What LEM is not

[Magnus Anderson]

According to your (incorrect) interpretation you are EITHER Magnus Anderson or Skepdick.
According to my (correct) interpretation you are JUST Magnus Anderson.

You're repeating yourself.

[Skepdick]

You're repeating yourself.

So are you, but what's your point?

[Magnus Anderson]

You're repeating yourself.

So are you, but what's your point?

You're forcing yourself in this discussion unnecessarily. If you have nothing to add, which you clearly don't, leave.

[Skepdick]

You're repeating yourself.

So are you, but what's your point?

You're forcing yourself in this discussion unnecessarily. If you have nothing to add, which you clearly don't, leave.

If I have nothing to add, I will leave.

Alas... here I am.

[Magnus Anderson]

If I have nothing to add, I will leave.

Alas... here I am.

You're not an authority on that subject -- nor any other. But thanks for revealing to us more about yourself.

[Skepdick]

If I have nothing to add, I will leave.

Alas... here I am.

You're not an authority on that subject -- nor any other. But thanks for revealing to us more about yourself.

I didn't say I am an authority on the subject., nor do I need to be an authority to add something.

Are you the authority on who is and isn't an authority?
Are you the authority on who does; and doesn't have something to add to the subject?

Revealing indeed.

[Gary Childress]

If the axiom/law of the excluded middle is potentially wrong or questionable, then where does that leave logic? Does that mean that logicians can no longer use the axiom to construct or support their arguments? And how important is the LEM for constructing arguments? Can arguments be constructed without using it as a foundation?

[godelian]

I have already told you that it does not matter what these terms mean in modern logic.

Your argument is based on actively ignoring centuries of progress in logic (and mathematics). That is how you end up with your incorrect take on the matter.

[godelian]

If the axiom/law of the excluded middle is potentially wrong or questionable, then where does that leave logic?

The LEM is not "wrong" or "questionable". In some contexts, the LEM applies and in other ones, it does not.

Not every question is decidable.

The truth value of the LEM is equivalent to the truth value of David Hilbert's Entscheidungsproblem:

Can every yes/no question be answered with a yes or a no?

The question was finally answered in 1936 by Alonzo Church and Alan Turing:

https://en.m.wikipedia.org/wiki/Entscheidungsproblem

In mathematics and computer science, the Entscheidungsproblem (German for 'decision problem'; pronounced [ɛntˈʃaɪ̯dʊŋspʁoˌbleːm]) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928.[1] It asks for an algorithm that considers an inputted statement and answers "yes" or "no" according to whether it is universally valid, i.e., valid in every structure. Such an algorithm was proven to be impossible by Alonzo Church and Alan Turing in 1936.

The answer is a resounding "no".

There is no algorithm possible that can do this.

Does that mean that logicians can no longer use the axiom to construct or support their arguments? And how important is the LEM for constructing arguments? Can arguments be constructed without using it as a foundation?

You must first explicitly establish all system-wide premises and determine if the LEM is generally sustainable in that context. If not, then the LEM may still be sustainable for some questions but not for other ones.

The constructivist concern is not about banning the use of the LEM. It is about carefully investigating if the LEM is sustainable in the context of a particular set of system-wide premises (a "theory").

If a theory is decidable, then you can use the LEM. If a theory is not generally decidable, but the particular question still turns out to be decidable, then you can still use the LEM.

The problem with quite a few philosophers is that they believe that reasoning in a context without system-wide premises is possible.

This is a misguided view. It always means that the person is simply not aware of the existence of these system-wide premises and deliberately ignores them.

Logic itself is already a set of system-wide premises. Systemless reasoning is therefore not even possible.

[Magnus Anderson]

Your argument is based on actively ignoring centuries of progress in logic (and mathematics). That is how you end up with your incorrect take on the matter.

You don't quite understand the modern developments in logic that you obsess over.

But most importantly, you do not really know what LEM is. A consequence of you not really knowing how to determine the meaning of a sentence.

[Magnus Anderson]

In some contexts, the LEM applies and in other ones, it does not.

It applies in all contexts, you merely don't understand what LEM is.

[godelian]

In some contexts, the LEM applies and in other ones, it does not.

It applies in all contexts, you merely don't understand what LEM is.

