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

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 seperate 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 seperate statement in your sense...

master_coda

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

Funny I should read this on the internet!

Who knows maybe for some creatures theres 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

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

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.