Trying to overcome the principle of non-contradiction with the "Liar's paradox" is a bit like throwing a broken toothpick at the Bull of Wall Street and expecting it to fall over. :grin:

Wouldn't you agree we must assume a liar to be a liar most of the time — Gregory

Depends on the definition of 'is a liar'.

Right. The so-called "Liar's paradox" seems quite silly, akin to something a third grader thought up at recess. — Leontiskos

Godel's incompleteness theorems use the same basic structure as The Liar's paradox. As such, it is worth understanding the principles of contradiction/paradox.

Part 1: There are no lies

It is impossible to definitively define what a lie is.

A lie is not the truth. The truth is not a lie.

All definitions are circular (A defines B and B defines A).

Part 2: Squiggles don't lie

Sentences do not have any inherent meaning.

All descriptions are of the form: X is not(Everything Else).

That is, The Liar's Paradox as a set of symbols with no connection to anything else has no inherent meaning.

When we read a sentence, we create meaning as part of the process of interpreting the symbols.

The perception of a lie doesn't exist in written words - it exists in your mind.

Silly

You are right that the Liar's paradox is a silly misattribution within which lies cannot be defined and meaning exists within the observer, not arbitrary symbols.

However, understanding exactly why the paradox is not very paradoxical illuminates the nature of understanding and has direct implications in our pursuit of knowledge.

It is impossible to definitively define what a lie is. — Treatid

It is true that a lie cannot be represented by formal logic, as it pertains to intention. Still, Augustine defined a lie 1500 years ago, "Locutio contra mentem."

The perception of a lie doesn't exist in written words - it exists in your mind. — Treatid

I would point out that a written word is not merely a set of squiggles.

However, understanding exactly why the paradox is not very paradoxical illuminates the nature of understanding and has direct implications in our pursuit of knowledge. — Treatid

I'm not convinced that it does.

Godel's incompleteness theorems use the same basic structure as The Liar's paradox. As such, it is worth understanding the principles of contradiction/paradox. — Treatid

I think it is worth understanding the concepts of contradiction and paradox, but I don't think the "Liar's paradox" is either one.

Some have argued that, <Godel's Incompleteness theorems are important, therefore the "Liar's paradox" is important>. I am not convinced of this either, and I'm not sure what "basic structure" they are both supposed to conform to. Granted, I have only formally worked through Godel's Completeness theorem.

Edit: Why do I think that the "Liar's paradox" is silly? Primarily because it is not a paradox, and it is silly to call a non-paradox a paradox. In particular I have run into individuals on TPF who think the "Liar's paradox" is so impressive that it justifies them in rejecting the principle of non-contradiction. Apparently such people call themselves "dialetheists." This is what I see as silly, and I don't think it has much to do with Godel. Studying someone else's mistake can always lead to insight, but I don't see this mistake as particularly helpful or important.

The observer may in fact determine that these five words are not part of a language, in that they are not a statement. — RussellA

I don't whole hog buy into your general view about language, but for the sake of argument, suppose these matters are observer dependent. May not another observer determine that it is a statement?

Kripke proposed that a statement that refers to itself cannot have a truth-value as not grounded in the world, and only statements that are grounded in the world can have a truth value. — RussellA

With admittedly only a cursory search, mainly what I find about Kripke in this regard are: a Wikipedia article and paper by Kripke about direct self reference and proving incompleteness.

Kripke's work is highly technical and I am not versed in it. However, the paper about incompleteness seems clear enough that not only does Kripke approve "This sentence is not provable" but he discusses how he arrives at such a sentence more directly (in a technical sense) than Godel did.

I don't trust Wikipedia, especially in technical matters in logic. So I don't trust that the very brief synopsis does justice to Kripke's view. Also 'grounded' is a very technical notion with Kripke and I don't know that it is properly reduced to a colloquial sense of 'grounded'. Also, while the Wikipedia article mentions 'contingent facts', I don't know that implies a colloquial sense of 'the world'; moreover as "the world" is not even mentioned in the Wikipedia article, and as we should keep in mind that with Kripke, and with modal logic in general, the notion of worlds also is technical. However, the Wikipedia article does refer to a paper by Kripke as its source, though I have not read the paper. I'm not claiming that Kripke doesn't align with you view, but rather that one should be cautious in claiming that he does.

If I am right then [the paradox] requires that there be no speaker at all, even implicit or hypothetical. — Leontiskos

I don't know of any such requirement.

What the proponent of the "Liar's paradox" fails to understand is that the two senses they attribute to the same sentence are mutually exclusive, and it is impossible for a speaker to intend or mean them both. — Leontiskos

I don't know what 'proponent of the liar's paradox' means, but usually discussants of the liar paradox quite understand that the liar sentence is a contradiction and that to assert the liar sentence is to assert a contradiction. That is at the heart of subject.

"To say, "Wow, but what if he is lying and telling the truth at the same time!?," is to fall into incoherence while pretending to be sophisticated." — Leontiskos

I don't know who says something like that. I think the most common view (which abides by the law of non-contradiction) is that "I am lying" implies a contradiction. We wouldn't say that it can be the case that the statements "I am lying" and "I am telling the truth" can both be true. Rather that the liar sentence implies that they are both true, so the liar sentence is inconsistent.

