On truths, falsehoods, and negation

[Sikz]

Every statement of a truth or a falsehood is a negation of its opposite.

"This tree is here."
This is a negation of "This tree is not here."

"There is not a tree here."
This is a negation of "There is a tree here."

Both of these can be understood by logic. More complicated concepts are negations as well. I shall put a statement and its negation (in actuality, they are each negations of each other) now in a list:

"Rabbits exist."
"Rabbits do not exist."

"Rabbits act like this."
"Rabbits do not act like this."

Notice that "Rabbits exist." depends upon "Rabbits do not exist."- it has absolutely no dependency upon "Rabbits do not act like this." The state of rabbits is not in question, but rather the existence of rabbits in any state.

So logically, any statement is understood as a negation of its opposite. No set of a statement and its opposite is possible:

"Rabbits exist AND Rabbits do not exist."
"The tree is here AND the tree is not here."

These are "logical impossibilites" of the higher order. "Logical impossibilities" can also denote a conclusion that is disproved by logic from a set of axioms- these are "logical impossibilities" of the lower order; they are not TRUE impossibilities due to logic, but due to the axioms chosen. Whenever the terms "logical impossibilities" and "logical impossibility" are used in this document it is implied that they are of the higher order.

Why are these logical impossibilities? Because they have no negations. What is the opposite of "Rabbits exist AND Rabbits do not exist."? There is none! Therefore the statement is meaningless.

Now we come to the reason why we cannot come to a logical understanding of reality in relation to the unreal or of the universe in relation to nothing.

"The universe exists." This statement is understood as the negation of "The universe does not exist." We can look at either of these states and examine them; but in our attempted study of their definitions in relation to each other, we find our progress blocked- we can only describe them as opposites, we cannot describe in any more detail the difference (although we can describe either one by itself in a large amount of detail). This is because the statement we are trying to understand has no negation:

"The universe exists AND the universe does not exist."

As you the reader have no doubt noted, this statement is a logical impossibility.

"Reality is and reality is not."

Another logical impossibility.

There is also an interesting mathematical idea dealing with two "polarities" that can be described but never reach each other, but I shan't go into that at the moment. Anyone have any ideas on what I've posted?

 

[phoenixthoth]

it all seems correct in binary logic, especially considering the law of the excluded middle.

in ternary logic and fuzzy logic, some statements are equivalent to their negations. if "God exists" has the third truth value, for example, then so would "God does not exist" and the conjunction of the two statements also has this third truth value.

the generalization of the excluded middle there is that every statement has to have exactly one truth value.

 

[master_coda]

Originally posted by Sikz
Why are these logical impossibilities? Because they have no negations. What is the opposite of "Rabbits exist AND Rabbits do not exist."? There is none! Therefore the statement is meaningless.

Just as a clarification, the negation of "Rabbits exist AND Rabbits do not exist" is "Rabbits do not exist OR Rabbits do exist".

This is a result of the fact that:

NOT( X AND Y ) = NOT( X ) OR NOT ( Y )

 

[phoenixthoth]

right.

~=not
^=and
v=or

one of de morgan's laws is what you stated is that
~(X ^ Y) is equivalent to ~X v ~Y.

this is a case where Y=~X and if you assume further that ~~X=X,

~(X ^ ~X) is equivalent to ~X v X, which you could say is the common form of the law of the excluded middle: ~X v X.

 

[Sikz]

The negation of "Rabbits exist AND Rabbits do not exist." is "Rabbits do not exist OR Rabbits do exist."?

Wouldn't you say that's more of in "inverse" than a negation? "Rabbits do not exist OR Rabbits do exist." is dealing with two separate statements ("Rabbits do not exist." and "Rabbits do exist.") and saying that one is "true" and one is "false"- which is the same as saying one of those statements alone. However, the "OR" also contains some element of uncertainty, which is inconsistent with the certainty statements I've been making thus far.

