And I replied:

When someone considers the claim, "Colourless green ideas sleep furiously," you will inform them that the statement they are considering is nonsensical.
— Leontiskos

You skipped my examples that are not of that kind.

But as to your example.

Just now, you referenced the sentence "colourless green ideas sleep furiously" without there being an implied speaker other than a hypothetical one. You were able to type the sentence, reference it, and still you are not the speaker of the sentence.

I'll do it again. Consider the following sentence that I am displaying but not asserting:

Colorless green ideas sleep furiously.

There is no implied speaker, especially not one asserting.

And previously:

For example, in a math book may appear sentences that were typed by an author but are not considered to be specific to any one person. For example, I can display the sentence, "Harry Truman was a president" and that sentence can be discussed no matter that its just typed by me.
— TonesInDeepFreeze

Also, consider the following sentence that I am not asserting but merely displaying so that we can talk about it:

This sentence has five words. — TonesInDeepFreeze

You brought this around full circle.

- And I of course replied to your reply. :roll:

Again:

1. Phil is a fool.
2. Colourless green ideas sleep furiously.

What is the difference between (1) and (2)? The relevant difference is simply that (1) has a possible (implicit or hypothetical) speaker whereas (2) does not. To merely assert that (1) has no hypothetical speaker is to ignore this difference between (1) and (2). The things that the OP is considering are like (2), not (1), ergo, "There is no possible speaker in such cases, and hence the "sentences" are nonsensical." — Leontiskos

You have replied to my recent post. But, as I said, I replied to your earlier post that you again linked to, thus you brought this full circle.

Phil is a fool.
Colourless green ideas sleep furiously.

What is the difference between (1) and (2)? The relevant difference is simply that (1) has a possible (implicit or hypothetical) speaker whereas (2) does not. To merely assert that (1) has no hypothetical speaker is to ignore this difference between (1) and (2). The things that the OP is considering are like (2), not (1), ergo, "There is no possible speaker in such cases, and hence the "sentences" are nonsensical." — Leontiskos

Of course "Colourless green ideas sleep furiously" can be spoken.

One could go out tomorrow with a bullhorn on the Spanish Steps to say a hundred times

"Colourless green ideas sleep furiously! Do you hear me people, colourless green ideas sleep furiously!"

And again, my point was to question the claim (not yours) that sentences may not refer to themselves.

And, I've answered the claim that there is always an implied speaker.

"This sentence has five words"

That sentence refers to itself. It has five words. My question is whether that sentence is true.

Of course "Colourless green ideas sleep furiously" can be spoken.

One could go out tomorrow with a bullhorn on the Spanish Steps to say a hundred times

"Colourless green ideas sleep furiously! Do you hear me people, colourless green ideas sleep furiously!" — TonesInDeepFreeze

Still playing dumb, then. I've yet to find anyone online who intentionally misrepresents their interlocutor as often as you.

The objection to such a consideration is always something like, "No one in their right mind would ever speak such a thing." To consider an utterance that has no possible speaker is to consider a nonsensical utterance. — Leontiskos

Also, in other threads I've commented on the matters of the incompleteness theorem in philosophy of mathematics. I don't have comments at this time on the incompleteness theorem in connection with science, epistemology, ontology and metaphysics. — TonesInDeepFreeze

In his 2002 lecture, "Gödel and the end of physics", Stephen Hawking made excellent comments on the connection with physics. He obviously left out technical details because those were irrelevant to the question at hand. People attending his lecture were simply not interested in the technical details. They just wanted to know what the value of Gödel's theorem is for physics. Hawking pointed out that positivism is simply futile. Laplace's demon is the wrong view on physics because it will never be possible.

I responded to:

What is the difference between (1) and (2)? The relevant difference is simply that (1) has a possible (implicit or hypothetical) speaker whereas (2) does not. To merely assert that (1) has no hypothetical speaker is to ignore this difference between (1) and (2). The things that the OP is considering are like (2), not (1), ergo, "There is no possible speaker in such cases, and hence the "sentences" are nonsensical." — Leontiskos

However, I grant that would be qualified by your earlier "in their right mind".

[An example where I was reading too fast, trying to keep up, but corrected myself immediately when it was made clear that I did not consider the full text. That is unlike Leontiskos who continues to deny that he lied about my view me regarding the main issue in a thread.]

