Also, context is missing.

A statement with no context is meaningless.

Put the statement in context and then we can evaluate what it means/says.

Here's context: A person walks up to a stranger on a city street and says, "I am lying", and then walks away.

But, ''this'' isn't like ''I''. If we stay true to the definition of the word then ''this'' doesn't apply to itself and it should for the liar paradox to be one.

Of course we could invent a self-referential word e.g. ''thes'' and define it as such and the paradox would appear.

If one were to be as exact as possible the definition of ''this'' doesn't include self-reference. It is grammatically incorrect (I'm not a linguistic expert).

However, people do use ''this'' as you have (''this Australian needs a bath'' :D) but note that such forms of language are classified as referring to oneself in the third person. It isn't completely an instance of self-reference. People would find it odd to hear someone refer to himself in the third person.

So, I still think the liar sentence is grammatically incorrect.

However, as I mentioned above we could invent a self-referential word like ''thes'' and the liar paradox still is a problem. — TheMadFool

"This sentence is false," is only self-referential on the sentence level. "This" on its own refers to nothing at all; it's a determiner in the noun-phrase "This sentence", and that nounphrase is also not self-referential (It can't be because a noun-phrase can't be the referent for "this sentence").

Finally, the syntax can only tell you that "this sentence" refers to a sentence that the speaker indicates. The sentence is not inherently self-referential. You could point to any other false sentence while saying this. There's nothing in the syntax, though, that prevents you from picking the sentence the nounphrase occurs in, making that sentence (but not the nounpharse itself, much less "this" alone) self-referential. The liar sentence is perfrectly grammatical, and the syntax is pretty much irrelevant, except that it allows the sentence to have a self-referential interpretation.

Formally, "This sentence is false," is self-referential under the liar-interpretation because the sentence's subject refers to the sentence it is a subject of. To be able to do this, the subject cannot refer to itself (and thus be self-referential on its own).

The liar sentence is perfectly grammatical.

That seems false. Just take the Liar and it gives you a grounded truth value to start with (namely falsity). — MindForged

Yes, but it doesn't ever give you a grounded truth-apt subject. To determine the truth of the liar sentence first requires determining the truth of the subject ("This sentence"). That requires substitution with the original liar sentence and so on ad infinitum. There is no final truth-apt subject to ground the liar sentence.

So the liar sentence fails to assert anything about a truth-apt subject and so isn't itself truth-apt. If you disagree, then what do you think is being asserted?

I have a gut feeling the liar paradox is important. It must mean something. I just don't know what it is. — TheMadFool

The liar sentence shows that not all sentences that appear to meaningfully assert something actually do so. It's the linguistic equivalent of a mirage.

Thanks. Dialetheism it is then.

Hah, it's not that simple but I confess I find the view persuasive.

Are you sure it can do that validly? The linked page states the lemma with a premise that restricts it to first-order languages, which I expect would rule out its use in a T-schema environment which I believe is higher order.

I was going to check the proof to see if that premise is actually used, but I got tired and didn't, so I'm hoping maybe somebody else did. It would be unusual to state a premise that was not used though.

I admit I'm too lazy (and busy, about to go to work) to check, so I just jumped on Wiki real quick and it seems to say the same:

Tarski proved a stronger theorem than the one stated above, using an entirely syntactical method. The resulting theorem applies to any formal language with negation, and with sufficient capability for self-reference that the diagonal lemma holds. First-order arithmetic satisfies these preconditions, but the theorem applies to much more general formal systems.

Yes, but it doesn't ever give you a grounded truth-apt subject. To determine the truth of the liar sentence first requires determining the truth of the subject ("This sentence"). That requires substitution with the original liar sentence and so on ad infinitum. There is no final truth-apt subject to ground the liar sentence.

That's just assuming the Kripke's solution and it fails for the same reason. This notion of groundedness can be just as easily used to restate the paradox:

This sentence is ungrounded.