Truth statements in my example are represented by a 1 mathematicly, falsehood statements a -1. You can understand 1 as the negation of -1, and you can understand -1 as the negation of 1. However, when you try to look at both of them they "cancel out" to 0, which has no negation. What you've done is this: If the number line extends as a one dimensional line and negations are the mirrored numbers relative to zero, you've said zero's negation is a number UNDER zero, on an axis of numbers perpendicular to our number line- which is not the same type of negation we had been using.

If you introduce this perpendicular line into our logic, it is to be inferred that numbers other than zero also have "negations" in your sense; that "The tree exists." has a negation in my sense of "The tree does not exist.", and a separate statement in your sense...

 

[master_coda]

Originally posted by Sikz
Wouldn't you say that's more of in "inverse" than a negation? "Rabbits do not exist OR Rabbits do exist." is dealing with two separate statements ("Rabbits do not exist." and "Rabbits do exist.") and saying that one is "true" and one is "false"- which is the same as saying one of those statements alone. However, the "OR" also contains some element of uncertainty, which is inconsistent with the certainty statements I've been making thus far.

No, that is the negation, by the definition of negation. It's also the "inverse" if you define "inverse" as negation. Otherwise, there is no definition of inverse when dealing with just logic. "inverse" is a term defined in some fields of mathematics built on top of logic, but is not an intrinsic part of logic itself, at least in standard constructions.

Yes, "Rabbits do not exist OR Rabbits do exist." is dealing with two separate statements, but so is "Rabbits do exist AND Rabbits do not exist". Both OR and AND are connectives which join two statements into a compound statement.

OR does not contain any more uncertainty than AND. If the statement X OR Y is true, then we have "uncertainty" in the sense that we do not know which one of the statements is the true one, or perhaps they both are true.

But on the other hand, AND contains the same uncertainty. If X AND Y is false, then we cannot determine which of the statements is the false one, or perhaps they both are false.

As for using numbers to represent logic truth and falsity...this is generally bad style when studying foundations. Our concepts of numbers are built on top of logic, so when studying the logic itself, it's best to avoid using the numbers.

 

[toasty]

Originally posted by phoenixthoth
it all seems correct in binary logic, especially considering the law of the excluded middle.

in ternary logic and fuzzy logic, some statements are equivalent to their negations. if "God exists" has the third truth value, for example, then so would "God does not exist" and the conjunction of the two statements also has this third truth value.

the generalization of the excluded middle there is that every statement has to have exactly one truth value.

Funny I should read this on the internet!

Who knows maybe for some creatures there's A or B or C or D or E or F ... ad infinitum.

Perhaps for others everything is just a shade of grey from the next, so the term 'or' would be nonsense to them, just like 'x is and is not at the same time' is to us?

 

[phoenixthoth]

that's what fuzzy logic's all about. ;)

 

[Canute]

Phoenix

in ternary logic and fuzzy logic, some statements are equivalent to their negations. if "God exists" has the third truth value, for example, then so would "God does not exist" and the conjunction of the two statements also has this third truth value.

What does this third value mean? Can you give an example of how using a third value as an answer to a question like that of God's existence helps to answer it.

 

[phoenixthoth]

Originally posted by Canute
Phoenix


What does this third value mean? Can you give an example of how using a third value as an answer to a question like that of God's existence helps to answer it.

you might as well be asking me to define truth or falsehood; same difference. simply and naively, a third truth value is neither true nor false, whatever those mean.

intuitively, though i don't know what I'm talking about, the third truth value may function as the concept of mu in eastern philosophy though perhaps you'd know more about that than me. well, that's what inspired me to work on ternary logic anyway, even if my intuition was misguided. hence why i use the letter M in my work; it ain't for "maybe" and i despise that connotation though i must admit that is one way of looking at what it means.

another way of looking at it may be that if a statement has the third truth value, it is rudimentarily independent and/or undecidable relative to things that are in the true and false categories. philosophically, this to me would imply that free will is alive and kicking in that one is free to believe in something that isn't false (or true, according to "the system" at least).

this website on many-valued logic may do a better job of explaining it than i did:
http://plato.stanford.edu/entries/logic-manyvalued/

 

[Canute]

I thought that this might be where you were coming from. Bear in mind that I'm not an expert but ...

