how do you avoid the problem of infinite recursion in a self-referential sentence? — RussellA

No argument has been sustained that there is such a problem.

"This sentence contains five words". — RussellA

For sake of continuity with the thread, I'll use:

This sentence has five words.

Non self-referential case
Let "this sentence" refer to the sentence "this sentence contains five words" — RussellA

Yes, please let it, since "This sentence" does refer to the sentence "this sentence has five words".

Then, "this sentence, the sentence "this sentence contains five words", contains five words". This is meaningful. — RussellA

Meaningful and false, since (1) "this sentence, the sentence "this sentence has five word", has five words" falsely claims that "this sentence, the sentence "this sentence has five word", has five words" is "this sentence has five words" and (2) "this sentence, the sentence "this sentence has five word", has five words" has twelve words not five.

the sentence "this sentence contains five words" is not the same sentence as "this sentence contains five words", even though the wording is identical. These are two completely different sentences. — RussellA

Wrong. "This sentence has five words" is "This sentence has five words". They are the same linguistic object. As RussellA himself says, the wording is identical. So they are the same sentence.

RussellA has reduced himself to explicitly contradicting the law of identity.

the truth of the sentence "this sentence contains five words" is independent of the truth of the sentence "this sentence contains five words". — RussellA

Since "this sentence has five words" is "this sentence has five words", the truth of "this sentence has five words" cannot possibly be independent of the truth of "this sentence has five words".

For any sentence S: S is true if and only if S is true; and S is false if and only if S is false.

RussellA has reduced himself to explicitly contradicting the the principle "P if and only if P".

Self-referential case
In the self-referential case, "this sentence, the sentence "this sentence contains five words", contains five words".
But we know that this sentence is the sentence "this sentence contains five words". — RussellA

What does the first occurrence of "this sentence" refer to in the sentence immediately above?

Two choices:

(a) "this sentence" refers to "we know that this sentence is the sentence "this sentence has five words""

And (a) is false, since we do not know that "we know that this sentence is the sentence "this sentence has five words"" is the sentence "this sentence has five words"; but rather, we know that it is not the case that "We know that this sentence is the sentence "this sentence has five words"" is "this sentence has five words".

(b) "this sentence" refers to "this sentence, the sentence "this sentence contains five words", has five words".

And (b) is false, since we do not know that "this sentence, the sentence "this sentence contains five words", has five words" is "this sentence has five words"; but rather we know that it is not the case that "this sentence, the sentence "this sentence contains five words", contains five words" is "this sentence has five words".

Therefore, "this sentence, the sentence "the sentence "this sentence contains five words" contains five words", contains five words".
Ad infinitum. Infinite recursion. Therefore meaningless. — RussellA

Non sequitur.

It is the case that:

"This sentence has five words" is true if and only if "This sentence has five words" has five words. (So, "This sentence has five words" is true.)

""This sentence has five words" is true" if and only if ""This sentence has five words" is true" is true. (So, ""This sentence has five words" is true" is true.)

ad infinitum - the value is true

But also:

"Florida is a state" is true if and only if Florida is a state. (So, "Florida is a state" is true.)

""Florida is a state" is true" is true if and only if ""Florida is a state" is true" is true.

ad infinitum - the value is true

That is different from the liar paradox:

"This sentence is false" is true if and only if "This sentence is false" is false, so "this sentence if false" is true.

""This sentence is false" is true" is true if and only if ""This sentence is false" is true" is true, so "this sentence is false" is false.

ad infinitum - the value alternates between true and false

These points - (a) it is not just self-referential sentences that provide "ad infinitum" and (b) mere self-reference doesn't provide paradox but rather the combination of self-reference and negation - have been pointed out to RusellA probably a half dozen times already

Note that in the self-referential case, the sentence "this sentence contains five words" is the same sentence. — RussellA

Same sentence as what?

So you assert.

Moreover, I don't know what you mean by a sentence "wanting" to say something.

Meanwhile, a counterargument has been given, and that counterargument has not been refuted.

He's been refuted at every point in every detail.

"This sentence has fifty words" does not seem to be a paradox.

That's not what has been at issue.

Rather at issue has been whether sentences such as "This sentence has fifty words" are meaningful.

Why has it taken nine pages for RussellA still to still stick with his refuted arguments for his claim:

"If "this sentence" is referring to "this sentence contains fifty words", it has no truth-value and is meaningless."

The answer is psychological.

I maintain that barbers are people who shave people who are in the world.......................Therefore, if a barber tries to shave himself, there is an inherent contradiction. — bongo fury

Makes sense to me. I've never understood any validity in the barber paradox. — RussellA

The barber sentence is not just invalid, it is logically false.

The expression ‘secondary employment’, also commonly referred to as ‘double jobbing’, simply describes a situation where an employee takes on a second job.

During the day, someone works as an engineer in an engineering works. In order to pay their rent, during the evening they work in a cafe as a barista.

No one would call someone who serves you coffee an engineer.

No one would call someone welding machinery a barista.

As you say, being a barber is what someone does, not what they are. — RussellA

Wow! How can someone so ignorantly miss the point!

The barber paradox doesn't at all rely on claiming that someone is a barber. Mentioning someone who is a barber is merely incidental and does not at all bear upon the logic.

We could just as well not use the word 'barber' or even 'person':

There is an x such that for all y, x shaves y if and only if y does not shave y.

For that matter, we don't need 'shave', which is merely illustrative:

For any 2-place relation R:

(1) There is an x such that for all y, Rxy if and only if it is not the case that Ryy.

And (1) is logically false.

Wow. RussellA has such strong opinions on the subject but doesn't know what the subject even is.

Barbers cannot shave themselves.

I maintain that barbers are people who shave people who are in the world.

If they must be shaved, the barbers must visit other barbers. Shaving involves a correspondence between an ideal of cleanliness and the state of affairs on an actual face. Therefore, if a barber tries to shave himself, there is an inherent contradiction. — bongo fury

Is that a joke? To shave is to cut hair from the skin. There is no requirement of an ideal of cleanliness or that a face is involved.

A barber is someone who is in the business of cutting hair and shaving. There is no contradiction that a barber shaves himself of herself; surely some barbers do.

Moreover, the barber paradox is merely an illustration. The relation of 'x shaves y' could be any 2-place relation R:

(1) There is an x such that for all y, Rxy if and only if it is not the case that Ryy.

And (1) is logically false.

My context here, unless mentioned otherwise, is ordinary mathematics, which is classical mathematics as found in calculus for the sciences, which is axiomatically formalized in set theory. I don't opine that that is the only context we should consider; only that when you talk about "real numbers", without qualifying that you don't mean other than the ordinary context, I surmise that you are talking about the ordinary context. And when I say "an object exists with property P" I mean that from the axioms we prove the theorem that we render in English as "There exists an x such that x has property P". I don't opine in this immediate context as to the philosophical aspects of such existence theorems.

We provide (with definitions from the primitives for membership (e) and identity (=), and from the axioms) such basics as set abstraction, subset, union, singleton, ordinal, function, sequence, finite, infinite, natural number, the set of all and only the natural numbers (w [read as 'omega']), denumerable, ordering and operations on the natural numbers, recursion, transfinite recursion, cardinal, aleph, irrational number, the set of all and only the rational numbers, ordering and operations on the rational numbers, Dedekind cut, equivalence class, and Cauchy sequence.