Which must be true as per the original Liar, and thus is true and not grounded. Or else it's false and not grounded. Or else we must say that this notion of groudedness does not do the work you would need it to, because it still throws up the contradiction in the "revenge" paradoxes. The predicate "is false" must be part of the project of determining the truth of the Liar sentence, otherwise your only recourse seems to be assuming that self-reference isn't an allowed move.

So the liar sentence fails to assert anything about a truth-apt subject and so isn't itself truth-apt. If you disagree, then what do you think is being asserted?

What's being asserted is that the sentence itself (falsity predicate included) is not the case.

I took the T-schema and so said that "True(x)" has the same truth-value as just asserting that "x". — MindForged

The T-schema is an attempt to define truth. What do we mean when we say that a sentence is true? To say that "'T' is true" is true if "T" is true, which you are claiming is what the T-schema is saying, doesn't answer this question.

What the T-schema is saying is that "'T' is true" means "T"; that "'it is raining' is true" means "it is raining". And what does "it is raining" mean? That can be explained by referring to some empirical state of affairs.

But take a sentence like "this sentence is true". We can use the T-schema to say that "'this sentence is true' is true" means "this sentence is true", but what does "this sentence is true" mean? Unlike "it is raining", we can't refer to some empirical state of affairs. The "is true" in "this sentence is true" isn't saying anything. Just as the "is false" in "this sentence is false" isn't saying anything. Truth-predication in these cases is a category error.

The T-schema is an attempt to define truth. What do we mean when we say that a sentence is true? To say that "'T' is true" is true if "T" is true, which you are claiming is what the T-schema is saying, doesn't answer this question.

Um, that's not what I said, even in your quote of me. I said:

I took the T-schema and so said that "True(x)" has the same truth-value as just asserting that "x".

IOW, that 'x' and "True(x)" are logically equivalent, they have the same logical value/truth-value according to the T-schema. That seems indistinguishable from what you said so I don't follow what you're saying here, we said the same thing as far as I can tell.

But take a sentence like "this sentence is true". We can use the T-schema to say that "'this sentence is true' is true" means "this sentence is true", but what does "this sentence is true" mean? Unlike "it is raining", we can't refer to some empirical state of affairs. The "is true" in "this sentence is true" isn't saying anything. Just as the "is false" in "this sentence is false" isn't saying anything. Truth-predication in these cases is a category error.

"This sentence is true" under the T-schema would be logically equivalent to saying the sentence is the case: True(x) <=> x

The same for the Liar under the T-schema. The sentence "This sentence is false (or untrue)" is logically equivalent (same truth-value) to the previous is the case: ~True(x) <=> x

Isn't this simply the Tarski Undefinability Theorem?

"This sentence is true" under the T-schema would be logically equivalent to saying the sentence is the case: True(x) <=> x — MindForged

And what does it mean for the sentence to be the case? Again, unlike something like "it is raining", it isn't explained by referring to some empirical state of affairs, and unlike something like "2 + 2 = 4", it isn't explained by referring to the axioms and definitions of mathematics.

It doesn't mean anything for sentences like "this sentence is true" or "this sentence is false" to be or not be the case, and so it doesn't mean anything for sentences like "this sentence is true" or "this sentence is false" to be true or false.

This sentence is the case
This sentence is not the case

They don't actually say anything.

Um, that's not what I said, even in your quote of me. — MindForged

You said that "True(x)" and "x" have the same truth value. I assume "True(x)" means "'x' is true"? So "'x' is true" and "x" have the same truth value. Which means that "'x' is true" is true iff "x" is true.

But the T-schema is saying more than this. It's trying to explain what it means to have a truth-value.

You said that "True(x)" and "x" have the same truth value. I assume "True(x)" means "'x' is true"? So "'x' is true" and "x" have the same truth value. Which means that "'x' is true" is true iff "x" is true.

But the T-schema is saying more than this. It's trying to explain what it means to have a truth-value.

