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 wish we knew more. — Truth Seeker

Yeah interesting stuff actually.

You are welcome my friend.   Having agreed the OP is not testable or verifiable in formal logic, we still can keep discussing it under philosophy of psychology.  People dream and have illusions, and it is also part of human mental events.

Under what circumstances do humans have illusions, hallucinations, simulations and dreams?
Some mental events seem to be conscious and self-induced nature such as the simulated perceptions, simulated thinking or making up the hypotheses for the scientific experiments in order to seek pleasure or as part of religious ceremonies or to find some scientific truths via coordinated experiments.

However, some mental events are unconscious and out of one's control such as dreaming, hallucinations or illusions.  These mental events seem to occur from drug inducements, hypnosis, during sleep or simple misunderstanding and misperception of the world objects, events and structures.

How does one's idea of self relate to the mental events?  Do some mental events have any deeper significance in religious meanings or esoteric nature?  Have some world famous artists' creations based on this type of mental events? 

There is a lot of meat in the OP from the perspective of philosophy of psychology.

Are you talking about the untestable hypothesis that our perceived world could be a simulation or hallucination or dream or illusion? Then they are untestable ideas. If we could test them then the results would be in the "matter of facts" but we can't test them. — Truth Seeker

Ok your point accepted. :cool:

{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.

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.

Science-fiction is fun. — Truth Seeker

Science-fiction must have a plot, and the characters as well as the storyline. The OP seems to lack all the essential elements of qualifying for SF.

If you read Hume, he says all human knowledge is either in the domain of "matters of facts" or "relation of ideas - geometry, arithmetic or algebra". In which category of human knowledge do simulation, illusion, hallucination, hypotheses and dreaming belong to?

It's not a belief. It's an imaginative exploration of a hypothetical situation. This is what science-fiction writers do. This is why I said "What if". — Truth Seeker

Science-Fiction sounds acceptable. :)

I agree. — Truth Seeker

:nerd: :ok:

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.

What if I exist as an immaterial soul that is experiencing the illusion of being in physical body on a physical planet in a physical universe? — Truth Seeker

If you could provide some evidence or arguments for your "What if" presumption and imagination, then we could further analyse if the what if claim is justified.

But without that, it would be just your unverified and irrational belief, which sounds like the beliefs or stories belonging to in the realm of esotericism, science fiction, or religious beliefs. It would be difficult to discuss under philosophical investigation.

No it is not. This is wrong. Confidence is informed by fear, yes. And fear is the patterns you are referring to as experience. But it also includes BEING in those situations. So, it can be hard to speak of single emotions rather than all together in experience. — Chet Hawkins

I disagree. How can a confidence backed by empirical experience and evidence can be related to fear? You might have fear when you assert something you don't have concrete knowledge, evidence or experience, so you don't know what you are talking about.

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}

Everything you claimed as possible i.e. dreaming, being under illusions, hallucinations and simulations are all possible, if and only if you physically existed in the real world. That is the precondition for all the possibilities you listed.

Exactly! I would say that I rest my case, but you are still not getting it.

Confidence IS NOT knowing. Firstly, it cannot be, because one cannot actually know. One only believes. So, confidence is exhibited as 'They who do not know, but believe strongly anyway'. Of course immoral fear types will chafe and call that incoherent. They are not really right, but this is the hubris of relatively high awareness or let's say a facility with awareness. — Chet Hawkins

Perhaps we are just keep missing each other's point. It is understandable that it can happen. After all we have different ways looking at things in the world, and I am not aware of the contents in your mind what is going on, and you must be the same.

Confidence in the linguistic expression is based on the empirical experience and evidence from the real world events and observations, hence it can be said with most certainty.

