite set, but to my understanding, there isn't an easily-recognized trivial/general formula, either. The notion seems second-order, a matter of tracking a complex relation between other sets and elements by a name that consolidates the terms of the relation. We want to translate, "If A is a set of elements B of some set C, then A is a subset of C and an element of some D (where D is the powerset of C)":

enter image description here

I think that if we don't emphasize A's sethood, it's harder to see how C itself will be a subset of itself, since C is not an element of C. Not impossible, though; the formula from Wikipedia works (I think), although maybe it's not as "clear" (to me, anyway).

  • The Trichotomy lemma starts with a set s and uses the subset axiom to constructs 3 subsets t, f and m where t=s, f=m={ }. Every set is a subset of itself. And the empty set is a subset of every set. Note that this is just a proof of the existence of a single trichotomy on s. This is not an attempt to construct every possible trichotomy on s. dcproof.com/LiarParadoxLemma.htm Aug 21, 2023 at 5:41
  • @DanChristensen that link starts out by saying, "For every set there exists 3 disjoint subsets on which a trichotomy rule holds," but there are two sets, the empty and unit sets, that don't have three subsets. Also, if f = m, then I'm even more lost (I thought f was the subset of false sentences???). Aug 21, 2023 at 5:57
  • I also don't understand why "a e s & a e s" occurs. Do you mean ((a e s) & (a e s)) or (a e (s & a e s))? You might have to take this to a logic professor to get a more detailed response, but from where I'm standing, the intended translation of the formalism doesn't seem to work. Aug 21, 2023 at 6:00
  • The lemma proof at dcproof.com/LiarParadoxLemma.htm is a proof about sets in general--nothing to do with sets of sentences, etc. Apologies for this confusion. This proof is used only to justify the existence of trichotomies on ANY given set. The proof starts with an arbitrary set s (nothing to do with sentences). Then it constructs, i.e. proves the existence of subsets t, u and v that are shown to have the trichotomy property. These subsets, too, have nothing to do with sentences. Aug 21, 2023 at 15:57
  • @DanChristensen but we know that not every set has three such subsets, because again, the empty and unit sets don't. So the proof is falsified by the first two cases for use in transfinite induction, isn't it? But again, between your interpretation of your formalism, the formalism itself, and the actual topic as historically understood, there seems to be a large gap, so it's hard to tell what you mean by "all sets have the trichotomy property." I have never seen a derivation of such a property in my studies, albeit brief, in partition theory, so I am wary of a claim to derive this property. Aug 21, 2023 at 17:42

It would help to get some terminology straight here. A sentence, in the ordinary sense of the term, is a sequence of words that obeys some grammar and is able to express a complete unit of meaning. Some sentences express propositions; others are used to express speech acts that are not propositions. Philosophers use the word 'proposition' to mean lots of different things, but the core meaning includes the fact that a proposition is a truth-bearer, i.e. it is capable of being true or false.

It is not appropriate to compare the liar sentence to a speech act. Speech acts are incapable of being truth-bearers under any circumstances. By contrast, the liar sentence obeys the grammar of ordinary English in stating a proposition. It should therefore be capable of having a truth value, but it is problematic because neither true nor false fit the bill.

The principle of bivalence states that every proposition is either true or false. It is a statement about the potential truth values of a proposition. Under the assumption of bivalence, a proposition can be true; it can be false; no other possibilities exist. Classical logic, at least in its standard form, assumes bivalence. There are logics that are not bivalent. Some of these are many-valued, i.e. they are allow for additional truth values. Others, like intuionistic logic, do not have any third value, but they are not bivalent because they allow that a proposition may fail to be either true or false and just have no truth value at all.

If you wish to allow that a proposition, such as the liar sentence, is neither true nor false, then you are abandoning bivalence, and with it classical logic. Your proof of:

((x is an element of the subset of true sentences) ↔ (x is an element of the subset of false sentences)) → (x is an element of the subset of sentences of indeterminate truth value)

misses the point that true and false do not just designate sets of sentences, they are themselves truth values. You could partition the set of propositions into subsets using arbitrary labels such as red, green and blue. It would then be straightforward to prove that

((x ∊ Reds) ↔ (x ∊ Greens)) → (x ∊ Blues)

But red, green and blue, are not truth values of propositions. True and false are. You are failing to include the background semantic information that

  • x is true iff (x is an element of the subset of true sentences)
  • x is false iff (x is an element of the subset of false sentences)
  • x is indeterminate iff (x is an element of the subset of indeterminate sentences)

Hence if the liar sentence is neither an element of the subset of true sentences nor an element of the subset of false sentences then it is neither true nor false, and so bivalence is lost. You either have a third truth value, or the possibility that a proposition lacks a truth value under some circumstances. Either way, we are off into the realms of non-classical logic. There are many potential resolutions of the paradox, some of which stick to classical logic while others do not. The SEP article has a good overview.

As an important corollary. Simple attempts to avoid the liar paradox by introducing an extra category such as indeterminate do not work because it is always possible to formulate an alternative version of the liar, such as

This sentence is either false or indeterminate.

This restores the paradox, since it is true if and only if it is false or indeterminate.

  • Again, sentences, as opposed to logical propositions, may be classed as true, or false or of indeterminate truth value. It seems to me that this distinction between sentences and logical propositions is key to a resolution of the Liar Paradox. Having a set divided into 3 disjoint subsets does not in itself introduce a third possible truth value. Aug 21, 2023 at 16:12
  • Sentences that are not propositions are neither true, false, nor indeterminate. They lack truth values entirely. Propositions may be true or false, and if we are willing to abandon bivalence, they may have other truth values, or possibly lack a truth value under some circumstances. One can say that the liar sentence fails to be a proposition: this is in fact one possible resolution of the paradox known as contextualism. plato.stanford.edu/entries/liar-paradox/#ContAppr
    – Bumble
    Aug 21, 2023 at 19:37

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