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It seems to me and many others that can we solve the Liar's Paradox by saying that the Liar Sentence "This sentence is false" doesn't express a proposition. However, both the IEP and the SEP claim that such solution to the Liar Paradox is defeated by a Strengthened Liar:

(i) This sentence is either false or meaningless. (IEP)

(ii) This sentence does not express a true proposition. (SEP)

One would presumably derive a contradiction analyzing, say, (ii), thus: if (ii) expresses a true proposition, then, since it says it doesn't express a true proposition, it follows that it doesn't express a true proposition. Contradiction. If (ii) doesn't express a true proposition, then, since it says it doesn't express a true proposition, it follows that it expresses a true proposition. Contradiction.

The fault in the analysis, in my view, is in bold: (ii) doesn't say anything! It is much like "How old are you?" or "weofjwojiajzoijfeowi". That's essentially what it means to not express a proposition.

It is also claimed in the linked IEP article that saying the Liar Sentence is meaningless simply because "otherwise we get a paradox" is an ad hoc remark and, therefore, not a solution. Yet any set theorist will give you precisely that explanation when asked "Why can't we form sets of the form {x: φ(x)} for arbitrary formulas φ?". And people seem to be satisfied with that answer.

Just like the naive notion of forming sets by unrestricted comprehension is abandoned because it leads to contradiction, so should the naive notion that the Liar Sentence expresses a proposition.

Why isn't it enough to say the Liar Sentence doesn't express a proposition?

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    One problem is that this claim seems plainly false: the (modified) Liar Sentence [LS] is meaningful in a way in which ‘weofjwojiajzoijfeowi’ isn’t. We can understand LS just as much we can understand e.g. ‘This-and-this non-liar sentence does not express a true proposition’. So, you’d need to say what it is that we (seem to) understand or grasp here, if it's not a proposition. (I feel able to form the belief that the LS does not express a true proposition. In turn, if belief is a relation between believer and proposition, to what proposition am I relating here?) – MarkOxford Feb 4 '18 at 22:39
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    A closely related problem comes from the Principle of Compositionality. Very roughly, the PoC says that if you take meaningful words and string them together in a grammatically correct way, you get out something meaningful again (and ‘meaningful’ here is tantamount to expressing a proposition). In turn, all the words in LS are meaningful, and it’s grammatically well-formed. Hence, claiming that LS expresses no proposition would force you to give up Compositionality – which we’d rather not do. – MarkOxford Feb 4 '18 at 22:41
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    I see no reason why we cannot say the Liar sentence is meaningless. It looks meaningless to me, just a muddle of words. It is just as meaningless if we say 'This sentence is true'. We could say 'This word is difficult to understand' and everybody would agree this sentence is meaningless. . – PeterJ Feb 5 '18 at 12:41
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    @PeterJ Of course ‘This sentence’ doesn't have a truth-value: only sentences do, while sub-sentential expressions have referents / denotations / ‘semantic values’ of some other kind. In (ii), ‘this sentence’ simply denotes (ii) itself. To deny this would mean to meddle with the semantics of demonstratives, and that's methodologically unwise. (Also, a sentence can arguably be meaningful but have no truth value: ‘Harry Potter is a wizard’.) If the demonstrative really bothers you, consider this version of the Liar: “Let ‘L’ be a name for the following sentence: ‘L is false’”. – MarkOxford Feb 8 '18 at 15:55
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    @MarkOxford So is L false or true? And how can you included the sentence 'L' as part pf the longer sentence 'L is false' when L is supposed to name the whole sentence? – PeterJ Feb 9 '18 at 13:28
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[Since the OP found my comments helpful, I decided to expand them to a fuller answer.]

1. Preliminary remark

In the original post, the (unmodified) Liar Sentence was stated thus:

(L) This sentence is false.

Note the demonstrative ‘this sentence’. In turn, since mathematical languages don’t usually include demonstratives, this version of the Liar Sentence can’t easily be formalised, which may be a disadvantage. The main alternative would be to use ‘L is false’ as L, thus generating the self-reference by letting L use its own name. Yet ‘L is false’ is non-paradoxical per se, and paradoxical only if it is named ‘L’. So, does ‘L is false’ show something about truth, or about our use of names? Let’s leave that discussion and let’s go with (L) as it stands.

2. The Proposal

The solution advocated in the original post is to claim that L fails to express a proposition. Thus, the proposal is to endorse P:

(P) Sentence L does not express a proposition.

I think the rationale behind P is: Neither the claim that L is true, nor the claim that L is false, is tenable. So, we must avoid both claims. However, we don’t just want to say that L expresses a proposition that is neither true nor false; for, then the paradox can be restated as ‘This sentence is not true’. So, instead, we’ll say that L does not express a proposition at all. If it doesn’t, the question of whether it is true or false does not even arise – no more than the question of whether the Eiffel Tower is true.