Most pertinently here, we define the property "is a real number", and we prove the existence of the set of all and only the real numbers (R), a particular ordering on that set (<), two particular elements (0 and 1), and two particular operations (+ and *). We also define "is a completed ordered field" and we prove that the system <R < 0 1 + *> (I have to use the symbol '<' for both less-than and the opening bracket) is a complete ordered field and that all complete ordered fields are isomorphic with one another. Thus we have the number system of real numbers that is unique within isomorphism. (Then we also define subtraction (-), negative, positive, non-negative, non-positive, division (/), absolute value (| |) and intervals (( ), [ ), ( ], [ ]).

And we define "the continuum" as R along with <. That is the ordered pair with R and <. So, the continuum is the set of real numbers considered vis-a-vis the standard ordering of the set of real numbers. Since we proved the existence of the set of all and only the real numbers and we proved the existence of the standard ordering on that set, we proved the existence of the ordered pair, and thus are entitled to name it, as we name it "the continuum". The continuum exists.

Usually "is a real number" is not rigorously defined by "is an integer followed by a decimal sequence". Rather, an integer followed by a decimal sequence is taken to be merely a representation of a real number.

The two most common rigorous definitions of "is a real number" are:

Df. r is a real number if and only if r is an equivalence class of Cauchy sequences

or

Df. r is a real number if and only if r is a Dedekind cut.

But we can define "is a real number" as a certain kind of sequence as long as we eschew infinitely many consecutive 9s:

Df. r is a real number
if and only if
r is a sequence whose domain is w and
r(0) is an integer and
for n>0, r(n) is in {0 1 2 3 4 5 6 7 8 9} and
for all n in w, there is an m in w such that n<m and f(m) not= 9

So, r(0) is the integer part, then for each n>0, r(n) is the nth digit in the infinite expansion, and there are not infinitely many consecutive 9s.

They key point is that all three - the equivalence class of Cauchy sequences definition, the Dedekind cut definition, and the decimal sequence definition - all provide a complete ordered field, and all complete ordered fields are isomorphic with one another.


But what about infinitesimals? A possible definition:

i is an infinitesimal if and only if 0 < |i| and there is no real number strictly between 0 and i.

or

i is an infinitesimal if and only if 0 not= i and there is no real number strictly between 0 and i.


Then it is easy to prove that no real number is an infinitesimal.

But the fact that no real number is an infinitesimal doesn't prevent us from formulating another system in which there do exist infinitesimals.

Perhaps the three main two approaches to having a number system with infinitesimals are:

(1) In set theory, we use models. Thus we have two different number systems to use: The real number system and a different number system that has infinitesimals.

(2) In set theory, we use ultrafilters. Thus we have two different number systems to use: The real number system and a different number system that has infinitesimals.

(3) To set theory we add a primitive unary predicate "is standard" and additional axioms. In that theory we also have the real number system and a different number system with infinitesimals.

In any case, having a different number system doesn't disprove the existence of the other number system and doesn't disprove the existence of the continuum.

/

Your terminology about gaps conflates the predicate "is a gap" with the nouns "G1" and "G2".

Let's fix that with clear definitions of predicates:

Df. g is a G1-gap if and only if g is an interval

Df. g is a G2-gap if and only if g is an interval between two different points q and r such that there is no point strictly between q and r.

We prove that in the real number system there are no G2-gaps.


You propose to have 0.0...01 serve as an infinitesimal.

We could as easily write that as 0.0...1.

But two things:

(1) We need to be clear what 0.0...1 is.

(2) You need to define an ordering that has your supposed infinitesimal in the field of the ordering and to prove the needed theorems.

Regarding (1):

First:

A sequence is a function whose domain is an ordinal.

Every cardinal is an ordinal, but not every ordinal is a cardinal.

Ordinal addition is different from cardinal addition. But we use the same symbol '+' for ordinal addition and cardinal addition. So, we consider context to see whether '+' is being used for ordinal addition or cardinal addition. (And '+' for addition of real numbers is different also.)

So I'll use '+' for addition of real numbers, '#' for ordinal addition, and '+' [in bold] for cardinal addition.

The least infinite ordinal is w. w is also a cardinal, referred to as 'aleph_0' when we emphasize it as a cardinal.

{n | n is a natural number} = N = w = aleph_0 = card({n | n is a natural number}).

(aleph_0)+1 = aleph_0 = w.

w#1 not= w.

w#1 not= aleph_0.

Second:

Lets look at three things you've mentioned:

(a) 0.0...1

(b) 0.0...2

(c) 0.0...11

(a) and (b) represent sequences whose domain is w#1.

(c) represents a sequence whose domain is w#2.

Regarding (2):

You've not defined any such ordering.

Moreover, though you are free to have such things as (a), (b) and (c) in a number system that you may formulate, they do not represent real numbers as we have represented real numbers. And if they represent infinitesimals, then they are not in any number system isomorphic with the set of real numbers. So, your argument is profoundly ill-conceived. Even if you went on to fulfill your own formulation of a number system with infinitesimals, then that would not be the real number system nor isomorphic with the real number systems, thus it is ludicrous to say, as you do, that there is an interval in the reals between two different points q and r such there is no real strictly between q and r.

Moreover, your profoundly ill-conceived argument does not in any way support your titular assertion that the continuum does not exist.

.

"This sentence has five words" is true IFF this sentence has five words — RussellA

You should stop right there. You continue to blatantly assert the same confusion. You should go back and actually read the explanation that has been given you about this five times already.

For about the fifth time you have skipped the point that you are not taking into account that pronouns are contextual. So, again I suggest that you consider:

"This sentence has five words" is true IFF this sentence has five words.

The first occurrence of "This sentence" refers to the sentence "This sentence has five words"; but the second occurrence of "this sentence" refers to the sentence ""This sentence has five words" is true IFF this sentence has five words".

"This sentence has five words" has five words.

"This sentence has five words" is true IFF this sentence has five words" does not have five words.

Your argument that blatantly ignores the context of pronouns is blatantly fallacious.

Consider:

"This guy is in love with Lani" is true if and only if this guy is in love with Lani.

The first occurrence of "This guy" refers to Herb Alpert; but the second occurrence of "this guy" refers to TonesInDeepFreeze.

Herb Alpert is in love with Lani. TonesInDeepFreeze is not in love with Lani.

/

That you blatantly skip this point over and over is intellectual dishonesty.

/

The more you are corrected on each point, the more you continue to propagate both old and new speciousness.

A neat review:

@RussellA offers three options for "This string has five words":

(1) "This string" refers to "This string".

(2) "This string" refers to some other unrelated string such as "The cat is black".

(3) "This string" refers to "This string has five words".

Each one:

(1) is false. "This string" doesn't refer to the string "This string" but rather to the string "This string has five words", which is the string that has five words.

(2) is false and ridiculous.

(3) is true. But he claims that then "This string has five words" is meaningless due to self-reference. But why does self-reference make it meaningless? His answer is that self-reference is meaningless because the would-be referrer doesn't refer to something in the world. But why is a string not something in the world? His answer is that because strings are in the mind not outside the mind. But then not just self-referring strings are in the mind not in the world. (Moreover, he agrees that strings are not mere inscriptions but that the same string may have more than one inscription.) So if strings are not things in the world, then the string "Caesar was a Roman emperor" is not in the world. So ""Caesar was a Roman emperor" has five words" must also be meaningless. But then his default is to merely reassert that it is self-referring strings that are meaningless. And around in circles it goes, back to him reasserting his three options.

Meanwhile, the Pentastring argument has not been refuted.

The song that is titled "Your Song" has the lyric:

This is the song. It may be quite simple.

"This" refers to the song itself, which is the song that is titled "Your Song". "It" refers to the song itself, which is the song that is titled "Your Song".

Within the song, there is reference to the song. It seems to me that it is meaningful when the song says of itself that it may be quite simple.

