I want to say that (B∧¬B) and (B∧C) are not meta-logically equivalent, and because of this the truth tables are misleading. — Leontiskos

What are 'B' and 'C'?

If 'B' and 'C' are atomic sentences, then
(B∧¬B)
and
(B∧C)
are not equivalent.

Buy if 'B' and 'C' meta-variables ranging over sentences, then we note that it is not the case that for all sentences B and C, we have that (B∧¬B) and (B∧C) are equivalent. But that doesn't entail that there is no sentence C such that (B∧¬B) and (B∧C) are equivalent. For example, we may instantiate both 'B' and 'C' to a sentence P (where 'P' is not a variable but is an atomic sentence), and (P & ~P) and (P & ~P) are not equivalent and indeed the same. Or, we could instantiate 'B' to P and 'C' to Q & ~Q (where 'Q' is not not a variable but rather an atomic sentence), and (P & ~P) is equivalent with (P & (Q & ~Q)).

Truth tables are not "misleading" in this regard.

Assertions, assertions, and more assertions. — Leontiskos

What rule is used? What rule is used? What rule is used?
Questions, questions, questions.

RAA. RAA. RAA.
Answers, answers, answers.

Prove RAA from MT. Prove RAA from MT. Prove RAA from MT.
Challenges, challenges, challenges.

Proof of RAA from MT and LNC. Proof of RAA from MT and LNC. Proof of RAA from MT and LNC.
Challenge met, challenge met, challenge met.

serious philosophical engagement — Leontiskos

Leontiskos is to serious philosophical engagement as salmonella is to a seriously good dinner.

as Tones' claim demonstrates. — Leontiskos

Leontiskos uses my name for a link to his own post. That deserves a link to my reply to his post:

https://thephilosophyforum.com/discussion/comment/922505

A nifty little editorial there, some about the subject, but a lot, pejoratively, about me.

You can't define "mentions". — Treatid

The sense in which 'mention' is used in context of the use-mention distinction:

To mention a word is to speak about the word.

On the other hand, a word may be used to refer to something else.

For example:

"London" has six letters.
The word is spoken about.

London is populous.
The word is used to refer to the city not to the word.


It should be easy to see:

London is a city. (true)

London is populous. (true)

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

London has exactly six letters. (false - London is not a word and does not have a number of letters)

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

"London" is populous. (false - "London" is not a city and does not have a population)

"London" is a word. (true)

"London" has exactly six letters. (true)

You are trying to assert a set of invariant linguistic rules. — Treatid

I'm just observing that if we don't keep straight the difference between use and mention, then we get whacky results, and that some of the other poster's argument don't hold up on account of conflating a phrase itself with the thing the phrase refers to.

you are trying to browbeat RussellA into believing that your interpretation of words is the one and only true interpretation. — Treatid

I was interested in his argument and interested in what may be his reply to criticisms of it. I took a good amount of time to study his argument. Then I saw errors in it; and explained why they are errors. And I presented a counterargument that seems correct to me, though I was interested in what criticisms there might be to the counterargument. And I offered him information and explanation of a common and well known notion. I haven't tried to "browbeat" anyone into believing anything.

every single person who reads a sentence interprets it as they will. — Treatid

True. And interesting that the other poster's view is that "This sentence has five words" can't be meaningfully understood. But it does seem understandable to me. Yet, I don't accuse the other person of trying to "browbeat" me to believe that is not understandable.

The idea that your personal interpretation of, say, the Liar's paradox is correct and everyone else is wrong is a level of hubris even I don't aspire to. — Treatid

Pretty much all I've said about the liar paradox is:

(1) To state what it is.

(2) To correct the the misunderstanding that results if we don't recognize that it stipulates "all and only" and not just "only".

(3) To point out that reading various writers on the subject informs us as to why it is a subject of interest in mathematics, logic and philosophy.

(4) The video that was mentioned argues erroneously by conflating "refers to" with "equals".

