"This sentence is not true" is not a truth bearer thus not a proposition thus cannot be included in any Boolean logic system. — PL Olcott

A physical analog would be a digital logic inverter (NOT gate) with its output connected back to its input. Such a circuit forms an oscillator, with the output continually swinging back and forth between 0 and 1.

This seems your source of misunderstanding. In propositional logic, you would day "This sentence is not true." But in predicate logic, it can be translated into "Some sentence is not true."
In FOL it can be translated into "X is not true." which are all perfectly true or false depending on the truth criteria of the quantifiers and variables. — Corvus

The Variables of propositional logic and every other order of bivalent logic must have a Boolean value. Any variables that cannot possibly be true or false must be excluded from every bivalent logic system. https://en.wikipedia.org/wiki/Three-valued_logic can have the values {True, False, Nonsense}.
"This sentence is not true" has the semantic value of {nonsense}.

The predicate Is_Not_True(X) summarize by the operator "~" is fine unless
X is defined as X := ~True(X). Then it is the oscillator that wonderer1 referred to.

A physical analog would be a digital logic inverter (NOT gate) with its output connected back to its input. Such a circuit forms an oscillator, with the output continually swinging back and forth between 0 and 1. — wonderer1

Good job that is a perfect analogy!

The Variables of propositional logic and every other order of bivalent logic must have a Boolean value. — PL Olcott

Variables in propositional logic is for the whole sentence, not the elements in the sentence, hence its limitation. You are still talking under the propositional logic domain only.

When you widen the scope into predicate logic, FOL and HOL, the concept of truth and falsity has multifaceted nature. FOL enables you employ the variables for the individuals and subjects. HOL can deal with the variables for the relations, operators and properties within the sentence.

When you widen the scope into predicate logic, FOL and HOL, the concept of truth and falsity has multifaceted nature. FOL enables you employ the variables for the individuals and subjects. HOL can deal with the variables for the relations, operators and properties within the sentence. — Corvus

None-the-less in every bivalent system of logic we must be able to reduce every variable to a Boolean value. Your reply did not seem to understand that. Your reply merely stated that variables in higher orders of logic represent more complex things than in Propositional logic.

It seems that you are simply failing to understand this:
In logic, a three-valued logic (also trinary logic, trivalent, ternary, or trilean,[1] sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating true, false, and some third value. This is contrasted with the more commonly known bivalent logics (such as classical sentential or Boolean logic) which provide only for true and false.
https://en.wikipedia.org/wiki/Three-valued_logic

A three-valued logic system that can easily handle self-contradictory expressions would have the values of: {True, False, Nonsense}.

None-the-less in every bivalent system of logic we must be able to reduce every variable to a Boolean value. Your reply did not seem to understand that. — PL Olcott

Your reply merely stated that variables in higher orders of logic represent more complex things than in Propositional logic. — PL Olcott

You are still under confusion, or don't want to see the real point. We have not been only talking about bivalent system of logic here. If you can recall the OP is about HOL. Not 2000 year old propositional logic. Hence it was necessary and relevant considering and looking into the multifaceted nature of truth, which are in the domains of FOL and HOL.

This is contrasted with the more commonly known bivalent logics (such as classical sentential or Boolean logic) which provide only for true and false.
https://en.wikipedia.org/wiki/Three-valued_logic — PL Olcott

You have been reading too much Wiki pages, and they can lead you to the wrong places unfortunately.

A three-valued logic system that can easily handle self-contradictory expressions would have the values of: {True, False, Nonsense}. — PL Olcott

If some thing is Nonsense, then it is equivalent to False. In FOL HOL, truth values can be far more than just 3 above you listed. : {True, False, Unknown, Neutral, Contradiction}

You have been reading too much Wiki pages, and they can lead you to the wrong places unfortunately. — Corvus

I always verify that the essential reasoning is correct.
The key elements of my system come straight from me figuring them out and
no one ever wrote them down before.

If some thing is Nonsense, then it is equivalent to False. In FOL HOL, truth values can be far more than just 3 above you listed. : {True, False, Unknown, Neutral, Contradiction} — Corvus

A currently unknown Boolean value is still a Boolean value.
No such thing as a neutral Boolean value. "What time is it?" has no Boolean value.
Contradiction proves False.

"This sentence is not true"
(a) If it was false that would make it true and
(b) If it was true that would make it false,
thus it takes on the third value of nonsense.
{Nonsense} is reserved for expressions that cannot be true or false.

A currently unknown Boolean value is still a Boolean value.
No such thing as a neutral Boolean value. — PL Olcott

Boolean values are applicable up to FOL, but FOL cannot express the full complexities in the world. Hence you are going up to HOL, which has the extended truth values, and can describe more complex states of the real world.