"It may be quite simple" is true if and only if the song itself (which is titled "Your Song") may be quite simple.

Consider this botched version:

"It may be quite simple" is true if and only if it may be quite simple.

The first instance of "it" refers to the song.

The second instance of "it" refers to the sentence "It may be quite simple".

But the song is not the sentence "It may be quite simple".

We cannot sensibly ignore the context of pronouns as @RussellA so ignorantly does.

Prediction: @RussellA will reply by (1) making yet another bizarrely false and stupid extension of his already existing bizarrely false and stupid arguments and by (2) finally resting yet again on his argument by mere assertion that self-referring expressions are meaningless.

@RussellA proffers the possibility that:

"This sentence has five words" is true if and only if this sentence has five words.

But that is wrong and stupid. It is wrong and stupid as he is not distinguishing the two different contexts of 'this'.

In the first instance "This sentence" refers to "This sentence has five words".

In the second instance, "this sentence" refers to ""This sentence has five words" is true if and only if this sentence has five words."

It is foolishness to ignore that distinction.

"This guy is in love with Lani" is true if and only if Herb is in love with Lani.

It is not the case that "This guy is in love with Lani" is true if and only if this guy is in love with Lani.

In the first instance "This guy" refers to Herb.

In the second instance "this guy" refers to TonesInDeepFreeze.

Herb is in love with Lani. TonesInDeepFreeze is not in love with Lani.


"This sentence has five words" has five words.

""This sentence has five words" has five words" does not have five words. (where "has five words" is meant as "has exactly five words")

/

Suppose there is a billboard out on Highway 61 in Bobsville, Arizona that says:

This billboard has five words.

"This billboard" refers to the billboard out on Highway 61 in Bobsville, Arizona. That is, "This billboard" refers to itself.

"This billboard has five words" is true if and only if the billboard out on Highway 61 in Bobsville, Arizona has five words.


"The Pentastring" refers to "This string has five words".

The Pentastring is "This string has five words".

The Pentastring is true if and only if "This string has five words" has five words.

The Pentastring is true if and only if "This string has five words" is true. (where "this string" refers to the string "This string has five words")


"Einstein's famous formula" refers to "E=MC^2".

Einstein's famous formula is "E=MC^2".

Einstein's formula is true if and only if E=MC^2.

Einstein's formula is true if and only if "E=MC^2" is true.


"JiffyJeff" refers to "All men are created equal".

JiffyJeff is "All men are created equal".

JiffyJeff is true if and only if all men are created equal.

JiffyJeff is true if and only if "All men are created equal" is true.


"Herb'sVow" refers to "This guy is in love with Lani".

Herb'sVow is "This guy is in love with Lani".

Herb'sVow is true if and only if Herb is in love with Lani.

Herb'sVow is true if and only if "This guy is in love with Lani" is true. (where "this guy" refers to Herb)


"BobsvilleMessage" refers to "This billboard has five words".

BobsvilleMessage is "This billboard has five words".

BobsvilleMessage is true if and only if the billboard out on Highway 61 in Bobsville, Arizona has five words.

BobsvilleMessage is true if and only if "This billboard has five words" is true. (where "this billboard" refers to the billboard out on Highway 61 in Bobsville, Arizona)

It is not the case that "This billboard has five words" is true if and only if this billboard has five words. Especially since the second instance of "this billboard" does not even properly refer.

You are again repeating your previously refuted arguments, as you skip right past the substance of the refutations. And then you argue yet again by mere assertion.

I agree that
1) "snow is white" is true IFF snow is white
2) "New York is in France" is true IFF New York is in France
3) "This sentence has five words" is true IFF this sentence has five words — RussellA

Regarding (3): You agree with yourself. You don't agree with me.

"This sentence has five words" is true IFF this sentence has five words" is ridiculous. You skip my previous response in which it is pointed out to you that you are ignoring the way pronouns work.

The problem with 3) is what exactly are "this sentence" and (this sentence) referring to? — RussellA

That's been answered. You skip my previous response.

in a non-self-referential case, "this sentence" could be referring to the sentence "New York is in France" — RussellA

In "This sentence has five words", obviously "This sentence" does not refer to "New York is in France".

It is pointless for you to even offer that non-possibility.

That's been mentioned. You skip my previous response.

(this sentence) could be referring to (New York is in France).
The non self-referential case is meaningful. — RussellA

Maybe what you mean is that "This sentence" refers to the claim that New York is in France.

Again, it doesn't.

It is pointless for you to even offer that non-possibility.

in a self-referential case, "this sentence" could be referring to the sentence "this sentence has five words", and (this sentence) could be referring to (this sentence has five words). The self-referential case is meaningless. — RussellA

"This sentence" refers to "This sentence has five words".

The problem with the self-referential case, is that the content of a sentence contains no information about the form of the sentence.

The content of the sentence "this sentence has five words" is that "this sentence has five words". The form of the sentence "this sentence has five words" is that the sentence "this sentence has five words" has five words. — RussellA

The "content" of "This sentence has five words" is the claim that "This sentence has five words" has five words.

The "form" of "This sentence has five words" is that "This sentence has five words" is the string that is the sequence of the five words "this", "sentence", "has", "five" and "words".

The content of a sentence can say nothing about the form of the sentence. It cannot self-refer. — RussellA

That's argument by mere assertion.

The "content" of "This sentence has five words" is the claim that "This sentence has five words" has five words. And having five words is an aspect of the form of "This sentence has five words".

As the sentence "New York is in France" says nothing about how many words are in the sentence "New York is in Paris, the sentence "this sentence has five words" says nothing about how many words are in the sentence "this sentence has five words" — RussellA

You repeat that bizarre and stupid analogy again!

I agree when you say:

"New York is in France" makes no mention of the number of words in "New York is in France".
— TonesInDeepFreeze

From the same logic, "this sentence has five words" makes no mention of the number of words in "this sentence has five words". It makes no mention of the fact that "this sentence has five words" has five words. — RussellA

"This sentence" refers to the sentence "This sentence has five words" and makes the claim that that sentence has five words.

Any similarity in expression is purely accidental. — RussellA

Similarity between what and what? And what is meant by "accidental" in this context?

Content cannot refer to its own form. — RussellA

Again, argument by mere assertion. And not even coherent. The content does not refer. The content is the claim that "This sentence has five words" has five words. That claim does not refer. With the sentence "Jack is tall", the sentence makes the claim that Jack is tall. What refers are the noun phrase "Jack", which refers to Jack, and the predicate "is tall", which refers to the property of being tall. With the sentence "This sentence has five words", the sentence makes the claim that "This sentence has five words" has five words. What refers is the noun phrase "This sentence", which refers to "This sentence has five words", and the predicate "has five words", which refers to the property of having five words.

"This sentence has five words" is true IFF "this sentence has five words" has five words.

If this were the case, then it would follow that:

"New York is in France" is true IFF "New York is in France" has five words. — RussellA

That is one of the most bizarre arguments I've ever heard.

"New York is in France" is true if and only if New York is in France.

"New York is in France" makes no mention of the number of words in "New York is in France".

From the fact that "This sentence has five words" is true if and only if "This sentence has five words" has five words it does not follow that "New York is in France" is true if and only if "New York is in France" has five words!

You are bizarre. It is amusingly disturbing about you that you are willing to enter increasingly outlandish arguments after each of your previous confused and outlandish claims is defeated.

You haven't "tackled" anything. You've fallen on your behind, dizzy from spinning while chasing yourself in circles of your own confusions.