I get that you are trying to establish a common ground for productive dialogue. But you can't do it. — Treatid

Of course, there is never a perfect commonality among people. That doesn't entail that we cannot reach enough commonality in certain situations. We don't have to have perfect commonality just to have a productive exchange. Moreover, at least the other poster does know more about use-mention now, so that he can use it going forward, revise it for himself, or reject and ignore it if he likes.

only person who interprets things identically to you - is an identical copy of you. — Treatid

The only person who interprets identically to me is me. But identical understanding is not required. Meanwhile, the only person who interprets things identically to you is you. But I'm not disqualifying anything you say on that consideration.

for someone apparently sure of their position - you are peddling a whole lot of BS. Actually, [...] — Treatid

Actually, I explicitly stated that I am not sure of the argument I've given in favor of the view "This sentence has five words" is meaningful and true. I said it seems to me that my argument is correct but that I am interested in seeing how it might not be correct. On certain other matters though, yes, I'm pretty sure. I'm pretty sure that London is a city and not a word, and that "London" is a word and not a city. And that in the video "This sentence" refers to "This sentence is false", and that "This sentence" does not equal "This sentence is false". And that the other poster's argument about "This string has five words" suffers from a crucial use-mention error. And you've not shown that anything I've said is "BS".

I struggle to understand how anyone with the slightest awareness of linguistics can turn around and proclaim a given interpretation to be definitive. And yet here you are trying to tell RusserlA that his interpretation is wrong.

For shame. — Treatid

My remarks were about his claim that self-referring sentences are not meaningful. He gave an argument for his claim. I gave a plausibility counter-argument and showed errors in his own argument. And I carefully studied each line of his posts. I don't think that's shameful.

do you genuinely believe that you can define... anything? — Treatid

I use many words in their ordinary sense, and some words in certain contexts in more academic senses. In some contexts, rigorous definitions may be provided, while in other contexts I would just point to an ordinary dictionary, and while recognizing that dictionaries are ultimately circular in that certain words are defined with other words whose definition circles back to the word being defined. And that certain words are just so basic that they are ultimately understood contextually or ostensively. And sometimes I use words idiosyncratically for effect, hoping that they'll be understood in context and by some tolerance of the reader. And sometimes I mistakenly misuse words.

Meanwhile, I guess you have your own notions about the way you use words, such as with the words you use in your fusillade against me.

You may do an Internet search on 'use-mention' for guidance. Meanwhile, the notion of use-mention is prevalent in the literature of logic, and is explained with examples in a lot of texts including these:

'Introduction To Mathematical Logic' - Alonzo Church, pg 61

By the way, the Introduction of this book is the best overview of starting considerations for logic that I have found.

/

'Elementary Logic - 2nd ed' - Benson Mates, pg 21

By the way, this is one of the best intro books logic that I have found - rigorous, clear and concise.

/

'Introduction To Logic' - Patrick Suppes, pq 121

By the way, this book has the best treatment on formal definitions that I have found.

/

'Methods Of Logic - 4th ed' - W.V. Quine, pg 50

/

'An Introduction To Formal Logic' - Peter Smith, pg 83

/

'The Logic Of Book - 4th ed' - Merrie Bergman, James Moor, Jack Nelson, pg 67

I guess you mean that the examples are instances where the word 'mention' applies to describe them.

Anyway, your argument that self-referring strings are not meaningful sentences failed.

And the argument in the video you endorsed about the liar paradox fails.

By the way, I looked at Mates's 'Elementary Logic' (a great book) where his system uses non-intuitionistic MT (If G |- ~A -> ~B and G |- B, then G |- A), which is actually the more common primitive form for classical systems anyway.

So, RAA must be derivable from non-intuitionistic MT without even invoking LNC. Though, of course, non-intuitonistic MT does entail LEM, while my proof does not.

Mates lists LNC as a theorem in his system with non-intuitionistic MT. So, with that theorem, we would continue with the proof I gave. A full proof would have to show the details in deriving LNC.

the more purely formal a system is, the less this discontinuity of reductio ad absurdum is able to be recognized. — Leontiskos

The more formal the system, the more precisely we can determine exactly what is and what is not permitted by it.

The formalization doesn't overstep the ordinary, informal argument form of reductio ad absurdum. Rather, the formalization requires that the form be adhered to strictly.

Meantime, look up Euclid and Pythagoras to see that the argument form does go back to the ancients. Moreover, take a look at the quotes from the articles about the argument form.

All you [another poster] need to do is provide such a derivation. — Leontiskos

I did.

And his is okay too; all it needs is to be generalized as I did with G.

