What LEM is not

[Skepdick]

Bullshit.

Bullshit to your bullshit.

You were asking me tell you to the location of the concept attached to the word "is".

In fact, you went so far to ask me to tell you its GPS coordinates.

Obviously, you were suggesting that this is necessary to do in order to prove the physical existence of a thing.

Yes. Location is necessary property of physical things.
If it's physical then it has a location.

If you aren't justifying the physicality of the concept "is" via its location, then how are you justifying it?

You didn't ask, "How do you know it's physical?" You asked, "Where is it? Tell me its GPS coordinates."

Naturally. Because location is a necessary property of physical things.

You are welcome to furnish a sufficient property, if you so choose.

An extremely imbecilic thing to do. But then again, you ARE an extreme imbecile.

Yeah, but I am always a lesser imbecille than you.

So you disagree that concepts are physical?

I am not disagreeing. I am merely not agreeing.

There is no involution between agreement and disagreement.

Three values, you see {agree, disagree, neither}

Do they exist at all?

Non-sequitur.

not having physical location doesn't imply non-existence.
It implies non-physicality.

[Magnus Anderson]

Fuzzy-logic. Continuous values between 0 and 1.

Predictable.

In fuzzy logic, every truth value is either "true" ( i.e. 1 ) or "false" ( i.e. less than 1. )

I already explained this here in this thread long time ago.

Three-valued logic: {true, false, undefined}

The value they call "undefined" is not a truth value.

I already explained this here in this thread long time ago.

Sure. In fuzzy logic that would mean half-true/half-false

It means "false". "False" is defined as "not true". Thus, in fuzzy logic, every value less than 1 is equivalent to false.

I already explained this here in this thread long time ago.

That's not generally true in all logics.

Irrelevant.

We're talking about what LEM means. And in the context of LEM, "false" means "not true". How other systems of logic define this term is completely irrelevant for the purpose of trying to understand what LEM is saying.

That's ONLY true in a bivalent logic with involution.

Nonsense.

Contradiction. P is true or the NEGATION of P is true.

Dummy, LEM is not saying that negation is involvutive. It is saying that either P is true or involutive negation of P is true. Learn to make a difference between the subject of a proposition ( the described portion of reality ) and the predicate ( the description of that portion. ) LEM says ABSOLUTELY NOTHING about negation.

S is P. Subject is predicate. Either P is true or involutive negation of P is true. Subjects are "P" and "involutive negation of P". Predicates are "true" and "true". "Involutive negation of P" is not the same thing as "negation". Learn some English language.

Contradiction. Real values between 0 and 1 are degrees of truth.

And?

You are claiming that 0 is false, and >0 is true.
That's not true.

I am telling you that you're a master at misinterpretation. That's why you never learn anything.

I didn't say that "> 0" is "true". I said that "< 1" is "false".

English "true" maps to fuzzy logic's "1" and English "false" maps to fuzzy logic's "less than 1."

[Skepdick]

In fuzzy logic, every truth value is either "true" ( i.e. 1 ) or "false" ( i.e. less than 1. )

I already explained this here in this thread long time ago.

That's not true. You are attempting to collapse a continuous value [0,1] into a discrete one. [0,1) U {1}

The value they call "undefined" is not a truth value.

Yes, it is.

Is x/0=y true, false or undefined?

I already explained this here in this thread long time ago.

And you were wrong.

It means "false". "False" is defined as "not true". Thus, in fuzzy logic, every value less than 1 is equivalent to false.

That's not true.

I already explained this here in this thread long time ago.

And you were wrong.

Irrelevant.

That's false.

We're talking about what LEM means. And in the context of LEM, "false" means "not true". How other systems of logic define this term is completely irrelevant for the purpose of trying to understand what LEM is saying.

Yes, but in the context of logic LEM is not always true.

Nonsense.

Your inability to make sense of it doesn't make it nonsense. It makes you incapable of reason.

Dummy, LEM is not saying that negation is involvutive.

Yes, it is. LEM does not hold without involutive negation.

It is saying that either P is true or involutive negation of P is true.

Contradiction.

And?

And anything >0 is not false. Only 0 is false.

I didn't say that "> 0" is "true". I said that "< 1" is "false".

