P=P is a Contradiction

[Skepdick]

Umm...I'm pretty sure you don't know what "special pleading" is, then.

Mathematics is a language.
Logic is a language.

That's your best logic yet: it looks like this...

Dogs are animals
Cats are animals.
Therefore dogs are cats, or else you're "special pleading"?

P1. The law of identity is a linguistic matter.
P2. Mathematics is linguistic.
C. The law of identity concerns mathematics.

Again, the subject matter was the law of identity. It was linguistic, not mathematical.

^^^^^ Special pleading.

Mathematics is a formal language.

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

[Averroes]

The "P1" and "P2" is of a different mathematical nature than the "P" in "P∧¬P".

No, in what I was talking about, neither has a "mathematical" nature. They both have a linguistic nature. That's the real point.

If by "linguistic" you mean the language of sentential logic then it's mathematics. But if by "linguistic" you mean natural languages such as English or French or Arabic, then you are wrong! In natural languages words refer to the outside world but in logic the propositional variables such as "P" or "Q" have no such reference! They are purely mathematical objects. Now, this is here that the beauty of mathematics can be seen to shine, in that natural language can use mathematics to model its functioning! But there is a gap which must first be filled! The mathematical objects "P" or "Q" have to be given references for it to model natural languages.

Likewise, in Aristotle's law of non-contradiction, written in symbols as ¬ P (P∧¬P), it's a linguistic placeholder, not a mathematical one.

As I said previously, "¬ P (P∧¬P)" is not a well formed formula (wff) in LSL, it is nonsense in LSL. The law of non contradiction is expressed as " ¬(P∧¬P)" in LSL. You have to drop the first "P" for it to express LNC.

Again, the subject matter was the law of identity. It was linguistic, not mathematical.

There is a subtle point that you must first grasp about logic before we get into deeper waters, otherwise you will keep drowning each time.

[Immanuel Can]

You have to drop the first "P" for it to express LNC.

Whoops. Yes, you are correct about that.

But now, you write:

In natural languages words refer to the outside world but in logic the propositional variables such as "P" or "Q" have no such reference! They are purely mathematical objects.

Right. And that's all the difference in the world.

To refer to mathematics as a "language" is not deceptive, but is ambiguous, because "language" is a word we use to refer to broad systems of symbols (as in "computer languages," for example), and yet in other contexts, a word we use specifically to indicate what you call "natural languages," things like English or Arabic, which are not composed of symbols each of which stands for a fixed quantity or property, but are rather formed into chains with empirical referents.

Mathematics takes the law of identity as a given. One might say it's so intrinsic to maths that without assuming the law of identity there would be none. Fine. But in empirical situations, there are no absolute and fixed quantities implicated. Rather, there are sets of linguistic markers which have to be compared for their relative coherence -- a much less precise matter, but no less important than the mathematical operations. And it is in this second situation that the law of identity is questioned...not in maths.

So what mathematics has to do with all this? I'm still not at all certain. So far as I know, no mathematician even questions the law of identity.

[Immanuel Can]

P1. The law of identity is a linguistic matter.
P2. Mathematics is linguistic.
C. The law of identity concerns mathematics.

Amphiboly Fallacy. Twice.
You went from "is" to "concerns." That one's obvious.
The second, less so; you went from "linguistic" as in an actual language to merely "symbolic" in premise 2, which is not the same use of the word "language."

[Skepdick]

Amphiboly Fallacy.

Fallacy fallacy

You went from "is" to "concerns." That one's obvious.
The second, less so; you went from "linguistic" as in an actual language to merely "symbolic" in premise 2, which is not the same use of the word "language."

Further special pleading.

I am using the symbols "language" to refer to both English and Mathematics.

I use English for exactly the same purpose I use Mathematics for: self-expression.

And if you insist that you are a better narrator for my use of symbols than I am... fuck off?

[Immanuel Can]

Further special pleading.

