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.

Hmmm I see what you are saying now, but I don't agree with you. "Beyond" in the example of "It is beyond imagination" has nothing to do with the concept of space. It is just the way people use the word in the context.

If we ask people is this sentence true: "What time is it?" the smartest ones will say type mismatch error. Those that have less insight will simply be confused. — PL Olcott

What time is it? can be true or false depending on what was the criteria of the truth.
If the criteria of truth was whether someone asked the question or not, and someone asked the question, then it was true that the question was asked.

Also it can be inferred and reasoned that the person who asked the question didn't know the time. If the criteria of truth was, if there was anyone who didn't know the time, then it was true that the questioner didn't know the time at the time of asking the question.

If you didn't have the criteria of truth, then you wouldn't have the information regarding true or false on the question, which sounds obvious then you don't know what the context of the question was about, and you had no knowledge of what the criteria of truth of the expression was.

Often the smartest sounding folks can be the dumbest. You have to learn to think differently from others. If you sound the same as the others on these issues, then it wouldn't prove anything apart from that you have spent all your life browsing the internet. If you think differently and come up with different ideas, you may get told that you have a problem in logic by the crowds, but you know that you are thinking with your own mind, not just parroting or agreeing and yearning to be accepted to the group of the crowds.

Basically I am saying that self-contradictory expressions such as the epistemological antinomies that Gödel refers to are not truth bearers (neither true nor false) thus must be excluded from formal systems and never any part of any formal proof. — PL Olcott

Every truth or falsity must be derived from some facts in the world or the known axioms which are self evidently true. The paradox starts with the obscure sentence whose truth falsity value no one knows where or what it was derived from. Therefore there is no point for you progressing into the If then arguments or inferencing. That is my point.

"This sentence is false."

If it is false, then it is true.
If it is true, then it is false.

The If parts need reference (under what ground it is false or true) to claim it is either false or true.
There is no indication of what the reference for presuming it is false or true.
Hence the arguments are invalid.

An expression of language that is both a question and a statement would also have
to be rejected until it is translated into one or the other. — PL Olcott

But people use the expression all the time in daily ordinary communications. Why reject?

The sentence: "Did you lie?"
is not a truth bearer thus would be rejected by a correct Truth Predicate. — PL Olcott

It wasn't "Did you lie?" we were talking about. It was "You lied, didn't you?" That was the original sentence. It cannot be chopped into two sentences. It is one sentence, which is both declarative and questioning form. It means, you lied, and it is true.

"Isn't it a beautiful day?" Another example saying that it is true that it is a beautiful day, but in questioning form.

“it is beyond me” refers to something which is spatially separate from yourself; so, no, this is not an example of a different meaning of ‘beyond’ that is aspatial. — Bob Ross

From your post, it appears that you might not be a native speaker of English language actually. When you said "It is beyond me", it sounds like "It is behind me in space." literally, but it actually means, you are "unable to understand". "Beyond imagination" would mean "unable to imagine". It has nothing to do with physical space in this context.

"Beyond" would only indicate physical space, if you were talking about a placement of a physical object in your sentence.

See my point? Even a simple word, "beyond" has different meanings depending on the context, and how you use it.

No, I am not a Hindu. I am an agnostic atheist materialist monist. — Truth Seeker

For an agnostic atheist materialist monist, having all the

simulation/hallucination/dream/illusion — Truth Seeker

sounds like one's claim that she is a vegetarian, but loves eating beef, pork, chicken, lamb, and enjoys BBQ. :grin:

What do you mean by "So the OP is not the actual case."? — Truth Seeker

It means the OP is under some sort of suppositional or imaginary scenario rather than based on the fact. When you say "It is possible that", it must have some degree of plausibility with the factual evidence for being real life cases. Without it, "It is impossible that" has the same plausibility too.

I did answer the second question by editing my initial answer as I had initially forgotten to answer the second question. — Truth Seeker

Not too worry.

Do you have a religion? — Truth Seeker

If religion is a belief system, then no. No religion is my religion.

I suppose, theoretically, I could have my brain removed and put in a jar that keeps it alive, and is wired to sensory apparatus so I could still perceive what's near me. My guess is I would still be conscious, and still myself. My brain is where my consciousness lies. I can lose any number of body parts, and still be my self. — Patterner

Removing the brain from the body would make them both die immediately. Medically, scientifically and realistically it is unimaginable. Brain can only function properly in the body intact from the birth of the agent, and then naturally having been nurtured by the parents, growing experiencing and interacting with the real world and other members in the society.