And that's wrong. Anything above 0 is not false. because ONLY 0 is false.

In fact, your choice to assign <1 as false is completely arbitrary.
It's just as arbitrary to assign >0 as true.

Any attempts to collapse the continuous range [0,1] into bivalence are equally wrong.

English "true" maps to fuzzy logic's "1" and English "false" maps to fuzzy logic's "less than 1."

That's not true. English's true maps to fuzzy logic's 1. English's false maps to fuzzy logic's 0.

This leaves (0,1) as truth values with no direct equivalents in a bivalent logic.

[Magnus Anderson]

Yes, but in the context of logic LEM is not always true.

There are no such things as statements that are SOMETIMES true and SOMETIMES false.

If a statement is true ( or false ) once, it is always true ( or false. )

And that applies to LEM just as well.

And LEM happens to be true.

The problem here is that you're playing word games ( your favorite activity. )

The value they call "undefined" is not a truth value.

Yes, it is.

It's true that x/0 is undefined.

And that just happens to be a very embarrassing example.

The statement that you used as an example is "x/0 is undefined" and the truth value that you assigned to it is "true".

A proper example would be something like "This sentence is false". In a three-valued logic such as Łukasiewicz logic they would say that the truth value of that statement is "undefined".

But 'undefined" is a truth value merely in name. It's not proper truth value because a truth value is a value that describes the extent to which a description corresponds to the portion of reality it is describing.

Here we have a statement that's not describing anything -- a statement with no proposition attached to it, i.e. a non-propositional statement. Such statements have no truth value. And what "undefined" indicates is that the statement is not a proposition, i.e. it has no truth value.

The reason they do this is because they operate on sentences that do not necessarily represent propositions.

It's like saying the truth value of "cat" is "undefined". Of course it is, it's not a proposition.

But tht's not a truth value.

The fact that you can assign just about any value to a sentence does not mean that value is a truth value.

You are attempting to collapse a continuous value [0,1] into a discrete one. [0,1) U {1}

Not true. I am merely saying that every truth value in fuzzy logic can be either classified as "true" or as "false" which is precisely what LEM is saying. In other words, there are no truth values that can neither be classified as "true" nor as "false". All numbers less than 1 can be classified as "false" ( since the word "false", in the context of LEM, as well as in the context of English language, does not mean "completely false" but merely "not true" ) and number 1 itself can be classified as "true".

Yes, it is. LEM does not hold without involutive negation.

No, it is not. You have to learn what it means for a statement to be saying something.

LEM does not have to be expressed using the concept of negation. It can also be expressed as, "For every proposition P, P is either true or false". That statement says the same exact thing as the statement, "For every proposition P, either P is true or negation of P is true". How is that possible if LEM is describing negation?

And anything >0 is not false. Only 0 is false.

LEM:
For every proposition P, P is either true or false.

Here, the word "false" is defined as "not true". It does not mean "completely false" as ignoramuses think.

Thus, what LEM is saying is, "For every proposition P, P is either true or not true."

In fuzzy logic, "true" is represented by "1". As such, "not true" would be any value that is not 1. Since there are no values greater than 1 in fuzzy logic, that means "not true" is equivalent to any value that is less than 1.

Thus, when you apply LEM to fuzzy logic, what you get is, "For every proposition P, P is either 1 or it is less than 1."

And that is obviously true.

The end.

What you and your friends are doing is misinterpreting and redefining LEM.

In fact, your choice to assign <1 as false is completely arbitrary.

False.

That's not true. English's true maps to fuzzy logic's 1. English's false maps to fuzzy logic's 0.

Wrong. Chat GPT knows better than you.

[Skepdick]

There are no such things as statements that are SOMETIMES true and SOMETIMES false.

If a statement is true ( or false ) once, it is always true ( or false. )

And that applies to LEM just as well.

And LEM happens to be true.

Nonsense.

The statement "Today is Tuesday" is only true on Tuesdays.
The statement "It's raining outside" is true when it's raining and not true when it's not raining.

The problem here is that you're playing word games ( your favorite activity. )

And your problem is that you are playing word games without understanding the rules.

The value they call "undefined" is not a truth value.

And that just happens to be a very embarrassing example.