I referred to the last clause in this quote as it is still posted:

Now, the conclusion that I arrived at is that "A does not imply a contradiction" is not an accurate statement about ¬(A→(B and ¬B)), it would be a true statement about (A→¬(B and ¬B)) instead. — Lionino

"A does not imply a contradiction" is not a true statement about "(A→¬(B and ¬B))".

For example, proving the deduction theorem (thus deriving the rule of '-> introduction') for a Hilbert style system. That's one of the key topics in an early chapter of just about any intro text in mathematical logic that uses a Hilbert style system. Cf., e.g., Enderton's 'A Mathematical Introduction To Logic'.

Laws of deduction are not usually derived from one another. — Banno

It's done frequently.

I'll invite you to derive RAA from MT as a way to engage with what I've already written. — Leontiskos

RAA derived from MT and LNC. Done.

It becomes primarily a way to elaborate and extend a system. — Leontiskos

No, in a natural deduction system it is not a mere "elaboration" nor "extension". It is crucial for proving negations; it is key to having a system that deals with negation. Without such rules, the system would not be complete, in the sense that there would be validities not provable.

Laymen and logicians alike are on occasion apt to say, "An absurdity? A contradiction? So what? 'I contain multitudes'." — Leontiskos

What is supposed to be the point of that? Classical logic doesn't excuse contradiction.

creative attempts to justify reductio in classical propositional logic. — Leontiskos

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

It's merely a matter of showing that if G along with P proves a contradiction, then there are no interpretations in which the all the members of G are true and P is true.

Proving that is actually rather routine and dull, hardly creative.

and part of the difficulty here is that an absurdity and a contradiction are not synonyms in the historical senses of reductio ad absurdum. Metaphysical and logical absurdities are both utilized historically under that name.) — Leontiskos

Read the articles.

Look up Pythagoras to start.

whether RAA can be derived from MT, and this is not at all apparent. — Leontiskos

It is apparent that RAA can be derived from MT and LNC. (Among non-dialetheists, LNC should be uncontroversial):


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

MT:
If G |- A -> B and G |- ~B, then G |- ~A

An instance of MT:
If G |- P -> (Q & ~Q) and G |- ~(Q & ~Q), then G |- ~P

LNC:
{} |- ~(Q & ~Q)


To derive RAA from MT and LNC:

Show:
If G |- P -> (Q & ~Q) and G |- ~(Q & ~Q), then G |- ~P
and
{} |- ~(Q & ~Q)
implies
If Gu{P} |- Q & ~Q, then G |- ~P

Suppose:
If G |- P -> (Q & ~Q) and G |- ~(Q & ~Q), then G |- ~P
and
{} |- ~(Q & ~Q)

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

Suppose: Gu{P} |- Q & ~Q
So G |- P -> (Q & ~Q)
{} |- ~(Q & ~Q), so, a foritori G |- ~(Q & ~Q)
So G |- ~P


The other direction drives MT from RAA:

Show:
If Gu{P} |- Q & ~Q, then G |- ~P
implies
If G |- P -> Q and G |- ~Q, then G |- ~P

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

Show: If G |- P -> Q and G |- ~Q, then G |- ~P

Suppose: G |- P -> Q and G |- ~Q

{P} |- P
So Gu{P} |- Q & ~Q
So G |- ~P

It would be hard to dispatch Tones' army of strawmen. — Leontiskos

It's more than hard. It's impossible. Because it's impossible to dispatch what doesn't exist

I almost guarantee that Aristotle will see a reductio as a metabasis eis allo genos — Leontiskos

Your guarantees or "almost" guarantees are as worthless as a day pass to a rickety ride park closed for serial safety violations fifty years ago.

RAA directly leverages the LEM in an entirely unique way. — Leontiskos

That was addressed long ago in this thread.

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

makes no use of LEM.

However

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

does require LEM.

You don't know Jack Kennedy about this subject.

Pythagoras's proof that the diagonal of a square is not commensurate with a side is a quintessentially famous example of reductio. And, if I recall correctly, so is Euclid's proof of the irrationality of the square root of 2.

Internet Encyclopedia Of Philosophy (if it is correct):

"As indicated above, this sort of proof [Pythagoreas's proof] of a thesis by reductio argumentation that derives a contradiction from its negation is characterized as an indirect proof in mathematics. (On the historical background see T. L. Heath, A History of Greek Mathematics [Oxford, Clarendon Press, 1921].)