Trying to overcome the principle of non-contradiction with the "Liar's paradox" — Leontiskos

Paraconsistent logicians may eschew non-contradiction. But ordinarily, discussion of the liar paradox is not aimed at rejecting non-contradiction.

Ah, but later you wrote:

I have run into individuals on TPF who think the "Liar's paradox" is so impressive that it justifies them in rejecting the principle of non-contradiction. Apparently such people call themselves "dialetheists." — Leontiskos

That does put your remarks in better perspective. You're talking about some posters in a forum such as this, not necessarily logicians who discuss the liar paradox not toward trying to deny non-contradiction?

This is what I see as silly, and I don't think it has much to do with Godel. — Leontiskos

A connection with Godel is that paraconsistent logic does not yield the incompleteness theorem.

Studying someone else's mistake can always lead to insight, but I don't see this mistake as particularly helpful or important. — Leontiskos

At least one way in which discussion of the liar paradox has been productive is that it suggests methods in mathematics that are similar to the liar paradox but that don't engender the absurdity of the liar sentence.

Godel's incompleteness theorems use the same basic structure as The Liar's paradox. — Treatid

But we must keep in mind that there are crucial differences between the liar sentence and the Godel sentence.

Some have argued that, <Godel's Incompleteness theorems are important, therefore the "Liar's paradox" is important>. — Leontiskos

I'd like to know those arguments in context.

On the other hand, in reverse, of course it is often mentioned that consideration of the liar paradox helps understanding the incompleteness proof. But, again, the liar paradox is not used in the incompleteness proof, but rather a similar, but crucially different construction is used.

If the barber shaves those and only those and all those who do not shave themselves then he doesnt shave himself because of the "not shave themselves" part and this leaves others who do shave themselves who the barber doesn't shave because he doesn't shave those who shave themselves. Maybe nobody shaves the barber. Or someone or everyone shaves him. Surely someone is shaved because it's about a barber. I think you are making this a tar baby toward no genuine purpose

If the barber shaves those and only those and all those who do not shave themselves then he doesnt shave himself — Gregory

Yes, the supposition that he shaves all and only those who do not shave themselves implies that he does shave himself. But that supposition also implies that he doesn't shave himself.

Suppose B shaves all and only those who don't shave themselves.

If B shaves himself, then he doesn't shave himself, which is impossible (B can't both shave himself and not shave himself), so B doesn't shave himself.

If B doesn't shave himself, then he does shave himself, which is impossible (B can't both not shave himself and shave himself), so B shaves himself.

Therefore, the supposition "someone shaves all only those who do not shave themself" is absurd.

I think you are making this a tar baby toward no genuine purpose — Gregory

I'm just explaining the barber paradox.

Your premise 1 is oddly stated. There is no "if he does.. then he doesn't". He just doesn't shave himself because he shaves only those who do NOT shave themselves. Premise 2 is just plain wrong

• If the Barber shaves them, then they don't shave themselves
• If they don't shave themselves, then the Barber shaves them

An absurdity arises when "they/them" includes the Barber. The commonsensical reading would say that the Barber is not included in the set "they/them."

If we include the Barber in the set and then replace they/them with "the Barber," we get:

• If the Barber shaves the Barber, then the Barber doesn't shave himself
• If the Barber doesn't shave himself, then the Barber shaves the Barber

He just doesn't shave himself because he shaves only those who do NOT shave themselves. — Gregory

You seem to be missing that it's not just that B shaves only those who do not shave themselves, but also that B shaves all those who do not shave themselves.

The premise is "B shaves all and only those who do not shave themselves".

The rest of the argument follows to show that the premise is absurd.

Premise: B shaves all and only those who do not shave themselves. So:

B shaves all those who don't shave themself. So, if B does not shave himself, then B shaves himself. But it is impossible that both B does not shave himself and B does shave himself. So, by modus tollens, B does shaves himself.

B shaves only those who don't shave themself. So, if B shaves himself, then B does not shave himself. But it is impossible that both B shaves himself and B does not shave himself. So, by modus tollens, B does not shave himself.

But it is impossible that B shaves himself and does not shave himself. So it is impossible that B shaves all and only those who do not shave themself.

Oh so youre including the barber in "all". So this is just Russell's paradox in a simple form.? All that is needed to solve it is more information it seems to me. Not enough is given to make anything out of it that is normal and helps us grow. It's too narrow of a puzzle then for me to find interesting. The right and left hemispheres or some crap get too equally balanced to see it clearly. I doubt anyone can see that clearly



Thanks for the clarification.

So this is just Russell's paradox in a simple form.? — Gregory

It's Russell's paradox illustrated anecdotally.

It's actually a theorem schema of first order logic, whether 'shaves', 'is an element of' or any other 2-place predicate:

For any 2-place predicate S:

~ExAy(Syx <-> ~Syy)

and

~ExAy(Sxy <-> ~Syy)

I didn't read the whole speech but Hawking said this about incompleteness — TonesInDeepFreeze

The remainder of the speech is mostly about research headaches in (advanced) theoretical physics, probably his pet peeves.