My initial reply is that whether in right mind or not, it can be said. Moreover, my examples are not refuted by that example, as I argued.

However, I grant that would be qualified by your earlier "in their right mind". — TonesInDeepFreeze

Yes, good.

My initial reply is that whether in right mind or not, it can be said. — TonesInDeepFreeze

You were replying to my argument about speakers, and to disregard "whether in right mind or not" is to misrepresent my argument. It does little to help your case to note that someone who is not in their right mind might agree with you, and might say something that someone who is sane would not say.

Indeed, a reply to your argument should not have overlooked your qualification 'in their right mind', so when you noted that, I immediately recognized that you did qualify that way. Now, if you would only recognize that you were wrong to continue to claim I took a position, when I had posted at least a few times that I take the opposite of that position, and hopefully to desist from misrepresenting me that way. I pointed out to you that you misrepresented me, but I did not say you lied until you continued to misrepresent me even after I pointed it out to you.

Indeed, a reply to your argument should not have overlooked your qualification 'in their right mind', so when you noted that, I immediately recognized that you did qualify that way. — TonesInDeepFreeze

Okay, I appreciate that.

Again, trying to get to the point:

• If no one in their right mind would say or mean X, then X is nonsensical
• No one in their right mind would say or mean what the liar of the "Liar's paradox" is supposed to say
• Therefore, the utterances attached to the "Liar's paradox" are nonsensical

(I am understating this given that a contradiction is more than merely nonsensical in the way of these other statements.)

Now, if you would only recognize that you were wrong to continue to claim I took a position, when I had posted at least a few times that I take the opposite of that position, and hopefully to desist from misrepresenting me that way. — TonesInDeepFreeze

What position is that?

What position is that? — Leontiskos

The one in connection with the incident that you posted about in this thread earlier today, when you said that I conflate a mere falsehood with a lie! Wow, in this thread, you brought up the incident from another thread, and yet you don't even remember what it was about. Actually, not surprising.

In the other thread, I think about three times, you claimed that I conflate material implication with everyday use of "if then", or that I that I consider material implication to be the only correct understanding of "if then", or that I insist that "if then" can only be considered as material implication. Something to that effect.

But I had said probably at least three times that material implication does not represent many everyday senses of "if then", and not just everyday senses, but also material implication is not compatible with other approaches to "if then" in logic. And I said that I do not claim that material implication is the only way that "if then" should be understood.

And what makes your lies even more egregious is that they were not about mere interstitial points, but about the central subject in the thread.

(I added to my previous post in edit while you were posting your latest.)

Also, in other threads I've commented on the matters of the incompleteness theorem in philosophy of mathematics. I don't have comments at this time on the incompleteness theorem in connection with science, epistemology, ontology and metaphysics.
— TonesInDeepFreeze

In his 2002 lecture, "Gödel and the end of physics", Stephen Hawking made excellent comments on the connection with physics. He obviously left out technical details because those were irrelevant to the question at hand. People attending his lecture were simply not interested in the technical details. They just wanted to know what the value of Gödel's theorem is for physics. Hawking pointed out that positivism is simply futile. Laplace's demon is the wrong view on physics because it will never be possible. — Tarskian

Of course, one needs to leave out a lot of details in a talk like that or in a post. But again, as you keep skipping:

(1 Your post didn't just leave out details, but it mischaracterized the theorem terribly, and in a way that readers would think it says something it does not, and was even about something it is not about. And that is critical in that discussion, since overstatement of the theorem feeds to bad inferences about its implications.

(2) It requires only very easy and basic technical terms (such as 'consistent', 'axiomatization') to correctly state the theorem.

(3) You often enter technical details anyway.

(4) Your reply doesn't relate to what you quoted of me.

Let me jump in somewhere: if a liar says he is lying, the foundation is shaky enough to allow either that he is telling the truth or that is he not (telling the truth). What exactly is he lying or not lying about? That's unclear. He is the premise, the substance, that is relative in the equation, so anything he says can either be true or false and we can't know because he, a priori, is an unreliable being. From his own perspective he may have some dialectic that tells him when and where he lies, but in our eyes we can assume nothing about what he says except that he lies most of the time.

I'm probably missing some detail or atomization

- I've addressed this before.

• "You habitually do such-and-such." (e.g. "you are under the spell of material implication")
• "No I don't!"
• "I think you do."
• "I've told you I don't, so now you're lying."