I was right. You don't know what that is.

I am using the symbols "language" to refer to both English and Mathematics.

I know. That's what amphiboly is: using one word for two distinct situations or referents, within the same syllogism, and not realizing the difference. It gave you what's called a "shifting middle term."

I should have pointed out, too, that the structure of your proposed syllogism was also fallacious. You affirmed the consequent. See line 2. It's a formal fallacy, that also renders the argument unsound.

If you made the structure correct, it should have read something like:

English is a language
Languages are (all) mathematics.
Therefore, English is mathematics.

But you wouldn't have made so obvious a mistake if you had formed your syllogism formally-correctly. Hence, the importance of not affirming the consequent.

[Skepdick]

I was right. You don't know what that is.

I was righter.

You keep committing fallacies and can't tell.

I know. That's what amphiboly is: using one word for two distinct situations or referents, within the same syllogism, and not realizing the difference. It gave you what's called a "shifting middle term."

Clearly you are confusing distinctions with differences.

I should have pointed out, too, that the structure of your proposed syllogism was also fallacious. You affirmed the consequent. See line 2. It's a formal fallacy, that also renders the argument unsound.

If you made the structure correct, it should have read something like:

English is a language
Languages are (all) mathematics.
Therefore, English is mathematics.

But you wouldn't have made so obvious a mistake if you had formed your syllogism formally-correctly. Hence, the importance of not affirming the consequent.

if English is different to Logic, then why are you interpreting my English expression through the formal lens of "soundness" ? Why are you viewing English as syllogistic?

This is sheer philosophical idiocy. Even your pretension at disagreement is half-assed.

[Immanuel Can]

if English is different to Logic, then why are you interpreting my English expression through the formal lens of "soundness" ? Why are you viewing English as syllogistic?

Oh, boy. Where do I start?

Well, firstly, I did not say "English is different to logic." I said "English is different from mathematics." Those are two very different claims.

Secondly, logic is syllogistic. Logic is also linguistic. There is no contradiction there. Syllogisms can be formed using words or of symbols. Either way, formal fallacies of structure can pertain to them.

Thirdly, the reason logicians distinguish between formal and informal fallacies is very simple: there are two broad categories of logical fallacy. You made both kinds. You amphibolized two words, and you also made an error of affirming the consequent.

Lastly, "soundness" means that a syllogism has no fallacies of either kind in it. And yours had both.

[Averroes]

You have to drop the first "P" for it to express LNC.

Whoops. Yes, you are correct about that.

Yes, I know.

So what mathematics has to do with all this?

Everything!

But first of all let me remind you that you made a mistake in writing the following paragraph is the reason I am writing these clarifying posts. You said:

P.S. -- It occurs to me, after the fact, that I may have to point out (or perhaps explain for the first time, if you didn't happen to know already) that "P" in logic stands for either "premise" or "predication." Thus it is not at all the same as a mathematical placeholder like "X" or "Y." It refers to a specific linguistic utterance, rather than to mathematical formulation or quantity.

As we have seen already in the logic proposition "¬(P∧¬P)", contrary to what you wrote, "P" is indeed a mathematical placeholder(or as we say a sentential variable).

So far as I know, no mathematician even questions the law of identity.

Of course! Denying the law of identity results in a contradiction. I already addressed this before on the forum some years ago but I fear you will get more confused by the explanation than be enlightened. I have struggled to explain to you basic logical stuff up to this point, which I hope you understood. Getting into the proof of the law of identity requires that you have a solid grasp of logic which I fear you are lacking at this point. Please do not be offended by what I said. If you are interested in a version of that proof, then I refer you to the initial part of the transcendendal deduction of the categories of Kant where he talks about the transcendental apperception! This is the proof of the law of identity! There are others as well who expressed this proof in other words, but it really requires a solid understanding of logic to be convinced by the proof.
The interesting thing about Kant's argument is that he talks about mathematics in this argument! Spot on the subject! Let me quote it for you and let you ponder over it. Do not take that argument lightly.