"G can not be demonstrated from the axioms of mathematics." That's really bad and it is the kind of thing that leads people (who don't know the theorem) to make unfortunate inferences about the theorem. — TonesInDeepFreeze

Hawking was talking to other physicists. For most of them, I guess that life is often simplified to just one elusive further unspecified set of axioms in mathematics that they do not even explicitly name, because that is irrelevant to what they are doing. In fact, even mathematicians may operate like that:

https://en.wikipedia.org/wiki/Axiomatic_system

In practice, not every proof is traced back to the axioms. At times, it is not even clear which collection of axioms a proof appeals to.

one elusive further unspecified set of axioms in mathematics that they do not even explicitly name, because that is irrelevant to what they are doing. — Tarskian

Of course. Meanwhile, it is crucial not to say, "G can not be demonstrated from the axioms of mathematics", since that is plainly false. And the lecture botches the key point that it's not that there is a true sentence such that for all theories, the sentence is unprovable", but rather, ifor all theories (of a certain kind), there is a true sentence not provable in the theory. That is not a mere "detail" and it is even more critical in context of what may be the scientific or philosophical gleanings from the theorem.

https://en.wikipedia.org/wiki/Axiomatic_system

In practice, not every proof is traced back to the axioms. At times, it is not even clear which collection of axioms a proof appeals to.

So what? The incompleteness theorem has nothing to do with that, since the incompleteness theorem regards formal theories in which the axioms are explicit and such that theorems are strictly from explicit axioms.

Meanwhile, it is crucial not to say, "G can not be demonstrated from the axioms of mathematics", since that is plainly false. — TonesInDeepFreeze

For Hawking's audience of physicists, the term "axioms of mathematics" refers to PA or ZFC.

A mathematical theory in which Gödel's incompleteness does not apply -- because it cannot even do arithmetic -- is probably not even in use anywhere in science.

Therefore, what Hawking said, may be technically false, but in all practical terms it will never lead to problems.

So what? The incompleteness theorem has nothing to do with that, since the incompleteness theorem regards formal theories in which the axioms are explicit and such that theorems are strictly from explicit axioms. — TonesInDeepFreeze

That is indeed the case from the standpoint of mathematical logic.

From the point of view of science and engineering, since they always operate in PA and/or ZFC, I am confident that practitioners of science or engineering are not even conscious about their implicit choice, if only, because it actually never matters to them.

Gödel always applies to the default context in their typical environment.

I don't whole hog buy into your general view about language, but for the sake of argument, suppose these matters are observer dependent. May not another observer determine that it is a statement? — TonesInDeepFreeze

Yes. Every observer of a set of words interprets the same set of words differently, because it is the observer that gives the set of words meaning. It is not the "squiggles" that the observer sees on the screen that give the "squiggles" meaning.

So I don't trust that the very brief synopsis does justice to Kripke's view. — TonesInDeepFreeze

I agree, but we have to start somewhere.

Though my belief remains that the basic problem with the statement "this statement is false" is that it is not grounded in the world. In the same way, any set of words, such as the statement "a b c", is meaningless until grounded in the world.

The word "a" may be defined as "d e f"
The word "b"may be defined as "g h i"
The word "c" may be defined as "j k l"

Continuing, the word "d" may be defined as "m n o", etc

This may give us a coherent language, but will remain meaningless until sooner or later a word corresponds with something in the world.

IE, a language in order to be useful must correspond with the world in addition to being coherent. IE a useful language must be grounded in the world , which the statement "this statement is false" isn't.

"This sentence is false"

You regard that sentence as meaningless on the basis that self-reference is meaningless when applied to a sentence.

But is it the case that all self-referential sentences are meaningless? If they are not, then, if that sentence is deemed meaningless, there must be a basis other than that it's self-referential.

But there's another aspect to the sentence, which is negation. So I ask about a self-referential sentence that does not involve negation:

"This sentence has five words"

"This sentence has five words" has five words. The meaning of the sentence is that the predicate (has five words) holds for the subject ("This sentence has five words"); and its truth value is 'true'.

It's not the case that in general self-reference using the pronoun 'this' is meaningless:

"This Guy's In Love With You" [a song title]

It's not the case that a sentence referencing a sentence is meaningless:

""This sentence has five words" has five words" is meaningful and true.

So, why would "This sentence has five words" be meaningless?

If it's meaningless, then it's not because it's self-referential nor that its subject is a sentence, but rather that it's both self-referential and its subject is a sentence.

But why does being both self-referential and having a sentence as its subject make it meaningless?

As I understand, your argument is that the sentence does not pertain to "the world".

(1) It would help to have an explanation of what you mean by 'the world'.