What if I am in a special dream which lasts six hours of actual time but 100 years of experiential time? What if I am an alien with this technology that lets me dream I am a human? — Truth Seeker

If you provide some evidence for your presumption, and if they were convincing enough, then I might say "ok maybe, but are you sure?" Without the evidence or arguments, I would say "Nah, impossible" :D

I am not convinced. I think that it is extremely unlikely that my perceived reality is a simulation or hallucination or dream or illusion but it does not mean that there is not even a one-in-infinity chance of it being a simulation or hallucination or dream or illusion. — Truth Seeker

Logically possible, metaphysically maybe, but physically impossible.

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.

I am surprised that you have not watched The Matrix, Predestination and The Truman Show. They are such excellent movies for philosophical discussions. You can watch all of them on YouTube. — Truth Seeker

But watching movies bores me to death. I would rather read or go for a walk, or do workouts :D

What if I am in a dream that lasts my entire lifetime? What if my death is the only way to wake up from the dream? — Truth Seeker

It is impossible for dreams last more than a night, because you would have to wake up in the morning for the real life. Biologically and physically, if your dreams lasted more than 3 days, then you would get exhaustion not having consumed the real food and drinks, your body will not last in the real world, making your dreams evaporate.

What if I am in an illusion that lasts my entire lifetime? What if my death is the only way to break the illusion? — Truth Seeker

No illusions or dreams last one's life time. If it did, then it would be reality. Not illusions or dreams.

I will either find out when I die or I will just cease to exist when I die. — Truth Seeker

What if, death and dyings are illusions and dreams?

If your experience lasted more than a day or two, then you can safely conclude that it was a reality, not dreams or illusions.

Every increment of HOL above FOL quantifies over expressions of the next lower order. FOL quantifies over propositions, thus propositions are the ground level of all every order of logic. — PL Olcott

This seems to be 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.

Your claim that FOL quantifies over propositions doesn't make sense. FOL can use the variables in the individuals and subjects in the sentence. Not over the propositions.

Your point is locked up in the propositional logic only, not seeing further into the other domains of Logics.

3. Inception — Truth Seeker

I am not a great fan of movies, but recall watching Inception many year ago. It was an interesting movie right enough.

Here is an evidence all we are experiencing is not dreams, illusions, hypotheses or hallucinations. If it were, you wouldn't be asking the questions or doubting the reality.

If your experience of reality were not real, then it wouldn't last your life time. You would wake up from the illusion or dreams, and get back to your reality within a day or two. Not doing so proves your experience is real.

I am surprised too. Will add it into my reading list. :)

Will have reads and get back to your point on this part later.

If it can't be a proposition then it must be rejected by any logic system
from propositional logic on up to higher order logic. — PL Olcott

It seems that you are not able to tell the difference between propositional logic, predicate logic and HOL. What you were saying is confined to propositional logic. But once you are in the realm of predicate logic and upwards, the concept of truths becomes multifaceted nature.

The sentence, "This sentence is not true." can be true, unknown, false or contradictory depending on the condition of truth.

Just because they are logical possibilities it does not mean they are actual possibilities. — Truth Seeker

I think you are talking about the Modality and Possible world in Logic, which is interesting subject. Sure, we can talk about the possible world and its scenarios, inferences and reasonings with Modal Logic.

I am familiar with Christianity. I am an ex-Christian. — Truth Seeker

Wow cool. I never imagined you could have been an ex-Christian.

Have you read the whole Bible? — Truth Seeker

No, I haven't read the Bible at all. All I know about the Bible is the 1 quote. It goes something like "God said, Let there be light, and there was light. God was jolly happy and satisfied with the light."

I have. — Truth Seeker

Cool. I know who to ask with any queries with the Bible then.

It's the most evil book I have ever read. — Truth Seeker

Really? Interesting.

If it is true that makes it untrue if it it false that makes it true. This proves that it is neither true nor false. — PL Olcott

In FOL or PL, "X is not true" depends on the content of X.
In the traditional propositional logic, there is no option for that, hence it is only true in grammatical form of the sentence. Some folks insist it is still true. Likewise "What time is it now?" is true in the form of grammar. So is, "There are the Martians living in Mars."

