"And Richard Zach reminds us that "Peirce, Wittgenstein, and Post are commonly credited with the truth-table method of determining propositional validity.""

It seems doubtful to me that anyone who sees that a truth table tests validity would not remark that some sentences are true in all rows and some sentences are false in all rows, whether such sentences are given the names 'logical truth' or 'logical falsehood' respectively. Indeed, to say that we test for validity is to say that we test whether the sentence is true in all rows (i.e. whether the sentence is a logical truth (aka 'tautology')).

It seems amazing that it wasn't invented a lot earlier. Such a simple idea by now. It shows how much we take for granted in intellectual products.

I edited. Not 'tautology' there. I meant 'the number of letters'.

I'll maintain that the cardinal step, to using truth tables as a device for determining tautology and contradiction, was taken by Witti. — Banno

That might be the case; but hardly clear that it is.

I don't know exactly what the author meant by "anticipated"

Meanwhile, may I take it that the point is made about the number of letters? He wouldn't have to display a truth table with n number of letters for every natural number n for us to grasp that there is no finite bound on the number of letters in a truth table.

The author says, "But the discovery by Zellweger of Peirce’s manuscript of 1902 does permit us to unequivocally declare with certitude that the earliest, the first recorded, verifiable, cogent, attributable and complete truth-table device in modern logic attaches to Peirce, rather than to Wittgenstein’s 1912 jottings and Eliot’s notes on Russell’s 1914 Harvard lectures." Whether the author properly makes the case for that would deserve more scrutiny of the paper.

Number of letters: "he noted there that, for a proposition having n-many terms, there would be 2^n-many sets of truth values."

Tautologies: "For many years, commentators have recognized that Peirce anticipated the truth-table method for deciding whether a wff is a tautology.”

/

Aside from the question of invention, some salient and useful points about sentential logic:

(1) Sentential logic is, in a certain exact sense, isomorphic with the Tarski-Lindenbaum algebra.

(2) There are 16 binary truth functions. And, for all n, any n-ary truth function can be reduced to a binary truth function.

(3) All 16 binary functions can be derived from just one binary function (either Sheffer stroke or Nicod dagger).

(4) The truth table method can be formulated algorithmically. Sentential logic is decidable. And since sentential logic is decidable, it suffices to have just one inference rule: If G tautologically implies P then G proves P (e.g. 'A Mathematical Introduction To Logic' by Enderton, though he does it by saying that all tautologies are axioms).

(5) Soundness and completeness.

One source says Peirce came up with truth tables in 1902. If that is correct, then why rule out that he didn't also see that we can use them to make inferences and infer that a sentence is a logical truth or logical falsehood or neither? He had to have been a pretty smart guy, so it is unlikely that he would look at truth tables and not notice that we can use them to make inferences and check whether a sentence is logically true or logically false or neither.

He's hopeless.

Several times he was told that the rule used was RAA, and the proofs were stated in situ as being RAA, and yet he keeps demanding that it hasn't been said what rule was used.

That's just for starters.

Why do you claim that the notions of logically true and logically false were original from Wittgenstein?

Why do you reject the claim that Peirce came up with truth tables? Why do you omit Post?

Yeah.

Another source (from search of 'history of sentential logic') says, "The truth table system for Sentential Logic was invented in 1902 by the American logician Charles Peirce to display how the truth of some sentences will affect the truth of others. Truth tables were rediscovered independently by Ludwig Wittgenstein and Emil Post."

One source (I don't know whether reliable) says Peirce invented truth tables, then later Wittgenstein and Post independently. Of course, Boole invented Boolean algebra (though maybe there were precursors?).

(The fact that you think this sort of thing can be adjudicated by a truth table is proof that non-truth-functionality is in your blind spot.) — Leontiskos

The rule and truth tables agree. They agree and are independent. They are independent in the sense that are formulated separately without reference to each other. But there is the theorem that connects them with an equivalency:

Definition: An inference from G to P is valid if and only if there are no rows in the which all the members of G are true and P is false.

Theorem: An inference is allowed by the rules if and only if the inference is valid.

