Whether tangential or not, it raised an important and interesting point. The reason yours was not a correct translation is instructive. And it wasn't a matter of definitions, let alone a definition identified with me. Meanwhile, my translation was utterly obvious, therefore dull, but the point of such a translation is not to entertain but to be as pinpoint faithful to the formula as possible. If it had been a creative writing exercise then I would have tried to come up with something a lot more colorful.

If I recall correctly, I was thinking of specifically my post with the several examples compared line by line which was what finally established that that translation was not possible. — Lionino

Actually, this:

"If A implies B & ~B, then A implies a contradiction" is true, but it is a statement about the sentences, not a translation of them.
— TonesInDeepFreeze

Yes, granted. I used the word "translation" wrong in basically all of my posts. I meant "is a true statement about..." instead. — Lionino

So far, so good. There you're saying what I had spent a few posts explaining.

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

That last clause is wrong, obviously. (Maybe you corrected it subsequently.)

When it comes to ¬(A→(B and ¬B)), as it is the same as (¬A→(B and ¬B)), "not-A implies a contradiction" is a true statement about it. — Lionino

Right.

Leontiskos was further saying that the RAA is not strictly logical because it does not tell you which side of the conjunct to rule out. I disagreed in the last post of page 21. — Lionino

That's good. I disagreed with it many pages ago, as I saw immediately that it's wrong. (Not too very bumptious of me to say. winky face emoji whatever.)

More like the point Leontiskos is making. — Lionino

I'm referring to your correct point (shared by you, Leontiskos, Banno, and me) that instead of refuting P, we could refute one of the other premises. You adapted that point to my notation. I just pointed out that your notation doesn't suit the way I was using the notation and so I rephrased your point so that it does accord with my notation.

I couldn't tell since you often jump into a conversation that happened several pages before the last post of the respective thread. — Lionino

I linked exactly to your post. No matter how many intervening posts, all you had to do was click on your name under my quote of you.

But you chose to (1) Make a false claim that I was commenting on a post that was made before I came into this thread, (2) Get the conversation completely wrong by falsely saying that I had not explained why your translation was incorrect, (3) Falsely claim that my point about the translation was a nitpick. (4) Try to get a cheap advantage out the fact that I corrected myself in a formulation, even though that is unrelated to the subject of the translation.

alright however you wanna fly it — Lionino

It's not how I choose to see it. It actually is as I showed.

if we reach a contradiction we know there is a false assumption — Banno

We know that the assumptions are inconsistent, not merely that one of them is false. But, of course, if a set is inconsistent, then for any interpretation, at least one of the members is false in that interpretation.

You got it wrong. I know what I meant with my posts. "We" there refers to me, I was not talking about anyone else. The specific post you quoted did not help sort out the issue, specifically the nitpick on "translation", which is why I had to make a whole new thread for that topic specifically. — Lionino

Ugh^ugh (that's ugh to the power of ugh).

I lightly jibed, aiming at myself as much as anyone; and you turn it a thing. But since your revisionist attack on me is ill-premised I respond.

(1) I haven't claimed that you don't know what you mean. (2) I haven't claimed that you used the word 'we' in any particular way.

(3) You can look at the posts, and I as I mentioned some of them in my previous reply:

You asked me for a translation of "~(A -> (B & ~B))". So I obliged your request.

You claimed that "A does not imply a contradiction" is a translation.

I explained why that is not a translation of "~(A -> (B & ~B))", a few times. As you still didn't see the point, I gave even more details, to which you replied:

"Yes, granted. I used the word "translation" wrong in basically all of my posts."

(4) The matter of the translation is not a nitpick. You posted the incorrect translation at least a few times and presumably it was important enough to you to do that. So I was right on the money to note that it is not a translation and to explain why.

In fact in my thread you corrected yourself about something midway through the discussion — Lionino

Yes, to my credit, I corrected myself as soon as I realized I erred in a formulation. But that was not related to my correct explanations as to why "A does not imply a contradiction" is not a translation of "~(A -> (B & ~B))".

(5) In my previous post, I pointed out that you falsely claimed (why?) that my reply "You're welcome for that" was in response to a post from before I entered the threads. But my reply was from 13 days ago; I entered the thread 19 days ago. More importantly, my explanations as to why "A does not imply a contradiction" is not a translation of "~(A -> (B & ~B))" were also before that reply.