"This sentence is not true" is called the strengthened Liar Paradox and is its best form. — PL Olcott

"This sentence is not true." can be true in the form of the sentence X is not true in grammar. Nothing wrong with that. But the content of the sentence is unclear. It doesn't say which sentence it is talking about, and "not true" in what sense. So, it is both true and unclear.

In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. https://en.wikipedia.org/wiki/Undecidable_problem — PL Olcott

This wiki document needs to be verified, the wiki says. But going back to the OP, you need to bring out some arithmetic sentences or expressions, which proves Tarski's undefinability is correct or incorrect. And then we will try them under HOL, and see if it is still valid.

Like you, I too do not have a religion. — Truth Seeker

The Christian doctrines might have all the answers you are seeking for. Would you not be interested in reading about them?

I am just aware of the possibility that my perceived reality could be a simulation or hallucination or dream or illusion. It does not mean that I am convinced this is the case. If I were convinced, I would have said that I am convinced. — Truth Seeker

What are the evidences for the possibilities that you perceived? What makes you not convinced?

:roll: — Bob Ross

My point was crystal clear.

You just unknowingly contradicted yourself. "Over the boundary" is the idea that there are two things in space (at least conceptually) and one is beyond the "boundary" of the other. — Bob Ross

There was no contradiction in "boundary", apart from your misunderstanding on it. "Boundary" can be physical as well as mental. Again your inability in understanding the concept stems from your lack of understanding the context in linguistic expression.

Natural language cannot be accurately evaluated until it is translated into some totally precise form. An expression that is both a statement and a question cannot be properly evaluated by any Boolean True(L, x) predicate. — PL Olcott

If you say, natural language must be translated into some totally precise form (I take it to axiomatic or formal language.), then questioning sentences can be translated into declarative or descriptive form, which can make them truth bearers.

All linguistic expressions are usually interpreted into the hidden or suggestive meanings in practical conversations. For example, if you say "Are you a non-native English speaker?", to mean polite way of saying "You must have flunked your English in school.", or "What time is it now?" to suggest, "It is time to prepare some dinner. I am bloody hungry the now." ... etc.

:up:

I agree. — PL Olcott

:ok:

Not when the entire notion of undecidability depends on these things. In that case we use your first quote as the basis of True(x). From this we derive False(x) ≡ True(¬x) and by this process the whole notion of undecidability utterly ceases to exist. — PL Olcott

In "This sentence is false", whether "is false" or "is true" referred to the subject of the sentence "The sentence" or the whole sentence "This sentence is false" was obscure. Would this be part of the undecidability? Or is it for something else? If for something else, then can you give a few example of the undecidability?

The idiom "it is beyond me" cannot be made sense of, conceptually, without the idea of space. — Bob Ross

It is easy enough to understand this, when one tries to describe "it is beyond me" without using spatially-loaded terms like "beyond": they can't. It loses meaning. — Bob Ross

I cannot imagine anyone relating the idea of space with the word or concept "beyond". It is just an idiom of the linguistic expression which we use habitually to mean over the boundary of something. The something can be physical or mental depending on the context.

This is demonstrated quite well by the multiply by zero effect. You are always safer saying a thing is unlikely than you are to say it is impossible. — Chet Hawkins

Not making sense at all. "It is unlikely that" sounds you are lacking confidence on what you are saying, or just being evasive. "It is impossible that" sounds far more declarative and certain of what you are saying.

I disagree. No matter how implausible something is, it is nowhere near the same thing as saying something is objectively impossible. That again partakes of a dangerous misunderstanding of what perfection is. Perfectly impossible is probably just that, as in do not talk about it at all because (re-read this sentence until you get it). — Chet Hawkins

"It is possible that " is based on guessing, illusion, unfounded optimisms of low probability of the cases manifesting in reality. "It is impossible that " is based the empirical experience on the cases backed by the highest probability of something not happening in reality.

They are at the opposite end of the scale in the statements. The point was that "It is impossible that" has far more plausibility than "It is possible that".

Do you think there is some physical space in your imagination or around your understanding? I don't get it.