(2) But no matter what you mean by 'the world', sentences may be subjects of sentences and be meaningful, so, it seems your argument should allow that sentences are in "the world". I surmise you would agree. But you draw the line at sentences that refer to themselves. But such sentences are in "the world". "This sentence has five words" is right in front of us in "the world". If one argued that it's not in "the world" because it refers to itself, then that would be petitio principii.

meaningless until sooner or later a word corresponds with something in the world. — RussellA

I think the following is right:

Suppose we define 'the Magna Carta' as "the charter assented to by King John' and we define 'is old' as 'dates from the long past'.

So, we have a subject in the world, viz. the Magna Carta.

So, "The Magna Carta is old" is meaningful.

To determine whether "The Magna Carta is old" is true, we determine whether the charter assented to by King John dates from the long past.


Suppose we define the Witness Statement as "The Chevy ran a light".

So, we have a subject from the world, viz. the Witness Statement.

So, "The Witness Statement has five words" is meaningful.

To determine whether the "The Witness Statement has five words" is true, we determine whether the Witness Statement has five words.


Suppose we define 'the Minma Senta' as "This sentence has five words".

So, we have a subject from the world, viz. the Minma Senta.

So, "The Minma Senta has five words" is meaningful.

To determine whether the Minma Sentence is true, we determine whether the Minma Sentence has five words.


In "This sentence has five words", 'this sentence' refers to the Minma Senta, which is in the world. And "This sentence has five words" is equivalent with "The Minma Senta has five words", in the sense that each is true if and only if the Minma Senta has five words. So, "This sentence has five words" is meaningful.

To determine whether "The Minma Senta has five words" is true, we determine whether the Minma Senta has five words, which is to determine whether "This sentence has five words" has five words. To determine whether "This sentence has five words" is true, we determine whether "This sentence has five words" has five words. The determination of the truth value of the Minma Senta is exactly the determination of the truth value of "This sentence has five words".

Meanwhile, it is crucial not to say, "G can not be demonstrated from the axioms of mathematics", since that is plainly false.
— TonesInDeepFreeze

For Hawking's audience of physicists, the term "axioms of mathematics" refers to PA or ZFC. — Tarskian

(1) According to you, they are not versed in foundations of mathematics. So, by what basis do you claim that they take first order PA to be "the axioms of mathematics"? If one were not versed in foundations, then it's likely as not that they know of ZFC (since it is so often cited as the foundational theory) but know nothing or very little of PA (since PA does not axiomatize the mathematics for the sciences but only axiomatizes study of the natural numbers, which is subsumed by ZFC).

(2) As far as I know, "the axioms of mathematics" is ordinarily understood to mean ZFC as axioms sufficient for the mathematics for the sciences in the language of set theory in which analysis (calculus done right), topology, abstract algebra, etc. can be expressed, and not mere PA.

(3) By the way, Godel's own proof was not about PA nor ZFC but about a system P he formulated to simplify Whitehead and Russell's PM.

(4) Since G can be proven to be true in ZFC, "G can not be demonstrated from the axioms of mathematics" is horribly misleading, as it falsely suggests that ordinary mathematics cannot prove that G is true.

(5) Again, as you keep skipping this, saying "G can not be demonstrated from the axioms of mathematics" is not only horribly misleading, but it misses the crucial point that it is not the case that there is a sentence G that can't be proven in any theory, but that for any theory, there is a sentence such that neither it nor its negation are provable in that theory.

That is crucial to talking about scientific or philosophical implications of incompleteness. For example, some people argue that the human mind trumps computation because there are sentences that the mind sees to be true but for which there is no computation of their truth. But that argument is wrong, or at least need revision, since incompleteness very much does not show that there are such sentences, but rather that for any given consistent theory adequate for arithmetic, there are true sentences not provable in that theory. That is crucial to understand in order not to make wrong inferences about science or philosophy vis-a-vis incompleteness.

A mathematical theory in which Gödel's incompleteness does not apply -- because it cannot even do arithmetic -- is probably not even in use anywhere in sciences. — Tarskian

(6) I wouldn't rule out there being sub-theories without arithmetic that are useful.

(7) The claim you just made goes against your own argument. Indeed, if arithmetic weren't included then it wouldn't be in general adequate for science. But if the real numbers weren't included then that would not be adequate for the sciences. And the reals come from ZFC not PA. So PA is not an axiomatization of the mathematics for the sciences.

what Hawking said, may be technically false, but in all practical terms it will never lead to problems. — Tarskian

It is not merely "technically" false, it is both technically false and fundamentally false. Being fundamentally false and widely disseminated is a problem in and of itself. It terribly messes up the theorem in mathematics and is a welcome mat to misapplying the theorem in areas other than mathematics.

In practice, not every proof is traced back to the axioms. At times, it is not even clear which collection of axioms a proof appeals to.

So what? The incompleteness theorem has nothing to do with that, since the incompleteness theorem regards formal theories in which the axioms are explicit and such that theorems are strictly from explicit axioms. — TonesInDeepFreeze

That is indeed the case from the standpoint of mathematical logic. — Tarskian

You miss the point again. I don't doubt that physicists don't care about tracking back to the mathematical axioms, but that's irrelevant to the subject Hawking is talking about when he is talking about what the axioms prove and don't prove. Yes, physicists don't bother with knowing how all the math was derived from ZFC or even what the ZFC axioms are. But their disinterest doesn't vitiate that the subject of incompleteness very much pertains to axioms and especially as in the lecture Hawking is directly talking about axioms.