In ordinary mathematics, from axioms we define 'is a real number' and we prove that there is a set whose members are all and only the real numbers. And we prove that there is a total ordering on the set of real numbers such that every non-empty subset with an upper bound has a least upper bound (equivalently, every non-empty set with a lower bound has a greatest lower bound). We call that 'the standard ordering'. Then we define 'is a complete ordered field'. And we prove the existence of operations on the set of real numbers such that, along with the standard ordering, we have a complete ordered field. Then we prove that all complete ordered fields are isomorphic with the system of real numbers.

There are different notions of 'the continuum' and 'a continuum'. An ordinary mathematical notion is that the continuum is the set of real numbers along with the standard ordering of the real numbers; then a continuum is any set and ordering on that set that is isomorphic with the continuum.

But you say that you use the terminology to mean that "a continuum is a set of distinct points with no abrupt change of gap between the points".

So what are the definitions of "abrupt change" and "gap between"? But even more basically, what are your axioms?

You do say:

By a gap, I mean an interval. — MoK

There is however either a gap between all pairs of points of the continuum or there is no gap — MoK

Between any two different real numbers there is an interval.

But the ordinary definition of continuum is not "there are no intervals between points". It is only by you personally redefining 'continuum' that you infer that there is no continuum.
One could as easily define 'is a mammal' by 'is an omnipotent animal' to then conclude that there are no mammals.

We are dealing with the same point of the continuum if there is no gap between a pair of points — MoK

[x x] is the interval whose only member is x.

(x x) is the empty interval.

Therefore there is a gap between all pairs of distinct points of the continuum — MoK

Of course, between any two different real numbers there is a non-empty interval.

Therefore, the continuum does not exist — MoK

We have:

If x < y, then (x y), (x y], [x y) and [x y] are non-empty intervals.

[x x] is a singleton interval.

From that it follows that there is an interval between any real numbers.

But it is only your ersatz definition of 'continuum' that leads you to infer that there is not a continuum from the fact that between any two points there is an interval. We might as well define 'is a computer' by 'can make infinitely many calculations in finite time' to conclude that there are no computers.

people claim that continuum is the real number. — MoK

There are uncountably many real numbers. There is not just one real number that is to be called "the real number".

Maybe you mean that we say that the continuum is the set of real numbers. I would rather be more exact in saying that the continuum is the set of real numbers along with the standard ordering of the set of real numbers.

The real number, however, is constructed from two parts, an integer part and a decimal part. — MoK

Again, there is not a real number that is called "the real number".

The usual constructions of real numbers are as either a Dedekind cut or an equivalence class of Cauchy sequences. Then we prove that every real number is representable as an integer along with a denumerable sequence (binary, decimal or whatever, as suits). And we prove that every integer along with such a sequence corresponds with a unique real number.

Infinitesimal can be constructed as follows: 0.0...01 by "..." I mean Aleph_0 0 — MoK

I guess you mean that there are denumerably many 0s followed by 1. There are rigorous treatments of nonstandard analysis in which there are infinitesimals. And there is the sequence {<x y> | x in w+1 & (x in w -> y = 0) & (x = w -> y = 1)}. But you are merely handwaving. It doesn't follow that such a sequence is an infinitesimal or represents an infinitesimal. Rather, you have to define 'is an infinitesimal' then prove that such sequences are or represent infinitesimals. Then define operations and orderings that include infinitesimals. To see how that is actually done, look up 'nonstandard analysis'. Moreover, that there are systems with infinitesimals doesn't contradict that there are other systems without infinitesimals.

How can "London" be a city?
— RussellA

What? "London" is not a city. No one said it is. — TonesInDeepFreeze

Still would like to know why you ask "How can "London" be a city?" when no one has said that "London" is a city.

Wait, I think I do know why. You still are confused about the distinction between use and mention.

the question was about the truth of the words - this sentence has five words.

The question was not about the truth of the words - "this sentence has five words". — RussellA

"This sentence has five words" is the sentence in question. It is true if and only if "This sentence has five words" has five words.

As regards "this sentence has five words", it all depends on what "this sentence" is referring to. — RussellA

"This sentence" refers to "This sentence has five words".

If it is self-referential, then it is meaningless — RussellA

And we're full circle about the third time, as your support for your assertion comes down to you just reasserting the assertion.

__________

"this sentence" is referring to "this sentence". — RussellA

Wrong. And it's been pointed out to you that it's wrong. "This sentence" refers to "This sentence has five words". "This sentence" and "This sentence has five words" are not the same expression.

__________

"This string has five words" was named "The Pentastring", and "This string has five words" is the Pentastring.
— TonesInDeepFreeze

No problem that "this string has five words" was named "the Pentastring"

I agree when you say:

"London" is a city. (false - "London" is a word, not a city)
— TonesInDeepFreeze

Then how can "this string has five words" be the Pentastring? — RussellA

Because I gave "this string has five words" the name "The Pentastring".

You might as well as how ask how Samuel Clemens can be Mark Twain. Samuel Clemens is Mark Twain because Samuel Clemens was given the name "Mark Twain". Or, suppose I give "All men are created equal" the name "JiffyJeff". Then "All men are created equal" is JiffyJeff.

__________

You think Mark Twain was someone other Samuel Clemens?
— TonesInDeepFreeze

My problem is:

That baby was named "Samuel Langhorne Clemens" and was Samuel Clemens
— TonesInDeepFreeze

Sense and reference
"Mark Twain" and "Samuel Clemens" both refer to the same thing in the world, although the names have a different senses — RussellA

Right.

in that "Mark Twain" was an author whereas "Samuel Clemens" wasn't. — RussellA

"Mark Twain" was not an author!

"Mark Twain" is the name of an author.

Mark Twain was an author.

You still don't understand use-mention.

As regards reference, "Mark Twain" and "Samuel Clemens" are both referring to the same thing in the world. Let this something be both Mark Twain and Samuel Clemens. In this event, Mark Twain is Samuel Clemens. — RussellA

Now you're back to sanity.

As regards sense, "Mark Twain" is referring — RussellA

You just conflated sense and reference. What even does "regards sense, X is referring" mean?

"Mark Twain" refers to Mark Twain.

"Mark Twain" refers to Samuel Clemens.

"Samuel Clemens" refers to Samuel Clemens.

"Samuel Clemens" refers to Mark Twain.

Mark Twain is Samuel Clemens.

The denotation of "Mark Twain" is the denotation of "Samuel Clemens".

"Mark Twain" is not "Samuel Clemens".

The sense of "Mark Twain" is not the sense of "Samuel Clemens".

Not only do you not understand quotation marks, use-mention or pronouns, you don't understand sense-reference.

Back to the point, contrary to your claim, it is not in doubt that Mark Twain is Samuel Clemens.

That "Samuel Clemens" is Samuel Clemens would give rise to logical contradictions. — RussellA

Of course "Samuel Clemens" is not Samuel Clemens.

If person A, born in Hannibal, is named "Samuel Clemens" then that person becomes Samuel Clemens. If person B, born in New York is also named "Samuel Clemens" than that person also becomes Samuel Clemens. In logic, the law of identity states that each thing is identical with itself, in this case, that Samuel Clemens is Samuel Clemens. But this means that person A born in Hannibal is person B born in New York. Something is wrong. — RussellA

What is wrong is your failure to distinguish between formal logic and everyday usage.

In formal logic, given a particular interpretation of the language, a name refers to one and only one object. But in every day usage, sometimes names often have different referents. You didn't notice that there many thousands of people named "Jane Smith" and that all of them are Jane Smith, though they are different Jane Smiths?