The proposal faces the following problem: What shall we say about L+, below? If we extend P to this sentence and say that L+ does not express a proposition, either, then it looks as though L+ is true: after all, what it says is the case. Yet if L+ is true, then it does not express a true proposition, in which case it can’t be true. Finally, if L+ is false, it does express a true proposition, in which case it is true again.

(L+) This sentence does not express a true proposition.

In the original post, this problem is answered by holding that L+ is never true. To motivate this, note that e.g. the Eiffel Tower is never true, either – simply because the Eiffel Tower isn’t the kind of thing that can be true (or false). Likewise, L+ also isn’t the kind of thing that can be true or false – because it does not express a proposition. Thus, even under the assumption that L+ does not express a proposition, L+ is not true, contrary to the first step in the argument.

3. Objection

As the proposal stands, it is not very convincing because (a) there is a proposition that L+ seems to express, and (b) we have a reason for thinking that it does express this or some other proposition. Let’s start with (a), by noting that OP wants to endorse P+, which entails Q+.

(P+) Sentence L+ does not express a proposition.

(Q+) Sentence L+ does not express a true proposition.

Presumably, OP thinks that Q+ expresses a proposition, viz. the proposition that L+ does not express a true proposition. Let’s speak of PROP to mean that proposition. So, what reason is there for thinking that L+ doesn’t also express PROP? On the face of it, L+ is the subject of both Q+ and L+, and both claim of their subject that it does not express a true proposition. So, aren’t they saying the same thing? Don’t both express PROP? If not, what’s the difference?

Suppose I point at L+ and say: ‘OP believes that this sentence does not express a true proposition’. That seems like a true belief report, and it seems to express that the belief-relation holds between OP and PROP. Yet for that to happen, my use of L+ = ‘this sentence does not express a true proposition’ somehow singles out PROP. If so, why can’t those words express PROP when they occur on their own, viz. as L+? (Here, the preliminary remark, and the nature of demonstratives, may become relevant.)

The upshot is that we need to be told some story about why L+ does not express PROP. A Tarskian / Contextualist may have such a story; but we can’t just claim that L+ does not express a proposition and consider the Liar Paradox solved. Further, whatever story we do tell, it needs to say something about the Principle of Compositionality.

(PoC) For all complex expressions e, the meaning of e is determined by the meanings of e’s constituents, together with e syntactic structure.

Given that the constituents of L+ are meaningful, and given that L+ is syntactically non-defective, PoC entails that L+ is meaningful. That’s not quite the same as expressing a proposition, but it’s close enough to make trouble. In particular, if the constituents of Q+ compose to express PROP, why don’t the constituents of L+? Individually, all the constituents seem to have the right semantic values.

4. The ad hoc concern

As I said, we need a reason for denying that L+ expresses a proposition. The OP’s suggestion was that the paradox itself provides such a reason: L+ couldn’t express a proposition; because if it did, there would be a paradox. To support this, OP points out that Naïve Comprehension [NC] was rejected because it led to a paradox (viz. Russell’s). And that was the only reason for rejecting it (according to OP). So why isn’t it a good-enough reason to reject that L+ expresses a proposition?

One answer, I think, is that Russell’s Paradox directly challenges NC. NC says: ‘For every F, there’s a set of all Fs’, and the paradox then asks: ‘What about F = does not contain itself as element?’ Since NC can’t answer this question, it looks like NC is directly responsible for our troubles. (Though Dummett argued otherwise?) By contrast, there is no one principle that generates the Liar Paradox, where we’d say: ‘Yeah, that’s the culprit!’ The Liar Paradox has a number of ingredients, so that it really would be ad hoc to just pick one and discard it.

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Liar paradox variants

It seems to me and many others that can we solve the Liar's Paradox by saying that the Liar Sentence "This sentence is false" doesn't express a proposition.

It depends on what you mean by "proposition", but indeed with the right view it is perfectly defensible.

Both the IEP and the SEP claim that such solution to the Liar Paradox is defeated by a Strengthened Liar:

(i) This sentence is either false or meaningless. (IEP)

(ii) This sentence does not express a true proposition. (SEP)

Note that IEP and SEP are not necessarily accurate or precise, because each article is often written by a single person and not peer-reviewed. In this case, they are only correct if you impose classical logic on sentences (i) and (ii).

The fault in the analysis, in my view, is in bold: (ii) doesn't say anything! It is much like "How old are you?" or "weofjwojiajzoijfeowi". That's essentially what it means to not express a proposition.

I would say that you partly got it, but not very clearly, so let me explain.

Firstly, there is a valid objection against the liar paradox that it is not a valid definition. Logically, one cannot refer to something that one has not defined. In this case, any variant of the liar paradox that uses "this sentence" is referring to something that has not yet been defined! This is equivalent to the following nonsense:

??? Let P be a boolean sentence such that P is equivalent to ¬P.

If it is not clear why this is illogical, consider the following:

??? I want to talk about the integer that is one more than itself.

The correct objection is that we cannot talk about something we have not defined, and we cannot talk about something that satisfies some description unless we have shown that there is such a thing to begin with!