I don't know if it means that, because notice how the T-schema is setup:

True(x) <=> x

If it were simply to explain the meaning if truth within logic, wouldn't the equality sign be used (such statements are in some sense metalogical)? The T-schema uses a biconditional (an operator within a logical system), which just means the proposition and the proposition predicated as true have the same truth-value. After all, it goes both ways. "True(x) <=> x" and "x <=> True(x)". In a sense, they are the same, but really I think it's simply that they are inter-derivable.

Tarski proved a stronger theorem than the one stated above, using an entirely syntactical method. The resulting theorem applies to any formal language with negation, and with sufficient capability for self-reference that the diagonal lemma holds. First-order arithmetic satisfies these preconditions, but the theorem applies to much more general formal systems. — Wiki

That quote is from a different article (this one), and what it refers to as 'the one stated above' is not the Diagonal Lemma.

The article goes on to say that Tarski's Undefinability Theorem - the theorem the article is about - says there cannot be a formula in the relevant language L that defines T*, the set of true formulas in (L,N). I interpret that as implying that the 'Liar sentence' is not expressible in a formal language, even one with 'sufficient self-reference', because there is no 'True(...)' formula or predicate.

At the end of the article it says

An interpreted language is strongly-semantically-self-representational exactly when the language contains predicates and function symbols defining all the semantic concepts specific to the language. Hence the required functions include the "semantic valuation function" mapping a formula A to its truth value ||A||, and the "semantic denotation function" mapping a term t to the object it denotes. Tarski's theorem then generalizes as follows:No sufficiently powerful language is strongly-semantically-self-representational. — wiki

I am interested in exploring this more. I do not have much familiarity with Tarski's work.

I don't have time to do a proper response at this moment but just a note. You said the quote was from a different article but you literally linked the same Wiki article (and even the same section) as I did. Maybe I misunderstood your newest post (I did have to skim it), but too busy at work, lol.

The same for the Liar under the T-schema. The sentence "This sentence is false (or untrue)" is logically equivalent (same truth-value) to the previous is the case: ~True(x) <=> x — MindForged

Introducing "...is the case" adds nothing, but gives the illusion of explanation. In particular it gives an unwanted air of correspondence to Tarski's schema.

There's two things you can do with T-sentence.

If you know that the subject sentence on the left and on the right are the same, you can use it as a definition of "...is true".

So "it is raining" is true IFF it is raining

can give us a definition of truth; but only if we have a clear understanding of meaning. Easy when the sentences are the very same. What of:

"Hab SoSlI' Quch!" is true IFF your mother has a smooth forehead

That's the other use. If you are not so sure of the meaning of "Hab SoSlI' Quch!", you can take truth as given and use a T-sentence to set out the translation.

That's Davidson's plan.

What you can't legitimately do is use a T-sentence to define both meaning and truth - at least, not for the same sentence.

In particular it gives an unwanted air of correspondence to Tarski's schema. — Banno

Tarski actually intended for that. From The Semantic Conception of Truth:

We should like our definition to do justice to the intuitions which adhere to the classical Aristotelian conception of truth-intuitions which find their expression in the well-known words of Aristotle's Metaphysics:

To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, or of what is not that it is not, is true.

If we wished to adapt ourselves to modern philosophical terminology, we could perhaps express this conception by means of the familiar formula:

The truth of a sentence consists in its agreement with (or correspondence to) reality.
(For a theory of truth which is to be based upon the latter formulation the
term "correspondence theory'' has been suggested.)

If, on the other hand, we should decide to extend the popular usage of the term "designate" by applying it not only to names, but also to sentences, and if we agreed to speak of the designate of sentences as "states of affairs," we could possibly use for the same purpose the following phrase:

A sentence is true if it designates an existing state of affairs.

I was by no means surprised to learn (in a discussion devoted to these problems) that in a group of people who were questioned only 15% agreed that "true" means for them "agreeing with reality," while 90% agreed that a sentence such as "it is snowing" is true if, and only if, it is snowing. Thus, a great majority of these people seemed to reject the classical conception of truth in its "philosophical" formulation, while accepting the same conception [my emphasis] when formulated in plain words.