Gödel always applies to the default context in their typical environment. — Tarskian

Incompleteness applies to any consistent formal theory adequate for arithmetic. And the theorem is not that G is unprovable in ZFC. And the theorem is not that there is a sentence, such as G, that is not provable in any theory for a "default context" but rather that for any such theory, there is a sentence such that neither it nor its negation is provable in that theory.

You may not be interested in that point, but it is basic for understanding incompleteness, and avoidance of understanding it is an open door to bad misunderstanding of incompleteness vis-a-vis science and philosophy.

.

There is a lot to work through in your post, and to do justice to the points you have made, I am working through them one by one.
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But is it the case that all self-referential sentences are meaningless? — TonesInDeepFreeze

There are sets of words, such as "colourless green ideas sleep furiously", that are meaningless, yet are not self-referential.

If I am correct in my belief that any set of words that is self-referential must be meaningless, then this set of words shouldn't be called a "sentence", as a sentence is a syntactic unit in language that does have a meaning.

From the Merriam Webster definition of "sentence"

a word, clause, or phrase or a group of clauses or phrases forming a syntactic unit which expresses an assertion, a question, a command, a wish, an exclamation, or the performance of an action, that in writing usually begins with a capital letter and concludes with appropriate end punctuation, and that in speaking is distinguished by characteristic patterns of stress, pitch, and pauses

The expression "self-referential sentence" is itself a paradoxical contradiction, in that if a set of words is self-referential then it cannot have meaning, and if cannot have meaning then cannot be a sentence.
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"This sentence has five words" has five words. The meaning of the sentence is that the predicate (has five words) holds for the subject ("This sentence has five words"); and its truth value is 'true'. — TonesInDeepFreeze

Difference between form and content
Suppose there are marks on a screen. The screen is in the world. Both an Italian speaker and English speaker in looking at these marks observe the same form, that of five distinct marks, but only the English speaker knows that the marks are words and the content of these words is "this sentence has ten words".

The content of these marks, that there are ten marks, is independent of the form of these marks, that there are five marks.

This means that the form of the words cannot be determined from the content of the words

The sentences "this sentence has ten words", "this sentence has five words" and "this sentence has fifty words" all have different contents, but are all five words.

The number of words in a sentence cannot be determined from the content of that sentence. IE, the content of a sentence does not refer to the form of that sentence.

The form of these marks exists in the world, whilst the content of these marks only exists in the mind of a sentient observer.

The sentence ""This sentence has five words" has five words" is ungrammatical, and therefore without truth-value.
On the screen I see the sentence "this sentence has ten words"

I can then write on the same screen "the sentence "this sentence has ten words" has five words"

The predicate "has five words" is referring to "the sentence "this sentence has ten words""

The predicate "has five words" is not referring to "this sentence has ten words".

The sentence "the sentence "this sentence has ten words" has five words" has the truth-value of being true.

The sentence ""this sentence has ten words" has five words" is ungrammatical, and therefore meaningless, and therefore without any truth-value.
===============================================================================

It's not the case that in general self-reference using the pronoun 'this' is meaningless: "This Guy's In Love With You" — TonesInDeepFreeze

I agree that there is nothing ungrammatical about the sentences "this sentence has five words" and "this guy is in love with you".

However, as the pronoun "this" is external to both "the sentence" and "the guy", the pronoun isn't being self-referential.

The problem arises when the sentence is being self-referential, in the event that "this sentence has five words" is referring to itself and "this guy is in love with you" is referring to itself.
===============================================================================

It's not the case that a sentence referencing a sentence is meaningless: — TonesInDeepFreeze

True. The sentence "the sentence "this sentence has ten words" has five words" has the truth value of being true.
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So, why would "This sentence has five words" be meaningless? — TonesInDeepFreeze

It depends what "this sentence" refers to. If it refers to the sentence "this sentence has five words", then it has a truth-value, but if it refers to "this sentence has five words", then it has no truth-value.
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It would help to have an explanation of what you mean by 'the world'. — TonesInDeepFreeze

As an Indirect Realist, I perceive things through my five senses. My belief is that these perceptions have been caused by something outside me, and this something outside me I call "the world".
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it seems your argument should allow that sentences are in "the world". I surmise you would agree — TonesInDeepFreeze

I agree that marks exist in the world, but only a sentient being can attach a meaning to these marks. Only a sentient being knows when a set of marks is a part of a language. Only a sentient being knows when a set of marks is a sentence, meaning that sentences only exist in the mind.

Sets of marks exist in the world. Sentences exist in the mind.

If I am correct in my belief that any set of words that is self-referential must be meaningless, then this set of words shouldn't be called a "sentence", as a sentence is a syntactic unit in language that does have a meaning. — RussellA

Throughout my previous post, inside the strings, instead of 'sentence' we may say 'string' (and we could do that throughout :

'This string has five words'

Is that a sentence?