A group of Modernists name a painting "good", meaning that the painting is good. A group of Post-Modernists name the same painting "bad", meaning that the same painting is bad. But this means that good is bad, which breaks logic. — RussellA

You did it again! You conflate a name with a predicate. You simply ignored that I already pointed out that error.

Naming a painting "Good" is very different from asserting that the predicate "is good" applies to the painting. If you make some squiggles and name that product "Good", of course it doesn't then follow that the squiggle drawing is good. If you named one of your posts "Reasoned Argument", of course it doesn't follow that your post is reasoned argument.

"Mark Twain" and "Samuel Clemens" exist in the mind not the world. If there were no minds, then neither "Mark Twain" nor "Samuel Clemens" would exist. — RussellA

You equivocate. First you say that expressions do exist in the world. And you agreed that expressions exist in the world not just as particular inscriptions. Now you say they don't exist in the world.

Mark Twain is Samuel Clemens.
— TonesInDeepFreeze

Open to doubt. — RussellA

You think Mark Twain was someone other Samuel Clemens?

is this something X in the world Samuel Clemens, or has the something X in the world been named "Samuel Clemens"? — RussellA

Both.

"This string has five words" was named "The Pentastring", and "This string has five words" is the Pentastring.

As naming something in the world "a cat" doesn't make that something a cat, in that I could name a horse "a cat", naming something in the world "Samuel Clemens" doesn't make that something Samuel Clemens. — RussellA

If you name something (other than the author of 'Roughing It') "Samuel Clemens" then, of course, that thing doesn't become the person who wrote 'Roughing It'. But "Samuel Clemens" may also refer to something other than the person who wrote 'Roughing It'. If someone today has the last name "Clemens" and names her baby "Samuel", then the name "Samuel Clemens" refers to her baby as well as, in other contexts, it refers to the author of 'Roughing It'.

My naming that tall tower in Paris in the 7th Arr of Champs de Mars "a kangaroo" doesn't make that something in the world a kangaroo. — RussellA

You're still mixed up. You ignore what has been pointed out to you. I didn't name "This string has five words" with "a Pentastring". And I didn't define a predicate "is a Pentastring". Rather, I made up the name "Pentastring" and used it to name "This string has five words".

Giving something in the world a name doesn't make that something into what has been named. — RussellA

When the great trumpet player William Alonzo Anderson was nicknamed "Cat", of course, he didn't become a cat, but he is Cat.

Just for fun, some more in the menagerie of great jazz artists:

Hawk is Coleman Hawkins though he was not a hawk.

Bird is Charlie Parker though he was not a bird.

Rabbit is Johnny Hodges though he was not a rabbit.

Bunny is Roland Bernard Berigan though he was not a rabbit.

The Frog is Ben Webster though he was not frog.

Chick is William Henry Webb though he was not a chicken.

The Lion is William Henry Joseph Bonaparte Bertholf Smith though he was not a lion.

Pony is Norwood Poindexter though he was not a horse.

Just because something in the world has been named "Samuel Clemens", that doesn't mean that Samuel Clemens exists in the world. — RussellA

The baby born on November 30, 1835 in Florida, Missouri existed no matter what his name would be. That baby was named "Samuel Langhorne Clemens" and was Samuel Clemens. If another person is named or nicknamed "Samuel Clemens" then that person too is Samuel Clemens though not the same Samuel Clemens who wrote 'Roughing It'.

Although "Samuel Clemens" and "Mark Twain" exist in language, as neither Samuel Clemens nor Mark Twain exist in the world, then it is not correct to to say that Mark Twain is Samuel Clemens. — RussellA

What? Samuel Clemens, who is Mark Twain, exists as a deceased person. You're really utterly captiously quibbling about this? Wow. But, to accommodate even your most ridiculous quibbles, we'll use an example of a living person:

Ben Kingsley is Krishna Pandit Bhanji.

"Ben Kingsley" and "Krishna Pandit Bhanji" are two names of the same person.

"The Pentastring" is a name for the expression "This string has five words".
— TonesInDeepFreeze

No problem, setting aside what "this string has five words" means, and treating it as a set of words such as "a b c d e", and ignoring any meaning that it may or may not have. — RussellA

I named the string itself. But then I also discussed meaning.

The Pentastring is "This string has words".
— TonesInDeepFreeze

Open to doubt. — RussellA

Not reasonable doubt.

As before, my assumption has been that because "This string has five words" is in quotation marks, this means that "This string has five words" is an expression in language, and because the Pentastring is not in quotation marks, this means that the Pentastring is something that exists in the world.

The problem is, you are not saying that "this string has five words" is the name of the Pentastring, you are saying that "this string of five words" is the Pentastring. — RussellA

If I recall, a while ago you agreed that expressions are things in the world.

And you've not shown any problem. (1) Of course I am not saying that "This string has five words" is a name of the Pentastring. I stipulated that "The Pentasting" is a name of "This string has five words". (2) The Pentastring is "this string has five words". You've only adduced your own confusions in trying to fight that.

If A is B then B is A. If "this string has five words" is the Pentastring, then the Pentastring is "this string has five words". — RussellA

Indeed that is an instance of the symmetry of identity. But what's your point?

How can an expression in language be something in the world? — RussellA

If I recall, you said that it could. Moreover, you said "thing in the world" means [paraphrase:] "thing observed outside oneself". Well, I observe expressions outside myself. When someone says, "Today's soup special is split pea", I observe that there is the expression ""Today's soup special is split pea" and that it is outside myself.

How can "London" be a city? — RussellA

What? "London" is not a city. No one said it is.

This is not a side issue, as crucial to your argument that a self-referencing expression can be meaningful. — RussellA

It's not any kind of issue, since no one says that "London" is a city. No, actually, it is an issue. The issue is why you would come to such a bizarre conclusion that it is an issue.

You said that this is okay:

"The Pentastring" is a name for the expression "This string has five words"

You say you get lost at:

The Pentastring is "This string has five words"

Why are you lost?

Use-mention:

(1)

"Mark Twain" is a name for the person Samuel Clemens.

Mark Twain is Samuel Clemens.

Samuel Clemens is the person named by "Mark Twain".

Samuel Clemens is Mark Twain.

(2)

"Dangerfield's catchphrase" is a name for the expression "No respect".

Dangerfield's catchphrase is "No respect".

"No respect" is the expression named by "Dangerfield's catchphrase".

"No respect" is Dangerfield catchphrase.

(3)

"The Pentastring" is a name for the expression "This string has five words".

The Pentastring is "This string has words".

"This string has five words" is the expression named by "The Pentastring".

"This string has five words" is the Pentastring.

If I said "this sentence" is "this sentence". this would be meaningless. — RussellA

It's meaningful and true.

"this sentence" is a phrase. It is the same as itself.

I said "this sentence has five words" is "this sentence has five words", this would also be meaningless. — RussellA

It's meaningful and true.

"this sentence has five words" is a phrase. It is the same as itself.

such self-referential expressions cannot have any meaning. — RussellA

Ding ding ding ding ding! RussellA gets the prize! RussellA gets the prize for the most times arguing by repetition of an assertion. Congratulations, RussellA! Enjoy your all expenses paid trip to the luxurious Rabbit Hole Hotel.

are there any examples in language where a linguistic expression that refers to itself has a meaning? — RussellA

The Pentastring?

As regards usage, as more than one Pentastring exists in the world, the expression "The Pentastring" is not referring to one particular Pentastring, but is being used to refer to a general class of objects. — RussellA

Why do you ignore what is actually posted?

When I first introduced the term "The Pentastring", I used it as a name not an adjective.

I said that The Pentastring is "This string has five words". ("The Pentastring" is a name for the expression "This string has five words".)