You seem to be unfamiliar with David Hilbert's Entscheidungsproblem, 'Do all yes/no questions have a "yes" or a "no" answer?', and also unfamiliar with Alonzo Church and Alan Turing's answer to David Hilbert's question that a general algorithm to answer yes/no questions is not possible.

This is equivalent to stating that the LEM is generally undecidable.

I have already mentioned Robert Harper's notes in which he explains exactly that:

https://www.cs.cmu.edu/~rwh/courses/clo ... ts/lem.pdf

Neither Provable nor Refutable

From a constructive point of view, the judgement A ∨ ¬A true asserts that the problem expressed by the proposition A is decidable in that we either know a proof of A or we know a proof of ¬A (i.e., we can refute A). On this interpretation, it is obvious that not every proposition is decidable. Indeed, an open problem is precisely a proposition for which we have neither a proof nor a refutation.

Since there are (and always will be) open problems, we cannot expect LEM to be true, constructively speaking.

Since some problems are, in fact, decidable, we cannot expect LEM to be refuted either. That is, we cannot expect ¬(A ∨ ¬A) true to be evident in
general. In fact we can show something even stronger: constructively logic positively denies the falsehood (refutability) of (every instance of) LEM.

Your views on the LEM are not just contrary to modern logic. You systematically reject centuries of in-depth investigation on the matter. What you are claiming is simply in violation of mainstream uncontested knowledge. Seriously, who else sees things like you do? Your opinion is obviously plain wrong.

[Magnus Anderson]

I have already mentioned Robert Harper's notes in which he explains exactly that:

https://www.cs.cmu.edu/~rwh/courses/clo ... ts/lem.pdf

Neither Provable nor Refutable

From a constructive point of view, the judgement A ∨ ¬A true asserts that the problem expressed by the proposition A is decidable in that we either know a proof of A or we know a proof of ¬A (i.e., we can refute A). On this interpretation, it is obvious that not every proposition is decidable. Indeed, an open problem is precisely a proposition for which we have neither a proof nor a refutation.

Since there are (and always will be) open problems, we cannot expect LEM to be true, constructively speaking.

Since some problems are, in fact, decidable, we cannot expect LEM to be refuted either. That is, we cannot expect ¬(A ∨ ¬A) true to be evident in
general. In fact we can show something even stronger: constructively logic positively denies the falsehood (refutability) of (every instance of) LEM.

There is nothing in that Robert Harper's document that I haven't previously addressed.

1. Harper does not understand what LEM is. Nor do constructivists.

2. The constructivist version of LEM isn't really LEM. It's LEM only in name. You can use the word "unicorn" to speak of horses all you want, and you can even talk about how similar unicorns and horses are, but it won't change the fact that horses aren't unicorns. That's what's taking place here.

3. LEM really only applies to propositional sentences, i.e. sentences that represent actual propositions. The constructivist version of LEM goes further than that -- it also applies to non-propositional sentences, i.e. sentences that do not represent propositions, which is why they have a need for another value beside "true" and "false". That third value, however, is not a truth value.

4. Constructivists confuse veracity with provability. They define their version of LEM in terms of provability. But LEM isn't about what can be proven or disproven.

All of this is a consequence of a serious disregard for definitions and language in general ( language being the foundation of all thought. )

Definitional logic completely escapes them.

Your views on the LEM are not just contrary to modern logic. You systematically reject centuries of in-depth investigation on the matter. What you are claiming is simply in violation of mainstream uncontested knowledge. Seriously, who else sees things like you do? Your opinion is obviously plain wrong.

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

[godelian]

1. Harper does not understand what LEM is. Nor do constructivists.

2. The constructivist version of LEM isn't really LEM. It's LEM only in name. You can use the word "unicorn" to speak of horses all you want, and you can even talk about how similar unicorns and horses are, but it won't change the fact that horses aren't unicorns. That's what's taking place here.

3. LEM really only applies to propositional sentences, i.e. sentences that represent actual propositions. The constructivist version of LEM goes further than that -- it also applies to non-propositional sentences, i.e. sentences that do not represent propositions, which is why they have a need for another value beside "true" and "false". That third value, however, is not a truth value.