The statement that you used as an example is "x/0 is undefined" and the truth value that you assigned to it is "true".

I didn't use statements. I use a question. What is the truth-value of x/0?

But 'undefined" is a truth value merely in name. It's not proper truth value because a truth value is a value that describes the extent to which a description corresponds to the portion of reality it is describing.

Which portion of reality is being described by x/0?

Here we have a statement that's not describing anything -- a statement with no proposition attached to it, i.e. a non-propositional statement. Such statements have no truth value. And what "undefined" indicates is that the statement is not a proposition, i.e. it has no truth value.

Using the exact same logic - what portion of reality is LEM describing? If no portion of reality is being described then LEM has no truth-value.

Q.E.D

The reason they do this is because they operate on sentences that do not necessarily represent propositions.

x/y=z is a proposition. Such propositions are generally true; or false. Except when y=0.

You can re-state this as a question: "What is the truth-value of x/y=z?"
Or you can re-state it as a statement" The truth-value of "x/y=z is either true, false or undefined".

Not true. I am merely saying that every truth value in fuzzy logic can be either classified as "true" or as "false" which is precisely what LEM is saying.

That's not true.

In other words, there are no truth value that can neither be classified as "true" nor as "false".

0.5

All numbers less than 1 can be classified as "false" ( since the word "false", in the context of LEM, as well as in the context of English language, does not mean "completely false" but merely "not true" ) and number 1 itself can be classified as "true".

That is an arbitrary interpretation. All numbers greater than 0 can be classified as true.

In the English language that would mean 0 as completely false, and >0 as partially true.

LEM does not have to expressed using the concept of negation. It can also be expressed as, "For every proposition P, P is either true or false".

Distinction without a difference. You keep insisting that "not true" means false; and "not false" means true.

That statement says the same exact thing as the statement, "For every proposition P, either P is true or negation of P is true". How is that possible if LEM is describing negation?

Because true and false are involutive in Classical logic.

not False = True.
not True = False
not not True = True
not not False = False

LEM:
For every proposition P, P is either true or false.

Here, the word "false" is defined as "not true".

Q.E.D Involution

It does not mean "completely false" as ignoramuses think.

There's no difference between false and "completely false" in a bivalent system.

You have only two options: true, false. Completely false is false.

In fuzzy logic, "true" is represented by "1". As such, "not true" would be any value that is not 1. Since there are no values greater than 1 in fuzzy logic, that means "not true" is equivalent to any value in fuzzy logic that is less than 1.

Thus, when you apply LEM to fuzzy logic, what you get is, "For every proposition P, P is either 1 or it is less than 1."

And that is obviously true.

It's so "obviously true" it's false.

False.

It's false that it's false.

Wrong. Chat GPT knows better than you.

Why are you lying?

Q: In fuzzy logic is it true that 1 maps to true and <1 maps to false?

ChatGPT said:

In fuzzy logic, the mapping isn't as binary as that. Unlike classical logic, where statements are strictly true (1) or false (0), fuzzy logic allows for a continuum of truth values between 0 and 1. Here’s how it works:

True (1) typically represents absolute truth, but any value between 0 and 1 can represent a degree of truth. This means that 0.7, for instance, could denote a statement that's "mostly true" or "somewhat true."

False (0) represents absolute falsity, but values close to 0 are not strictly false. For example, 0.3 could imply that something is "mostly false" or "barely true."

In this system, truth is seen as a gradient rather than an either-or, making fuzzy logic useful for situations where information is uncertain, ambiguous, or partially true. So, to answer your question, values less than 1 do not map directly to "false" but rather indicate varying degrees of truth.

[Magnus Anderson]

Nonsense. The statement "Today is Tuesday" is only true on Tuesdays.

False.

It's true if it was stated on Tuesday. And it's false if it was NOT stated on Tuesday.

In both cases, its truth value is constant and unchanging. If it's true, it was and it will always be true. And if it's false, it was and it will always be false.

But if you don't know WHEN it was stated, you can say that its truth value is unknown. That much can be accepted. But "unknown" is not a truth value. It has nothing to do with veracity and everything to do with knowledge. If you want to make it about veracity, you can substitute "unknown" with "either true or false".

I didn't use statements. I use a question. What is the truth-value of x/0?