I didn't say that a Pentrastring is a string with five words. (I didn't say "The Pentastring" is an adjective for the property: is a string with five words.)

And I recently wrote:

No, I didn't define a predicate "is a Pentastring". Rather, I defined a name "The Pentastring". — TonesInDeepFreeze

It is bizarre foolishness that you tried to falsely twist my naming a particular string into an adjective. Either you're hopelessly confused or hopelessly dishonest.

Said another way:

I could have used any phrase, whether "The Pentastring" or "ForumFinExample" or whatever to name the expression "This string has five words". That's not an adjective for the property of being a string with five words.

I would have thought that the formula E=MC^2 shouldn't be in quotation marks. For example, science is culturally important, and "science" has seven letters. Similarly, E=MC^2 is famous, and "E=MC^2" has six characters. — RussellA

Wrong. You still don't understand use-mention and quote marks.

I am referring to the literal string of symbols. The formula is a string of symbols. The string of symbols expresses a scientific principle, but the string of symbols is itself a linguistic entity.

Go back to the examples:

Define "Einstein's famous formula" as "E=MC^2". So Einstein's famous formula is "E-MC^2".

Define "The EqualityClause" as "All men are created equal". The EqualityClause" is "All men are created equal".

Define "JFK's'IconicMaxim" as "Don't ask what your country can do for you. Ask what you can do for your country". JFK'sIconicUtterance is "Don't ask what your country can do for you. Ask what you can do for your country".

Define "The Pentastring" as "This string has five words". The Pentastring is "This string has five words".

Pentastring is "This string has five words."
— TonesInDeepFreeze

This is grammatically incorrect, as an object in the world is not an expression in language. — TonesInDeepFreeze

Expressions exist. The expression, "All men are created equal" exists.

Consider:

Jefferson's most famous quote is "All men are created equal".

"Jefferson's most famous quote" refers to "All men are created equal".

Those are grammatical.

/

Meanwhile, one after another of your arguments have been refuted. Your replies are yet more straw-grasping, foolishly specious, smoke blowing confusions that are refuted.

You don't understand use-mention, quote marks, pronouns, the difference between a name and an adjective. And you show that you don't read the posts to which you reply and that you don't know to refrain from obvious speciousness.

An expression that refers to itself can never have a meaning

An expression can only have a meaning if it refers to something outside itself. — RussellA

That's your claim, which you try to support with arguments that have been shown to be specious.

We are given the expression "this sentence has five words", yet we both agree that the expression "this sentence" has two words.

So how can the same expression have both two words and five words? — RussellA

No expression can.

"This sentence" refers to "This sentence has five words".

"This sentence" is not "This sentence has five words".

It can only be that the expression "this sentence" in the first instance of its use is not referring to the second instance of its use. — RussellA

Consider:

This sentence has five words.

Recognize:

"This sentence" is not "This sentence has five words".

"This sentence" refers to "This sentence has five words".

"This sentence" doesn't refer to "This sentence".

You wrote that your belief is that some self-referring expressions can be meaningful, and give the Pentastring example — RussellA

I said that it seems to me that there are self-referring expressions that are meaningful but that I'm open to be being convinced otherwise and that I'm interested in finding any flaws there might be with the Pentastring argument.

Cleaning up your mess:

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

Why? If it did, then "this string has ten words" would assert that "this string has ten words" has ten words. — RussellA

So what? "This string has ten words" is false. That doesn't make it meaningless. "London is in France" is false. That doesn't make it meaningless.

Why are you wasting our time on such points?

So it seems "This string has five words" is a sentence as it fulfills the two requirements: syntactical and meaningful.
— TonesInDeepFreeze

Not necessarily. It depends what "this string" refers to. If it refers to either "this string" or "this string has five words", then it is self-referential and meaningless. — RussellA

That's just you again reasserting you claim! It's not an argument. You keep doing that: Responding to my argument by just reasserting you claim. Meet Mr. Ouroboros.

"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.
— TonesInDeepFreeze

Then it would follow that "the cat is grey" is true if "the cat is grey" has four words. That the sentence "the cat is grey" has four words doesn't make it true that the cat is grey. — RussellA

What in the world? How in the world do you come up with such a bizarre non sequitur?

we define 'the Pentastring' as the [string] "This string has five words".
— TonesInDeepFreeze

No problem, let's define 'the Pentastring' as the "This string has five words". This sounds very similar to defining 'Big Ben' as "the bell inside the clock tower". — RussellA

There you ignore that I mentioned that an expression can refer to an expression.

Define "Einstein's famous formula" as "E=MC^2". So Einstein's famous formula is "E-MC^2".

Define "The EqualityClause" as "All men are created equal". The EqualityClause" is "All men are created equal".

Define "JFK's'IconicMaxim" as "Don't ask what your country can do for you. Ask what you can do for your country". JFK'sIconicUtterance is "Don't ask what your country can do for you. Ask what you can do for your country".

But we know that "the Pentastring" has been defined as "This string has five words". Therefore "The Pentastring has five words" means that "this string has five words has five words". But this doesn't seem grammatical, and if not grammatical, then meaningless — RussellA

No, "The Pentastring has five words" means "This string has five words" has five words.

You speciously (either from dishonesty or ignorance) dropped the quote marks.

I didn't say that the Pentastring is this string has five words. I said the Pentastring is "This string has five words"

How transparently specious or stupid to drop the quote marks!

objects existing in the world, such as Big Ben and the Pentastring have no truth value,they can be neither true not false. — RussellA

Big Ben is not a linguistic object. The Pentastring is a linguistic object. Just as TheEqualityClause is a linguistic object.

The sentence "the Pentastring has five words" has five words. It is not the Pentastring that has five words. — RussellA

Again, the Pentastring is "This string has five words". The Pentastring has five words, they are: 'this', 'sting', 'has', 'five' and 'words.

Just as the EqualityClause have five words.

the sentence "The Pentastring has five words" means that "this string has five words has five words" — RussellA

It's not enough for you to post that speciousness. You have to post it twice.

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

[...] is not an example of self-reference. A Pentastring is a string of five adjacent words existing in the world. — RussellA

Again, the Pentastring is "This string has five words".

it is true that the sentence "this string has five words" has five words. It is also true that the sentence "the cat is grey" has four words.

The fact that the sentence "the cat is grey" has four words is irrelevant to whether the cat is grey. Similarly, the fact that the sentence "this string has five words" has five words is irrelevant to whether this string has five words. — RussellA

That is one of the most bizarrely irrelevant analogies I've ever read.

"The cat is grey" says nothing about number of words.

"This string has five words" does say something about number of words.

"The cat is grey" is true if and only if the cat is grey.

"This string has five words" if and only if "This string has five words" has five words.

sentence "the cat is grey" is true if the cat is grey. Similarly, the sentence "this string has five words" is true if this string has five words. — RussellA

That's nonsense concocted by you by ignoring that pronouns are contextual.

Consider:

"This guy is in love with Lani" means this guy is in love with Lani.

That's false.

The first occurrence of 'this guy' refers to Herb Alpert. The second occurrence of 'this guy' refers to me. Herb Alpert is in love with Lani. I'm not in love with Lani.

"This guy is in love with Lani" in context is true if and only if Herb Alpert is in love with Lani. It's not the case that "This guy is in love with Lani" in that same context is true if and only if TonesInDeepFreeze is in love with Lani.

"This string has five words" in context is true if and only if "This string has five words" has five words. It's not the case that "This string has five words" is true if and only if this string has five words.

As a Pentastring is a string of five words — RussellA

No, I didn't define a predicate "is a Pentastring". Rather, I defined a name "The Pentastring".