This is the same conflation of falsehoods with lies that I brought up. You can have a habit without realizing it, and I can believe this to be true without lying. You may as well accuse a doctor of lying when he tells you that you have a tumor and you tell him that you do not.

Let me jump in somewhere: if a liar says he is lying, the foundation is shaky enough to allow either that he is telling the truth or that is he not (telling the truth). What exactly is he lying or not lying about? That's unclear. He is the premise, the substance, that is relative in the equation, so anything he says can either be true or false and we can't know because he, a priori, is an unreliable being. From his own perspective he may have some dialectic that tells him when and where he lies, but in our eyes we can assume nothing about what he says except that he lies most of the time. — Gregory

Yes, but it seems to me that in this case we are considering the assertion, "I am a liar," or, "I always tell lies," rather than, "I am lying." To lie requires a statement about which to lie, whereas to be a liar does not (at least in the same proximate way). The OP tends to pivot on a statement that is supposed to simultaneously be a lie and a non-lie. Nevertheless...

If we know that someone is a liar then, as you say, their utterances are thrown into question. These utterances wouldn't be both true and false; they would merely be questionable.

Another basic thing to note is that knowledge of a lie or a falsehood is composite, not simple. There must be both the utterance that is false or a lie, and also the claim that it is false or a lie. Thus for such a thing to obtain there must be two things, or at least two aspects of the same thing. "This is a lie," or, "This is a falsehood," provides only one thing, not two. There can be no lie or falsehood without some claim that is apt to be lied about or to be understood as false.

e.g. you are under the spell of material implication — Leontiskos

That's not "e.g." since it is not what you said - it is clearly weaker.

And, if I recall, you didn't merely say that you "think" I have that position.

You're trying to ameliorate your lies.

And it's not even a matter of what one thinks. I explicitly stated my view, at least a few times, that is the opposite of what you what you claimed about me. And I stated it again at least a couple of timse after you misrepresented it, and then lied about it as you persisted after I pointed out to you. Explicitly and clearly. The posted record is there. And brought to your attention, but you continued.

You may as well accuse a doctor of lying when he tells you that you have a tumor and you tell him that you do not. — Leontiskos

That is a stupid analogy.

I did not merely claim that I posted the opposite of your claim about me. It is posted record that I said the opposite of your lie about me.

If someone said, "Leontiskos says that JFK was a secret member of the Ku Klux Klan" then you would be quite right to say that you did not say that, and if the person persisted, you would be quite right to say they're lying.

You lied about me about then and you're trying to weasel out of it now.

That's not "e.g." since it is not what you said - it is clearly weaker. — TonesInDeepFreeze

It seems that you have no clear idea what I am supposed to have said:

In the other thread, I think about three times, you claimed [...], or [...], or that [...]. Something to that effect. — TonesInDeepFreeze

Which is wonderful: you run around accusing people of lying and you have no idea what they are even supposed to have said. :roll:

That is a stupid analogy. — TonesInDeepFreeze

It's not.

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

"Godel's theorem is proved using statements that refer to themselves. Such statements can lead to paradoxes. An example is, this statement is false. If the statement is true, it is false. And if the statement is false, it is true. Another example is, the barber of Corfu shaves every man who does not shave himself. Who shaves the barber? If he shaves himself, then he doesn't, and if he doesn't, then he does. Godel went to great lengths to avoid such paradoxes by carefully distinguishing between mathematics, like 2+2 =4, and meta mathematics, or statements about mathematics, such as mathematics is cool, or mathematics is consistent. That is why his paper is so difficult to read. But the idea is quite simple. First Godel showed that each mathematical formula, like 2+2=4, can be given a unique number, the Godel number. The Godel number of 2+2=4, is * (sic). Second, the meta mathematical statement, the sequence of formulas A, is a proof of the formula B, can be expressed as an arithmetical relation between the Godel numbers for A- and B. Thus meta mathematics can be mapped into arithmetic, though I'm not sure how you translate the meta mathematical statement, 'mathematics is cool'. Third and last, consider the self referring Godel statement, G. This is, the statement G can not be demonstrated from the axioms of mathematics. Suppose that G could be demonstrated. Then the axioms must be inconsistent because one could both demonstrate G and show that it can not be demonstrated. On the other hand, if G can't be demonstrated, then G is true. By the mapping into numbers, it corresponds to a true relation between numbers, but one which can not be deduced from the axioms. Thus mathematics is either inconsistent or incomplete. The smart money is on incomplete."