This is why I asked you before if your view was closer to Ramsey's or Tarski's. Tarski's is a correspondence theory, just rephrased to be more precise and clear, whereas Ramsey's is a true redundancy theory.

Yes, it was the same as in your latest post. What I meant was that it was different from the article you linked in introducing this sub-thread, which was about the diagonal lemma. But I don't think that matters, and I wonder if we have been talking past each other, while really agreeing all along. My reading of that latest article is that Tarski used the diagonal lemma to show that, if there is a 'True(...)' formula in a strongly self-referential language L then a contradiction must arise. It follows from that that such a language cannot have a 'True(...)' formula, so that the Liar sentence cannot be expressed in it. Hence, any string of symbols purporting to implement the Liar sentence must fail to do so, either syntactically (it is syntactically invalid) or semantically (it implements some other meaning).

In other words, Tarski proved that sentences that look like Liar sentences are really just meaningless confusions arising from a failure to be sufficiently formal.

Do you agree with that interpretation?

This sentence.

I think I agree. The article does say the Liar cannot be expressed in a strongly self-referential language. So any language the Liar appears is not "strongly-semantically-self-representational". That's why Tarski (as per the article) thought the Liar was a reductio of such expressively powerful languages, to the extent that he argued that natural languages were in fact inescapably incoherent and should be replaced with similar but formal languages which avoid such paradoxes. However, I do not agree that the point is the Liar is meaningless or cannot be expressed. It depends on how expressive the language is. If it's "strongly-semantically-self-representational" then it's not expressive enough to make a Liar.

That's just assuming the Kripke's solution and it fails for the same reason. — MindForged

I think Kripke grants that the liar sentence is a meaningful assertion but that it just lacks a truth value (and so therefore has some third value). Whereas I am claiming that the liar sentence isn't a meaningful assertion at all because it fails to meet the logical criteria for one. A bit like the sentence "the tree is false".

This notion of groundedness can be just as easily used to restate the paradox:

This sentence is ungrounded. — MindForged

That sentence fails for the same reason as the liar sentence. We can all agree that that sentence is ungrounded. But, being ungrounded, the sentence itself doesn't meet the logical criteria required for a meaningful assertion. So you can't then treat it as if it does.

That is, the sentence appears to be asserting something about itself. But it is not, despite surface appearances. Whereas our assertions about the sentence are truth-apt as long as we're not asserting that the sentence is true or false.

That is the sense in which the liar, truth-teller and revenge paradoxes are like a mirage. There appears to be water there, and it makes us think about water, but appearances are sometimes deceiving. There's no water there.

This sentence. — Luke

Not even wrong. :-)

...is not a sentence because it lacks a verb.

I think Kripke grants that the liar sentence is a meaningful assertion but that it just lacks a truth value (and so therefore has some third value). Whereas I am claiming that the liar sentence isn't a meaningful assertion at all because it fails to meet the logical criteria for one. A bit like the sentence "the tree is false".

Kripke is the one who came up with this groundedness solution as far as I know, you seem to have presented his analysis of it. It is not stated to have a 3rd value, it's interpreted as a failure of the Law of the Excluded Middle. And that comparison seems disanalogous. Trees don't even have the appearance of a truth-apt object, whereas even you seem to agree that the Liars at least appear as if they are truth-apt.

That sentence fails for the same reason as the liar sentence. We can all agree that that sentence is ungrounded. But, being ungrounded, the sentence itself doesn't meet the logical criteria required for a meaningful assertion. So you can't then treat it as if it does.

That is, the sentence appears to be asserting something about itself. But it is not, despite surface appearances. Whereas our assertions about the sentence are truth-apt as long as we're not asserting that the sentence is true or false.

That is the sense in which the liar, truth-teller and revenge paradoxes are like a mirage. There appears to be water there, and it makes us think about water, but appearances are sometimes deceiving. There's no water there.