You did -- before you modified your post. You said "It's true that x/0 is undefined." Own it. Don't lie.

Which portion of reality is being described by x/0?

The concept attached to the symbol "x/0".

Using the exact same logic - what portion of reality is LEM describing? If no portion of reality is being described then LEM has no truth-value.

Q.E.D

The concept of truth value.

x/y=z is a proposition. Such propositions are generally true; or false. Except when y=0.

Not every statement is a proposition. Just because it has the form "A is B" or "A = B" does not mean it's a proposition.

0.5

0.5 is false.

That is an arbitrary interpretation. All numbers greater than 0 can be classified as true.

LEM still applies. Every number is either true or false. In this partcular language of yours, "0" would be "false" and every number greater than zero would be "true".

LEM's true indicates "complete correspondence" and LEM's false indicates "not complete correspondence".

The end.

You keep insisting that "not true" means false; and "not false" means true.

Yes.

Because true and false are involutive in Classical logic.

not False = True.
not True = False
not not True = True
not not False = False

They are involutive everywhere.

not less than 1 = 1
not 1 = less than 1
not not 1 = not less than 1 = 1
not not less than 1 = not 1 = less than 1

Why are you lying?

Not lying. I already posted my conversation with Chat GPT here in this thread. You're late to the party. Read it.

[Skepdick]

Nonsense. The statement "Today is Tuesday" is only true on Tuesdays.

False.

It's true if it was stated on Tuesday. And it's false if it was NOT stated on Tuesday.

Contradiction. You said that there are no statements that are SOMETIMES true and SOMETIMES false.

Yet you went onto agree that the truth of the sentence is conditional.

In both cases, its truth value is constant and unchanging. If it's true, it was and it will always be true. And if it's false, it was and it will always be false.

Contradiction. The truth-value of the statement depends on the time when the sentence being uttered.
It's not inherent to the sentence.

But if you don't know WHEN it was stated, you can say that its truth value is unknown.

Q.E.D you are saying the sentence "Today is Tuesday" has no inherent truth-value.

Its truth-value is determined by factors other than the contents of the statement.

You are failing to distinguish between "The sentence claiming today to be Tuesday was uttered on a Wednesday" and "Today is Tuesday"

You did -- before you modified your post. You said "It's true that x/0 is undefined." Own it. Don't lie.

Q.E.D The statement's truth-value changed. Before and after.

The concept attached to the symbol "x/0".

The symbol expresses a concept - it doesn't describe it. Which portion of reality is the concept of "dividing x by 0" describing?

The concept of truth value.

The concept of truth-value doesn't inherintly imply bivalence.

Not every statement is a proposition. Just because it has the form "A is B" or "A = B" does not mean it's a proposition.

It's literally called called propositional equality.

https://ncatlab.org/nlab/show/propositional+equality

0.5 is false.

Why are you lying?

If x=0 then X is false.
If x !=0 then X is not false.
0.5 !=0 then 0.5 is not false.

LEM still applies. Every number is either true or false.

That's false. Only 1 is true. Only 0 is false. That's not "every number"

In this partcular language of yours, "0" would be "false" and every number greater than zero would be "true".

Which contradicts your particular language where any number below 1 is false.

But that requires splitting the continuum [0,1] into two.

LEM's true indicates "complete correspondence" and LEM's false indicates "not complete correspondence".

Why? Why not 0 as "NO correspondence" and >0 as SOME correspondence?

Yes.

Q.E.D that's an involution.

They are involutive everywhere.

not less than 1 = 1
not 1 = less than 1
not not 1 = not less than 1 = 1
not not less than 1 = not 1 = less than 1

Is 0.999... less than 1; or 1?

Not lying. I already posted my conversation with Chat GPT here in this thread. You're late to the party. Read it.

You mean the conversation where you misled ChatGPT into the wrong answer?

Yeah.. you are lying.

[Magnus Anderson]

You said that there are no statements that are SOMETIMES true and SOMETIMES false.

Yes.

If a proposition is true, it was always true and will always be true.

If a proposition is false, it was always false and will always be false.

Yet you went onto agree that the truth of the sentence is conditional.

You misunderstood.

