What LEM is not

[Magnus Anderson]

Pure reason is blind. In pure reason, proof is the only method that we can use to discover truth. In pure reason, there simply is no alternative to Soundness theorem: If it is provable, then it is true in all its interpretations.

This is indeed a weakness of pure reason. This is the reason why we cannot discover any truth that is unprovable. Concerning true but unprovable statements, we know of their existence. It actually took until 1931 for Kurt Gödel to prove their existence with his incompleteness theorems. We even know that these true but unprovable statements dominate mathematical truth, but we simply have no way of discovering them. Yanofsky extensively writes about this very problem in his paper "True but unprovable":

http://www.sci.brooklyn.cuny.edu/~noson ... ovable.pdf

Outside pure reason, the LEM simply does not apply. There is no logical truth outside pure reason. In pure reason, the LEM only applies to truth that we can discover, i.e. provable truth. Constructivists do not confuse veracity with provability. On the contrary, on top of Kurt Gödel's work, a very extensive body of understanding has been built that sharply distinguishes between provable truth and unprovable truth.

You're still talking about provability as if it has anything to do with LEM.

Anyone who thinks with their own mind and does not merely parrot popular views.

I am critical about pretty much everything but not about provable theorems in mathematics. I consider them to be pretty much incontrovertible in their given context. I personally sense that they are the wrong target for criticism.
[/quote]

You don't question their understanding of LEM, just as you don't question yours.

[godelian]

You're still talking about provability as if it has anything to do with LEM.

So, how do you know that a proposition is logically true?

[Magnus Anderson]

You're still talking about provability as if it has anything to do with LEM.

So, how do you know that a proposition is logically true?

That's beyond the scope of this thread.

We don't have to know how we come to accept that a proposition is true.

The task is to understand what LEM is.

To do that, all we have to figure out, and know, is what the words "statement", "proposition", "true" and "false" mean. And specifically, how they were used in the original formulation of LEM.

Statements are symbols such as "This statement is false" and "This statement is true" that may or may not represent propositions.

If a statement represents a proposition, it's a propositional statement.

If a statement does not represent a proposition, it's a non-propositional statement.

Most symbols are non-propositional, e.g. words such as "cat", "dog", "unicorn" and so on. Words, on their own, are not representations or descriptions of an aspect of reality.

The two statements that I mentioned, "This statement is true" and "This statement is false", are also non-propositional. Another example would be Godel's statement.

Non-propositional statements have no truth value. They are, figuratively speaking, neither true nor false ( I say figuratively because literally speaking "neither true nor false" is an oxymoron. )

A proposition is an idea that a portion of reality exists in certain state. Propositions aren't symbols. Rather, they are what is attached to symbols. One and the same proposition can be attached to multiple different symbols, i.e. one and the same proposition can be represented differently in different languages.

Every proposition is made out of two components: the subject and the predicate. The subject is a referenced or described portion of reality. The predicate is a description of the referenced portion of reality. In order for a proposition to be a proposition, it must consist of these two components. If at least one of them is missing, we're not dealing with a proposition.

"2 + 2 = 4" is a statement representing a proposition that is saying "The number represented by the symbol '2 + 2' is equal to the number represented by the symbol '4'". The subject is the relation between the numbers attached to symbols "2 + 2" and "4". The predicate is that the numbers are equal.

Compare that to a non-propositional statement such as "This statement is true". Obviously, the subject of this statement is the truth value of some proposition ( indicated by "this statement" which means "the proposition attached to this statement" ) and the predicate is "true". But the proposition that is referenced is actually missing. What is it describing? "Itself." What is this "itself" describing? "I told you, itself!" But what is that "itself" describing? "ITSELF!" It's circular. It does not actually tell you what's being described. It just screams, "Itself, itself, itself!" It's like asking someone "Who is this man?" and they answering "It's him!" over and over again.

The truth value of a proposition is the degree to which the referenced portion of reality can be represented by the predicate. Only propositions have truth value.

There is an infinite number of truth values but we generally prefer to think in terms of 2 categories, namely, "true" and "false". The word "true" represents the highest truth value, i.e. the highest degree of correspondence between the subject and the predicate, whereas the word "false" is reserved for all other truth values.

LEM merely states the obvious, namely, that for every proposition P, the truth value of P is either true or false.

That's true by definition.

Key points:

1) LEM is really only concerned with propositional statements.

2) LEM is stating that each propositional statement describes the referenced portion of reality with an accuracy that is either perfect or imperfect.

That's really all it does.

[godelian]

You're still talking about provability as if it has anything to do with LEM.

So, how do you know that a proposition is logically true?

That's beyond the scope of this thread.
We don't have to know how we come to accept that a proposition is true.

This is essential to understanding the LEM. This is how we come the modern take on the LEM:
If the proposition is decidable then the proposition is true or the proposition is false.

