What LEM is not

[Magnus Anderson]

You defined a proposition as a logic sentence representing a state in physical reality. Let's call this a "physicalist" sentence.

The LEM itself is not such proposition, as it is about other propositions and not about a state in physical reality. Hence, in your take on the matter, the LEM does not even apply to itself.

Do you agree that propositions physically exist e.g. as ideas inside people's heads ( as beliefs, for example )?

If so, do you agree that LEM is about every oonceivable proposition?

If so, do you agree that what is true for every conceivable X is also true for every X that existed, exists or will exist?

Ultimately, these so-called analytic statements can be said to be about concepts. Do you agree that concepts physically exist e.g. inside people's brains?

This sounds like a psychologically defined set. Which person would we take as the benchmark for the "psychological conceiving"? How do you guarantee that the proposition that one person can conceive, is also conceivable to another person?

I don't see the relevance of your remark. Of course, there might be people who, for whatever reason, cannot imagine some propositions.

The concept attached to the word "proposition", or in plain terms, the meaning of the word "proposition", determines the set of all conceivable propositions.

If there are people who cannot conceive of some propositions, they are simply mentally deficient.

Furthermore, the "for each" quantifier is not supported for every set. On what grounds do you believe that this psychological set of propositions is recursively enumerable? What algorithm is capable of enumerating this set?

Well, it is your position that "for each" quantifier isn't supported for every set. You would have to prove that first.

As far as I am concerned, "for each" has nothing to do with enumeration and everything to do with signifying that a rule applies to every member of a set.

I can't enumerate every member of the set of all natural numbers because that would entail an infinite number of tasks to be performed and completed. But does that mean that I can't say, "For each natural number x, x is either even or odd" ?

[godelian]

Do you agree that propositions physically exist e.g. as ideas inside people's heads ( as beliefs, for example )?
Do you agree that concepts physically exist e.g. inside people's brains?

They can exist in people's brains. However, they can also exist as a string in computer memory.

Of course, there might be people who, for whatever reason, cannot imagine some propositions.
If there are people who cannot conceive of some propositions, they are simply mentally deficient.

The term "conceivable" is a poor approach to defining the term "proposition".

Well, it is your position that "for each" quantifier isn't supported for every set. You would have to prove that first.

It is actually possible to quantify over non-denumerable sets. However, the proof cannot visit every element of an non-denumerable set, while it is still required to prove that the property holds for every element:

ChatGPT: Is induction supported over non-denumerable sets?

Induction, particularly the principle of mathematical induction, is typically applied to well-ordered sets, such as the natural numbers. For non-denumerable sets, like the real numbers, traditional induction doesn't directly apply because these sets cannot be put into a one-to-one correspondence with the natural numbers.

However, there are generalizations of induction that can be used in certain contexts involving non-denumerable sets. For example, transfinite induction is a method that extends the principle of induction to well-ordered sets that may be larger than the natural numbers, including sets of ordinals.

In summary, while standard induction doesn't apply to non-denumerable sets, extensions like transfinite induction can be used in specific cases.

Proving a property for each element of a non-denumerable set is still possible but substantially harder. The set of "conceivable" propositions may even be undefinable, since it depends on unspecified psychology.

But does that mean that I can't say, "For each natural number x, x is either even or odd" ?

The natural numbers are denumerable. So, in that context, induction is perfectly possible.

[Magnus Anderson]

The set of "conceivable" propositions may even be undefinable, since it depends on unspecified psychology.

Actually, it has nothing to do with psychology. You merely misunderstood.

There are things that exist.

Examples: Horses, cats, people, etc.

Everything that exists is also conceivable. It does not matter if anyone can actually conceive these things.

Then there are things that do not exist but that can be conceived. It also does not matter if anyone can actually conceive these things.

Examples: Unicorns, dragons, etc.

And then there are things that do not exist and that cannot be conceived at all. Square-circles, for example.

Three rules:

1. Anything that is not an oxymoron is conceivable.

2. Only that which is conceivable can exist.

3. That which is conceivable does not necessarily exist.

If you don't like the word "conceivable", you can substitute it with "possible" or "has the ( minimum ) possibility to exist".

1. For every proposition P. ( Short version. )

2. For every conceivable proposition P. ( Long version. You say this if you want to emphasize that you're talking about every proposition that can possibly exist not merely the ones that exist. )

3. For every possible proposition P. ( Same as above. )

Of course, someone else might complain that even the word "possible" is vague in that there are different kinds of possibilities. Whenever we say, "X is possible", we are saying "X is possible under these conditions".

Here, I am talking about "possible under any conditions". Square-circles, for example, are not possible under any condition ( not counting redefining the term, for which one can say it's not really a condition. )

Some people are never satisfied and will complain about the slightest ambiguity that they can spot.

Some will even go so far as to invent ambiguity if they can't spot one just so that they can criticize.