The truth-value of the statement depends on the time when the sentence being uttered.
It's not inherent to the sentence.

It's inherent to the sentence. "Today" refers to the day the sentence was spoken.

But if you don't know WHEN it was stated, you can say that its truth value is unknown.

Q.E.D you are saying the sentence "Today is Tuesday" has no inherent truth-value.

No, I am saying that, if you don't know WHEN the statement was uttered, you can say that you don't know its truth value.

Not knowing when the statement was uttered means that you don't know what the word "today" refers to.

It simply means YOU DON'T KNOW WHAT'S BEING SAID.

And when you don't know what's being said, you can't establish the truth value of what's being said.

The statement is either true or false, and whatever is the case, it was always the case and will always be the case. Its truth value is constant and unchanging. You not knowing what that proposition is about, and thus not being able to determine its truth value, does not change that fact.

Its truth-value is determined by factors other than the contents of the statement.

Yes, such as what the contents of the statement, such as words, actually mean.

But that does not prove your point in any way.

You have to learn how English language works for once.

When people speak of the truth value of a statement, they are talking about the truth value of a statement that was made by someone at some point in time using some sort of language. The truth value of such a statement is equal to the truth value of the proposition that was attached to it by its speaker. Such statements are not sometimes true and sometimes false. Their truth value is constant. If they are true, they were ALWAYS true and they will ALWAYS be true. If they are false, they were ALWAYS false and they will ALWAYS be false. And that's because we're dealing with a single, concrete, specific proposition.

You, on the other hand, are not talking about a particular statement that was made by someone at some point in time using a language of some sort. Instead, you're talking about a category of propositions. When you say "Today is Tuesday", you do not have in mind someone saying that on certain day. Instead, you just have a category of propositions in mind, a category that consists of propositions such as "Today is Tuesday. ( Said on Monday. )", "Today is Tuesday. ( Said on Tuesday. )", and so on. But categories of propositions have no truth value themselves. They are just sets of propositions which means all you can do is speak of things such as the percentage of its propositions that are true. If you represent them using templates, such as "Today is Tuesday.( Said on X. )" where "X" is a day in a week, you can also talk about what conditions must be met in order for an instance of that template to be true or false.

What you have to get inside your head is that LEM is NOT a category of propositions.

It is a SPECIFIC proposition.

[Gary Childress]

In fuzzy logic, every truth value is either "true" ( i.e. 1 ) or "false" ( i.e. less than 1. )

I already explained this here in this thread long time ago.

That's not true. You are attempting to collapse a continuous value [0,1] into a discrete one. [0,1) U {1}

The value they call "undefined" is not a truth value.

Yes, it is.

Is x/0=y true, false or undefined?

That's an interesting contention. If something is 'undefined', then is it YET be true or false = pending definition and once it is given a definition it is subject to the LEM?

[godelian]

Is x/0=y true, false or undefined?

That's an interesting contention. If something is 'undefined', then is it YET be true or false = pending definition and once it is given a definition it is subject to the LEM?

The sentence "x/0=y" is a request for a model, i.e. an interpretation, for which the sentence is true.

So, the answer is a set { (x1,y1), (x2,y2) ... (x[n],y[n]) } with each two-tuple (x[k],y[k]) an individual interpretation that makes the sentence true.

Therefore, the sentence "x/0=y" is not true, false or undefined.

The sentence has a model, i.e. a legitimate interpretation, or does not have one.

https://plato.stanford.edu/entries/model-theory

Model theory began with the study of formal languages and their interpretations, and of the kinds of classification that a particular formal language can make.

Sometimes we write or speak a sentence S that expresses nothing either true or false, because some crucial information is missing about what the words mean. If we go on to add this information, so that S comes to express a true or false statement, we are said to interpret S, and the added information is called an interpretation of S. If the interpretation I happens to make S state something true, we say that I is a model of S, or that I satisfies S, in symbols ‘I ⊨S’.

So, we are looking for the set:

{ (x1,y1), (x2,y2) ... (x[n],y[n]) } ⊨ ( x/0=y )

Which happens to be empty.

[Skepdick]

That's an interesting contention. If something is 'undefined', then is it YET be true or false = pending definition and once it is given a definition it is subject to the LEM?

