What LEM is not

[Magnus Anderson]

ChatGPT rejects your claim that undecidable propositions are not propositions:

But I never said that.

[Magnus Anderson]

Sorry, what I mean to say is that the LEM is basically reducible to P=A or else P = not the case A (that's two alternatives and it must be one or the other). There is no B or third alternative. Correct?

If A stands for a truth value, and if you mean that the truth value of P is either A or not-A, and that the truth value of P cannot be neither A nor not-A, then I agree.

Is that what you meant?

So for example the LEM would not be P = True, False, or else neither True nor False. That would be 3 alternatives.

Is that correct?

Yes. P is either true or false. "Neither true nor false" is an oxymoron. Since "false" means "not true", it translates to "Neither true nor not true" which in turn translates to "Not true and not not true" which in turn translates to "False and not false" which is an obvious contradiction in terms.

And it's not the case that the LEM is identical to P = A or B. Is that correct? By that I mean, the LEM is not the same as saying a proposition is either red or green. Red is not the opposite of green and green is not the opposite of red. In contrast, true is the opposite of false. It's not just a random arbitrary second alternative in the sense that red and green are to each other.

Yes, that's most definitely correct.

[godelian]

ChatGPT rejects your claim that undecidable propositions are not propositions:

But I never said that.

ChatGPT:

What is the truth value of an undecidable proposition?

An undecidable proposition does not have a truth value; it cannot be definitively classified as true or false within a given formal system. This means that, based on the axioms and rules of that system, there is no way to prove or disprove the proposition. Examples of undecidable propositions often arise in mathematical logic and set theory, such as those highlighted by Gödel's incompleteness theorems.

So, in that case, how can an undecidable proposition be true or false, since it has no truth value?

[Magnus Anderson]

So, in that case, how can an undecidable proposition be true or false, since it has no truth value?

A proposition is said to be undecidable within a formal system if its truth value cannot be deduced within that formal system.

Basically, what this means is that, given a set of initial premises P and a set of inference rules R, a proposition is undecidable if it cannot be deduced from P using R.

But the fact that the truth value of a proposition cannot be deduced within a formal system does not mean the proposition has no truth value.

The other thing that you have to take into account is that different people define the term "proposition" in different ways. Some people use the word to mean the same thing as "statement". A statement is a linguistic entity, a sequence of words with words being a sequence of letters. When defined that way, a proposition can indeed lack truth value. An example would be, "This statement is false". But other people define the term more narrowly to refer to an idea that a portion of reality exists in certain state. Defined this way, propositions aren't statements, they are merely ideas that can be represented by statements. Moreover, defined this way, every proposition has truth value, i.e. there are no propositions that have no truth value.

The question is merely whether LEM is concerned with propositions in the linguistic sense of the word or whether it's concerned with propositions in the non-linguistic sense of the word?

And my claim is that LEM is solely concerned with propositions in the non-linguistic sense.

[godelian]

But the fact that the truth value of a proposition cannot be deduced within a formal system does not mean the proposition has no truth value.

So, you are effectively rejecting chatGPT's answer.

But other people define the term more narrowly to refer to an idea that a portion of reality exists in certain state.

If it is about a portion of physical reality, it never has a logical truth value. It can only have a probabilistic truth value.

Moreover, defined this way, every proposition has truth value, i.e. there are no propositions that have no truth value.

That amounts again to rejecting the existence of undecidable propositions by referring to statements about physical reality which cannot have a logical truth value to begin with. Physical reality is always probabilistic.

And my claim is that LEM is solely concerned with propositions in the non-linguistic sense.

In your take on the matter, the LEM is not a claim about logic. At the same time, it cannot be a claim about physical reality either because it is not probabilistic.

[Magnus Anderson]

So, you are effectively rejecting chatGPT's answer.

Not necessarily. It depends on what Chat GPT means when it says that "An undecidable proposition has no truth value within a formal system within which it is decidable".

If it's merely saying that the proposition is undecidable within that formal system, then I am not rejecting its claim.

In that case, Chat GPT would be merely using a language that is potentially misleading and unnecessarily so.

If it is about a portion of physical reality, it never has a logical truth value. It can only have a probabilistic truth value.

Can you explain the difference between "logical" truth values and "probabilistic" truth values?

I am not aware of any such truth values.