We don't say: If Gu{P} |- Q and Gu{P} |- ~Q, then ~P |- G.
— TonesInDeepFreeze

I made a mistake. I meant to say:

"The bulk of the debate here between Banno and Leontiskos (and me interjecting sometimes) is why say G |- ~P instead of P |- ~G." — Lionino

You can use the notation however you wish, but in my formulations, G is a set of formulas not a formula, and on the left side of the turnstile is a set of formulas while on the right side of the turnstile is a formula.

The rule:

(1) If Gu{P} |- Q and Gu{P} |- ~Q, then G |- ~P.

For concision, we can formulate that as:

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

So, the point you are making could be stated ('\' stands for set difference):

(3) If Gu{P} |- Q & ~Q, then there could be an R in G such that

(G\{R}) u {P} |- ~R
and
(G\{R}) u {P} is consistent.

That is a formal way of saying what, as you mention, has been correctly said in this thread for a long time:

If ~P is derived by RAA, then it might be the case that there could be a premise R, different from P, such that we derive ~R instead.

And I gave specific examples of that in exact RAA formulation. My point in that regard has always been in agreement with anyone who correctly makes that observation.

But I don't take that to be a problem with RAA.

As I mentioned, RAA, along with the rest of the natural deduction rules, provide all and only the valid inferences (that is the completeness and soundness theorems).

[From previoulsy and adding to it:] Of course, the choice of axioms or added premises is subject to whatever criteria we wish to subject them to. That we have that option is not a fault in RAA. It's not a matter of a fault in RAA but rather we have the obvious fact that we are free to choose whichever axioms or premises we want to choose. It is good that natural deduction allows us to work from whatever axioms or premises we want to choose. The purpose of the deduction system is not to dictate our starting assumptions, but rather merely to permit all and only the valid inferences from whatever assumptions we choose.

Banno — Lionino

((P & R) -> (Q & ~ Q)) |- (P -> ~R) & (R -> ~P)

That is the idea with a pair of formulas P, R; while my formulation above is way of saying the same idea but generalized with any set of formulas G.

Indeed, his idea of looking at conjunction elimination, not just RAA, in this regard is insightful.

It has happened before in the history of science where we had to reject G when finding out that Gu{P} is contradictory, because P was so evidently true. — Lionino

I would think so. And that is what this basic logic provides.

The cranks says, "I have no idea what Tones has been talking about"

I do like when occasionally the crank speaks truthfully.

And characteristic of him to not know what is being talked about even when what is being talked about is what he's been talking about.

For starters, the crank throws around the word 'platonism' but he doesn't even know what it is.

Then the crank says, "Our language is very much platonist."

Right, it was Plato, Platonists, and platonists who installed the words 'it', 'thing' and 'object' and the need for them.

And notice no response to:

I'd like to see the crank try to write mathematics in English without referring to sets, numbers, etc. as if they are things of some kind. Specifically, that requires avoiding the word 'it' to refer to things. — TonesInDeepFreeze

or

What is the crank's definition of 'a fiction'? — TonesInDeepFreeze

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

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

'This string has five words'

Is that a sentence?

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

No string that is self-referential is a sentence.

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

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

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

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

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

Or:

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

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

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

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

Put this way:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The tower Big Ben is Big Ben.

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

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

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

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

You wrote:

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

"This string has five words"

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

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

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

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

Regarding axioms in proofs:

I'll call both of the two forms below 'RAA' (though different writers label them in different ways):

(1) If Gu{P} |- Q and Gu{P} |- ~Q, then G |- ~P

(2) If Gu{~P} |- Q and Gu{~P} |- ~Q, then G |- P [not intuitionistic]

In a natural deduction proof, any formula can be entered on a line and charged with that line number. It is not required that the formula be a member of a given axiom set. But some lines can be "discharged" so that even though the line was used, it is no longer counted on as being required for the inference. That is the essence of natural deduction. For example, an RAA:

(1) P -> (Q & ~ Q) {1}
(2) P {2}
(3) Q & ~Q {1,2}
(4) ~P {1}

Line (2) was discharged at line (4).