There's is enough information in the formulation to assert it as undefined.

x/y=z is true, false or undefined.
x/0=z is undefined.

[Skepdick]

{ (x1,y1), (x2,y2) ... (x[n],y[n]) } ⊨ ( x/0=y )

Which happens to be empty.

There is a model.

https://xenaproject.wordpress.com/2020/ ... ory-a-faq/

[Magnus Anderson]

That's an interesting contention. If something is 'undefined', then is it YET be true or false = pending definition and once it is given a definition it is subject to the LEM?

Suppose that division is defnied as the inverse of multiplication so that "x/y" means "the only number that gives x when multiplied by y".

In that case, "x/0" would be a number that gives "x" when multiplied by "0". If "x" is not 0, then no number can give "x" when multiplied by 0, meaning that, when x =/= 0, "x/0" is a contradiction in terms, an oxymoron. If "x" is equal to 0, any number can give "x" when multiplied by 0, meaning that, when x = 0, there's no such thing as "the only number that gives x when multiplied y" making "x/0" a contradiction in terms, an oxymoron. Thus, in all cases, "x/0" is an oxymoron.

Suppose now that "x = y" means "The only number that can be represented by the symbol x is the same as the only number that can be represented by the symbol y".

In that case, "x/0 = y" would mean that the only number that can be represented by the symbol "x/0" is the same as the only number that can be represented by the symbol "y". But "x/0" being an oxymoron means that no number can be represented by "x/0" meaning that the subject of the statement itself is an oxymoron. And when the subject of the statement is an oxymoron, the statement is not describing anything, which means, it's a non-propositional statement.

Therefore, "x/0 = y" is a non-propositional statement, i.e. a statement that has no truth value.

There are systems of logic, such as Lukasiewicz logic, that operate on sentences that are not necessarily propositional, statements such as "Square-circles are squares". Because of that, they had to introduce a third value, a value that indicates that the statement has no truth value. They call that a third value even though it's really not. They would say that the truth value of "x/0 = y" is "undefined". And that would be correct. There is nothing wrong with that. But when they say that LEM is not compatible with that, it is then that they make a mistake.

[Magnus Anderson]

THE MEANING OF THE WORD "FALSE"
( in the context of natural languages as well as LEM )

Suppose that John is a tall porter.

Consider the proposition, "John is a short porter."

That statement is one part true because John is indeed a porter. But it's also one part false because John is not short.

Thus, the proposition is half-true; or numerically speaking, its truth value is 0.5.

In English language, and in fact, in every natural language, it will be said that the proposition is false. If one part of it is wrong, then the entire proposition is wrong. That's it. How much of it is wrong, i.e. whether it's entirely wrong or merely partially wrong, is completely irrelevant.

If "false" means "completely false", then the above statement is neither true nor false. But noone will say that.

Similarly, if "false" means "completely", then the above statement isn't isn't exactly false but merely approximately false ( as well as approximately true, so one could also say it's true. ) But noone will say that.

They will simply say it's false.

And that is because the word "false" is generally understood to mean "not completely true" rather than "completely false".

And that's how the word is used in the context of LEM.

[Magnus Anderson]

THE MEANING OF THE WORD "NEGATION"
( in the context of natural languages as well as LEM )

In the context of English language, as well as every other natural language; but more importantly, in the context of LEM, the symbol "not X", which is an example of negation, means "anything other than X".

In this context, negation is no more than a way to create new concepts from existing ones by literally inverting their meaning.

But this concept of negation is quite different from the ones used in fuzzy logic and Godel logic.

In fuzzy logic, "not 0.7" would be equivalent to "0.3".

In Godel logic, "not 0.7" would be equivalent to "0".

In natural languages, as well as in the context of LEM, "not 0.7" would be equivalent to "anything other than 0.7". Thus, any number other than 0.7, including 0 and 1, would be able to be represented by "not 0.7".

When evaluating the truth value of a statement, it's important to understand the words it uses as they are meant to be understood. You are not free to understand them any way you want. As such, if a statement is using the word "negation", you have to understand what kind of concept of negation it is talking about.

But more importantly, as we shall see in the next post, one has to understand that LEM is NOT an open statement, i.e. one containing free variables, but a closed one.