If you say it's not a sentence because it is self-referential, then you need to demonstrate the claim:

No string that is self-referential is a sentence.

You've argued for that claim, but my previous post is a response, and in the strings, we may use 'string' instead of 'sentence'.

If I am correct in my belief that any set of words that is self-referential must be meaningless, then this set of words shouldn't be called a "sentence", as a sentence is a syntactic unit in language that does have a meaning. — RussellA

At least at first blush, "The string has five words" seems syntactic. A noun phrase, "This string" followed by a predicate, "has five words".

So you need to demonstrate that it is meaningless. But meanwhile, perhaps see if there is an error in the reasoning I gave for why we may take it to be meaningful. That reasoning could be wrong, but if it is, then I'd be interested to know how.

"This string has five words" asserts that "This string has five words" has five words. That seems meaningful. So it seems "This string has five words" is a sentence as it fulfills the two requirements: syntactical and meaningful. And "This string has five words" is true if "This string has five words" has five words, which it does; so "This string has five words" seems to be true. So, "This string has five words" seems to be true sentence.

Or:

Suppose we define 'the Pentastring' as the "This string has five words".

So, we have a subject from the world, viz. the Pentastring.

So, "The Pentastring has five words" is meaningful.

To determine whether the Pentastring is true, we determine whether the Pentastring has five words.

Put this way:

In "This string has five words", 'this string' refers to the Pentastring, which is in the world. And "This string has five words" is equivalent with "The Pentastring has five words", in the sense that each is true if and only if the Pentastring has five words. So, "This string has five words" is meaningful.

To determine whether "The Pentastring has five words" is true, we determine whether the Pentastring has five words, which is to determine whether "This string has five words" has five words. To determine whether "This string has five words" is true, we determine whether "This string has five words" has five words. The determination of the truth value of the Pentastring is exactly the determination of the truth value of "This string has five words".

The form of these marks exists in the world, whilst the content of these marks only exists in the mind of a sentient observer. — RussellA

For sake of argument, let's say that is true. By what argument does it follow that the content of "This string has five words" is not meaningful? Do you claim that no observer can see it as contentful? An observer may reasonably and correctly say "I see it as contentful. The content is the claim that "This string has five words" has five words."

On the screen I see the sentence "this sentence has ten words"

I can then write on the same screen "the sentence "this sentence has ten words" has five words"

The predicate "has five words" is referring to "the sentence "this sentence has ten words"" — RussellA

Wrong. It's referring to the sentence "this sentence has ten words", which is to say that it is referring to "this sentence has ten words".

The sentence "this sentence has ten words" is "this sentence has ten words".

The sentence "this sentence has ten words" is not "The sentence "this sentence has ten words"".

And if your argument is supposed to be addressing mine, then no matter anyway, since I didn't use a construction "the sentence "this sentence has five words", and even if I had, your argument would be wrong since:

The sentence "this sentence has five words" has five words
is not saying
"The sentence "this sentence has five words"" has five words

It's not the case that in general self-reference using the pronoun 'this' is meaningless: "This Guy's In Love With You"
— TonesInDeepFreeze

I agree that there is nothing ungrammatical about the sentences "this sentence has five words" and "this guy is in love with you"

However, as the pronoun "this" is external to both "the sentence" and "the guy", the pronoun isn't being self-referential.

The problem arises when the sentence is being self-referential, in the event that "this sentence has five words" is referring to itself and "this guy is in love with you" is referring to itself. — RussellA

I don't know what you mean by 'external' there.

If by "referring to itself", then what is referring to "This sentence has five words" is 'this', in the sense that 'this' is referring to the sentence "This sentence has five words".

But with "This guy is in love with you", 'this' is referring to the guy who is the speaker of the sentence.

I mentioned "This guy's in love with you" only to put out of the way any objection that might come up to use of the pronoun 'this' to refer to the speaker of the sentence. Actually, it's not relevant here anyway.

So, why would "This sentence has five words" be meaningless?
— TonesInDeepFreeze

It depends what "this sentence" refers to. If it refers to the sentence "this sentence has five words", then it has a truth-value, but if it refers to "this sentence has five words", then it has no truth-value. — RussellA

The sentence "this sentence has five words" is "this sentence has five words".

The tower Big Ben is Big Ben.

As an Indirect Realist, I perceive things through my five senses. My belief is that these perceptions have been caused by something outside me, and this something outside me I call "the world". — RussellA

I agree that marks exist in the world, but only a sentient being can attach a meaning to these marks. Only a sentient being knows when a set of marks is a part of a language. Only a sentient being knows when a set of marks is a sentence, meaning that sentences only exist in the mind.

Sets of marks exist in the world. Sentences exist in the mind. — RussellA

I don't see a good argument so far here that "This string has five words" cannot be only a set of marks and not exist in the mind.

You wrote:

but will remain meaningless until sooner or later a word corresponds with something in the world. — RussellA

"This string has five words"

The words seem to me to correspond with things in the world.

'this string' corresponds with the string "This string has five words".

'has five words' corresponds with the property of a string having five words, which is something that I observe some strings to have.

