The constructivist animosity against the law of the excluded middle

[Veritas Aequitas]

My philosophical stance is that of Constructivism [Kantian]. The LEM is only useful for classical logic which is limited.
ALL of reality is conditioned upon a continuum from 0.001% to 99.99%

Here is AI's [wR] on the why LEM is limited;

The Law of the Excluded Middle (LEM) states that for any proposition P, either P is true or not-P is true. There's no third option. Constructivists, however, take issue with this principle in the context of mathematics. Here's a breakdown of their critique:

1. Emphasis on Construction: Constructivists believe mathematics is about building and manipulating objects. A proof, for them, isn't just logical deduction but a constructive procedure that demonstrates the existence of a mathematical entity or the validity of an operation.

2. Undecidable Propositions: Not all propositions might be demonstrably true or false. LEM forces a decision, but what if we can't construct a proof for P or not-P? This could be due to the limitations of our current knowledge or the inherent nature of the proposition.

3. The Case of Infinite Sets: Classical mathematics freely uses LEM for infinite sets. But constructivists argue that we can't necessarily verify the truth or falsity of statements about infinite objects. We can't check every element in an infinite set, so how can we be sure a property either holds or doesn't for all elements?

4. Alternative Logic: Intuitionistic logic, a foundation for constructive mathematics, rejects LEM. It focuses on proofs rather than truth values. If we can't construct a proof for P or not-P, the proposition remains undecided within the system.

Analogy: Imagine you have a box full of real numbers. LEM says every number must be either rational or irrational. Constructivists argue that with our current tools, we might not be able to definitively categorize every number we pull out. There could be numbers that resist classification for now.

By rejecting LEM, constructivists develop a more cautious and step-by-step approach to mathematics.
They focus on what can be demonstrably constructed rather than relying on the principle of having to pick a side.
This alternative viewpoint has led to interesting developments in constructive mathematics, offering a different lens to explore mathematical concepts.

[night912]

My philosophical stance is that of Constructivism [Kantian]. The LEM is only useful for classical logic which is limited.
ALL of reality is conditioned upon a continuum from 0.001% to 99.99%

Here is AI's [wR] on the why LEM is limited;

The Law of the Excluded Middle (LEM) states that for any proposition P, either P is true or not-P is true. There's no third option. Constructivists, however, take issue with this principle in the context of mathematics. Here's a breakdown of their critique:

1. Emphasis on Construction: Constructivists believe mathematics is about building and manipulating objects. A proof, for them, isn't just logical deduction but a constructive procedure that demonstrates the existence of a mathematical entity or the validity of an operation.

2. Undecidable Propositions: Not all propositions might be demonstrably true or false. LEM forces a decision, but what if we can't construct a proof for P or not-P? This could be due to the limitations of our current knowledge or the inherent nature of the proposition.

3. The Case of Infinite Sets: Classical mathematics freely uses LEM for infinite sets. But constructivists argue that we can't necessarily verify the truth or falsity of statements about infinite objects. We can't check every element in an infinite set, so how can we be sure a property either holds or doesn't for all elements?

4. Alternative Logic: Intuitionistic logic, a foundation for constructive mathematics, rejects LEM. It focuses on proofs rather than truth values. If we can't construct a proof for P or not-P, the proposition remains undecided within the system.

Analogy: Imagine you have a box full of real numbers. LEM says every number must be either rational or irrational. Constructivists argue that with our current tools, we might not be able to definitively categorize every number we pull out. There could be numbers that resist classification for now.

By rejecting LEM, constructivists develop a more cautious and step-by-step approach to mathematics.
They focus on what can be demonstrably constructed rather than relying on the principle of having to pick a side.
This alternative viewpoint has led to interesting developments in constructive mathematics, offering a different lens to explore mathematical concepts.

That's a false analogy. An accurate analogy would be this:

Imagine you have a box full of real numbers. LEM stipulate that there are two known propositions here, and it applies to each individually.

Proposition 1: Every number is either rational or not rational.

Proposition 2: Every number is either irrational or not irrational.

This analogy demonstrates that LEM is not limited in this situation.