Moreover, the truth table method is algorithmic, thus sentential logic is decidable. So, actually the truth table method, if appropriately formulated, can itself be used as a deduction system.

I don't get to say:

P→Q
P
~Q
∴ Q {See truth table for 1, 2} — Leontiskos

First, you don't need '~Q' there.

And I didn't say "see truth table" in the proof.

There are two separate things: the deduction system, (such as natural deduction) which is syntactical, and truth evaluation (such as truth tables), which is semantical.

But we have the soundness and completeness theorem that states that a formula P is provable from a set of formulas G in the deduction system if and only if there is no row in the truth table such that all the members of G are true and P is false.

The proof would be this:

1. P -> Q {1}
2. P {2)
3. Q {1, 2}

The rule applied there is modus ponens:

From P and P -> Q, infer Q and charge it with all lines charged to P and to P -> Q.

The truth table would be:

P true, Q true ... P -> Q true

P true, Q false ... P -> Q false

P false, Q true ... P -> Q true

P false, Q false ... P -> true

There are two rows in which both P and P -> Q are true, and Q is true in both of those rows. There is no row in which both P and P -> Q are true but Q is false, so modus ponens is valid.

But if you want to include ~Q:

P→Q
P
~Q
∴ Q {See truth table for 1, 2; avert eyes from 3 at all costs. I repeat: do not allow 3 a seat at the truth table!} — Leontiskos

There's no "avert eyes", "don't allow 3 a seat"

1. P -> Q {1}
2. P {2}
3. ~Q {3}
4. Q {1, 2}

or if we are required to use (3):

1. P -> Q {1}
2. P {2}
3. ~Q {3}
4. Q {1, 2}
5. Q & ~Q {1, 2, 3}
3. Q {1, 2, 3}

Truth table:

P true, Q true ... P -> Q true ... ~Q false

P true, Q false ... P -> Q false ... ~Q true

P false, Q true ... P -> Q true ... ~Q false

P false, Q false ... P -> Q true ... ~Q true

There are no rows in which P -> Q, P, and ~Q are all true and Q is false. The argument from {P -> Q, P, ~Q} to Q is valid. ~Q is in the truth table.

If you want to bring clarity you should explain what inference you used to draw (4). — Leontiskos

I explicitly said that they are examples of RAA. And the examples when given were earlier explicitly said as examples of RAA. And they were given this time in direct response to questions about RAA.

Again, as I said, for concision we may state RAA [emphasis added] without conjunction elimination:

If Gu{P} |- Q and if Gu{P} |- ~Q, then G |- ~P
is equivalent with
If Gu{P} |- Q & ~Q , then G |- ~P

If Gu{~P} |- Q and if Gu{~P} |- ~Q, then G |- P
is equivalent with
If Gu{~P} |- Q & ~Q, then G |- P

So, in this case:

(version 1)
1. A -> (B & ~B) {1}
2. A {2}
3. B & ~B {1, 2}
4. ~A {1}

is equivalent with

(version 2)
1. A -> (B & ~B) {1}
2. A {2}
3. B & ~B {1, 2}
4. B {1, 2}
5. ~B {1, 2}
4. ~A {1} — TonesInDeepFreeze

For the millionth time:

RAA. RAA. RAA.

Can you not read the words 'RAA' when they appear over and over?

So many of your claims have already been debunked in this thread. The truth-table approach to reductio was dispatched almost ten pages ago! — Leontiskos

For the half-dozenth plus one time:

The poster is doing it again! Trying to discredit interlocutors by painting them with a brush "truth-functional", even after I had at least a few times addressed that.

For about the half-dozenth time:

I am not a "truth functionalist". I study and enjoy classical logic, and appreciate its uses. But I am interested in other logics. I do not say that classical logic is the only logic that can be studied, enjoyed and used.

But when classical logic is being discussed, especially critiqued, it is crucial to say what actually is the case with classical logic. And in bringing clarity to what classical logic actually is, one needs to explain. Providing such explanations does not make one a "truth functionalist". — TonesInDeepFreeze

1. A -> (B & ~B) {1}
2. A {2}
3. B & ~B {1, 2}
4. ~A {1}
— TonesInDeepFreeze

The poster continues to assert his baseless arguments without answering the question and providing the rule of inference he purports to use in order to arrive at conclusion (4). — Leontiskos

The proof quoted is exactly correct. I've answered every central question. It's not my fault that you are unwilling to read the answers or are incapable of understanding them though they are clear and exact.