And, as I mentioned, I see how "This string has five words" is meaningful, so that it is a sentence. And by the same reasoning, mutatis mutandis, "This sentence has five letters" is also a sentence.

'This string has five words' Is that a sentence? — TonesInDeepFreeze

On my screen I see the set of words "This string has five words".

What is a sentence? Is the meaningless set of words "colourless green ideas sleep furiously" a sentence? According to the Merriam Webster Dictionary, a sentence is a group of words that expresses a complete thought and has a subject and a verb. I would therefore suggest that a meaningless set of words cannot be classed as a sentence.

I am making use of Steve Patterson's video "How to Resolve the Liar's Paradox"

The question is, is the set of words "This string has five words" a sentence or not.

The problem is in knowing what "this string" refers to.

Possibility 1) If "this string" is referring to a string of characters existing in the world, such as the characters on my keyboard, then the set of words "This string has five words" is meaningful, is a sentence, has a truth-value and can be either true or false.

Possibility 2) If "this string" is referring to itself, then it is an empty reference, and the set of words "this string has five words" is meaningless, isn't a sentence and has no truth-value.

Possibility 3) If "this string" is referring to "This string has five words", then the expression "this string" can be replaced by "this string has five words".
We then get: ((this string has five words) has five words).
Continuing, we get: (((this string has five words) has five words) has five words).
But as this will go on ad infinitum, meaning that the set of words "this string has five words" is meaningless, isn't a sentence and has no truth-value.

Whether the set of words "this string has five words" is a sentence or not depends on what "this string" is referring to.
===============================================================================

"This string has five words" asserts that "This string has five words" has five words. That seems meaningful. — TonesInDeepFreeze

If "This string has five words" did assert that "This string has five words" has five words
then "This string has ten words" would be asserting that "This string has ten words" has ten words, which is not the case.

Therefore, "This string has five words" cannot be asserting that "This string has five words" has five words.
===============================================================================

"This string has five words".................'has five words' corresponds with the property of a string having five words, which is something that I observe some strings to have. — TonesInDeepFreeze

I see on my screen the set of words "this string has five words", and I see that there are five words in this set of words.

I see on my screen the set of words "this string has ten words", and I see that there are five words in this set of words.

I see on my screen the set of words "Diese Zeichenfolge besteht aus fünf Wörtern", and I see that there are six words in this set of words.

I go into a shop and buy five apples and notice that the time is exactly five pm. There is no logical link between the fact that I bought five apples and the fact that the time is five pm. That both involve the number five is accidental.

Similarly, that the content of the set of words "this string has five words" and the form of the set of words involves five is also accidental.

"This string has five words"
The words seem to me to correspond with things in the world. — TonesInDeepFreeze

This is just to expand on RussellA ’s response. The underlying issue here is that there is no information in the 5 words which lets us know that it is self referential. The words “This string” (or “This sentence”) could be pointing to a different sentence, say, “Two plus two equals four”.

You say the the words seem to you to correspond with things in the world - and that may very well be the case - but in order to make that conclusion we need to rely on additional information NOT in those 5 words.

Suppose we did this:
A) “The sentence identified by the letter A in this post has thirteen words”

This works, but we need to rely on information not in the sentence.

Alternatively, I could hand you a piece of paper and on that piece of paper would be the words “The sentence on this piece of paper I just handed you has fourteen words”

That also works, but again we are relying on information in the world.
- - - - - - - - - - - -

I do have a slight different take than RussellA on what combination of words constitutes a sentence. Consider poetry:

“The quality of mercy is not strained”
“The moon was a ghostly galleon”
"Make a joyful noise unto the Lord"
etc

I consider these to be sentences. They are grammatically correct and they evoke images and/or emotions in my mind. However they do not take a truth value since they are not asserting anything specific about the world. Well OK - “The moon was a ghostly galleon” in theory makes a statement about the world, so literally interpreted it is false, but we all recognize that it is poetry and not to be taken literally.

So in this sense “This sentence has five words” is a legitimate sentence, it just does not take a truth value unless we make an assumption about the specific meaning of the first two words.

I hope this helps clarify things.

You skipped that I caught a false claim by you:

On the screen I see the sentence "this sentence has ten words"

I can then write on the same screen "the sentence "this sentence has ten words" has five words"

The predicate "has five words" is referring to "the sentence "this sentence has ten words""
— RussellA

Wrong. It's referring to the sentence "this sentence has ten words", which is to say that it is referring to "this sentence has ten words".

The sentence "this sentence has ten words" is "this sentence has ten words".

The sentence "this sentence has ten words" is not "The sentence "this sentence has ten words"".

And if your argument is supposed to be addressing mine, then no matter anyway, since I didn't use a construction "the sentence "this sentence has five words", and even if I had, your argument would be wrong since:

The sentence "this sentence has five words" has five words
is not saying
"The sentence "this sentence has five words"" has five words — TonesInDeepFreeze

If you skip refutations, then we won't get anywhere.

You skipped my argument, for the second time (as now revised to use 'stirng' instead of 'sentence'):

Suppose we define 'the Pentastring' as the "This string has five words".