The use of such reductio argumentation was common in Greek mathematics and was also used by philosophers in antiquity and beyond. Aristotle employed it in the Prior Analytics to demonstrate the so-called imperfect syllogisms when it had already been used in dialectical contexts by Plato (see Republic I, 338C-343A; Parmenides 128d). "

/

Stanford Encyclopedia Of Philosophy (if it is correct):

"Both Zeno of Elea (born c. 490 BCE) and Socrates (470–399) were famous for the ways in which they refuted an opponent’s view. Their methods display similarities with reductio ad absurdum, but neither of them seems to have theorized about their logical procedures. Zeno produced arguments (logoi) that manifest variations of the pattern ‘this (i.e. the opponent’s view) only if that. But that is impossible. So this is impossible’."

/

Wikipedia (though I don't always trust Wikipedia):

"This argument form traces back to Ancient Greek philosophy and has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate."

and

"Reductio ad absurdum was used throughout Greek philosophy. The earliest example of a reductio argument can be found in a satirical poem attributed to Xenophanes of Colophon (c. 570 – c. 475 BCE).[8] Criticizing Homer's attribution of human faults to the gods, Xenophanes states that humans also believe that the gods' bodies have human form. But if horses and oxen could draw, they would draw the gods with horse and ox bodies.[9] The gods cannot have both forms, so this is a contradiction. Therefore, the attribution of other human characteristics to the gods, such as human faults, is also false.

Greek mathematicians proved fundamental propositions using reductio ad absurdum. Euclid of Alexandria (mid-4th – mid-3rd centuries BCE) and Archimedes of Syracuse (c. 287 – c. 212 BCE) are two very early examples.[10]

The earlier dialogues of Plato (424–348 BCE), relating the discourses of Socrates, raised the use of reductio arguments to a formal dialectical method (elenchus), also called the Socratic method.[11] Typically, Socrates' opponent would make what would seem to be an innocuous assertion. In response, Socrates, via a step-by-step train of reasoning, bringing in other background assumptions, would make the person admit that the assertion resulted in an absurd or contradictory conclusion, forcing him to abandon his assertion and adopt a position of aporia.[6]

The technique was also a focus of the work of Aristotle (384–322 BCE), particularly in his Prior Analytics where he referred to it as demonstration to the impossible (Greek: ἡ εἰς τὸ ἀδύνατον ἀπόδειξις, lit. "demonstration to the impossible", 62b).[4]"

MP and MT are commensurable with ancient (and colloquial) logic in a way that RAA is not. — Leontiskos

RAA is derivable from MT, and MT is derivable from RAA. [Edit: To be precise, RAA is derivable from MT and LNC, and MT is derivable from RAA.]

Moreover, the next post:

(1) The worst thing about you is that you lie about me. And that you tried to wiggle out of that with a specious point about lying, to which I've responded to twice but which you ignore.

(2) The second worse thing about you is that you reply without understanding and, often, not even recognizing that your arguments based in ignorance and confusion had already been dispatched. How ludicrous that you kept asking "by what rule?" when over and over and over you had been told "by RAA".

You did it again in your latest post. You continue to complain that I posted many posts in a row when I've already answered: I was not posting in the the thread during a time when you were posting prolifically (including posts quoting me), so I caught up later, and at a mere fraction of the number of posts as yours. The fact that others weren't posting when I was catching up is not a fault of mine. There's no rule, and there should be no rule, that a poster can't later catch up in a thread. How hypocritical and juvenile of you to demand that you can go post a ton of stuff, including quoting another poster, but that the other poster bears fault for catching up later.

(3) The third worst thing about you is that you don't know Jackson Browne about this subject yet you roar your ignorant, over-opinionated confusions about it.

I mostly ignore users who run into a thread shitting on everyone in sight who is not a mod, and that's what I largely did when Tones entered. — Leontiskos

Check the actual record of the posts.

Actually, there've been other first insulters in this thread.

Go back and see. — Leontiskos

Indeed! Look at the actual posts to see who "comported themself" how.

Referring to another poster, Leontiskos wrote:

That rare combination of hubris and senility. Gotta love it. — Leontiskos

Don't gotta love the crude, disgusting ageism there, no matter what the other poster's age is.

I'm replying to a bot programmed to not understand anything about this subject, not even to understand the inference rules of sentential logic nor how to read a truth table, and to skip recognition of replies already given.