And for the 1000000th time, the rule is RAA.

How many times do people do have to tell you?:

The rule is RAA. The rule is RAA. The rule is RAA.

Do you not get it?

The rule is RAA.

The poster seems to suffer from psychological delusions and grandiosity. When faced with simple questions he retreats into himself, opting for 3rd-person rhetorical strategies and failing to engage in inferential reasoning. — Leontiskos

(1) I am hardly grandiose as I've said several times in this forum that I have only an intermediary knowledge of this subject and that I do make (corrected) mistakes.

(2) I have head on addressed the main contentions in this thread. Your replies often blatantly skip the key information, explanations and corrections given him.

(3) My reasoning has not been shown to be incorrect. On the other hand, your arguments are so often a jumble of ignorance, confusions and illogic.

(4) I've referred to you in the third person because you deserve to be referred to that way.

The poster continues to evidence a significant difficulty in using fairly basic forum features, such as quotes. — Leontiskos

Oh please! People often get snagged copying/pasting quote brackets. I almost always correct in edit though. That you try to make any kind of deal out that shows that you're quite the fundament.

The poster continues to substitute rhetoric for argument, utterly failing to engage in rational argumentation or inferential reasoning. — Leontiskos

I've given exact information, and clear explanations, demonstrations, reasoning. There is no lacuna in rationality there. The fact that I also mention that the poster is ignorant, confused and specious doesn't vitiate the on topic content I provide.

Annotations do help to follow along in the proofs. But, depending on the formulation of the system, annotations may not be necessary. In the proofs I gave, annotations are not necessary. And that goes along with the fact that the statements of the rules do not require mentioning 'premise', 'assumption', 'supposition' or 'contradiction'. Unfortunately, the other poster understands none of this and knows jack about this subject.

I'll state this caveat again:

When we are talking about classical logic, we need to be clear as to what classical logic is and what is the case about it in terms of its formulations and definitions. So, for example, when I say 'valid' in this context, I mean 'valid' in the sense of classical logic.

If one wishes to use terminology, such as 'valid', in some other sense, then that is fine, and we can discuss in that context too. But we need to be clear in any instance which context we're in.

Thus, since this discussion has focused on classical logic and a critique of classical logic, unless otherwise stated, my remarks pertain to what is the case with classical logic given its formulations and definitions. And, again, that is not a claim or attitude that only classical logic is admissible, but rather that when we are examining classical logic, we need to at least start by knowing what it is - what its formulations are definitions actually are.

From a different angle, Tones says:

1. A -> (B & ~B) {1}
2. A {2}
3. B & ~B {1, 2}
4. ~A {1}
— TonesInDeepFreeze

If one looks at previous posts by me, one would see that I also directly, explicitly and formally addressed the matter that RAA also provides:

1. A -> (B & ~B) {1}
2. A {2}
3. B & ~B {1, 2}
4. ~(A -> (B & ~B)) {2}
— TonesInDeepFreeze

Tones thinks that ¬(1) and ¬(2) both follow from (1, 2, 3). — Leontiskos

Look at the proofs exactly. They show that ~(A -> (B & ~B)) follows from (2), and ~A follows from (1).

Of course, (1) and (2) together are inconsistent, so both ~(A -> (B & ~B)) and ~A follow from (1) with (2).

Meanwhile, it's not needed to mention (3) since it comes merely by inference from (1) and (2).

From a different angle, Tones says:

1. A -> (B & ~B) {1}
2. A {2}
3. B & ~B {1, 2}
4. ~A {1}
— TonesInDeepFreeze

If one looks at previous posts by me, one would see that I also directly, explicitly and formally addressed the matter that RAA also provides:

1. A -> (B & ~B) {1}
2. A {2}
3. B & ~B {1, 2}
4. ~(A -> (B & ~B)) {2}
— TonesInDeepFreeze — Leontiskos

It goes without saying that there is no rule of inference that forces one rather than the other. — Leontiskos

Yes, in the exact sense that there is no inference rule that dictates what set G must be.