(1) Contrary to your claim about the speech, Hawking does go into some technical details.

(2) "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. Ordinarily, one would take 'the axioms of mathematics' to refer to ZFC*. But it is provable in ZFC that G is true. Moreover, it's important to note that it's not the case that there are sentences of mathematics that are unprovable; for any sentence, there is a theory in which the sentence is a theorem. What the theorem does say is that, for any theory of a certain kind, there is a true sentence (such as G) that is not provable in that theory. It's a matter of quantification: It's not "There is a true sentence such that for all theories, the sentence is unprovable". Rather, it's "For all theories (of a certain kind), there is a true sentence not provable in the theory".

That is important, since people take such claims as "mathematics does not prove" to argue incorrectly about incompleteness.

(3) "Suppose that G could be demonstrated. Then the axioms must be inconsistent because one could both demonstrate G and show that it can not be demonstrated."

We need to not miss the distinction between the meta-theory and object theory. Let's take the object theory PA for example. It is the object theory PA that does not prove G and does not prove ~G, and it is not the case that PA proves that PA does not prove G and that PA does not prove G. Rather, it is in the meta-theory that we prove that PA does not prove G and does not prove ~G.

* I am not claiming that ZFC is the only axiomatization or that it is the best one. Only that if there is an axiomatization considered to be "the axioms of mathematics', then ordinarily it would taken to be ZFC. And, of course, that refers to classical mathematics.

if a liar says he is lying, the foundation is shaky enough to allow either that he is telling the truth or that is he not (telling the truth). — Gregory

Call this person 'L'. L says "I am lying". Now it could be the case that L lies always, or it could be the case that L lies only sometimes. But the puzzle asks whether he is lying or telling the truth when he says "I am lying".

What exactly is he lying or not lying about? — Gregory

He's talking about his lying. If he's lying, then he's lying that he's lying. If he's not lying, then he's telling the truth that he's lying. Both disjuncts are absurd.

he is an unreliable being. — Gregory

It's an interesting question how unreliable one would regard. Every person is unreliable at least to some degree. And a person who says he never lies would be taken as unreliable immediately, since (at least virtually) every one tells some lies. But a person who says "I am lying" is setting up a puzzle at least. I'm not sure what I would make of his reliability otherwise.

we can assume nothing about what he says except that he lies most of the time. — Gregory

How do you draw that inference?

All we know is that he has made one very bizarre statement that we can't determine to be a lie or not a lie, while arguably it's neither as it is merely nonsense. How would you infer from that that he lies most of the time?

It seems that you have no clear idea what I am supposed to have said:

In the other thread, I think about three times, you claimed [...], or [...], or that [...]. Something to that effect.
— TonesInDeepFreeze

Which is wonderful: you run around accusing people of lying and you have no idea what they are supposed to have even said. — Leontiskos

You are too much!

Here's what I wrote:

In the other thread, I think about three times, you claimed that I conflate material implication with everyday use of "if then", or that I that I consider material implication to be the only correct understanding of "if then", or that I insist that "if then" can only be considered as material implication. Something to that effect. — TonesInDeepFreeze

(1) Note that I did not put those in quotes, unlike when you quoted yourself - "you are under the spell of material implication" (an actual quote?) as if it represented the times you directly claimed that I had made a claim, even though I claimed the opposite. As if "you are under the spell of material implication" is really all you said.

(2) It is correct that you said things to the effect I mentioned. And I did not overstate about you, while you now understate about yourself.

(3) About three times in the thread, I directly addressed what you did say about me.

(4) The posts are there; even though have not quoted them verbatim here. But perhaps I should take even more of my time and energy to again clean up after you by again quoting you verbatim.

(5) 'people'. There are a few people I've found to lie. And when I comment that they do, I cite their specific instances, just as I did with you in the other thread.

(6) You remind me that not only did you lie about me, but you persistently tried to wedge an argument against me by painting me as a "truth functionalist" while I countered each time by explaining why that is so off-base.