The thing is that Mu doesn't mean that there is an arbirtrary third category of answer that has no clear meaning. It is used when there is no answer to the question because the question contains false assumptions. (As in 'have you stopped beating your wife?' for instance).

Within a formal system of theorems there are truths, falsities and undecidables relative to that system. It is only the undecidables that cause a problem. In a non-dual view of reality all these systems, without exception, are incapable of truly describing reality, since they can contain only relative truths and because they must, by defintion, exclude anything that lies beyond the system, of which there is always at least one thing. You might say that in Buddhism all questions are undecidable ex hypothesis and for good reason, whereas in other systems of reasoning our inability to prove anything is treated as an epistemelogical nuisance with no explanation.

To assert a truth is to assert that the system within which it is a truth is true. As reality lies beyond all such systems, (it is inevitably the meta-system), no (absolute} truth can be asserted within any formal system of truths and falsities. That is, there is no absolute truth within any formal system of reasoning since those systems can contain only relative proofs. Also they they make use of axioms which are either false or arbitrary.

However proofs of truth or falsity made within such systems can be considered to be true or false inasmuch as they are provable in the system. There is no suggestion that within the system there is a possible third truth value. It is only when a theorem makes claims about what lies outside the system that the trouble starts. (An example would be any scientific assertion about materialism or idealism).

Within any formal system all theorems that are not relatively true or false must be undecidable. Therefore all assertions about reality are undecidable. Therefore all such systems produce false accounts of reality.

Argh - it always gets in a muddle. Briefly, Mu is not an expression of a third type of answer, distinct from true or false. It is a dismissal of the question, usually invoked to counteract questions derived from dualistic assumptions and ways of thinking. In teaching situations sometimes a whack with a stick is used instead.

 

[phoenixthoth]

funny, i was just reading a book and they mentioned the slapping of one's sandle as a teaching technique.

yeah, i said i didn't know what mu was. i think i had also heard that it means "the question is absurd" which is closer to what you said than what i said.

thanks for the info.

 

[Dissident Dan]

I must admit that I am not well-read on many-valued logic systems,
but I am rather confident in saying that they do not describe reality, although they may be useful tools.

The thing about true and false is that they are defined such that:

true = not false
false = not true

This leaves no room for other truth values.

While we can imagine a system with 12 million truth values, there is no reason to think that such a system makes any sense.

 

[phoenixthoth]

true or false: i am tall?

 

[Dissident Dan]

Ah, tricky, you are.

Tallness is a relative term. That means that it requires a comparison. It will be true or false depending on what you compare yourself to.

Of course, sometimes you get an absurd question that doesn't even have a truth value to it because it doesn't even make sense in the first place.

 

[phoenixthoth]

the point being that fuzzy logic models tallness better. in binary logic, you define "i am tall" to be true at some arbitrary value, say x, and "i am tall" is true if my height is at least x. if you graph this it looks like a step function; at one point it suddenly jumps from false to true. in fuzzy logic, one is free to assign the truth value to "i am tall" or "i should apply brakes now" in any way one wishes. this way, it's not suddenly true that "i am tall" after a certain height and rather than have a step function, it could be a smooth curve or line.

just go to a search engine (eg google) and search for
application "fuzzy logic"
and you'll get many hits.

i'm not clear on what you mean by fuzzy logic not making sense. what do you mean?

 

[Dissident Dan]

My point in the "12 milllion truth values" comment is that we can imagine or make up any sort of system, but that doesn't mean that there's any reason to think that it's representative of reality.

Fuzzy logic is good for representing how people think, but that doesn't mean that it's representative of reality. In reality, there is no such thing as "tall", it's only a mental concept.

Tallness is a value that better fits a number than true/false.