So, we have a subject from the world, viz. the Pentastring.

So, "The Pentastring has five words" is meaningful.

To determine whether the Pentastring is true, we determine whether the Pentastring has five words.

Put this way:

In "This string has five words", 'this string' refers to the Pentastring, which is in the world. And "This string has five words" is equivalent with "The Pentastring has five words", in the sense that each is true if and only if the Pentastring has five words. So, "This string has five words" is meaningful.

To determine whether "The Pentastring has five words" is true, we determine whether the Pentastring has five words, which is to determine whether "This string has five words" has five words. To determine whether "This string has five words" is true, we determine whether "This string has five words" has five words. The determination of the truth value of the Pentastring is exactly the determination of the truth value of "This string has five words". — TonesInDeepFreeze

If you skip my main argument, then we won't get anywhere.

/

The glaring sophistry in that video is the claim that "this sentence" equals "this sentence is false." No, "this sentence" refers to "this sentence is false" but they are not the same. One is a noun phrase, and the other is a noun phrase and a predicate. One has two words, the other has four words. One refers to the other but they are not the same, they are not equal to one another.

Possibility 1) If "this string" is referring to a string of characters existing in the world, such as the characters on my keyboard, then the set of words "This string has five words" is meaningful, is a sentence, has a truth-value and can be either true or false. — RussellA

I distinguish between physical inscriptions and sentences:

The teacher writes on the blackboard, "Caesar was a Roman emperor". A student writes in her notebook, "Caesar was a Roman emperor". The physical inscription on the blackboard is made of chalk. The physical inscription in the notebook is made of pencil lead. There are two inscriptions. But there is only one sentence involved.

Possibility 2) If "this string" is referring to itself, then it is an empty reference, and the set of words "this string has five words" is meaningless, isn't a sentence and has no truth-value. — RussellA

So you say. The ball is in your court to support that claim. Also, I've given an argument why it is a meaningful, true sentence. That argument could be wrong, but what do you claim is wrong with it? Mind you, any counter-argument you give cannot invoke the premise that self-reference is meaningless, since that would be petitio principii.

Possibility 3) If "this string" is referring to "This string has five words", then the expression "this string" can be replaced by "this string has five words". — RussellA

Rule out that possibility. It's the specious argument that the video makes. "X refers to Y" doesn't entail "X can be replaced by Y".

If "This string has five words" did assert that "This string has five words" has five words
then "This string has ten words" would be asserting that "This string has ten words" has ten words, which is not the case. — RussellA

That's wrong.

It's not the case that "This string has ten words" has ten words; but it is the case that "This string has ten words" asserts that "This string has ten words" has ten words.

'this string' corresponds with the string "This string has five words".

'has five words' corresponds with the property of a string having five words, which is something that I observe some strings to have. — TonesInDeepFreeze

I see on my screen the set of words "this string has five words", and I see that there are five words in this set of words.

I see on my screen the set of words "this string has ten words", and I see that there are five words in this set of words.

I see on my screen the set of words "Diese Zeichenfolge besteht aus fünf Wörtern", and I see that there are six words in this set of words.

I go into a shop and buy five apples and notice that the time is exactly five pm. There is no logical link between the fact that I bought five apples and the fact that the time is five pm. That both involve the number five is accidental. — RussellA

Okay.

Similarly, that the content of the set of words "this string has five words" and the form of the set of words involves five is also accidental. — RussellA

I don't know what you mean by "the content of the set of words". But, yes, in form, "This string has five words" has five words. Anyway, I've made no argument that mentions "not accidental". I don't see how you infer that what I wrote is not correct:

'this string' corresponds with the string "This string has five words".

'has five words' corresponds with the property of a string having five words, which is something that I observe some strings to have. — TonesInDeepFreeze

"This string has five words"
The words seem to me to correspond with things in the world.
— TonesInDeepFreeze

The underlying issue here is that there is no information in the 5 words which lets us know that it is self referential. The words “This string” (or “This sentence”) could be pointing to a different sentence, say, “Two plus two equals four”. — EricH

Any pronoun could be referring to something different. "This ball is light" could be referring to any ball anywhere. But I don't think that way. It is by context that we regard the reference of a pronoun. There is no reason to arbitrarily think that 'this string' refers to "two plus two equals four" rather than "this string has five words" in which "this string" occurs. Or, if that's not good enough, we can stipulate that it does.

You say the the words seem to you to correspond with things in the world [emphasis in yours] — EricH

I said 'seem' because I am open to arguments to the contrary.

- and that may very well be the case - but in order to make that conclusion we need to rely on additional information NOT in those 5 words. — EricH

Denotations are not usually in words themselves. Rather, we stipulate denotations. It's not in the word 'ball' that it stands for the main objects in sports games.

A) “The sentence identified by the letter A in this post has thirteen words”

This works, but we need to rely on information not in the sentence.

Alternatively, I could hand you a piece of paper and on that piece of paper would be the words “The sentence on this piece of paper I just handed you has fourteen words”

That also works, but again we are relying on information in the world. — EricH

Take a look at my argument with the Pentastring.