{1} in line (4) is the set of formulas that are charged to the conclusion ~P. So, proving ~P in this case depends only on line (1) and not on line (2). Line (2) was discharged.

And it is well understood that this also is correct:

(1) P -> (Q & ~ Q) {1}
(2) P {2}
(3) Q & ~Q {1,2}
(4) ~(P -> (Q & ~ Q)) {2}

Line (1) was discharged at line (4).

{2} in line (4) is the set of formulas that are charged to the conclusion ~P. So, proving ~P in this case depends only on line (2) and not on line (1). Line (1) was discharged.

If we want only inferences from a given set of axioms, then each member of the set of formulas that are charged to the conclusion of an argument must be a member of that set of axioms. But the discharged formulas do not have to be members of that set of axioms.

Of course, having (1) as an axiom and (2) as not is different from having (2) as an axiom and (1) as not. Of course, it is optional which we choose to work with.

And of course, one is free to choose which axioms to work with, or to add to add premises of interest. But the discharged lines are not counted in the "bottom line". And yes, of course, the choices of axioms or added premises are subject to whatever criteria we wish to subject them to. That we have that option is not a fault in RAA. It's not a matter of a fault in RAA but rather it's the obvious fact that we are free to choose whichever axioms or premises we want to choose.

You're welcome for that. (Not too very bumptious of me.)
— TonesInDeepFreeze

The post you quoted there was before you joined these threads. So there is no connection to you. "We" there simply means "I" — not bumptious of me, the greatly humble person I am. — Lionino

Ugh.

From the post I quoted:

We have established that "A does not imply a contradiction" is not a good reading of ¬(a→b∧¬b). — Lionino

That was from 13 days ago. I entered the thread 19 days ago.

So, "The post you quoted there was before you joined these threads" is false. And my remark "You're welcome for that" is apposite in the timeline:

From 14 days ago, though my comments about the point went further back, a detailed, informative post:

Do you think it is correct to translate this as: when it is not true that A implies a contradiction, we know A is true?
— Lionino

Tones replied that that is not true for all contradictions but for some interpretations.
— Lionino

That's not what I said.

If I recall correctly, you said that "A -> (B & ~B)"* may be translated as "A implies a contradiction". (*Or it might have been a related formula; not crucial since my point pertains to all such examples.)

That is not the case as follows:

(1) The sentence has a sub-sentence that is a contradiction, but the sentence itself does not mention the notion of 'contradiction'.

(2) To say "a contradiction" is to implicitly quantify: "There exists a contradiction such that A implies it". And that quantifies over sentences. If we unpack, we get: "There exists a sentence Q such that Q is a contradiction and A implies it".

A translation of "A -> (B & ~B)" is:

If A, then both B and it is not the case that B.

and not

"A implies a contradiction".

(3) "B & ~B" is a particular contradiction, not just "a contradiction". Even though all contradictions are equivalent, a translation should not throw away the particular sentences that happened to be mentioned.

(4) If we have that A implies B & ~B, then of course, we correctly say "A implies a contradiction". But that is a statement about A, not part of a translation.

"If A implies B & ~B, then A implies a contradiction" is true, but it is a statement about the sentences, not a translation of them. — TonesInDeepFreeze

After that post, you finally understood the point (I had explained it previously) as you wrote:

"If A implies B & ~B, then A implies a contradiction" is true, but it is a statement about the sentences, not a translation of them.
— TonesInDeepFreeze

Yes, granted. I used the word "translation" wrong in basically all of my posts. I meant "is a true statement about..." instead. — Lionino

So, celebrating that information had been understood, I replied:

Thank you for recognizing my point. — TonesInDeepFreeze

Later you wrote:

We have established that "A does not imply a contradiction" is not a good reading of ¬(a→b∧¬b). — Lionino

Then I said, "You're welcome for that". And you are.

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

Ok, that is the derivation. The source I quoted at least is correct when abriding it. The RAA however is not how it was being presented in this thread by others before, which is what I was trying to confirm. — Lionino

You asked:

This is the RAA, innit?
(S∧¬P)→(B∧¬B)
S
∴ P — Lionino

So I answered correctly that it is not. And I showed an actual RAA.