 

[confutatis]

Originally posted by Dissident Dan
My point in the "12 milllion truth values" comment is that we can imagine or make up any sort of system, but that doesn't mean that there's any reason to think that it's representative of reality.

So which system is representative of reality? And why?

Fuzzy logic is good for representing how people think, but that doesn't mean that it's representative of reality. In reality, there is no such thing as "tall", it's only a mental concept.

Sure, but isn't that the case with binary logic as well? Aren't "true" and "false" mental concepts as well?

 

[confutatis]

Originally posted by Sikz
Now we come to the reason why we cannot come to a logical understanding of reality in relation to the unreal or of the universe in relation to nothing.

[...]

"The universe exists AND the universe does not exist."

As you the reader have no doubt noted, this statement is a logical impossibility.

That was an interesting point of view, but I would suggest giving it a little twist to try and get rid of the problem. I'd say that reality is in fact a paradox, and at some basic level two contradictory statements can be true. I should point out that the idea that paradoxes cannot be real is just an assumption, and it might turn out to be an invalid one.

A paradox is just a mental concept. It means we can't reconcile two ideas we believe to be true. When faced with paradoxes, we always do the same thing: change our point of view and state the two ideas in a slightly different way, so that their truth is preserved but the contradiction disappears. And that process gives rise to the experience we call "time".

Now think about that for a while. Isn't "time" something which makes false statements become true, and true statements become false? "Today is Wednesday" - "Today is not Wednesday". "I did not have lunch yet" - "I had lunch already". "I do not know that person" - "I know that person very well". If you think outside of time, reality is full of paradoxes, and everything we think of that "changes with time" is both true and false (I would say "true and false" at the same time, but we're outside of time...)

It's quite a jump, I know, to stop thinking of time as something purely physical, and start seeing it as purely mental. But that's just part of the paradox: "reality is non-mental" - "reality is mental". I believe both statements are true, "time" being that which brings reality from the mental domain (the future) into the non-mental domain (the present) then back to the mental domain again (the past)

 

[Canute]

Originally posted by confutatis
That was an interesting point of view, but I would suggest giving it a little twist to try and get rid of the problem. I'd say that reality is in fact a paradox, and at some basic level two contradictory statements can be true. I should point out that the idea that paradoxes cannot be real is just an assumption, and it might turn out to be an invalid one.

I agree, and disagree. The existence of the universe is a paradox according to science, hence science's lack of a coherent metaphysic. The 'something/nothing' problem prevents any full explanation of the universe as long as we assume that nothing comes from nothing.

In a way the problem is similar to that of wavicles. If it a wave and a particle at the same time then what is it?

Sikz summed up the case IMO

Now we come to the reason why we cannot come to a logical understanding of reality in relation to the unreal or of the universe in relation to nothing.

"The universe exists AND the universe does not exist."

As you the reader have no doubt noted, this statement is a logical impossibility.

I believe this is precisely correct, almost.

In a non-dual view it is true, in an everyday sense, that the universe exists and not-exists at the same time, and always has and always will.

However this is not a paradox (remember wavicles). It only appears to be one because we do not understand the problem of 'essence', what philosophers call the 'problem of attributes'. To cut a long story short this is the ancient and current problem of what remains of an object once its external attributes and aspects have been removed.

It appears to be an unsolvable problem, since one must conclude either that there is something underlying all attributes which does not exist, or that the universe is mysterious and cannot be understood rationally.

These problmes are solved by assuming that at the limit something is the same as nothing. However in itself this statement seems to make no sense because it is usually misinterpreted.

It should be taken to mean that there is 'something' which is neither something or nothing, and not half and half. This is an issue that confuses even fairly advanced Buddhists. The Buddha warns the practiser of getting confused by it in the Sutras. Here he is talking in the Surangama Sutra.

“As the practiser looks exhaustively into the fourth aggregate and finds that it ends in emptiness, he reasons that its end is annihilation and its manifestation is creation and so infers that each living being partly exists and partly does not, and so concludes that all things in the world are half finite and half infinite. By so discrimination between the finite and the infinite, he falls into the heresy which screens his Bodhi nature.”