That is a stupid analogy.
— TonesInDeepFreeze

It's not. — Leontiskos

I said exactly why it is stupid.

it seems to me that in this case we are considering the assertion, "I am a liar," or, "I always tell lies," rather than, "I am lying." — Leontiskos

I think we can work with any of these:

"I always lie"

"I am lying now"

"I am lying"

"This statement by me is a lie"

"This statement is a lie"

"This is a lie"

There are difference among them that might affect the analysis, but I think they are all worthy of the subject.

I took the poster at face value when he used "I am lying".

Consider these two sentences:
1. “This sentence has five words.” Or
2. “The sentence ’this sentence has five words’ has five words in it.”

Sentence number 2 is about something. Number one isn’t. The reader has to make 1 be about itself. In the sentence “This sentence has five words” you don’t know which sentence the speaker is taking about without being the speaker and pointing back to the sentence. Number 2 tells you what it is referencing, tells you what it is about.

Number 1 is a puzzle game, with missing pieces you have to bring with you to play; Number 2 is about counting words.

“This sentence is false” isn’t about anything that can be true or false. “Grammar is false” similarly isn’t about anything that can be true or false.

“Punctuation is true.”

What?

Number one isn’t. — Fire Ologist

It's about the sentence.

“This sentence has five words” you don’t know which sentence the speaker is taking about — Fire Ologist

First, there doesn't have to be a speaker. We can consider the sentence in and of itself.

Second, I do know which sentence is the subject of the sentence. The subject of the sentence is the sentence "This sentence has five words".

“Grammar is false” similarly isn’t about anything that can be true or false.

“Punctuation is true.”

What? — Fire Ologist

Yes, those don't make sense, since presumably what are true or false are sentences, and 'grammar' and 'punctuation' are not sentences. But "this sentence has five words" is a sentence, and 'this sentence' refers to it, as 'this sentence' is the subject of the sentence.

Does "this sentence has five words" have five words? Yes, it does. And does "this sentence has five words" claim that "this sentence has five words" has five words? Yes, it does. So does it not state a truth? So is it not true?

.

If a liar says he is lying, his utterance does need to be compared to the substance of the utterance, to the lying-ness of the speaker. The "claim" of the utterance relates to he who utters before it hits our ears. To the liar, he knows where he stands. To us, when he says "i always lie", we must understand the language "game" involved. Either he HAS always lied and he is owning up to it or he is lying that he always lies, wherein he must have at least once spoken the truth.. The latter seems to be where the trouble is

When L says he is lying, he hasn't specified what he is lying about. It's like the barber paradox. Not enough information is given so we must assume he grow his hair to hippie length. The liar may be lying that he lies about lying, but the reality of his truth or falsehoid falls somewhere. It seems different if we say "this sentence is false" because it necessarily is taken out of context and merely implies a spurious infinity and, hence, is a big fat Zero in terms of truth

Wouldn't you agree we must assume a liar to be a liar most of the time, or is redemption possible in this logic game?

This sentence has five words. Not true? — TonesInDeepFreeze

Yes, true.

Language communicates

As not every animal is a cat, not every set of words is part of a "Language".

The Merriam Webster Dictionary describes "Language" as i) the words, their pronunciation, and the methods of combining them used and understood by a community ii) a systematic means of communicating ideas or feelings by the use of conventionalized signs, sounds, gestures, or marks having understood meanings iii) the means by which animals communicate

As the core feature of a "Language" is its ability to communicate, presumably a set of words that is not able to communicate is not a part of what can be called a "Language".

A computer could be programmed to collect together random words, but a set of words does not of necessity make a "Language".

The meaning of words is not fixed

As prose, the set of words "colourless green ideas sleep furiously" is a nonsensical proposition, and because it doesn't communicate any idea, cannot be considered as part of a proper language. However, the meaning of words is not fixed, and the same words could be interpreted in a poetic or metaphorical way. If that were the case, one could possibly gleam some sensible meaning out of this particular set of words.

But for the sake of this post, I am only considering a set of words as prose.

Not all expressions in Language have a truth-value

Some expressions in language can be true or false, such as the expression "the Eiffel Tower is 1,200m in height", but other expressions, such as the command "bring me a coffee" or the exclamation "ouch!" are neither true nor false.

The author of a set of words is irrelevant to the meaning of the set

I see five words on the screen in front of me, and the particular words are "this", "sentence", "has", "five" and "words". I see five crayons on my desk next to the computer screen, and the particular colours are red, green, blue, yellow and orange.