Again:

Every row in which "A -> (B & ~B)" is true is a row in which "~A" is true.

Every row in which "A" is true is a row in which "~(A -> (B & ~B))" is true.

Every row in which both "P -> Q" and "P" are true is a row in which "Q" is true.

And the third example is not vitiated by the fact that (1), (2), (3) together are inconsistent:

(a) The inference is from (1) and (2) to (4). The fact that (3) is inconsistent with (1) and (2) does not entail that (1) and (2) do not imply (4).

(b) I previous showed a truth table that does include a column for ~Q. And again, in that truth table:

Every row in which both "P -> Q" and "P" are true is a row in which "Q" is true.

Moreover, indeed, there are no rows in which "P -> Q", "P" and "~Q" are all true, so, vacuously, every row in which "P -> Q", "P" and "~Q" are all true is a row in which "Q" is true.

Moreover, "Every row in which all the Xs are true is a row in which Y is true" is equivalent with "The is no row in which all the Xs are true and Y is false". In this case, there is no row in which "P -> Q", "P" and "~Q" are true and "Q" is false.

Leontiskos, if you're going to critique basic beginning logic, then know what you're critiquing!

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

The reason these are not RAA is because there is no supposition taking place — Leontiskos

The proof is RAA since it fulfills the definition of RAA. I've shown that several times already.

/

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

My reply was and is:

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

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

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).

/

A truth table does not adjudicate between (1) and (2). It does not perform the and-elimination of the reductio for us. What Tones is doing is just arbitrarily ignoring inputs to the truth table: — Leontiskos

My reply to that was and is:

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. — TonesInDeepFreeze

And, regarding a different example (modus ponens) given by the poster, but applicable to the current examples:

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.[/quote]

/

when I ask Tones why he drew one conclusion rather than the other, he tells me to look at the truth table — Leontiskos

Actually, I've given the poster a full explanation. I'll give it again, since he is unwilling to read what he even responds to:

One conclusion is from one set and the other conclusion from another set.

~A is syntactically inferred from the set {A -> (B & ~B)} per RAA.

~(A -> (B & ~B)) is syntactically inferred from the set {A} pre RAA.

Those are two different inferences, and both of them are valid.

To see that they are valid, we can look at the truth tables.

Every row in the truth table in which the member of {A -> (B & ~B)} is true is a row in which "~A" is true.

Every row in the truth table in which the member of {A} is true is a row in which "~(A -> (B & ~B))" is true.

/

Wrong. By the definition of 'valid' in context of classical logic, they are valid.
— TonesInDeepFreeze

According to what definition are both proofs valid? — Leontiskos

By the definition I posted in this thread probably at least three times. Again:

An inference from a set of formulas G to a formula P is valid
if and only if
every interpretation in which all the members of G are true is an interpretation in which P is true.

For sentential logic, that is equivalent with:

An inference from a set of formulas G to a formula P is valid
if and only if
Every row in the truth table in which all the formulas in G are true is row in which P is true.

Hey, Leontiskos, read!

all that can be agreed, and yet we still hold that Anellis has not carried his case. — Banno

As far as I can tell, Anellis outlines the case but does not make it fully. I have not claimed that Peirce developed the notion of truth tables sufficiently to deserve being called 'the inventor of the truth table regarding its use for testing validity of formulas with any finite number of letters'. Only that it is arguably the case that he did per Anellis's conclusion that is based on grounds adduced by Anellis but that seem to me to require more evidence from Peirce's writings. So I wouldn't so strongly conclude that Peirce did not develop the notion of truth tables sufficiently to deserve being called ''the inventor of the truth table regarding its use for testing validity of formulas with any finite number of letters'.

TonesInDeepFreeze
But a truth table determines validity.
— TonesInDeepFreeze
Does Anellis show explicitly that Peirce used a truth table in this way? — Banno

He does say:

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

and

"Lane [...] [tells us] that, “For many years, commentators have
recognized that Peirce anticipated the truth-table method for deciding whether
a wff is a tautology.”"

Again, I'd need to see the writings.

Now in the absence of further evidence, it is reasonable to supose that Wittgenstein was the first to do this. — Banno

And reasonable to be more cautious pending finding out more about Peirce's writings.