I would simply say that both of these proofs are invalid. — Leontiskos

Wrong. By the definition of 'valid' in context of classical logic, they are valid. If you have a different definition 'valid', then of course, anything goes.

There is no rule of inference to justify (4) on either count. — Leontiskos

Wrong. Both proofs provide valid inferences. The first proof validly infers ~A from (1), and the second proof validly infers ~(A -> (B & ~B)) from (2)

This all goes to the misunderstandings of reductio ad absurdum in this thread — Leontiskos

The misunderstanding in this thread is yours, as you don't even know what RAA is, despite that I've exactly formulated it for you and explained that exact formulation.

and in particular to Tones' recent claim that there is no need to advert to a difference between an assumption/premise and a supposition. — Leontiskos

It's not merely a claim. I showed you exactly, with exact examples, several times now.

Again and again the simple questions go unanswered:

What rule of inference do you think you used to draw (4)?
— Leontiskos — Leontiskos

It has been answered again and again and again. The answer is:

RAA

A proposition can be invalid qua conclusion — Leontiskos

If you gave a definition of 'valid' in your sense, then we could evaluate your claims about it.

Meanwhile, in ordinary formal logic the common definition is the one I've stated.

Of course, we may discuss relative to different definitions. I don't at all insist that the ordinary definition is the only one that we may use. But when the discussion is about classical logic, especially a critique of classical logic, then we need to at least see what happens in classical logic with its definitions. RAA is valid in context of classical logic. But, again, if you would provide your definition, then, of course, we may find that RAA is not valid in that different context.

Also, with RAA, as with any rule, validity pertains to the relation between a set of formulas and a formula. But there is another sense 'valid' in ordinary logic too, which is that a sentence is valid if and only if it is true in all interpretations.

Your example didn't mix styles. It was fine. All I did was show how to formulate the proof without 'premise', 'assumption', 'supposition' or 'contradiction'.

The argument doesn't draw (4) from (1) and (2). The argument draws (4) from (1) as (2) is discharged.
— TonesInDeepFreeze

Heh. Why is (2) "discharged" and not (1)? — Leontiskos

https://thephilosophyforum.com/discussion/comment/922432

And "discharged" in scare quotes is silly and juvenile.

The poster is doing it again! Trying to discredit interlocutors by painting them with a brush "truth-functional", even after I had at least a few times addressed that.

For about the half-dozenth time:

I am not a "truth functionalist". I study and enjoy classical logic, and appreciate its uses. But I am interested in other logics. I do not say that classical logic is the only logic that can be studied, enjoyed and used.

But when classical logic is being discussed, especially critiqued, it is crucial to say what actually is the case with classical logic. And in bringing clarity to what classical logic actually is, one needs to explain. Providing such explanations does not make one a "truth functionalist".

It is valid.
— TonesInDeepFreeze

But it's not. — Leontiskos

The poster continues to indicate that he does not know what validity is in this context and that he is unwilling to read the posts to which responds. He skips that I just stated exactly why the argument is valid. If he won't look at a truth table as suggested, then there's little hope he'll understand anything here.

All you are saying is, "ρ→¬μ," but this does not make the proof valid. — Leontiskos

The poster seems to not know what validity is and that he is unwilling to read the post to which he responded. He skips that I stated exactly why the argument is valid. If he won't look at a truth table as suggested, then there's little hope he'll understand anything here.

What rule of inference do you think you used to draw (4)? (4) adjudicates the and-elimination. — Leontiskos

The poster seems to not know what RAA is and that he is unwilling to read the posts to which he responds. He asks what rule is used, when the rule used is RAA, exactly as the rule is formulated.