The point is that to say that ultimate reality (essence) exists and not-exist at the same time is not paradoxical, it is merely difficult to understand.

The 'problem of attributes' virtually proves that 'existence' is not quite what we think it is, for if it is what we normally think it is then the problem of attributes, of what underlies physical phenomena, is logically unsolvable. Yet here we all are, so it must be solvable.

The problem of attributes implies that materialism is not just a metaphysical theory, it is also a mystical theory, for it must leave 'essence' (and thus matter) forever unexplained.

 

[phoenixthoth]

These problmes are solved by assuming that at the limit something is the same as nothing. However in itself this statement seems to make no sense because it is usually misinterpreted.

what if there are no axioms. truths, yes, but no axioms?

The point is that to say that ultimate reality (essence) exists and not-exist at the same time is not paradoxical, it is merely difficult to understand.

that's it exactly. and if our puny piles of organic matter locked in our skulls could for a minute understand that ultimate reality, it would or may probably cook.

 

[Canute]

Originally posted by phoenixthoth
what if there are no axioms. truths, yes, but no axioms?

If axioms are by definition the answers to unanswerable questions (within the system that derives from the axioms) then there are no absolutely true axioms. Ergo there are no axioms worth bothering with.

To reach truth we therefore have no choice but to take our own experience being axiomatically true, and work from there. (As Aristotle, Popper and co. have always argued, saying that certain knowledge necessitates the knower becoming one with the known. Popper's 'Problem of Induction' paper is worth reading if you haven't, it's unmissable and online).

So I agree with you. There are truths but no absolutely true axioms.

that's it exactly. and if our puny piles of organic matter locked in our skulls could for a minute understand that ultimate reality, it would or may probably cook. [/B]

As Buddhists have been asserting for a long time. They also assert that anyone can know it.

 

[phoenixthoth]

To reach truth we therefore have no choice but to take our own experience being axiomatically true, and work from there. (As Aristotle, Popper and co. have always argued, saying that certain knowledge necessitates the knower becoming one with the known. Popper's 'Problem of Induction' paper is worth reading if you haven't, it's unmissable and online).

we hold these truths to be self-evident...?

kinda brings more meaning to the word self-evident if one is the truth, huh?

 

[confutatis]

Originally posted by Canute
The point is that to say that ultimate reality (essence) exists and not-exist at the same time is not paradoxical, it is merely difficult to understand.

Well, a paradox is the statement of an idea together with its opposite. To state that reality exists and does not is a paradox. But all paradoxes have solutions, the only problem is that the solution of an old paradox often creates new ones.

It's certainly possible to understand how reality can exist and not-exist at the same time, by simply redefining what 'reality', 'exist', and 'time' mean (notice how I attempted that myself!). What does not seem possible is a complete understanding of everything which does not harbor any paradoxes. I have never seen that, was never able to do it myself, and am fairly skeptical of anyone claiming to have done it.

 

[Canute]

I can see why you say that. But knowledge has greater 'explanatory reach' than proof, and there are no paradoxical or unanswerable metaphysical questions in the non-dual view of reality.

 

[phoenixthoth]

bingo!

 

[confutatis]

Originally posted by Canute
I can see why you say that. But knowledge has greater 'explanatory reach' than proof, and there are no paradoxical or unanswerable metaphysical questions in the non-dual view of reality.

Now that really went way over my head. The only way I know to get rid of unanswerable metaphysical questions is to stop asking them. Rational thought tends to confuse the mind if applied to certain domains. So maybe that's what you're talking about, maybe not.

 

[Canute]

Originally posted by confutatis
Now that really went way over my head. The only way I know to get rid of unanswerable metaphysical questions is to stop asking them. Rational thought tends to confuse the mind if applied to certain domains. So maybe that's what you're talking about, maybe not.

Roughly yeah. You said that there was no rational understanding of everything that was non-paradoxical. I was disagreeing.