Kant wrote:

• "Without consciousness that that which we think is the very same as what we thought a moment before, all reproduction in the series of representations would be in vain. For it would be a new representation in our current state, which would not belong at all to the act through which it had been gradually generated, and its manifold would never constitute a whole, since it would lack the unity that only consciousness can obtain for it. If, in counting, I forget that the units that now hover before my senses were successively added to each other by me, then I would not cognize the generation of the multitude through this successive addition of one to the other, and consequently I would not cognize the number; for this concept consists solely in the consciousness of this unity of the synthesis. [ A104, Critique of Pure Reason]"

Think about it. The emphasis in the quote was mine. If from this argument you can find the contradiction of denying the law of identity then formulate it concisely in contemporary words and that's it! And you will have a solid grasp of why mathematicians never question the law of identity!

[Skepdick]

Secondly, logic is syllogistic. Logic is also linguistic. There is no contradiction there. Syllogisms can be formed using words or of symbols. Either way, formal fallacies of structure can pertain to them.

Look! You are pretending to disagree again. Words (formal or otherwise) are made of alphabets.

https://en.wikipedia.org/wiki/Alphabet_ ... languages)

Logic (like Mathematics) is a formal language. It even has the word "language" in it.

And in the very first paragraph of this article it tells you about the connection to linguistics.

Formal language theory sprang out of linguistics, as a way of understanding the syntactic regularities of natural languages.

[Immanuel Can]

...in the logic proposition "¬(P∧¬P)", contrary to what you wrote, "P" is indeed a mathematical placeholder

Linguistic placeholder. Not mathematical.

So far as I know, no mathematician even questions the law of identity.

Of course! Denying the law of identity results in a contradiction.

Yes, I know. But that is precisely why this isn't a mathematical, but rather a linguistic and logical issue. Maths does not even regard the law of identity as potential of any controversy.

[Immanuel Can]

It even has the word "language" in it.

So does computer "language."

But to imagine that makes it the same as what A. is calling "natural language" is to mistake a metaphor for a reality. It's to make the fallacy of amphiboly, if you use it that way in a syllogism.

Maths is not a "language" in the same sense that Urdu, or Russian, or French is.

[Averroes]

...in the logic proposition "¬(P∧¬P)", contrary to what you wrote, "P" is indeed a mathematical placeholder

Linguistic placeholder. Not mathematical.

Given that "¬(P∧¬P)" is expressed in the language of sentential logic, and also that we have mathematical linguistic, so the linguistic part is fine. But the mathematical denial is highly problematic. Please, allow me to ask you: do you consider algebra to be part mathematics or not?

[Immanuel Can]

mathematical linguistic

No, it's either a mathematical placeholder OR a linguistic one. It's not both.

Maths is not a "natural language," just as you have already said. It's a false analogy to treat the two as if they were the same.

And algebra is not linguistic. It's mathematical. The "letters" do not form words, phrases, sentences, paragraphs and so forth, as they do in natural languages. In algebra, letters are used only as mathematical placeholders. In fact, they can be replaced with other symbols, with no loss of meaning, so long as the new symbols are agreed upon. So there, the letters only sub in for numbers and operations. They do not form utterances.

[Skepdick]

So does computer "language."

But to imagine that makes it the same as what A. is calling "natural language" is to mistake a metaphor for a reality.

I am not calling it a "natural language" - you are doing that.

I am calling it a language.

Maths is not a "language" in the same sense that Urdu, or Russian, or French is.

Urdu, Russian and French are not "the same" as each other either, but they are languages.

Maths is a language in the same sense that Urdu, or Russian, or French are all languages.

The distinction you are drawing between "formal" and "natural" languages doesn't render Mathematics a non-language.
It makes it a different kind of language.

But it's still a language.