What LEM is not

[Magnus Anderson]

A proposition is an idea that a portion of reality is such and such. It is made out of two components: a reference to an aspect of reality ( "subject" ) and a description of that aspect of reality ( "predicate". )

A proposition is said to be true, and thus an instance of truth, if and only if the predicate completely corresponds to the subject. Otherwise, it is said to be false, an instance of falsehood.

If one or both of these components are missing from an idea, then that idea is NOT a proposition.

I'll provide a simple example, one that does not involve Russel's set and square-circles.

Let X be a shape that is a square and at the same time not a square.

Is the statement "X is a square" true?

If it is true, then it is a square, which means that it is not true that it is not a square, which is a contradiction.

If it is false, then it is not a square, which means it is not a square, which is also a contradiction.

As such, you can't say it's true because it leads to a contradiction and you also can't say it's false because that too will lead you to a contradiction. But you also can't say it's neither true nor false because that too will lead to a contradiction ( "neither true nor false" translates to "both true and not true". )

The problem here is that the subject of the proposition is the set of all conceivable things that can be represented by the symbol "X", which is an oxymoron, a symbol of the form "this and not this", meaning that absolutely nothing can be represented by "X", making the set of all conceivable things that can be represented by "X" an empty one.

So what we're asking here is whether everything in an empty set is a square.

All of this indicates that the subject of the proposition "X is a square" is missing meaning that "X is a square" is not really a proposition.

[Skepdick]

meaning that absolutely nothing can be represented by "X"

That's an error. The symbol X can represent absolutely anything and everything.

Much like the symbols "apple" can represent anything and everything.

Symbols symbolize. What do they symbolize? Whatever you use them to symbolize.

All of this indicates that the subject of the proposition "X is a square" is missing meaning that "X is a square" is not really a proposition.

Of course it is. According to LEM the statement "An elephant is a square" is either true or false.

That's what a proposition is.

[Magnus Anderson]

meaning that absolutely nothing can be represented by "X"

That's an error. The symbol X can represent absolutely anything and everything.

Much like the symbols "apple" can represent anything and everything.

Symbols symbolize. What do they symbolize? Whatever you use them to symbolize.

You're missing the point. ( Not uncommon for you. )

Once you ascribe a meaning to a word, you can't use it any other way without first redefining it.

Since the meaning of "X" is "A square and not a square", and since that's an oxymoron, and since nothing can be represented by an oxymoron, it follows that nothing can be represented by that symbol.

That you can change the meaning of the symbol "X" is irrelevant.

Of course it is.

Of course it is not.

[Skepdick]

Once you ascribe a meaning to a word, you can't use it any other way without first redefining it.

Since the meaning of "X" is "A square and not a square", and since that's an oxymoron, and since nothing can be represented by an oxymoron, it follows that nothing can be represented by that symbol.

You are literally using it to represent an oxymoron. Oops?

[Magnus Anderson]

Once you ascribe a meaning to a word, you can't use it any other way without first redefining it.

Since the meaning of "X" is "A square and not a square", and since that's an oxymoron, and since nothing can be represented by an oxymoron, it follows that nothing can be represented by that symbol.

You are literally using it to represent an oxymoron. Oops?

Not an "oops" at all.

[Skepdick]

Once you ascribe a meaning to a word, you can't use it any other way without first redefining it.

Since the meaning of "X" is "A square and not a square", and since that's an oxymoron, and since nothing can be represented by an oxymoron, it follows that nothing can be represented by that symbol.

You are literally using it to represent an oxymoron. Oops?

Not an "oops" at all.

On a scale of 1 to 10 is your confusion at 15 or 20?

What does the word "oxymoron" represent if nothing can be represented with it?

[Magnus Anderson]

On a scale of 1 to 10 is your confusion at 15 or 20?

What does the word "oxymoron" represent if nothing can be represented with it?

I think you've already been informed that you're not welcome in this thread. So what are you doing here? I certainly did not ask you to be more active than others. You have to understand that your input is not appreciated. You're perceived as someone who spams, distracts and deceives -- and relatively aggressively so. You certainly spend more time responding than processing. So I will tell you one more time, leave this thread. You're not welcome here.

[Skepdick]

I think you've already been informed that you're not welcome in this thread. So what are you doing here? I certainly did not ask you to be more active than others. You have to understand that your input is not appreciated. You're perceived as someone who spams, distracts and deceives -- and relatively aggressively so. You certainly spend more time responding than processing. So I will tell you one more time, leave this thread. You're not welcome here.

Your output is exactly as appreciated as my input.

Yet here we are. Inputing and outputing stuff...

You are perceived as somebody who talks bullshit. And no amount of reason can convince you that you are talking bullshit. Your guise of passivity doesn't change the fact that your output is...bullshit.

I will tell you too. Shut up. But you won't. Will you?

You will continue spewing bullshit; and it's just in my nature to be unpleasant to people who spew bullshit.

[godelian]

"Decidable" and "undecidable" are not truth values, they are epistemic categories.

"Decidable" means that the truth value can be logically determined:

If P is decidable then P is true or P is false.

It means that the law of the excluded middle is applicable. If P is undecidable, then there is no truth value or a fundamentally undetermined one.

In a three-valued logic system, undecidable or undetermined or unknown is a truth value:

https://en.wikipedia.org/wiki/Many-valued_logic

Many-valued logic (also multi- or multiple-valued logic) is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false") for any proposition. Classical two-valued logic may be extended to n-valued logic for n greater than 2. Those most popular in the literature are three-valued (e.g., Łukasiewicz's and Kleene's, which accept the values "true", "false", and "unknown"), four-valued, nine-valued, the finite-valued (finitely-many valued) with more than three values, and the infinite-valued (infinitely-many-valued), such as fuzzy logic and probability logic.