A truth value is a degree to which a proposition accurately represents the referred portion of reality.

Truth values are values such as "true", "false", "completely false", "half-true", "30% true" and so on.

Probability is a subjective thing that has to do with certainty ( = confidence ) which itself has to do with beliefs.

Only beliefs, i.e. propositions that are held by people, have a degree of certainty associated with them. Certainty indicates how confident someone is in a belief.

When you're evaluating the truth value of a mathematical statement such as "2 + 2 = 4", you have to first understand what these symbols mean. To do so, you have to figure out what language was used to construct that statement. If the statement was made by someone other than you, you have to figure out what language was used by that person. That may involve looking outside of yourself. It may also involve probabilistic reasoning. If it was made by you, you can try to correctly recall what language you used. It may be enough, but not necessarily, to look inside yourself. But even then, you might have to use probabilistic reasoning. Moreover, both of these processes are physical. And at the end of both, you will end up with a degree of confidence in your belief that "2 + 2 = 4" has whatever truth value you came to think it has.

In your take on the matter, the LEM is not a claim about logic. At the same time, it cannot be a claim about physical reality either because it is not probabilistic.

All meaningful claims are about physical reality. Even mathematics is about physical reality. The difference merely lies in what aspects of reality are being described. Mathematics concerns itself solely with symbols and concepts attached to them. Both symbols and concepts are part of physical reality. Symbols exist both within and outside of minds. Concepts, on the other hand, exist only inside minds.

[godelian]

When you're evaluating the truth value of a mathematical statement such as "2 + 2 = 4", you have to first understand what these symbols mean. To do so, you have to figure out what language was used to construct that statement. If the statement was made by someone other than you, you have to figure out what language was used by that person.

Addition in Peano Arithmetic is recursively defined as:

[1] a + 0 = a
[2] a + S(b) = S(a+b)

So, the expression "2 + 2" gets reduced to S(S(2)), i.e. the second successor of 2, which is 4.

Peano Arithmetic is an objectively shared common understanding. There are no degrees of freedom as to what the symbols "2+2=4" mean. I do not have to figure out what language was used by the person who wrote this.

And at the end of both, you will end up with a degree of confidence in your belief that "2 + 2 = 4" has whatever truth value you came to think it has.

It is mechanically verifiable. There are no degrees of freedom in that matter.

Even mathematics is about physical reality.

No, mathematics is absolutely not about physical reality:

https://en.wikipedia.org/wiki/Formalism ... thematics)

In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules.

A central idea of formalism is that mathematics is not a body of propositions representing an abstract sector of reality.

According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all.

[Magnus Anderson]

I do not have to figure out what language was used by the person who wrote this.

If you're addressing their statement, you absolutely do. Otherwise, you are risking misinterpreting them.

If you know what language they are using, but you don't understand their language enough to understand what they are saying, but you're sure that their statement has been accurately translated to a language that you understand, then you may switch your attention away from their statement towards the translation that you can understand.

As an example, suppose you want to evaluate the truth value of "2 + 2 = 4". The first thing you have to do is to understand what language that statement is expressed in. This step can't be skipped. Once you do that, and understand that it is expressed in the standard language of arithmetic, you have an option to understand it directly or look for an accurate translation that you can actually understand. If you choose the latter, you have to make sure the translation is truly accurate. Otherwise, you will end up misinterpreting the statement. Suppose you understand the language of Peano Arithmetic. In PA, "2 + 2 = 4" can be accurately expressed as "S(S(0)) + S(S(0)) = S(S(S(S(0))))". As such, you are free to move your attention away from "2 + 2 = 4" towards the PA expression. But again, in order to understand the PA expression, you have to decode that expression so as to discover what it's actually saying. You do that by looking at the language of Peano Arithmetic and seeing what symbols such as "S(x)" actually mean and how they can be combined to create more complex meanings. Once you decipher the meaning of that statement, you can proceed to evaluate the truth value of the statement. The same exact process that I described earlier applies.

The point of this thread is that, in the case of LEM, LEM has been mistranslated. This can only be resolved by learning the language with which it was originally expressed. Focusing in how it was expressed in modern, formal, languages is pointless because the claim is that it wasn't correctly translated.

[Magnus Anderson]

No, mathematics is absolutely not about physical reality:

https://en.wikipedia.org/wiki/Formalism ... thematics)

In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules.