The body without a brain is a corpse, a brain without the body is just a biological organ. There have been no cases of brain transplanting in human history, and it is doubtful if it would ever be possible.

I didn't say it was actually the case. — Truth Seeker

I see. So the OP is not the actual case. You have been talking the whole lot under a simulation/hallucination/dream/illusion. Got you. Ok, please carry on. :wink:

Ok fair enough. Two questions.

1. I was not asking you to test the simulation/hallucination/dream/illusion hypothesis. But I was asking you to provide the evidence that your claim is true and real (not guessing and not imagination). That is totally different issue. Would you say so?

2. Are you a Hindu? Because your reply seem to be concretely based on the Hinduism in its principles.

I disagree. I could be a disembodied soul experiencing the simulation or hallucination or dream or illusion that I am in a human body, in a universe where there are other humans and other species. — Truth Seeker

Can you prove that is the case with some evidence? Or is it just your guessing or imagination?

It's possible that my body, the Earth, the universe, and all the other living things including you, are all part of a simulation or a hallucination or dream or illusion that I am experiencing. — Truth Seeker

For you to have been experiencing a simulation, hallucination, dream or illusion, you still need the physical body. No one can have all those experiences without having a biological body and brain. Therefore your reply is not making sense. Would you agree?

The statement: (a) "You lied." and the question: (b) Did you lie? When we break it down to its constituent parts (b) is still not a truth bearer. — PL Olcott

I am not sure if you are allowed to modify the given example sentence under the process of breakdown.
If it is allowed, then virtually all questions can be truth bearers.

For instance, "What time is it now?"
One could modify it into "It is true that she asked what time it is now.", hence one can say it is true.

"How are you doing?" - one can modify it again - "It is true that she asked him how he was doing." Therefore "How are you dong?" is a truth bearer.

Are modifications, inferences and breakdowns on the original sentence allowed for TF values?

questions cannot be {truth bearers} is known by everyone that knows what {truth bearers} are. — PL Olcott

Your statement here sounds nonsense. Some questions can be for true or false. For example, "You lied, didn't you?" This means you lied, and it is true. It is also to mean you should be aware of the fact that you lied.

We are not our body, but we appear to be embodied. I agree about the mind dying when the body dies. — Truth Seeker

What do you mean by "we appear to be embodied"? Can you imagine yourself existing without your body?

What is the true nature of the self? — Truth Seeker

Isn't one's physical body the true nature of the self? The moment the body dies, mind also dies truly and eternally.

Come to think of it, Afterlife sounds life keeps continuing after life ceased. Shouldn't it be called "Afterdeath"? :chin:

You missed the point: my linguistic expression of 'beyond' space is incoherent. 'Beyond' refers to something in space. — Bob Ross

Your keep parroting "You missed the point." in most messages you write wouldn't help you on understanding.

"Beyond" can mean other things too depending on the context. For example, "It is beyond me." - it does't mean something in space.

If your reader didn't understand what you wrote, then it is likely that your writing was not grammatically correct or it was out of context.

Gödel Incompleteness can only be implemented in systems that implement Boolean True(L, x) incorrectly. It cannot exist in systems where True(L, x) means that x is provable from L and False(L, x) means ~x is provable from L and for everything else x is simply untrue in L.

This same reasoning also conquers Tarski Undefinability.
When Tarski anchors his undefinability in the Liar Paradox: — PL Olcott

As I said before, will say again. The whole confusion with the paradox and undefinability have been originated from the single narrow perspective seeing the problems in propositional logic, which only allows a proposition must be either True or False.

If you think about the real world situations and objects, there are cases where things are neutral i.e. neither true nor false such as Number 0. And there are the real world cases where things are both True and False, read on QM or some Metaphysical topics.

If you open up the perspective wide and accept all these possibilities in the real world, it is quite normal for some cases to be either True or False, neither True nor False, or both True and False. If you apply FOL and HOL into your program languages bearing that point in your mind, and design the programs to deal with the particular cases in the real world examples, they deal all the case quite fine (Quine).