Most database systems in the world use three-valued logic by introducing the constant null. Databases that do not support this truth value are generally deemed to be practically unusable.

Moreover, if the truth value of a proposition cannot be known, it does not mean it does not exist. Albeit, in practice, everything can be known. All one has to do is randomly stumble upon truth.

If a proposition is logically undecidable then there is no method to figure out if it is true or false. You cannot randomly stumble upon the truth value of an undecidable proposition. Furthermore, "in practice, everything can be known" is very incorrect, and even provably so, because most true propositions are unprovable:

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

True But Unprovable
by Noson S. Yanofsky

Gödel’s famous incompleteness theorem showed us that there is a statement in basic arithmetic that is true but can never be proven with basic arithmetic. That is just the beginning of the story. There are more true but unprovable statements than we can possibly imagine.

The world that Gödel described can be extended. Of all the mathematical facts, only some are expressible with language. Any mathematical fact that is provable is automatically expressible because proofs must be done in a language. This figure is not drawn to scale because the expressible facts are miniscule within the collection of all facts.

With knowledge defined as justified true propositions (JTB), and with justification in mathematics provided by means of proof, you can see that the overwhelmingly vast majority of true propositions in mathematics cannot be known. Hence, there is solid mathematical proof against the claim that "in practice, everything can be known".

But most importantly, you can't argue against the claim that every proposition has truth value because it is so by definition.

Mathematical logic has come a long way from the days in which someone would still argue that every proposition is decidable. As I have mentioned before, claiming that "every proposition has truth value" is only sustainable in the context of zeroth-order logic:

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

The most thoroughly researched branch of propositional logic is classical truth-functional propositional logic,[1] in which formulas are interpreted as having precisely one of two possible truth values, the truth value of true or the truth value of false.[15] The principle of bivalence and the law of excluded middle are upheld. By comparison with first-order logic, truth-functional propositional logic is considered to be zeroth-order logic.[4][5]

This definition is completely unsustainable in first-order logic (and in higher-order logic).

[Magnus Anderson]

Your output is exactly as appreciated as my input.

Yet here we are. Inputing and outputing stuff...

You are perceived as somebody who talks bullshit. And no amount of reason can convince you that you are talking bullshit. Your guise of passivity doesn't change the fact that your output is...bullshit.

I will tell you too. Shut up. But you won't. Will you?

You will continue spewing bullshit; and it's just in my nature to be unpleasant to people who spew bullshit.

It's called "self-righteousness". Skepdick thinks that he's right and that others are wrong, so he feels justified in being unpleasant to others. Imagine what the forum would look like if everyone were like that.

Let me remind you that this is my thread.

[Skepdick]

It's called "self-righteousness". Skepdick thinks that he's right and that others are wrong, so he feels justified in being unpleasant to others. Imagine what the forum would look like if everyone were like that.

Let me remind you that this is my thread.

Idiot. You are applying LEM again.

You being wrong doesn't mean I am right.

ALL that I am right about is that YOU are wrong.

[Magnus Anderson]

"Decidable" means that the truth value can be logically determined:

"Decidable" means that the truth value of a proposition can be decided, i.e. determined, using a finite and determinsitic method of reasoning and a set of initial premises.

If P is decidable then P is true or P is false.

More accurately, if something is a proposition, then by definition, it has truth value. Conversely, if it has no truth value, then by definition, it is not a proposition, and at best, it merely looks like one.

Whether or not the truth value of a proposition is decidable does not change its status as a proposition.

It means that the law of the excluded middle is applicable. If P is undecidable, then there is no truth value or a fundamentally undetermined one.

If the truth value is undecidable, it means that you can't figure it out using a finite and deterministic method of reasoning and a set of initial premises.

It does not mean it does not exist. And if it does not exist, it merely means you're not dealing with a proposition.

In a three-valued logic system, undecidable or undetermined or unknown is a truth value:

It's truth value only in name. Just because they assign a value to a proposition, it does not mean it's a truth value.

I've covered Łukasiewicz's logic earlier in this thread and explained why the third value, "unknown", is not really a truth value.

I've also covered fuzzy logic.

Both systems of logic, like all other systems of logic, adhere to LEM.

You cannot randomly stumble upon the truth value of an undecidable proposition.

You can do so by flipping a coin. You have a 50% chance of getting it right. This is, of course, assuming that the proposition in question is truly a proposition and not a fake proposition such as "Square-circles are squares".

Mathematical logic has come a long way from the days in which someone would still argue that every proposition is decidable.

The problem is that you're confusing decidability with truth value. This thread has nothing to do with decidability.

[Skepdick]

"Decidable" means that the truth value of a proposition can be decided, i.e. determined, using a finite and determinsitic method of reasoning and a set of initial premises.

So what set of initial premises renders LEM decidable?

What makes "For every P either P or not P" true?

[Magnus Anderson]

"Decidable" means that the truth value of a proposition can be decided, i.e. determined, using a finite and determinsitic method of reasoning and a set of initial premises.

So what set of initial premises renders LEM decidable?

What makes "For every P either P or not P" true?

I explained that in the OP.

[Skepdick]

"Decidable" means that the truth value of a proposition can be decided, i.e. determined, using a finite and determinsitic method of reasoning and a set of initial premises.

So what set of initial premises renders LEM decidable?

What makes "For every P either P or not P" true?

I explained that in the OP.

No, you didn't.

You failed to explain how the truth-value of LEM is decided.