4. Constructivists confuse veracity with provability. They define their version of LEM in terms of provability. But LEM isn't about what can be proven or disproven.

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.

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.

[Veritas Aequitas]

The Law of Excluded Middle ( LEM ) is captured by the statement "For every proposition P, P is either true or it is false".
This should not be confused with the statement "For every conceivable thing T, T is either X or it is not X".
...........
In the case of LEM, the word "true" and "false" are very clearly defined as complementary.
So LEM remains true.
And binary classification ( i.e. the process of classifying a set of conceivable phenomena into two sets ) remains a separate thing from it.

Here's a note from WIKI;

The [LEM] principle should not be confused with the semantical principle of bivalence, which states that every proposition is either true or false.
The principle of bivalence always implies the law of excluded middle, while the converse is not always true.
A commonly cited counterexample uses statements unprovable now, but provable in the future to show that the law of excluded middle may apply when the principle of bivalence fails.
https://en.wikipedia.org/wiki/Law_of_excluded_middle

Here is a note from AI[wR];

Law of Excluded Middle (LEM) vs. Principle of Bivalence

Law of Excluded Middle (LEM):
Meaning: For any proposition, it must be either true or false, there is no third option.
Example: A given number is either even or odd. There's no other possibility.

Principle of Bivalence:
Meaning: Every proposition has a definite truth value, either true or false.
Example: The statement "The Earth is round" is either true or false.

Key Difference:
LEM: Focuses on the logical possibilities of a proposition.
Bivalence: Focuses on the semantic truth value of a proposition.

Why LEM Implies Bivalence but Not Vice Versa:
LEM to Bivalence: If a proposition must be either true or false (LEM), then it naturally has a definite truth value (bivalence).
Bivalence to LEM: While a proposition might have a definite truth value (bivalence), it doesn't necessarily mean there are only two logical possibilities. For example, in some non-classical logics, there might be a third truth value like "indeterminate" or "unknown".

Example:
LEM: "Either it will rain tomorrow, or it will not rain tomorrow."
Bivalence: The statement "It will rain tomorrow" is either true or false.
In this example, the LEM ensures that there are only two possibilities, while the bivalence principle ensures that one of these possibilities is true.

In essence: The LEM is a more restrictive principle that guarantees both the existence of truth values and the exclusion of other possibilities. The principle of bivalence is more general, ensuring only the existence of truth values.

I asked AI
Both the LEM and Bivalence must be conditioned or contingent upon a framework and system of reality or knowledge.

For example,
LEM: "Either it will rain tomorrow, or it will not rain tomorrow."
Bivalence: The statement "It will rain tomorrow" is either true or false.
must be conditioned upon the Logic & meteorological  Framework and System.

You're absolutely right that both the Law of Excluded Middle (LEM) and the Principle of Bivalence are contingent upon the framework and system of reality or knowledge we're operating within.

Here's a breakdown of why:

Framework of Reality:
Classical Logic: The LEM and bivalence are foundational principles in classical logic, which assumes a deterministic, objective reality with clear-cut true or false propositions.
Non-Classical Logics: In some non-classical logics (like fuzzy logic, paraconsistent logic, or intuitionistic logic), the LEM might be weakened or even rejected, as they deal with uncertainty, contradictions, or different notions of truth.

System of Knowledge:
Empirical Knowledge: In empirical sciences, the LEM is often assumed, as observations are typically categorized as either true or false based on evidence.
Theoretical Knowledge: In theoretical domains like mathematics or philosophy, the LEM might be challenged or questioned, especially when dealing with abstract concepts or hypothetical scenarios.

Your meteorological example is a perfect illustration:
The LEM and bivalence hold within the meteorological framework, assuming that weather can be accurately predicted and categorized as either raining or not raining.
However, if we were dealing with a situation where the weather prediction was highly uncertain or could involve multiple states (e.g., drizzle, light rain, heavy rain), the LEM and bivalence might not be as applicable.

In conclusion, the LEM and bivalence are powerful tools within their respective frameworks, but their applicability can vary depending on the specific context and the nature of the reality or knowledge being considered.

In other words, one cannot simply make statement like that of the OP without the imperative qualification to a specific human-based framework and system of reality and knowledge.