A central idea of formalism is that mathematics is not a body of propositions representing an abstract sector of reality.

According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all.

And formalists happen to be terribly wrong.

[Gary Childress]

Sorry, what I mean to say is that the LEM is basically reducible to P=A or else P = not the case A (that's two alternatives and it must be one or the other). There is no B or third alternative. Correct?

If A stands for a truth value, and if you mean that the truth value of P is either A or not-A, and that the truth value of P cannot be neither A nor not-A, then I agree.

Is that what you meant?

Yes.

[godelian]

No, mathematics is absolutely not about physical reality:

https://en.wikipedia.org/wiki/Formalism ... thematics)

In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules.

A central idea of formalism is that mathematics is not a body of propositions representing an abstract sector of reality.

According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all.

And formalists happen to be terribly wrong.

ChatGPT:

Is mathematical formalism wrong?

Mathematical formalism isn't inherently "wrong," but it's one perspective among several in understanding mathematics. Formalism emphasizes the manipulation of symbols and adherence to rules without necessarily interpreting the symbols in terms of real-world meaning or intuition.

Critics argue that this approach can sometimes overlook the intuitive and conceptual aspects of mathematics. Supporters, however, argue that it provides a rigorous framework that ensures consistency and clarity.

Ultimately, the effectiveness of formalism depends on the context in which it's applied and the goals of the mathematicians involved. Different mathematical philosophies—like Platonism, intuitionism, and formalism—each offer valuable insights into the nature of mathematics.

Mathematics is multifaceted. Formalism is necessary as a perspective in order to stamp out physicalism which would otherwise be rampant. While Platomism is another necessary perspective, in order to develop a deeper understanding of the abstract, Platonic world investigated by mathematics, physicalism is unacceptable.

[Magnus Anderson]

Mathematics is multifaceted. Formalism is necessary as a perspective in order to stamp out physicalism which would otherwise be rampant. While Platomism is another necessary perspective, in order to develop a deeper understanding of the abstract, Platonic world investigated by mathematics, physicalism is unacceptable.

If you say that mathematics isn't about physical reality, that's not merely a perspective, that's a claim, and a false one at that. ( Mathematics is about mathematical concepts and mathematical concepts are physical entities. )

If you say that mathematical symbols are meaningless, not merely outside of all formal systems, but within formal systems as well, that's not merely a perspective, it's a claim, and a false one at that. ( Mathematical symbols such as "2" are not meaningless and there are few things more obvious than that. If all you want to say is that what mathematical symbols mean can be ignored for a given purpose, that's one thing. Claiming that mathematical symbols have no meaning is another thing. )

If you say that the truth value of a mathematical statement is relative to a formal system and that the truth value of a statement is determined by whether that mathematical statement or its contradiction can be constructed within a given formal system, that's not merely a perspective, it's a claim, and a false one at that.

If you want formalism to be just a method, then you shouldn't be making claims about what mathematics is.

[Skepdick]

If you say that mathematics isn't about physical reality, that's not merely a perspective, that's a claim, and a false one at that. ( Mathematics is about mathematical concepts and mathematical concepts are physical entities. )

Which physical entity is the concept "is" when you say "X is Y" ?

[Magnus Anderson]

If you say that mathematics isn't about physical reality, that's not merely a perspective, that's a claim, and a false one at that. ( Mathematics is about mathematical concepts and mathematical concepts are physical entities. )

Which physical entity is the concept "is" when you say "X is Y" ?

Which physical entity is represented by the word "Skepdick"? I am not going to believe you exist until you tell me exactly where you live.

[Skepdick]

If you say that mathematics isn't about physical reality, that's not merely a perspective, that's a claim, and a false one at that. ( Mathematics is about mathematical concepts and mathematical concepts are physical entities. )

Which physical entity is the concept "is" when you say "X is Y" ?

Which physical entity is represented by the word "Skepdick"? I am not going to believe you exist until you tell me exactly where you live.

Equivocation.

The entity represented by "Skepdick" is me. Where I live (my home address) is not (currently) where I am.
Which physical entity is being represented by "is". You are claiming that the concept of "is" has a physical location - like I do.

Where is it?

Also, in your own nomenclature you are using "is" as a predicate...

2+2 is 5.
2+2 is 4.

So predicates are physical entities then?