The task is to understand what LEM is.

It is not possible to understand the LEM without understanding the notion of decidability.

To do that, all we have to figure out, and know, is what the words "statement", "proposition", "true" and "false" mean. And specifically, how they were used in the original formulation of LEM.

The original formulation of the LEM is as relevant today as the original Roman numerals. In fact, it is less relevant, because Roman numerals are consistent, while your take on the LEM is not.

https://www.calculatorsoup.com/calculat ... verter.php

Roman numerals have limitations:

454534444 -> Enter a valid Roman Numeral or Integer from 1 to 3,999,999.

Apparently, numbers larger than 3,999,999 are not supported.

Most symbols are non-propositional, e.g. words such as "cat", "dog", "unicorn" and so on. Words, on their own, are not representations or descriptions of an aspect of reality.

Logical truth is never about physical reality. It is always about an abstract, Platonic reality.

A proposition is an idea that a portion of reality exists in certain state.

There is no logical truth in physical reality. There is only probabilistic truth.

Compare that to a non-propositional statement such as "This statement is true". Obviously, the subject of this statement is the truth value of some proposition ( indicated by "this statement" which means "the proposition attached to this statement" ) and the predicate is "true". But the proposition that is referenced is actually missing. What is it describing? "Itself." What is this "itself" describing? "I told you, itself!" But what is that "itself" describing? "ITSELF!" It's circular. It does not actually tell you what's being described. It just screams, "Itself, itself, itself!" It's like asking someone "Who is this man?" and they answering "It's him!" over and over again.

"This statement is true" is undefinable in first-order arithmetic because of Tarksi's undefinability of the truth:

S ⇔ true(⌜S⌝)

The predicate true(n) is not definable.

https://en.wikipedia.org/wiki/Tarski%27 ... ty_theorem

Tarski's undefinability theorem: There is no L-formula True(n) such that for every L-sentence A, True (⌜A⌝) ⟺ A holds in N.
Informally, the theorem says that the concept of truth of first-order arithmetic statements cannot be defined by a formula in first-order arithmetic.

But then again, self-referential statements are otherwise perfectly allowed in first-order arithmetic. You seem to have problems working within a precise theoretical context and with constructs that may at first glance seem unusual but that are otherwise perfectly valid.

LEM merely states the obvious, namely, that for every proposition P, the truth value of P is either true or false.
That's true by definition.

No, in modern logic, the LEM is undecidable. You refuse to make progress beyond the 17th century. By rejecting modern logic and modern mathematics, your arguments become unconvincing at best, and outright faulty at worst.

[Skepdick]

To do that, all we have to figure out, and know, is what the words "statement", "proposition", "true" and "false" mean. And specifically, how they were used in the original formulation of LEM.

Some people are ideologically committed to staying in Plato's cave.

What does the word "mean" mean?
What does the word "word" mean?
How is the word "how" used in this question?
What does the the sentence "What does the sentence 'What does the sentence (What does the sentence ( What does the sentence ...)mean?) mean?' mean?" mean?
How is "used" being used" in the above post?

Is recursion recursively recursive?

Fixed point computations are so lame.

[Gary Childress]

To do that, all we have to figure out, and know, is what the words "statement", "proposition", "true" and "false" mean. And specifically, how they were used in the original formulation of LEM.

Some people are ideologically committed to staying in Plato's cave.

What does the word "mean" mean?
What does the word "word" mean?
How is the word "how" used in this question?
What does the the sentence "What does the sentence 'What does the sentence (What does the sentence ( What does the sentence ...)mean?) mean?' mean?" mean?
How is "used" being used" in the above post?

Is recursion recursively recursive?

Fixed point computations are so lame.

The words have their meanings or definitions in dictionaries, and they seem to be widely understood for the most part. Perhaps their concepts could be part of some universal human grammar, I don't know (not very knowledgeable about linguistics).

[Magnus Anderson]

This is essential to understanding the LEM. This is how we come the modern take on the LEM:

If the proposition is decidable then the proposition is true or the proposition is false.

That's the crux of the problem. It's not essential at all. It's a distraction.

It is not possible to understand the LEM without understanding the notion of decidability.

Not true at all.

The original formulation of the LEM is as relevant today as the original Roman numerals. In fact, it is less relevant, because Roman numerals are consistent, while your take on the LEM is not.

Actually, it's very much relevant. In fact, far more relevant than Roman numerals are.

You are misinterpreting LEM. As a consequence of that, you are promoting the destruction of thought. You might not be doing that intentionally but that's what follows from what your actions.

No, in modern logic, the LEM is undecidable.

No, in modern logic, they don't know what LEM is. They have a horse and they call it a unicorn.

You refuse to make progress beyond the 17th century.