No no, the notion of groundedness refers to, essentially, hacking off the truth-predicate. The predicate "is grounded" (and its negation) aren't truth predicates so it's not subject to the same criticism, unless you are arguing that self-reference is itself not an allowed thing to do in language. If you agree the sentence is ungrounded, that entails that it is true, which contradicts being ungrounded. This is why Kripke's solution isn't very popular nowadays (even if there is much to commend about his attempt). If our assertions about the sentence are truth-apt there is no functional difference between the sentence referring to itself and saying exactly what we say about it. Kripke, who developed this resolution, also agreed that his solution is probably subject to the revenge paradox I gave.

They aren't mirages, they're contradictions (I showed a rendition of the argument earlier in the thread). Also, the truth teller isn't really a paradox.

And that comparison seems disanalogous. Trees don't even have the appearance of a truth-apt object, whereas even you seem to agree that the Liars at least appear as if they are truth-apt. — MindForged

What they both have in common is that the full sentence appears truth-apt (since it has a subject and a predicate) until, of course, the content of the sentence is analyzed and the subject is found to not support the predication. It's a category mistake (as Michael earlier noted).

No no, the notion of groundedness refers to, essentially, hacking off the truth-predicate. The predicate "is grounded" (and its negation) aren't truth predicates so it's not subject to the same criticism — MindForged

Then we are using "grounded" in a different way. I mean that the subject is resolved and supports the predication, whatever it may be. In this case, the subject doesn't support the grounded predication and so the sentence isn't truth-apt. (BTW, this was essentially Gilbert Ryle's solution to the liar-style sentences rather than Kripke's.)

unless you are arguing that self-reference is itself not an allowed thing to do in language. — MindForged

Self-reference is generally fine. For example, "this sentence has ten words". The truth or falsity of this doesn't depend on the subject being truth-apt, only that its words can be counted. That is a valid predication and so the sentence is truth-apt.

If you agree the sentence is ungrounded, that entails that it is true, which contradicts being ungrounded. — MindForged

It doesn't entail that since the sentence doesn't support truth predication (because, in turn, the subject of the sentence doesn't support grounded predication). But you're treating it as if it does.

The liar sentence shows that not all sentences that appear to meaningfully assert something actually do so. It's the linguistic equivalent of a mirage. — Andrew M

Why is the mirage there in the first place?

The liar sentence is perfectly grammatical. — Dawnstorm

I'll accept that because ''this'' may be defined to self-refer.

Why is the mirage there in the first place? — TheMadFool

The form: [subject] [copula] [adjective]. X is Y. And, ordinarily, it is appropriate for the adjective ("true") to be predicated of the type of thing that the subject is (a sentence). It's just that in this particular case it isn't, given that predicating truth of a sentence is only meaningful when affirming some other predicate (like "is red" or "follows from the axioms"), whereas the recursive nature of the liar sentence means that it isn't (hence the term "ungrounded").

I'll accept that because ''this'' may be defined to self-refer. — TheMadFool

You can rephrase the liar sentence:

"The sentence I am uttering right now is false."

"What I'm in the process of saying right now is false."

What matters is that the subject of the sentence refers to the sentence it occurs in. No single component of the sentence need be self-referiatial by itself for that to happen.

I don't understand why you want to define "this" to self-refer.

What matters is that the subject of the sentence refers to the sentence it occurs in. — Dawnstorm

Not necessarily:

1. 2 is true
2. 1 is false

True, you can rephrase this in many ways. What I'm addressing is the connection between syntax and self-reference that TheMadFool is trying to establisch here:



See Number 3.

The difference between your example and the single-sentence versions lies in the type of reference, I think.

Your example is endophoric (1. is cataphoric and 2. is anaophoric). The single-sentece versions are exophoric: you reference an object in the real world, which just so happens to be the sentence in question. I'm not sure any of this makes a difference, but if it does, that would be *very* interesting, though.