Again, as I said, for concision we may state RAA without conjunction elimination:

If Gu{P} |- Q and if Gu{P} |- ~Q, then G |- ~P
is equivalent with
If Gu{P} |- Q & ~Q , then G |- ~P

If Gu{~P} |- Q and if Gu{~P} |- ~Q, then G |- P
is equivalent with
If Gu{~P} |- Q & ~Q, then G |- P

So, in this case:

(version 1)
1. A -> (B & ~B) {1}
2. A {2}
3. B & ~B {1, 2}
4. ~A {1}

is equivalent with

(version 2)
1. A -> (B & ~B) {1}
2. A {2}
3. B & ~B {1, 2}
4. B {1, 2}
5. ~B {1, 2}
4. ~A {1}

The problem is that this proof of yours was invalid:

1. A -> (B & ~B) {1}
2. A {2}
3. B & ~B {1, 2}
4. ~A {1}
— TonesInDeepFreeze — Leontiskos

I edited to provide more explanation. My edit did not alter the substance of what was there before the edit. And it was it was not "pointed out" at that time that the proof is invalid. My edit could not have been about a claim of invalidity, since there was no such claim of invalidity today.

It is valid.

Every interpretation in which "A -> (B & ~B)" is true is an interpretation in which "~A" is true. If the poster can't see that, then he can make a truth table to see it.

is no rule of inference that allows us to draw (4) from (1) and (2). — Leontiskos

[fixed quote in edit]

Still the poster doesn't know what RAA is.

The argument doesn't draw (4) from (1) and (2). The argument draws (4) from (1), as (2) is discharged.

The argument is an instance of RAA. And RAA is sound.

To be clear, I was not faulting your formulation, but rather only showing, contrary to the other poster, that it does not require an appeal to 'premise', 'assumption', 'supposition' or even 'contradiction'.

If one looks at previous posts by me, one would see that I also directly, explicitly and formally addressed the matter that RAA also provides:

1. A -> (B & ~B) {1}
2. A {2}
3. B & ~B {1, 2}
4. ~(A -> (B & ~B)) {2}

G is {A}
P is A -> (B & ~B)
Q is B

G u {P} |- Q & ~Q, so G |- ~P.

And just a few posts ago, I discussed the no one is disallowed from choosing what to refute with RAA, as long as the refutation is valid. That upholds the very purpose of the logic system: It shows proofs from sets of formulas but does not dictate from which set of formulas (whether called 'axioms' or 'premises' or not called anything other than a 'set of formulas') we may derive. That was discussed with Lionino, who does understand the point, in the specific sense of axioms. People are free to choose their axioms, and to refute formulas inconsistent with those axioms. If G is a set of axioms and P (not a member of G) is inconsistent with P, then G refutes P. And if H is different set of axioms and Q (not a member of H but a member of G) is inconsistent with H, then H refutes Q. That we have this choice is a good thing, not a flaw in RAA nor in any other natural deduction rule, as it permits working from different axioms.

Also, A is not made true in that derivation. A is true or not depending on a given model. There is no requirement that any line be true, not even the conclusion, in a given model. For that matter ~A could even be a logically true formula itself. A sound logic system only requires that any interpretation in which all the members of G are true is an interpretation in which P is true.

1. A→(B∧¬B) assumption
2. A assumption
3. B∧¬B 1,2, conditional proof
4. ~A 2, 3 reductio — Banno

Without the word 'assumption':

1. A -> (B & ~B) {1}
2. A {2}
3. B & ~B {1, 2}
4. ~A {1}

G is {A -> (B & ~B)}
P is A
Q is B

G u {P} |- Q & ~Q, so G |- ~P.

Understanding RAA doesn't require reference to 'premise', 'assumption', 'suppostion' or 'contradiction'.

Here is RAA in exact formulation:

If P, along with possibly other lines, shows a formula Q and a formula ~Q, then infer ~P and charge it with all the lines used to show Q and to show ~Q except the line for P.

If ~P, along with possibly other lines, shows a formula Q and a formula ~Q, then infer P and charge it with all lines used to show Q and used to show ~Q, except the line for ~P. [not intuitionistic]

Those rules are equivalent with (formulated equivalently this time with conjunction):

If Gu{P} |- Q & ~Q , then G |- ~P

If Gu{~P} |- Q & ~Q, then G |- P [not intuitionistic]

There is no mention of 'premise', 'assumption' or 'supposition' nor, for that matter, 'contradiction'. — TonesInDeepFreeze

What is "secure" supposed to mean? At least RAA is as secure as modus tollens in the sense that they are sound.