Yours is a valid inference, but it is not formulated as RAA.

With the non-intuitionistic form we can have the sentence on the last line be P.
— TonesInDeepFreeze

We are all speaking non-intuitionistically here, which is standard at least in amateur circles. — Lionino

I mention it because the proof itself ends with ~~P. To get P requires additional steps that are not intuitionistic. And I mention, as added information, that they are not intuitionisitc because that is interesting and good to know. I wouldn't always belabor the point, but in this context it is stark that there are two forms: classical and intuitionistic, and in the natural deduction system I presented, we need both to achieve all the classical inferences.

And sometimes people do speak on behalf of intuitionism in this forum, as well as there are likely readers of threads that don't post in them.

(1) If Gu{P} |- Q and Gu{P} |- ~Q, then G |- ~P
— TonesInDeepFreeze

The bulk of the debate here between Banno and Leontiskos (and me interjecting sometimes) is why say G |- ~P instead ~P |- G. — Lionino

Whatever the bulk the debate is about, I presented an exact formulation of the rule a a reference for whomever might want to know exactly what the rule is. And as part of my explanations why certain objections to the rule are off-base.

We don't say: If Gu{P} |- Q and Gu{P} |- ~Q, then ~P |- G.

We don't say that because |- is meant to be equivalent with |=.
And it is not the case that for all G and P, we have:
If Gu{P} |= Q and Gu{P} |= ~Q, then ~P |= G

Formulating a rule: "If Gu{P} |- Q and Gu{P} |- ~Q, then ~P |- G" would be stupid. And, believe it or not, logicians try not to offer stupidities that would thwart the very intent of presenting a formulation.

I'd like to see the crank try to write mathematics in English without referring to sets, numbers, etc. as if they are things of some kind. Specifically, that requires avoiding the word 'it' to refer to things. ... Confound it, I did it again, I used the word 'thing'! The English language has that nasty habit of making it virtually impossible not to use the word 'thing'. What's up with that? The language must have been invented by satanists...I mean...platonists.

.

Welcome to another episode of 'A Day In The Life Of Muddlefizzle Undergarment Internet Crank':

Huston Lover: I saw 'The Maltese Falcon' again last night. What a great film.

Muddlefizzle Undergarment: Never heard of it.

Huston Lover: It's terrific. Humphrey Bogart plays this detective Sam Spade whose partner was murdered.

Muddlefizzle Undergarment: What do you mean "he plays"?

Huston Lover: What do you mean?

Muddlefizzle Undergarment: You said, "he plays". What is that?

Huston Lover: Humphrey Bogart was an actor. He plays the character Sam Spade.

Muddlefizzle Undergarment: A character? What's that?

Huston Lover: A character. A fictional person.

Muddlefizzle Undergarment: Fictional people don't exist.

Huston Lover: Right. They don't exist like you and I. But they exist in the stories. Like the statuette, the Maltese Falcon in the story. It doesn't exist like the Stanley Cup exists, but it is a fictional object.

Muddlefizzle Undergarment: So you're a Platonist!

Huston Lover: I don't even know what that is. I was just trying to tell you about the movie.

Muddlefizzle Undergarment: As I've told the Platonists at The Philosophy Forum, there are no fictional objects; there are only actual objects. It is nonsense to talk about fictional objects. Like it is nonsense when mathematicians try to make you believe that numbers and sets are objects. That breaks the law of identity! So please don't try to make me believe that there are fictional people and fictional objects.

Huston Lover: Then how can I tell you who Sam Spade and Brigid O'Shaughnessy and Kasper Gutman are? I mean, they're not living people or dead people, so what else can I call them except 'fictional people'? What can I call the Maltese Falcon if not 'a fictional object'?

Muddlefizzle Undergarment: You may call them 'fictions'. Otherwise, I'll post that you're a Platonist.

Huston Lover: Would that be bad?

Muddlefizzle Undergarment: Very bad. Because then I can say that you're a slippery sophist dragging us all into the Platonic sewer. But good for me, because then I'd have the satisfaction of exposing you as a slippery sophist sewer bound Platonist.