[Veritas Aequitas]

My philosophical stance is that of Constructivism [Kantian]. The LEM is only useful for classical logic which is limited.
ALL of reality is conditioned upon a continuum from 0.001% to 99.99%

Here is AI's [wR] on the why LEM is limited;

The Law of the Excluded Middle (LEM) states that for any proposition P, either P is true or not-P is true. There's no third option. Constructivists, however, take issue with this principle in the context of mathematics. Here's a breakdown of their critique:

1. Emphasis on Construction: Constructivists believe mathematics is about building and manipulating objects. A proof, for them, isn't just logical deduction but a constructive procedure that demonstrates the existence of a mathematical entity or the validity of an operation.

2. Undecidable Propositions: Not all propositions might be demonstrably true or false. LEM forces a decision, but what if we can't construct a proof for P or not-P? This could be due to the limitations of our current knowledge or the inherent nature of the proposition.

3. The Case of Infinite Sets: Classical mathematics freely uses LEM for infinite sets. But constructivists argue that we can't necessarily verify the truth or falsity of statements about infinite objects. We can't check every element in an infinite set, so how can we be sure a property either holds or doesn't for all elements?

4. Alternative Logic: Intuitionistic logic, a foundation for constructive mathematics, rejects LEM. It focuses on proofs rather than truth values. If we can't construct a proof for P or not-P, the proposition remains undecided within the system.

Analogy: Imagine you have a box full of real numbers. LEM says every number must be either rational or irrational. Constructivists argue that with our current tools, we might not be able to definitively categorize every number we pull out. There could be numbers that resist classification for now.

By rejecting LEM, constructivists develop a more cautious and step-by-step approach to mathematics.
They focus on what can be demonstrably constructed rather than relying on the principle of having to pick a side.
This alternative viewpoint has led to interesting developments in constructive mathematics, offering a different lens to explore mathematical concepts.

That's a false analogy. An accurate analogy would be this:

Imagine you have a box full of real numbers. LEM stipulate that there are two known propositions here, and it applies to each individually.

Proposition 1: Every number is either rational or not rational.

Proposition 2: Every number is either irrational or not irrational.

This analogy demonstrates that LEM is not limited in this situation.

I am not too sure of your argument, so, from AI [with Reservation]:

The argument presented in the text is flawed and does not demonstrate that the Law of Excluded Middle (LEM) is limited in its application.

Here's a breakdown of the issues:

Misunderstanding of LEM: The LEM states that for any proposition, either the proposition is true or its negation is true, and there is no third option. It does not imply that there are only two possible propositions.

Redundancy in Propositions: The two propositions presented are essentially the same. "Every number is either rational or not rational" is equivalent to "Every number is either irrational or not irrational." This does not demonstrate any limitation of LEM.

Irrelevance of Analogy: The analogy of a box full of real numbers does not provide any additional insight into the applicability of LEM. The LEM applies to logical propositions, not physical objects.

Therefore, the argument does not support the claim that LEM is limited in its use.

[night912]

My philosophical stance is that of Constructivism [Kantian]. The LEM is only useful for classical logic which is limited.
ALL of reality is conditioned upon a continuum from 0.001% to 99.99%

Here is AI's [wR] on the why LEM is limited;

That's a false analogy. An accurate analogy would be this:

Imagine you have a box full of real numbers. LEM stipulate that there are two known propositions here, and it applies to each individually.

Proposition 1: Every number is either rational or not rational.

Proposition 2: Every number is either irrational or not irrational.

This analogy demonstrates that LEM is not limited in this situation.

I am not too sure of your argument, so, from AI [with Reservation]:

The argument presented in the text is flawed and does not demonstrate that the Law of Excluded Middle (LEM) is limited in its application.

Here's a breakdown of the issues:

Misunderstanding of LEM: The LEM states that for any proposition, either the proposition is true or its negation is true, and there is no third option. It does not imply that there are only two possible propositions.

Redundancy in Propositions: The two propositions presented are essentially the same. "Every number is either rational or not rational" is equivalent to "Every number is either irrational or not irrational." This does not demonstrate any limitation of LEM.

Irrelevance of Analogy: The analogy of a box full of real numbers does not provide any additional insight into the applicability of LEM. The LEM applies to logical propositions, not physical objects.

Therefore, the argument does not support the claim that LEM is limited in its use.