You refuse to think. All you're doing here is parroting other people's views. You present no reasoning whatsoever.

By rejecting modern logic and modern mathematics

I am not rejecting modern logic and modern mathematics.

[Magnus Anderson]

Some people are ideologically committed to staying in Plato's cave.

True. Skepdick, for example.

What does the word "mean" mean?
What does the word "word" mean?
How is the word "how" used in this question?
What does the the sentence "What does the sentence 'What does the sentence (What does the sentence ( What does the sentence ...)mean?) mean?' mean?" mean?
How is "used" being used" in the above post?

Is recursion recursively recursive?

Fixed point computations are so lame.

I agree that your objections are pretty lame.

[godelian]

You refuse to think. All you're doing here is parroting other people's views. You present no reasoning whatsoever.

I agree with the views of modern logic on the LEM. Besides that, I have nothing much original to say on the matter. There are lots of subjects on which I am critical, but modern logic is not one of them.

[Gary Childress]

On a side note, the philosophy professor who taught me what I know about continental philosophy pointed me one time to the phenomenon of trying to look for a word to use to explain something, that pause we take before we find the right word to use. I didn't understand his reference at the time but looking back, it does seem that it could hint toward a pre-linguistic concept existing prior to having a word to denote it. It's an interesting thought.

The professor's name was Wayne Froman. He taught at George Mason University for many years where he had tenure. I have since heard that he passed away not long ago. God bless him.

[Skepdick]

Some people are ideologically committed to staying in Plato's cave.

True. Skepdick, for example.

What does the word "mean" mean?
What does the word "word" mean?
How is the word "how" used in this question?
What does the the sentence "What does the sentence 'What does the sentence (What does the sentence ( What does the sentence ...)mean?) mean?' mean?" mean?
How is "used" being used" in the above post?

Is recursion recursively recursive?

Fixed point computations are so lame.

I agree that your objections are pretty lame.

Your perpetual inability to grasp the objection is a reflection of you externalizing your lameness onto me.

If LEM were true then the statement "Truth is valuable" would be either true or false.

If it's true - justify.
If it's false - justify.

[Skepdick]

The words have their meanings or definitions in dictionaries, and they seem to be widely understood for the most part. Perhaps their concepts could be part of some universal human grammar, I don't know (not very knowledgeable about linguistics).

If use is meaning then dictionaries cannot be authorities on the use of language.

This is the position the OP has burried his heels into. He insists that words only mean what he intends them to mean. Any meaning other than the intended meaning is a misinterpretation on part of his interlocutors.

Of course, he smuggled in a presupposition that he didn't tell you about.

He's pretending to be a mind-reader who has direct access to the intentions of the original author of LEM.

[Magnus Anderson]

He insists that words only mean what he intends them to mean. Any meaning other than the intended meaning is a misinterpretation on part of his interlocutors.

Exactly. That's as trivial and as banal as "2 + 2 = 4". But obviously, some people don't get it. Just as there are a few who don't get that "1" isn't equal to "0".

Of course, he smuggled in a presupposition that he didn't tell you about.

He's pretending to be a mind-reader who has direct access to the intentions of the original author of LEM.

There is no such thing as a direct access to anything other than whatever immediately surrounds you in your environment.

Everything else is indirectly accessed.

And that also includes what other people are saying.

You yourself have demonstrated on more than one occasion that you have hard time understanding statements written in English language.

[Magnus Anderson]

If LEM were true then the statement "Truth is valuable" would be either true or false.

If it's true - justify.
If it's false - justify.

I don't have to do that.

You have to learn that your demands won't be fulfilled unless the other side shares your opinion that it is necessary to fulfill them for the task at hand.

And you also have to learn that you thinking that something is necessary does not mean it's truly necessary ( which means you need to keep an open mind about it and discuss it if it proves necessary which it clearly is. )

You're a terrible person to have a conversation with. That's why I asked you to leave this thread long time ago ( which you refused to do, preferring to force yourself where your presence is absolutely undesirable, something that only a specific type of person would do. )

[Skepdick]

Exactly. That's as trivial and as banal as "2 + 2 = 4". But obviously, some people don't get it. Just as there are a few who don't get that "1" isn't equal to "0".

So banal you don't even grasp that the EXPRESSION "2+2=4" could be true; or NOT true.

So banal you don't even grasp that the EXPRESSION "1=0" has no implicit truth-value.

1=0 could be true. For some (unspecified) notion of equality.

There is no such thing as a direct access to anything other than whatever immediately surrounds you in your environment.

Everything else is indirectly accessed.

Access is access is access. Direct or indirect - you are still claiming to have access.

And that also includes what other people are saying.

So do you have access to what I am saying; or not?

That's a rhetorical question...

You yourself have demonstrated on more than one occasion that you have hard time understanding statements written in English language.

You have a hard time understanding. Anything.