[godelian]

Then there are things that do not exist but that can be conceived. It also does not matter if anyone can actually conceive these things.

If nobody can conceive them, then how do you know that they can be conceived?

1. Anything that is not an oxymoron is conceivable.

In model theory, this translates to:

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

It's a consequence of Gödel's completeness theorem (not to be confused with his incompleteness theorems) that a theory has a model if and only if it is consistent, i.e. no contradiction is proved by the theory.

In model theory, the "conception" is called the "theory", while the abstract reality that exists as a consequence of the existence of the theory, is called the "model".

2. Only that which is conceivable can exist.

Without theory, no model.

3. That which is conceivable does not necessarily exist.

The model always exists for every non-contradictory theory that exists.

2. For every conceivable proposition P. ( Long version. You say this if you want to emphasize that you're talking about every proposition that can possibly exist not merely the ones that exist. )

For every sentence P in the language L of theory T.

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

In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold).[1] The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954.[2]

Why invent your own take on the matter? I think that Tarski did a great job systematizing the subject in 1954.

[Skepdick]

...

What a long-winded way to say. For some formal language L : L ⊭ P ∨ ¬P ∧ L ⊬ P ∨ ¬P
Or simply ¬(∀L, L x ⊨ P ∨ ¬P) ∧ ¬(∀L, L x ⊢ P ∨ ¬P)

Where "⊨" means semantic entailment and "⊢" means syntactic entailment.

In fact I can make an even stronger claim: ¬∃L, L x ⊢ P ∨ ¬P.
There exist no formal language where LEM is syntactically entailed.

And if you want models of the above... Kripke models for intuitionistic logic.

[Magnus Anderson]

If nobody can conceive them, then how do you know that they can be conceived?

In order to know that X can be conceived, someone has to conceive it, i.e. construct a symbol that is free from contradictions and with which X can be represented.

But if X is conceivable it is conceivable regardless of whether anyone is aware of it, e.g. as a consequence of not being able to conceive it due to personal limitations.

In model theory, the "conception" is called the "theory", while the abstract reality that exists as a consequence of the existence of the theory, is called the "model".

Not quite. Conception is here understood as the act of creating a symbol free from contradictions with which a thing can be represented.

The word "cat" would be an example of a symbol that is free of contradictions and with which any given cat would be able to be represented.

Note that the word "cat" is not a proposition ( let alone a formal theory. )

[Skepdick]

Not quite. Conception is here understood as the act of creating a symbol free from contradictions with which a thing can be represented.

What's a contradiction? Can you conceive of it? What is it that you are representing with the symbol "contradiction" ?

How is it that you are conceiving something inconceivable?

[Magnus Anderson]

Not quite. Conception is here understood as the act of creating a symbol free from contradictions with which a thing can be represented.

What's a contradiction? Can you conceive of it? What is it that you are representing with the symbol "contradiction" ?

How is it that you are conceiving something inconceivable?

Do you understand the difference between a square-circle ( an unconceivable thing ) and a symbol such as "square-circle" that has a concept attached to it that says that the the term "square-circle" can only be used to represent shapes that are squares and circles ( i.e. not squares ) at the same time ( a conceivable thing ) ?

[Skepdick]

Do you understand the difference between a square-circle ( an unconceivable thing )

It has been conceived, even if your own powers of conception are limited.

https://en.wikipedia.org/wiki/Taxicab_geometry#Spheres

In any case you failed to explain what the symbol "contradiction" represents.

[Magnus Anderson]

It has been conceived, even if your own powers of conception are limited.

https://en.wikipedia.org/wiki/Taxicab_geometry#Spheres

Predictable. I already mentioned that nonsense earlier in the thread.

That's not a square-circle

Learn what the word actually means instead of treating it like a symbol full of free variables so that you can misinterpret it any way you want.

[Skepdick]

It has been conceived, even if your own powers of conception are limited.

https://en.wikipedia.org/wiki/Taxicab_geometry#Spheres

Predictable. I already mentioned that nonsense earlier in the thread.

That's not a square-circle

Learn what the word actually means instead of treating it like a symbol full of free variables so that you can misinterpret it any way you want.

Go ahead and tell us what meaning you've attached to the symbol "square-circle" then.

[Magnus Anderson]

In any case you failed to explain what the symbol "contradiction" represents.

Any symbol of the form "X and not X".

[Magnus Anderson]

Go ahead and tell us what meaning you've attached to the symbol "square-circle" then.

I'm afraid that's a futile endeavor.

[Skepdick]

Any symbol of the form "X and not X".

But any logical conjunction is of this form!!!

let X: = 1.
let Y: = 2
Y is not X
therefore 1 and 2 is exactly of the form "X and not X"

So is 1 and 3.
So is 1 and 4; or 5; or 6...

[Skepdick]

I'm afraid that's a futile endeavor.

You don't know what you mean when you use the symbol "square-circle"?

You could've just told us your powers of conception are vastly limited.