Instead of using an AI to answer irrelevant to what i said, how about you actually do some critical thinking and address what I actually said.

[Skepdick]

That's a false analogy. An accurate analogy would be this:

Imagine you have a box full of real numbers. LEM stipulate that there are two known propositions here, and it applies to each individually.

Proposition 1: Every number is either rational or not rational.

Proposition 2: Every number is either irrational or not irrational.

This analogy demonstrates that LEM is not limited in this situation.

It doesn't demonstrate any such thing. Your use of "either" implies an axiomatic presupposition of LEM which makes your argument circular and insufficient to make your own case.

The Either monad is a computational reformulation of LEM.

https://hackage.haskell.org/package/cat ... ither.html

Your propositions are unproven until you present a proof. A decision-procedure which determines which disjunct is true (Left or Right) for any given element taken from the box.

[night912]

That's a false analogy. An accurate analogy would be this:

Imagine you have a box full of real numbers. LEM stipulate that there are two known propositions here, and it applies to each individually.

Proposition 1: Every number is either rational or not rational.

Proposition 2: Every number is either irrational or not irrational.

This analogy demonstrates that LEM is not limited in this situation.

It doesn't demonstrate any such thing. Your use of "either" implies an axiomatic presupposition of LEM which makes your argument circular and insufficient to make your own case.

The Either monad is a computational reformulation of LEM.

https://hackage.haskell.org/package/cat ... ither.html

Your propositions are unproven until you present a proof. A decision-procedure which determines which disjunct is true (Left or Right) for any given element taken from the box.

Steelman my argument.

[Skepdick]

Steelman my argument.

That's precisely what I did.

I turned LEM from a mere axiom to a theorem. Pending proof.

[night912]

Steelman my argument.

That's precisely what I did.

I turned LEM from an axiom to a theorem. Awaiting proof.

In that case, you don't know what my argument is.

[Skepdick]

In that case, you don't know what my argument is.

Great! Formalize it then.

https://agda.readthedocs.io/en/latest/g ... -agda.html

[night912]

In that case, you don't know what my argument is.

Great! Formalize it then.

https://agda.readthedocs.io/en/latest/g ... -agda.html

Can you tell me what you think is being discussed.

[Skepdick]

Can you tell me what you think is being discussed.

"Propositions" lacking proofs.

https://ncatlab.org/nlab/show/propositions+as+types

In type theory, the paradigm of propositions as types says that propositions and types are essentially the same. A proposition is identified with the type (collection) of all its proofs

I imagine this was a giveaway...

Proposition 1: Every number is either rational or not rational.

Proposition 2: Every number is either irrational or not irrational.

This analogy demonstrates that LEM is not limited in this situation.

Now you just have to codify/formaize the types (present the collection of proofs) which make your propositions true.

[night912]

Can you tell me what you think is being discussed.

"Propositions" lacking proofs.

https://ncatlab.org/nlab/show/propositions+as+types

In type theory, the paradigm of propositions as types says that propositions and types are essentially the same. A proposition is identified with the type (collection) of all its proofs

I imagine this was a giveaway...

Proposition 1: Every number is either rational or not rational.

Proposition 2: Every number is either irrational or not irrational.

This analogy demonstrates that LEM is not limited in this situation.

That's not what was being discussed. Whether Veritas Aequitas's analogy is valid or not, is what is being discussed.

[Skepdick]

That's not what was being discussed. Whether Veritas Aequitas's analogy is valid or not, is what is being discussed.

I am literally explaining "validity" to you.

Validity in a paradigm where LEM doesn't hold axiomatically.

[night912]

That's not what was being discussed. Whether Veritas Aequitas's analogy is valid or not, is what is being discussed.

I am literally explaining "validity" to you.

Terms witnessing the "A implies B".

Apparently, there's no point in continuing our discussion further because you're not willing to know what was actually being discussed. Ironically, you tried to explain what validity is, while not giving any valid points to what is being discussed.

[Skepdick]

Apparently, there's no point in continuing our discussion further because you're not willing to know what was actually being discussed. Ironically, you tried to explain what validity is, while not giving any valid points to what is being discussed.

Ironically. You don't even know what irony is. Or validity.