For all sets of formulas G and formulas P, if G |- P then G |= P. That is:

For all sets of formulas G and formulas P, if G |- P then for all models M, if M is a model of G then M is a model of P. That is:

For all sets of formulas G and formulas P, if P is derivable from G, then any interpretation in which all the members of G is true is an interpretation in which P is true. That is:

Classical natural deduction (which includes RAA as a primitive rule, modus tollens as a derived rule and contrapostion as a derived theorem schema) does not permit inference of a false conclusion from true premises.

No matter whether we start with RAA as primitive and derive modus tollens and contraposition, or start with modus tollens as primitive and derive RAA, or start with contraposition and derive RAA and modus tollens, we arrive at the exact same set of allowable inferences.

The Aristotelians here don't cotton to it.

You were right when you said:

"a cat" and "a carnivorous mammal long domesticated as a pet and for catching rats and mice" are interchangeable.

You were wrong when you said:

"a cat" refers to "a carnivorous mammal long domesticated as a pet and for catching rats and mice".


This is right:

"a cat" refers to a carnivorous mammal long domesticated as a pet and for catching rats and mice.

"a cat" does not refer to "a carnivorous mammal long domesticated as a pet and for catching rats and mice".

That is, "a cat" refers to the animal that is a carnivorous mammal long domesticated as a pet and for catching rats and mice", but "a cat" does not refer to the phrase "a carnivorous mammal long domesticated as a pet and for catching rats and mice".


"Big Ben" refers to the famous clock tower in London.

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

That is, "Big Ben" refers to the object that is the famous clock tower in London, but "Big Ben" does not refer to the phrase "the famous clock tower in London".


The use-mention distinction can get into some technicalities, but for ordinary discussions in logic, it's pretty simple:

Use:
Big Ben is the famous clock tower in London. (True)

Mention:
"Big Ben" has six letters. (True)

Mention:
"Big Ben" refers to the famous clock tower in London. (True)


Mistake:
"Big Ben" is the famous clock tower in London. (False)

Mistake:
Big Ben has six letters. (False)

Mistake:
Big Ben refers to the famous clock tower in London. (False)

Mistake:
Big Ben refers to "the famous clock tower in London". (False)


In this discussion there is a bit of a wrinkle that might (?) be throwing you off:

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

In that example, the thing referred to is itself a phrase, so unlike with the "a cat" or "Big Ben" examples, there are quote marks on the right side not just the left side.

Another example of that:

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

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

So, the right side may itself be a phrase.


"This string" has two words. (True)

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

"This string" is "This string has five words". (False)

"This string" and "This string has five words" are interchangeable. (False)


"Einstein's famous formula" has three words. (True)

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

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

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

"Einstein's famous formula" and "E=MC^2" are interchangeable. (False)

By the way, your comments about dictionaries led me to discover that, unlike years ago, OED is free online. It definitely is a much richer resource than Merriam. Thanks for that. I'll be using them side by side from now on (and with my old print edition of unabridged Merriam).

It's a hard job. Some companies don't mind too much agents spending whatever time is needed on a call. Other companies hector agents to get off calls as absolutely quickly as possible, thus a lot of pressure.

But the quality of tech support and customer support in general has plummeted over the last several years. Even when agents aren't pressured to wrap up the calls quickly, some of them are atrocious - not listening to you, mindlessly giving you steps that couldn't be expected to address your particular problem, and giving misinformation and lying. And resisting to the last breath transferring you to a tier 2 tech. And of course, long wait times, calls cut off, and failed promises to continue pursuit of a resolution with a followup call. And then, after you've been through weeks of runaround in multiple calls, chats and emails, they have the gall to send you an email survey asking how they did, and keep pestering you with even more emails. On the other hand, there are some companies that provide sterling support; agents that will stay on the phone and give you all kinds of extra info and tips.

Neither of those, but what you say is interesting. I guess it is a thing.