Huston Lover: Okay, but if Sam Spade and the Maltese Falcon can only be called 'fictions' and not 'a fictional person' and 'a fictional object', then how do I refer to them so that you know what they are?

Muddlefizzle Undergarment: Okay, you can call him 'a fictional detective' and you can call it a 'fictional statuette'. As long as you never say they are a fictional person and fictional object. Because, just like at The Philosophy Forum, I've explained that there are no fictional objects or people.

Huston Lover: Thanks for that, Muddlefizzle. But isn't a detective a person, and a statuette an object? So a fictional detective is a fictional person and a fictional statuette is a fictional object.

Muddlefizzle Undergarment: No! Not unless you want to be dirty slippery sophist sewer seeking Platonist rat!

Huston Lover: Okay, Muddlefizzle. But you should check out the movie.

Muddlefizzle Undergarment: I will not be doing that. I only watch documentaries. They're about real things.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

"This sentence is false"

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

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

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

"This sentence has five words"

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

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

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

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

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

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

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

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

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

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

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

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

I think the following is right:

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

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

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

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


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

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

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

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


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

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

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

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


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

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

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

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

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

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

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

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

It's Russell's paradox illustrated anecdotally.

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

For any 2-place predicate S:

~ExAy(Syx <-> ~Syy)

and

~ExAy(Sxy <-> ~Syy)

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

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

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

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

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

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

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

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

What is the crank's definition of 'a fiction'?

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

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

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

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

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

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

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

I'm just explaining the barber paradox.

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

I'd like to know those arguments in context.

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

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

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

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

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

Ah, but later you wrote:

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

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

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

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

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

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

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

I don't know of any such requirement.

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

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

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

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

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

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

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

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

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

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

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

Depends on the definition of 'is a liar'.

Question: How does reductio work?

Answer: There are two related inference rules:

(1) 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.

(2) 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:

(1) If Gu{P} |- Q and Gu{P} |- ~Q, then G |- ~P

(2) If Gu{~P} |- Q and Gu{~P} |- ~Q, then G |- P [not intuitionistic]

Then why do I have 13 new replies from you today, 11 of which are in a single thread? You're a spammer and I don't have time for this stupid shit. Get someone else to teach you how a reductio works. Maybe they can also teach you how to interact without spamming. Adios! — Leontiskos

How predictable of Leontiskos! His hypocrisy is remarkable. Arguably he has posted more than anyone in this thread. A significantly greater number of posts and characters than I have. And he made several posts about my posts and about me during a time when I was not posting in this thread, just as recently made several during a time when he was not posting. He exercised his prerogative that way, and I exercised mine.

A poster is not wrong for catching up to several posts in a thread all at once. How in the world would it be unreasonable for a poster to exercise the prerogative to post and to reply to posts after, for whatever reason, not having posted in a thread during several days?

No one has to confine themselves to Leontiskos's own posting times to reply to him so as to refrain from catching up at some later time.

/

And as to teaching how reductio works, I posted an exact specification how it works, in two formulations. Leontiskos could read that post, or better yet, get an introductory textbook that uses natural deduction, because sure as shootin' he ain't got a clue now, as I showed.

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

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

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

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

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

said person shaves herself.

said person does not shave herself.

no one shaves said person.

someone shaves said person.

everyone shaves said person.

no one is shaved by anyone.

everyone is shaved by everyone.

some are shaved by someone

some are not shaved by anyone

etc.

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

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

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

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

a liar — Gregory

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

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

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

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

But that's not the puzzle.

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

Number one isn’t. — Fire Ologist

It's about the sentence.

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

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

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

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

“Punctuation is true.”

What? — Fire Ologist

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

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

.

lying is not the same as saying something that is false. — Leontiskos

When someone says a falsehood negligently, especially with the intent to discredit another, that too is a lie. And when someone says a falsehood that they have committed to belief by willful intent, that too is a lie. And when someone has clearly been shown that what they've said is false but continue to say it anyway, that too is a lie and an egregious lie.

And notice that I did not say that Leontiskos lied the first time he made the false claims about me. Rather, I called to his attention that I have said the very opposite of what he claims I've said. But then he continued to make the false claims. At that point I securely said they are lies.