To isolate the key point:

The Pentastring is "This string has five words." The expression "The Pentastring" refers to the expression "This string has five words".

Just as:

Einstein's famous formula is "E=MC^2". The expression "Einstein's famous formula" refers to the expression "E=MC^2".

There is no conflict with what I wrote previously.

"This string has five words" is an expression, whilst the Pentastring is something that exists in the world. — RussellA

Again, as I pointed out:

An expression, such as the expression "Big Ben" refers to the thing Big Ben.

But with other expressions, the thing an expression refers to may also be an expression.

Einstein's famous formula is "E=MC^2". The expression "Einstein's famous formula" refers to the expression "E=MC^2".

The Pentastring is "This string has five words." The expression "The Pentastring" refers to the expression "This string has five words".

Again, since the start, and as I have pointed out to you again, but now you ignore again:

I'll list the variations again so that, hopefully, you'll understand:

An expression refers to a thing:

Big Ben is the clock tower in London.
"Big Ben" refers to the clock tower in London.
"Big Ben" refers to Big Ben.
"Big Ben" is not "the clock tower in London".
"Big Ben" does not refer to "the clock tower in London".
"Big Ben" does not refer to "Big Ben".
Big Ben is not "the clock tower in London".
Big Ben is not "Big Ben".

An expression refers to a thing that is itself an expression:

Einstein's famous formula is "E=MC^2".
"Einstein's famous formula" refers to "E=M2^2".
"Einstein's famous formula" refers to Einstein's famous formula.
"Einstein's famous formula" is not "E=MC^2".
"Einstein's famous formula" does not refer to "Einstein's famous formula".
Einstein's famous formula is not "Einstein's famous formula".

An expression refers to a thing that is itself an expression:

The Pentastring is "This string has five words".
"The Pentastring" refers to "This string has five words".
"The Pentastring" refers to the Pentastring.
"The Pentastring" is not "This string has five words".
"The Pentastring" does not refer to "The Pentastring".
The Pentastring is not "The Pentastring".

This Pentastring is this string of five words - OK — RussellA

"This Pentastring is this string of five words" has more than five words, so it's false. Anyway, it's not something I ever wrote.

These are not something I wrote. I don't know the point of mentioning them:

The Pentastring is this string of five words - not OK
This Pentastring is the string of five words - not OK
"The Pentastring is this string of five words" - not OK
"This Pentastring is the string of five words" - not OK — RussellA

The crank still hasn't corrected his claim that platonism is the view that abstractions are objects, despite the fact that he has been given corrective references:

https://thephilosophyforum.com/discussion/comment/920596

As to whether a sentence which is seemingly self referential but instead points to the world is truly self referential or not? — EricH

"The Pentastring has five words" doesn't have the word 'this' so perhaps it seems not self-referential in the way of "This string has five words". But keep in mind that the Pentastring is "This string has five words".

"This string has five words" is self-referential. But the Pentastring is "This string has five words", so the Pentastring is self-referential.

To determine whether the Pentastring is true is to determine whether "This string has five words" has five words.

To determine whether "This string has five words" is true is to determine whether "This string has five words" has five words.

The Pentastring is meaningful if and only if "This string has five words" is meaningful, because the Pentastring is "This string has five words".

A self-referential expression cannot refer to something existing in the world. — RussellA

So sayeth RussellA. Contrary to demonstration.

RussellA: Your posts are runaway trains of confusions. Maybe I can catch up later. But for now:

Suppose we define 'the Pentastring' as the [string] "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

The sentence "the Pentastring has five words" is not self-referential, because we have been explicitly told that the Pentastring exists in the world, ie we have a subject from the world, viz. the Pentastring. — RussellA

Notice that there you left out that the Pentastring is "This string has five words". So "This string has five words" exists (in the world, or whatever rubric du jour). Applying the name "The Pentastring" to "This sentence has five words" allowed you to determine that the Pentastring is meaningful. But the determination of the truth value of the Pentastring is exactly the determination of the truth value of "This string has five words".

I agree that expression ("London" is a city) is ungrammatical. — RussellA

I said it is false; I didn't say it is ungrammatical.

However, in the expression ("London" is "a city"), as the linguistic expression "London" is being spoken about, this is also an example of mention. In this case that it is "a city". Note that "a city" is just a set of words, and is not referring to anything that may or may not exist in the world. — RussellA

(1) Just to be clear, the example I gave was not:

"London" is "a city"

It was:

"London" is a city.

(2) "a city" refers to that two word phrase.

"London" is "a city"

assets that the "London" (which is one word) is "a city" (which is two words), which is false.

As Treatid correctly points out: Me: I challenge you to define "Word". You: Words. Me: Define those words. You: More Words. Me: Define those words. Etc. You can choose infinite regression or circular definitions. — RussellA

(1) I never defined 'words'.

(2) It's not clear to me what his point is, but perhaps it's (a) We define 'word' by using words. So what? (b) Definitions are either circular or infinitely regressive (or some words are taken as undefined, i.e. primitive). So what? You and I are both using words and definitions. That doesn't disqualify either of our arguments.

/

The video that was mentioned argues erroneously by conflating "refers to" with "equals".
— TonesInDeepFreeze

The Liar Paradox

That the paradoxical expression "this sentence is false" is meaningless doesn't depend on the word "equals". The argument in the video is about meaning. — RussellA

"This sentence" refers to "This sentence is false".

But "This sentence" is not "This sentence is false".

"This sentence" has two words. "This sentence is false" has four words. Clearly, "This sentence" is not "This sentence is false".

[He uses gray, white and black text to separate out phrases. But in certain instances I take it that he is indicating mention as opposed to use. I'll just use quote marks. And I'll use italics to indicate what he says orally or on screen.]

He says: "This sentence" has to reference the entire sentence.

Indeed, "This sentence" references "This sentence is false"'.

But then he conflates 'references' with 'equals':

He says: "This sentence" equals "This sentence if false".

That's false. "This sentence" references "This sentence is false". But "This sentence" does not equal "This sentence is false".

Then he says: If "This sentence" is "this sentence is false"

It's not. So we're done really. Same as with 'equals', "This sentence" references "This sentence is false". But "This sentence" is not "This sentence is false".

He continues: then what the liar's paradox is really saying is [on screen]"This sentence is false" is false[/on screen] but this is a problem, because if you substitute "This sentence" for itself, you're left with [on screen]""This sentence is false" is false" is false[/on screen]

But he didn't substitute "This sentence" for itself. He substituted "This sentence is false" for "This sentence", based on his false claim that "This sentence" doesn't just refer to "This sentence" but that "This sentence" is "This sentence is false".

Consider:

"This sentence was typed at noon"
And suppose "This sentence was typed at noon" is true.

There, "This sentence" refers to "This sentence was typed at noon" but it is not the case that "This sentence" is "This sentence was typed at noon".

"This sentence was typed at noon" was typed at noon. (True)

Substituting "This sentence was typed at noon" for "This sentence":

""This sentence was typed at noon" was typed at noon" was typed at noon. (False)

So substituting "This sentence was typed at noon" for "This sentence" is not legitimate.

Substituting "This sentence is false" for "This sentence" is not legitimate.

I will repeat the argument — RussellA

Actually, you've introduced new arguments:

In the expression "this sentence is false", which sentence is "this" referring to?

There are several possibilities.

Possibility one

It could be referring to the sentence "the cat is grey". In this case, the sentence "this sentence is false" means that the sentence "this cat is grey" is false, which is meaningful.