Thus any sentence that contains "this sentence" is simply a string of words without meaning.

Quine's paradox

But there is another paradox that completely avoids any circularity. Consider the following sentence Q:

" preceded by the quotation of itself is not a true sentence." preceded by the quotation of itself is not a true sentence.

Q is a perfectly grammatical sentence that does not refer to itself, so one cannot invoke the circularity objection to the liar paradox variants. But Q still uses the notion of "truth", which can only be imbued with meaning by interpretation in the real world, and as explained in the linked post that is exactly where it fails.

To be more precise, Q cannot be justified to be a sentence about reality, and so cannot be justified to have a (boolean) truth value. If Q is a true sentence, then we can deduce a contradiction. If Q is not a true sentence, then we also can deduce a contradiction. But "Q is a true sentence" itself cannot be justified to have a truth value! So we cannot deduce an absolute contradiction.

Furthermore, "Q is a sentence about reality" cannot be justified to have a truth value either, so we cannot deduce "Q is not a sentence about reality", although if we wish we can 'go to the meta-level' and observe that we really are unable to deduce "Q is a sentence about reality".

For whatever reason, not many philosophers are aware of this resolution of the paradoxes. But "sentence about reality" shares a striking similarity to Kripke's notion of "grounded sentences", because any sentence about reality is literally grounded semantically in the real world. Of course, Kripke had extended grounded sentences to beyond sentences about reality, but that is a whole other topic.

Unrestricted set comprehension

Now let me address the side-remarks about set theory.

Just like the naive notion of forming sets by unrestricted comprehension is abandoned because it leads to contradiction, so should the naive notion that the Liar Sentence expresses a proposition.

There is a significant philosophical problem with the common viewpoint of many set theorists. Namely, the notion of "set" was supposed to capture the notion of "collection". If really there is some set-theoretic universe that satisfies ZFC, then that universe itself is a collection, and clearly the ZFC axioms do not correctly capture that. MK (Morse Kelley) set theory does not solve that, because again there is no class of all classes.

In any case, there is no non-circular philosophical justification for ZFC, so ZFC is in fact a red-herring in discussing Russell's paradox.

Discarding axioms on reaching contradiction

Finally, I want to point out that it is not viable to simply discard axioms that lead to contradiction. For a simple example, if PA is consistent then PA+¬Con(PA) is also consistent, but proves a false sentence (under the standard interpretation of natural numbers in the real-world). This clearly shows that mere consistency is nowhere near enough to make a logic or formal system meaningful, and we must have some kind of soundness. At the least, we ought to have arithmetical soundness (at least at human scales).

  • Comments are not for extended discussion; this conversation has been moved to chat. Please head to the chat for any further comments. – Philip Klöcking May 4 '18 at 20:24
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This is a comment that is too long to be posted as such.

user21820's solution to the (Strengthened) Liar seems reasonable to me: we can't prove the Liar is grounded; for that, we'd need to show there exists a grounded sentence L which is equivalent to the grounded sentence "L is not a true sentence". In general, circular definitions can't be assumed to be valid: this is akin to the fact that, in mathematics, recursive function definitions must be accompanied, at least implicitly, by grounding theorems.

Some philosophers would criticize such solution by saying that there are unproblematic circular sentences, such as 'This sentence has more than 2 characters'. But we can simply rewrite such sentence as

"'This sentence has more than 2 characters' has more than 2 characters".

Note that the inner string need not be grounded as a sentence in order for the outer string to be grounded as a sentence, for the latter simply makes a claim about the former's syntax. We can easily conclude from reasonable assumptions that the sentence is grounded. If we tried to do the same for the Liar, the inner string would need to be grounded first, but it would be precisely the Liar all over again. Furthermore, sentences such as "This sentence is true" would not be provably grounded, which is OK, since they don't seem of much use anyway.

There are circular sentences that are useful in practice and that don't seem to be groundable, though. Note how I said "This is a comment that is too long to be posted as such" above. It is not clear how such statement can be proved to be grounded.

We could reformulate it as "What follows is a comment that is too long to be posted as such", but that would slightly change the meaning. Also, problems would arise because I'm mentioning the statement here as well, making everything circular again.

Thoughts on this last point?

  • Yes you understood me correctly. To say that a circular sentence is meaningful, we would have to somehow justify that it is equivalent to some grounded sentence. And I agree with your example, where "This sentence has more than 2 characters." is merely a string that concretely has more than 2 characters, and by that deduction we can then choose (if we wish) to imbue that string with meaning. This is why we must distinguish between syntax and semantics. As for your last point, isn't it the same as your example? "This" refers to your post as a string, not its interpretation. – user21820 May 3 '18 at 8:16
  • @user21820 Hmm, you're right. The notion of comment is indeed a syntactic one, not semantic. And I was simply making a claim about its length. I wonder if there are examples of ungroundable but useful sentences "in the wild", though. – Detached Laconian May 3 '18 at 20:53

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