I have been ignoring your posts, and have only read a handful of them. — Leontiskos

But not ignoring them enough to not lie about them.

As Philosophim said:

Not exactly the model of a sage and wise poster. You came on here with a chip on your shoulder to everyone. I gave you a chance to have a good conversation, but I didn't see a change in your attitude.
— Philosophim — Leontiskos

My reply:

Not exactly the model of a sage and wise poster.
— Philosophim

You leave out that I went on to give a proof in two versions. And it is appropriate to ask whether a poster is really serious asking for something that is, as far logic is concerned, as simple as showing that 4 is an even number. If in a thread about number theory someone happened to write "4 is even", and then another said "Prove it", you think that would not be remarkable enough to reply "Are you serious? You don't know how how to prove it?", let alone to then go on to prove it anyway.

You came on here with a chip on your shoulder to everyone.
— Philosophim

Where is here? This thread? I came with no shoulder chip, not to anyone, let alone "everyone". If I permitted myself to do as you do - to posit a false claim about interior states - I would say that you do so from your own umbrage at having been corrected.

And my point stands that I did not insult you, whereupon you insulted me.

I gave you a chance to have a good conversation
— Philosophim

By saying "don't be a troll".

You can converse as you please. I'm not stopping you. And I have read your subsequent posts, even after your insulting "don't be a troll" and have given you even more information and explanation. I have not shut down any conversation. — TonesInDeepFreeze

I don't have time for silly spats and allegations. — Leontiskos

Ah, the time honored tactic of pretending to be above a dispute by perpetuating it. And pointing out that Leontiskos is lying when he says I said the opposite of what I actually said is not silly. Rather it is mighty proper.

If you can't answer simple questions without telling me that I am lying a dozen times then I will just put you back on ignore. — Leontiskos

Time, energy and interest allowing, I answer sincere and coherent questions as best I can.

When we do a reductio
A, A→¬B∧B ⊢ ¬A is valid

But A, A→¬B∧B ⊢ A is also valid

So the question is: how do we choose between either? — Lionino

Choose in what sense?

Do you mean whether we should claim A or claim ~A?

I wouldn't claim either. The premises are inconsistent, a fortiori the arguments are not sound. I would claim a conclusion only from an argument that is not only valid but is sound, and especially not from an argument in which the needed premises are inconsistent.

This is the RAA, innit?
(S∧¬P)→(B∧¬B)
S
∴ P — Lionino

No. Here is an RAA:

1. (S & ~P) -> (B & ~B) {1}
2. S {2}
3. ~P {3}
4. B {1,2,3}
5. ~B {1,2,3}
6. ~~P {1,2}

Or if the version of the rule is to go through a conjunction B & ~B instead of B and ~B on separate lines:

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

With the non-intuitionistic form we can have the sentence on the last line be P.

how do you prove that you may derive ~ρ from ρ→(φ^~φ)?
— Lionino

I consider it an open question as to whether this question is answerable. — Leontiskos

I answered it.

how do you prove that you may derive ~ρ from ρ→(φ^~φ)? — Lionino

By showing a derivation:

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

RAA is a rule. If it is a primitive rule, then there's no call to prove it. If it is a derived rule from primitive rules, then we prove that any inference with the derived rule can be formulated with the primitive rules only.

A natural deduction system would have RAA as primitive. Ordinarily an axiom system would have RAA as derived.

Or, you might ask "how do we know that a rule is valid"? Well, we prove:

With any application of the rules, any interpretation in which the premises are true is an interpretation in which the conclusion is true.

That is the soundness theorem.

TonesInDeepFreeze [has] chosen:

(a→(b∧¬b)) → ¬a — Leontiskos

I haven't "chosen" it except that it is:

a theorem of sentential logic

a tautology

a symbolization, in one formula form, of certain common reasoning

What I have consistently said is that reductio is not valid in the same way that a direct proof is. — Leontiskos

RAA is valid in the same way any other rule is valid:

A rule is valid if and only if application is truth preserving, which is to say that any interpretation in which the premises are true is an interpretation in which the conclusion is true.

We have established that "A does not imply a contradiction" is not a good reading of ¬(a→b∧¬b). — Lionino

You're welcome for that. (Not too very bumptious of me.)