As five crayons don't of necessity make a language, five words don't of necessity make a language.

I don't know who or what has determined these five words. It could have been a person or a machine. In fact, I don't need to know as long as I can find my own meaning in the words. The observer determines any meaning in any set of words. The meaning is not determined by the words themselves. Only the observer is a conscious being. The words themselves are inanimate, neither aware of themselves not anything outside themselves.

The meaning of a set of words is determined by an outside observer of the words

That the five words on the screen include the words "this statement" is meaningless, as it is not the words that determine whether the five words forms a statement or not, but rather it is the observer who determines whether or not these five words are part of a language. The observer may in fact determine that these five words are not part of a language, in that they are not a statement.

Truth-values must be grounded in the world

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.

It is true that the set of five words on my screen are grounded in the world, as are the five crayons on my table, but as both the five words and the five crayons are inanimate, neither being conscious nor self-aware, neither the five words nor five crayons are able to refer. Only a conscious outside observer of the five words and five crayons is able to refer.

A set of words independent of any observer can never have any meaning. A set of inanimate words cannot give themselves meaning. Meaning cannot give itself meaning. Meaning is not self-referential, and as such can never give itself a truth-value.

A set of words can only have meaning when given a meaning by an outside observer, and in order for a set of words to be given meaning by an outside observer, this set of words must exist in the world.

Summary

In summary, I see a set of words on my screen. I see that there are five words, and this is true. The five words happen to be "this", "sentence", "has", "five" and "words". I, as the observer, recognize a meaning in the five words as "this sentence has five words". Words being inanimate cannot refer. Only a conscious observer outside the words can refer. In the mind of this conscious outside observer, the words "this sentence" refers to the statement "this sentence has five words", which is true.

a liar — Gregory

What is the definition of 'is a liar' here?:

every statement is a lie?
some statements are lies?
many statements are lies?
more than half the statements are lies?
etc.

when he says "i always lie" [...]Either he HAS always lied and he is owning up to it or he is lying that he always lies, wherein he must have at least once spoken the truth.. The latter seems to be where the trouble is — Gregory

If only the latter is problematic, then there's no paradox, as we just conclude that he has always lied previously but is telling the truth now.

But that's not the puzzle.

When he says "I always lie", we take that to mean not just that he is saying that he has always lied but also that he's lying now. If I say "I always breathe", then I mean that I have always breathed and that I am breathing now.

If he says "i always lie" and he is being untruthful because he had once told the truth then his lie is now and his truth was once. "This sentence is false" doesn't say what the sentence is about so it's spurious but maybe not in infinite regress. I don't see how these paradoxes can be made into necessary paradoxes, ones entailed by logic itself

When L says he is lying, he hasn't specified what he is lying about. — Gregory

Does he need to? If I say "I am speaking", I don't need to say what I'm speaking about.

It's like the barber paradox. Not enough information is given so we must assume he grow his hair to hippie length. — Gregory

I take that to mean that no one shaves the barber.

The premise is that there is a person who shaves all and only those who shaves himself. That premise is inconsistent, so we may infer all of these:

said person shaves herself.

said person does not shave herself.

no one shaves said person.

someone shaves said person.

everyone shaves said person.

no one is shaved by anyone.

everyone is shaved by everyone.

some are shaved by someone

some are not shaved by anyone

etc.

The puzzle [we don't need to mention 'barber' or 'village']:

There is a person who shaves all and only those who do not shave themselves. Therefore, there is a person who both shaves herself and does not shave herself.

We can put it in purely abstract form, where 'S' stands for any 2-place relation:

There is an x such that, for all y, xSy if and only if it is not the case that ySy. Therefore there is an x such that both xSx and it is not the case that xSx.

To us, when he says "i always lie", we must understand the language "game" involved. Either he HAS always lied and he is owning up to it or he is lying that he always lies, wherein he must have at least once spoken the truth.. The latter seems to be where the trouble is — Gregory

Right, but is it possible that his self-same statement is both the truth and a lie? I say that it is not possible.

Someone may be telling the truth or lying, and this is not a paradox. The paradox requires something more. If I am right then it requires that there be no speaker at all, even implicit or hypothetical.

The paradox depends on the idea that there is a sentence with two different senses, and both senses are true at the same time. 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. If someone is lying then they are not telling the truth. If someone is telling the truth then they are not lying. 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.