There's more to it also:

There are these forms:

(1) [name] refers to [object]

"Big Ben" refers to the bell.

(2) [object] is [object]

Big Ben is the bell.

(3) [name] refers to [object that is itself an expression]

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

(4) [object] is [object that is itself an expression]

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

/

The video and you conflate 'refer' with 'is' (or 'equals').

(5) "This sentence" refers to "This sentence is false".

but it is not the case that

(6) "This sentence" is (equals) "This sentence is false".

I would say just
mention
not
"mention"

As the expression "the bell inside the clock tower" refers to the expression "Big Ben", not to Big Ben as a thing in the world, — RussellA

Wrong. Very wrong.

"the bell inside the clock tower" refers to Big Ben, not to "Big Ben".

Once you're clear about that, we can go back to this:

The video you suggested said that

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

That is plainly a falsehood.

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

I was under the impression that you were going to take a moment to understand use-mention, but still you haven't, as you make the same mistake yet again, despite the fact that I gave you multiple examples from which one could easily understand the point, which you've skipped. You are incorrigibly irrational.

regards use, Big Ben is the bell inside the clock tower. — RussellA

Right.

regards mention, "Big Ben" is "the bell inside the clock tower" — RussellA

Wrong. Very wrong.

"Big Ben" has two words.

"the bell inside the clock tower" has six words.

So "Big Ben" is not "the bell inside the clock tower".


One more time:

Big Ben is the bell inside the clock tower.

"Big Ben" refers to the bell inside the clock tower.

"Big Ben" refers to Big Ben.

"the bell inside the clock tower" refers to the bell inside the clock tower.

"the bell inside the clock tower" refers to Big Ben.

"Big Ben" is not the bell inside the clock tower.

"Big Ben" is not "the bell inside the clock tower."

Big Ben is not "Big Ben".

Big Ben is not "the bell inside the clock tower".


One more time:

Big Ben is a physical object.

"Big Ben" is an expression.

"the bell inside the clock tower" is an expression.

"Big Ben" and "the bell inside the clock tower" refer to the same physical object.

"Big Ben" and "the bell inside the clock tower" are not the same expression.


Do you understand now?

it is the expression "this sentence" that is interchangeable with the sentence "this sentence is false" — RussellA

"this sentence" is not "this sentence if false".

(in context) "this sentence" refers to "this sentence is false".


"this sentence" and "this sentence is false" are not interchangeable":


"this sentence" has exactly two words. (true)

"this sentence is false" has exactly two words. (false)

"this sentence" has exactly four words. (false)

"this sentence is false" has exactly four words. (true)


So, you see that "this sentence" and "this sentence is false" are not interchangeable.


"Mark Twain" and "Samuel Clemens" refer to the same person.

"Mark Twain" and "Samuel Clemens" are not interchangeable*:


"Mark Twain" has exactly nine letters (true)

"Samuel Clemens" has exactly nine letters (false)

"Mark Twain" has exactly thirteen letters (false)

"Samuel Clemens" has exactly thirteen letters (true)


So, you see that, in a context such as this, "Mark Twain" and "Samuel Clemens" are not interchageable.*

*In an extensional context, what is inside the quote marks of "Mark Twain" and "Samuel Clemens" are interchangeable, but the whole units including the quote marks are not interchangeable. For example:

Mark Twain was friends with Nikola Tesla
is interchangeable with
Samuel Clemens was friends with Nikola Tesla

"Mark Twain" has exactly nine letters
is not interchangeable with
"Samuel Clemens" has exactly nine letters

Yes, Mark Twain is Samuel Clemens. So, (in an extensional context) every statement true of Mark Twain is true of Samuel Clemens and vice versa. But it is not the case that every statement true of "Mark Twain" is true of"Samuel Clemens" and it is not the case that every statement true of "Samuel Clemens" is true of "Mark Twain".

the one hand there is i) "this sentence" and on the other hand there is ii) the expression "this sentence". These are different things. — RussellA

Wrong.

"this sentence" is "this sentence".

the expression "this sentence" is "this sentence".

"this sentence" is the expression "this sentence".

the expression "this sentence" is the expression "this sentence".

should negate your doubts regarding interchangeability. — RussellA

It affirms my knowledge that you haven't bothered to understand use-mention.

/

In the sentence "this sentence is false", what does "this sentence" refer to?