"What time is it?" has no Boolean value.
Contradiction proves False. — PL Olcott

In HOL, "What time is it?" would be translated into computable format, and can be processed for the proper truth values.

{Nonsense} is reserved for expressions that cannot be true or false. — PL Olcott

Nonsense is not a logic world. It is an ordinary linguistic expression to mean False with added stupidity and foolishness connotation.

Boolean values are applicable up to FOL, but FOL cannot express the full complexities in the world. Hence you are going up to HOL, which has the extended truth values, and can describe more complex states of the real world. — Corvus

Boolean values are properties of every Proposition
A proposition is a central concept in the philosophy of language, semantics, logic, and related fields, often characterized as the primary bearer of truth or falsity. Propositions are also often characterized as being the kind of thing that declarative sentences denote. https://en.wikipedia.org/wiki/Proposition

In HOL, "What time is it?" would be translated into computable format, and can be processed for the proper truth values. — Corvus

No not at all. When I ask you is this sentence true or false: "What time is it?"
you have no correct answer because the question has a type mismatch error

Nonsense is not a logic world. It is an ordinary linguistic expression to mean False with added stupidity and foolishness connotation. — Corvus

{Nonsense} is a stipulated term of the art of my formal three-valued formal system of logic
having values of {True, False, Nonsense} that only applies to expressions such as this:

...14 Every epistemological antinomy can likewise be used for a similar
undecidability proof...(Gödel 1931:43-44)

epistemological antinomy AKA self-contradictory expression that cannot possibly be true or false.
This guy seems to sum up my exact same position much more clearly.

The Strengthened Liar and Paradoxes of Incompleteness
https://www.youtube.com/watch?v=5LWQPGjAs3M

duplicate

Boolean values are properties of every Proposition
A proposition is a central concept in the philosophy of language, semantics, logic, and related fields, often characterized as the primary bearer of truth or falsity. Propositions are also often characterized as being the kind of thing that declarative sentences denote. — PL Olcott

A truck load of strawmen here. I didn't deny Boolean values, but I was simply saying that in FOL and HOL, you have the extended truth values including Boolean.

{Nonsense} is a stipulated term of the art of my formal three-valued formal system of logic
having values of {True, False, Nonsense} that only applies to expressions such as this: — PL Olcott

Nonsense !! Nonsense is just a colloquial expression saying, no you are bloody wrong mate.

The Strengthened Liar and Paradoxes of Incompleteness
https://www.youtube.com/watch?v=5LWQPGjAs3M — PL Olcott

Many Wiki pages and Youtube videos are rubbish. Don't trust and worship them as if they are the bible. Think with your own mind, and if it doesn't make sense, then you should be able to say "Nonsense mate. This is what I think, because of this and that." As I said before, they may slag you for saying what you think is true, but at least you know you have been thinking with your own mind, rather than parroting what the Wiki pages and Youtubers said, or joined the herd of the inauthentic comedians seeking pleasure out of attacking the authentic self thinking man.

A truck load of strawmen here. I didn't deny Boolean values, but I was simply saying that in FOL and HOL, you have the extended truth values including Boolean. — Corvus

I don't think that you can find any source that ever says anything like that for bivalent systems of logic. I don't think you can find any sources that say anything like that for (a) propositional logic (b) FOL, (c) SOL, (d) HOL.

https://en.wikipedia.org/wiki/Law_of_excluded_middle only works in bivalent logic.

Nonsense !! Nonsense is just a colloquial expression saying, no you are bloody wrong mate. — Corvus

I have found that line of reasoning ineffective so I switched. We have to resolve my prior reply before you can begin to understand my updated reasoning.

I don't think that you can find any source that ever says anything like that for bivalent systems of logic. I don't think you can find any sources that say anything like that for (a) propositional logic (b) FOL, (c) SOL, (d) HOL.

https://en.wikipedia.org/wiki/Law_of_excluded_middle only works in bivalent logic. — PL Olcott

You should read some good Mathematical Logic books, not the Wiki pages.
But think about it even with your common sense. The world contains more problems, structures, events and objects than things that are just True or False.

For simplest example, when you see a formula, X > 3, that is not true or false until you know the value of X. Until that moment, X > 3 remains unknown.

If you say, "It is raining now." then it could be True in your town, but it could be false for someone living in some other part of the world, because it could be sunny. So your statement is contradictory when looking from both areas of the world.

Some statements or formula depicting the real world structure, events or objects can be unknown, neutral or contradictory. You don't simply reject that as nonsense. You accept them as true, false, unknown, neutral or contradictory depending on the given formula, statements, and analysis.

I have found that line of reasoning ineffective so I switched. We have to resolve my prior reply before you can begin to understand my updated reasoning. — PL Olcott

Please reread my post above.