Possibility two
It could be referring to itself. In this case, the sentence "this sentence is false" means that the expression "this sentence" is false. But this is meaningless, and is similar to saying "this house" is false.

Possibility three
It could be referring to the sentence "this sentence is false". In this case, the sentence "this sentence is false" means that the sentence "this sentence is false" is false.

But we know that the sentence "this sentence is false" means that the sentence "this sentence is false" is false.

This means that the sentence "the sentence "this sentence is false" is false" is false

Ad infinitum. Therefore meaningless. — RussellA

One. "This" in "This sentence is false" doesn't refer to "the cat is grey".

Two. "This sentence is false" does not mean that the expression "this sentence" is false.

Three:

"This string has five words" is true.

""This string has five words" is true" is true.

ad infinitum

"Einstein's famous formula has five symbols" is true.

""Einstein's famous formula has five symbols" is true" is true.

ad infinitum.

Fermat's Last Theorem is true.

"Fermat's Last Theorem is true" is true.

ad infinitum

Even without self-reference, we can generate infinitely many sentences in the manner that you do.

I mentioned that a while ago, and you didn't address it.

And recall that your claim is not just that "This sentence is false" is meaningless but that all self-referential sentences are meaningless. So the ad infinitum argument doesn't work for you in that regard.

And what you and the video miss regarding "This sentence if false" is that the longer and longer sentences alternate ad infinitum between effectively claiming that "This sentence is false" is false and effectively claiming that "This sentence is false" is true. And that boils down to the paradox itself.

Also, he argues that "This sentence is false" is not defined. But what is the definition of any sentence? Usually we don't define a sentence. What is the definition of "The sky is dark"? Definitions of 'sky' and 'dark' sure. And we say such things as ""The sky is dark" is true if and only if the immediate atmosphere lacks light". But that's not a definition of "The sky is dark". Meanwhile, "This sentence is false" is true if and only if "This sentence is false" is false. That's the paradox, but not for failing to "define" "This sentence is false".

/

And you've still not addressed the Pentastring example. It doesn't lose effect just because you choose to ignore it.

Just to be clear: I haven't said that I deny that all self-referential sentences are meaningless. Rather, I only suspect that it is not the case that all self-referential sentences are meaningless, and I'm interested in what would be objections to an example I gave in which it seems there is a self-referential sentence that is not meaningless.

I was not a person who objected to defining falsum as 'B & ~B'.

There are some distinctions to be made however.

In the context of my post, falsum is a symbol, not a truth value. It is a sentential constant. The value 'false' is not a symbol.

But the truth value of falsum is false in all interpretations. Just as the truth value of 'B & ~B' is false in all interpretations.

They are different.

One is an inference of B from {A -> B, A}.

The other is an inference of B from {~A v B, A}

However, A -> B and ~A v B are equivalent, so the inferences are different but equivalent.

As to "directional", we'd need a definition of "directional".

What "rabbit hole" there is depends on the silly rabbit looking for real or imagined rabbit holes.

We can define a sentential constant 'f' (read as 'falsum'):

s be the first sentential constant:

f <-> (s & ~s)

That is not "gibberish".

Moreover, in some systems 'f' is primitive and '~' is defined by:

~P <-> (P -> f)

/

Again, if 'A' and 'C' are variables ranging over sentences, then

A -> (B & ~B) comes from A -> C by instantiation of 'B & ~B' for 'C'. There is nothing problematic with that.

(A→B)∧(A→C) and A→(B∧C) are the not the same formula.

They are equivalent formulas, but not the same formula.

[...] what it means for a formula to be invalid. — Leontiskos

A sentence S is invalid if and only if there is an interpretation in which the sentence is false.

(A∧C)↔C is invalid for any (non-A) substitution of C. That's just what it means for a formula to be invalid. Yet when we substitute (B∧¬B) for C it magically becomes valid. — Leontiskos

Not "magically".

If 'A' and 'C' are meta-variables ranging over sentences, then there are instances of

(A & C) <-> C that are invalid and instances of it that are valid.

But if 'A' and 'C' are atomic sentences, then, of course
(A & C) <-> C
is invalid.

There's no more "magic" to that than:
x = y
is true for some values of x and y and not for other values of x and y.

/

Bottom line: We must not conflate two kinds of usage of letters:

(1) Letters as meta-variables ranging over sentences.

(2) Letters that are themselves atomic sentences.

/

Instance after instance after instance in which Leontiskos thinks he's pointed out some problem with symbolic logic, it turns out the real problem is that he's ignorant of the basics of the subject.

RAA pertains to the boundary of the system, not the interior. — Leontiskos

How refreshing to find a topological analysis of RAA and natural deduction. Advanced stuff indeed.

But when we place a contradiction in the consequent of a conditional it is no longer conditional (e.g. (A→(B∧¬B)). So if it is a meta-principle of classical propositional logic that all conditionals are conditional, then allowing the contradiction has upended this meta-principle. — Leontiskos

What does "conditionals are conditional" mean?

I don't know what the poster has in mind, but, of course, with A -> (B & ~B), the consequent is not contingent. That doesn't "upend" any principle of classical logic.

Considering in context of much of everyday discourse:

(1) "If Smith wins the election, then there will be a recession."
(P -> Q in ordinary symbolic logic)

That is understood as asserting a relation (perhaps necessity, causality or other relevance) between the antecedent and consequent.

But it does agree with the material conditional in one sense: The second row of the truth table for P -> Q (Smith wins but there will not be a recession) is not the case. That is, if "If Smith wins the election, then there will be a recession" is true, then it is not the case that both "Smith wins the election" is true and "there will be a recession" is false.

(2) "It is not the case that if Smith wins the election then there will be a recession."
(~(P -> Q) in ordinary symbolic logic)

That is understood as denying that there is a relation (perhaps necessity, causality or some other relevance) between the antecedent and consequent.

But it seems not to suit the material conditional at all: No row of the truth table for ~(P -> Q) (equivalent with P & ~Q) pertains, since "It is not the case that there is a (necessary, causal, or other relevance) relation between Smith winning the election and there being a recession" would not be understood as being equivalent to "Smith will win the election and there won't be a recession".

(3) "If Smith wins the election, there will not be a recession."
(P -> ~Q in ordinary symbolic logic)

That is understood as asserting a relation (perhaps necessity, causality or other relevance) between the antecedent and consequent.

But it does agree with the material conditional in one sense: The first row of the truth table for P -> ~Q (Smith wins and there will be a recession) is not the case. That is, if "If Smith wins the election, then there will not be a recession" is true, then it is not the case that both "Smith wins the election" is true and "there will be a recession" is true.

Tabulating:

(1) implies not (2).
(1) implies not (3)

(2) implies not (1)
does (2) bear upon (3)?

(3) implies not (1)
does (3) bear upon (2)?

So:

(1) and (2) contradict each other.

(1) and (3) contradict each other.

(2) and (3) don't bear upon each other?

in everyday reasoning the truth of (3) requires the falsity of (1), even though P→~Q does not entail ~(P→Q), which indeed does seem to be a problem for material implication. — Srap Tasmaner

It's a problem for material implication if material implication were required to accord with much of everyday reasoning.

all, if P requires that ~Q, it can hardly require that Q. — Srap Tasmaner

That's true. But I don't see its connection with the analysis.

people do recognize the difference even in everyday reasoning, and would accept that (2) is the simple contradiction of (1) — Srap Tasmaner

Okay.

and that (3), while also denying (1) a fortiori, is a much stronger claim. — Srap Tasmaner

(3) denies (1). But what a fortiori are you referring to? In what sense is (3) stronger than (2)? Does 'stronger than' include that (3) implies (2)? Does it?