This stands and is central:

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

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

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

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

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

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

This stands, at least so far:

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

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

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

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

Put this way:

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

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

The ball is in your court to support that claim
— TonesInDeepFreeze

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

Like I said, the ball the ball is in your court to demonstrate that claim.
— RussellA

But if "this string" refers to itself, then it is impossible to know what it means, and if no-one knows what it means, then it becomes part of a meaningless set of words. — RussellA

Your argument for that, as with the video, is shot down at the git-go by mentioning that the argument relies on the false equation: "This string" equals "This string has five words", mutatis mutandis in the video.

The glaring sophistry in that video is the claim that "this sentence" equals "this sentence is false."
— TonesInDeepFreeze

"A cat" may be defined as "a carnivorous mammal long domesticated as a pet and for catching rats and mice". — RussellA

The video you suggested said that:

"this sentence" equals "this sentence is false".

That is plainly a falsehood. It would help if you not skip that point.

And the video's argument is based on that falsehood.


"Big Ben" denotes the famous clock tower in London.

"Big Ben" is not the famous clock tower in London.

"The famous clock tower in London" denotes the famous clock tower in London.

"Big Ben" and "the famous clock tower in London" are (extensionally) interchangeable.

"Big Ben" and the famous clock tower in London are not interchangeable, since the former is a name and the latter is a clock tower.


"This sentence" denotes "This sentence has five words".

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

""This sentence has five words"" denotes "This sentence has five words".

/

"cat" is a word.

A cat is an animal

"cat" denotes the animal.

"cat" is not "the animal".

"A cat" refers to "a carnivorous mammal long domesticated as a pet and for catching rats and mice". — RussellA

No, "A cat" denotes a carnivorous mammal long domesticated as a pet and for catching rats and mice. "A cat" does not denote "a carnivorous mammal long domesticated as a pet and for catching rats and mice."

"Big Ben" denotes the famous clock tower in London.

"Big Ben" does not denote "the famous clock tower in London".

Your arguments rely upon the fact that you are not careful to distinguish between, for example,

"a carnivorous mammal long domesticated as a pet and for catching rats and mice" and a carnivorous mammal long domesticated as a pet and for catching rats and mice.

the two expressions are not linguistically equal (one is two words long and the other is fourteen words long) [...] — RussellA

Yes, that is the point I made earlier.

(1) "a cat"
and
"a carnivorous mammal long domesticated as a pet and for catching rats and mice"
may be substituted for one another (in an extensional context).

(2) "a cat"
and
a carnivorous mammal long domesticated as a pet and for catching rats and mice
may not be substituted for one another.

(3) a cat
and
"a carnivorous mammal long domesticated as a pet and for catching rats and mice"
may not be substituted for one another.

The mistakes you made earlier are violations of the nature of (2) and (3)

[...] they are semantically equal, meaning that one expression can be replaced by the other. — RussellA

"a cat" denotes a carnivorous mammal long domesticated as a pet and for catching rats and mice.

"a carnivorous mammal long domesticated as a pet and for catching rats and mice" denotes a carnivorous mammal long domesticated as a pet and for catching rats and mice.

So, yes, the two are interchangeable (in extensional contexts; I'll leave that qualification tacit from now on).

I suggest sticking with 'This string has five words' for now.


Just to be clear:

"This string" denotes "This string has five words".

Of course, that is contextual.

In "This string has ten words", "this string" does not denote "This string has five words".


"This string has five words" is true.

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

ad infinitum

There's no problem with that.


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

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

ad infinitum.

There's no problem with that, and if there is, then it's a problem with nearly any sentence, not just self-referential sentences.


"This string" denotes "This string has five words".

"This string" is "This string".

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

"This string" does not denote "This string".


"Big Ben" denotes Big Ben.

"Big Ben" is "Big Ben".

"Big Ben" is not Big Ben.

"Big Ben" does not denote "Big Ben".

If in the expression "this sentence is false", "this sentence" refers to "this sentence is false", its self-referential nature means that no meaning can be determined within a finite time, meaning that it becomes meaningless. — RussellA

That assertion comes from the fact that you that you improperly use quote marks.