It could refer to the sentence "the cat is grey in colour". — RussellA

But it doesn't.

Or it could refer to the sentence "this sentence is false". — RussellA

That's better.

In which case the sentence "this sentence is false" means that the sentence "this sentence is false" is false. — RussellA

I think so.

Yes, the sentence "this sentence is false" means: the sentence "this sentence is false" is false.

we know that the sentence "this sentence is false" means that the sentence "this sentence is false" is false. — RussellA

That's merely a tautology from the previous. You're just saying again what you said:

the sentence "this sentence is false" means: the sentence "this sentence is false" is false.

This — RussellA

What does 'This' refer to? I guess it refers to: the sentence "this sentence is false" is false.

means that the sentence ""the sentence "this sentence is false" is false" is false — RussellA

""the sentence "this sentence is false" is false"

has an odd number of quote marks.

Maybe you mean:

"the sentence "this sentence is false" is false" means: the sentence "the sentence "this sentence is false" is false" is false.

So:

"this sentence is false"
means
the sentence "this sentence is false" is false.

"the sentence "this sentence is false" is false"
means
the sentence "the sentence "this sentence is false" is false" is false.

"the sentence "the sentence "this sentence is false" is false" is false"
means:
the sentence "the sentence "the sentence "this sentence is false" is false" is false" is false.

ad infinitum


"the sentence "Paris is a city" is true"
means
the sentence "the sentence "Paris is a city" is true" is true.

"the sentence "the sentence "Paris is a city" is true" is true"
means
the sentence "the sentence "the sentence "Paris is a city" is true" is true" is true.

"the sentence "the sentence "the sentence "Paris is a city" is true" is true" is true"
means
the sentence "the sentence "the sentence "the sentence "Paris is a city" is true" is true" is true" is true.

ad infinitum

/

You've skipped the point we were discussing. You claimed that self-referential sentences are meaningless.

I mentioned "This sentence has five words". Then, to accommodate any objection that saying "sentence" there is question begging, I provided, "This string has five words". Perhaps "This string has five words"doesn't withstand scrutiny for meaningfulness after all. But at least prima facie it is meaningful. It has a subject "this string" that refers to "this string has five letters" and a predicate "has five words" that refers to the property of having five words. And it is true if and only if "this string has five words" has five words. And "this string has five words" has five words. So, "this string has five words" is true.

Then, to obviate any objections about the use of the pronoun 'this', I provided:

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

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

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

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

Put this way:

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

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

If your reply to that is yet more of your use-mention confusion, then my guess is that there's little hope you'd ever think about it enough to understand it, though it doesn't take a lot of thinking.

Ludwig Wittgenstein is generally credited with inventing and popularizing the truth table in his Tractatus Logico-Philosophicus, which was completed in 1918 and published in 1921.[2] Such a system was also independently proposed in 1921 by Emil Leon Post.[3]
— Wiki. — Banno

Yet, the Anellis paper says: "[T]he 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."

Anellis might not have adequately made the case for that assertion, as perhaps we would need to see Peirces's papers in more detail. But at least it must be allowed that Anellis may be correct. And, at least we see that Peirce was using truth tables.

Then you moved to a different claim:

What is original is that Witti points out how to use a truth table to determine tautology or contradiction. — Banno

But a truth table determines validity. And Peirce was using truth tables to determine validity. And 'valid' and 'tautology' are synonymous for sentential formulas. So Peirce was determining tautologousness.

Then there's the question of "complete". But what is meant by 'complete' in this context? Does it mean observing that there is no finite bound on the number of letters in a truth table? Is it a given that Peirce didn't observe that and Wittgenstein did?

The paper shows a Peirce matrix with truth values.
— TonesInDeepFreeze
Sure, and that is where it seems to stop. — Banno

No, he also showed the 16 truth tables for the 16 binary Boolean functions. And he did more work with truth tables. And we don't know the extent of his work without reading all that he wrote.

Another exercise is proving the correctness of the everyday methods for addition, subtraction, multiplication and division. I have a book that shows some of it.

completeness and consistency — Banno

completeness and soundness.

consistency follows from soundness.

what is actually quoted in the literature from Pierce seems to be about listing permutations of Boolean operators rather than showing truth. — Banno

What do you mean by "showing truth"? The paper shows a Peirce truth table with truth values.

"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.