You should read some good Mathematical Logic books, not the Wiki pages.
But think about it even with your common sense. The world contains more problems, structures, events and objects than things that are just True or False. — Corvus

It is a Boolean valued system. When epistemological antinomies are involved they must be rejected
as a type mismatch error because that have no Boolean values.

For simplest example, when you see a formula, X > 3, that is not true or false until you know the value of X. Until that moment, X > 3 remains unknown. — Corvus

It is resolved to true or false on the basis of the value of X.
Epistemological antinomies cannot possibly ever be resolved to a value of true or false.

If you say, "It is raining now." then it could be True in your town, but it could be false for someone living in some other part of the world, because it could be sunny. So your statement is contradictory when looking from both areas of the world. — Corvus

Yet again can possibly be resolved to a value of true or false depending on location.

Some statements or formula depicting the real world structure, events or objects can be unknown, neutral or contradictory. You don't simply reject that as nonsense. You accept them as true, false, unknown, neutral or contradictory depending on the given formula, statements, and analysis. — Corvus

Only well formed declarative sentences of natural language can be true or false. Any expression of language that is not a proposition must be rejected as a type mismatch error for any formal bivalent system of logic.

A proposition is a central concept in the philosophy of language, semantics, logic, and related fields, often characterized as the primary bearer of truth or falsity. Propositions are also often characterized as being the kind of thing that declarative sentences denote. https://en.wikipedia.org/wiki/Proposition

I have found that line of reasoning ineffective so I switched. We have to resolve my prior reply before you can begin to understand my updated reasoning.
— PL Olcott

Please reread my post above. — Corvus

Instead of a three values system with {True, False and Nonsense} I have bivalent systems of logic that derive a type mismatch error for any expression that is not a proposition.

A proposition is a central concept in the philosophy of language, semantics, logic, and related fields, often characterized as the primary bearer of truth or falsity. Propositions are also often characterized as being the kind of thing that declarative sentences denote. https://en.wikipedia.org/wiki/Proposition

Instead of a three values system with {True, False and Nonsense} I have bivalent systems of logic that derive a type mismatch error for any expression that is not a proposition. — PL Olcott

It is a Boolean valued system. When epistemological antinomies are involved they must be rejected
as a type mismatch error because that have no Boolean values. — PL Olcott

You are confusing between HOL and Computer Programming. In HOL, there is no such things as Boolean values. There are {Truth, False, Unknown, Contradiction, Neutral}, and they are the values of logical interpretation.

Boolean is a type of data in programming language. Boolean type value is not True or False, but "1" or "0", and certainly never have anything to do with "Nonsense".

You are confusing between HOL and Computer Programming. In HOL, there is no such things as Boolean values. There are {Truth, False, Unknown, Contradiction, Neutral}, and they are the values of logical interpretation. — Corvus

They call functions of the Boolean type predicates in all orders of predicate logic. Functions of any other type are called functions. Predicate: >(5,2)==TRUE Function: +(5,2)==7

They call functions of the Boolean type predicates in all orders of predicate logic. Functions of any other type are called functions. Predicate: >(5,2)==TRUE Function: +(5,2)==7 — PL Olcott

For some unknown reasons, you changed the subject to Functions. There are differences in functions of math, and functions in computer programming. Can you explain the difference?

Predicate: >(5,2)==TRUE Function: +(5,2)==7 — PL Olcott

Could you please explain that in plain English? And how is it related to our discussion?

Could you please explain that in plain English? And how is it related to our discussion? — Corvus

All bivalent systems of predicate logic only have (by definition of bivalent) two
Boolean values of True or False with nothing in between. What you have been
saying is the same as saying 2 == 5.

All bivalent systems of predicate logic only have (by definition of bivalent) two
Boolean values of True or False with nothing in between. What you have been
saying is the same as saying 2 == 5. — PL Olcott

I never said that. You keep misinterpreting me.

Anyhow it shows you that bivalent logic is not useful and incapable for the real world uses in describing the complexities of the structures, events and objects.

Not sure if your previous post was about the function call in Prolog, but it didn't look like the standard way of using function calls in the other PLs, hence I asked you about the difference between math functions and programming functions.

Anyhow it shows you that bivalent logic is not useful and incapable for the real world uses in describing the complexities of the structures, events and objects. — Corvus

Not at all every declarative sentence is {True, False} or incorrect.

Not sure if your previous post was about the function call in Prolog, but it didn't look like the standard way of using function calls in the other PLs, hence I asked you about the difference between math functions and programming functions. — Corvus

Prolog does not simply assume that every statement is True or False, thus can screen
out epistemological antinomies that are simply incorrect.