A New System of Logic

1. Dec 17, 2003

Sikz

To understand logic we need to describe it. To describe something you must understand its opposite- in this case a total lack of logic. Therefore to understand logic we need a system that encompasses both logic and a lack of logic.

Logic is based on binary- something is logical or it is not logical. Something is a rabbit or it is not a rabbit.

Perhaps we need something based on trinary or quadrary (if those are the proper terms)? Any replies would be appreciated :)

2. Dec 18, 2003

Some electrical circuits use a tristate logic - true (on), false(off), or high impedence.

3. Dec 18, 2003

Guybrush Threepwood

you should do a search on fuzzy logic....

radagast: the high impedance cannot be used as input, so the logic is still boolean

4. Dec 18, 2003

Good point.

5. Dec 18, 2003

Hurkyl

Staff Emeritus
And any (known) multivalued logic can be modelled with boolean logic (and generally vice versa), so there isn't any real benefit in discarding the boolean stuff.

6. Dec 19, 2003

phoenixthoth

you mean like
T
~T
~( T v (~T) )
~( T v (~T) v (~( T v (~T) )))?

[0,1] can be modelled by sequences of binary digits so i can see how this would be done in an infinite way. how do you do it finitely?

7. Dec 19, 2003

Redfern

Absolutely you don't need anything other than binary logic.

Let X denote a statement, and let |X| denote the truth value of x.
Let 0 denote the truth value true, and let 1 denote the truth value false.

The basis of binary logic is this:

A. For any statement X: |X|=0 XOR |X|=1
B. If |X|=0 XOR |X|=1 then X is a statement.

A and B are both statements, and what is being said is this:

"A and B" is a true statement.

From that it follows that any system of logic other than binary, will contradict a true statement.

8. Dec 19, 2003

phoenixthoth

there are equally valid alternatives. valid in their non self contradictory nature.

that |X|=0 XOR |X|=1 is just as well replaced by the following alternate axiom:

V is a function from a domain containing such X to [0,1], called the veracity function.

the exclusive or aspect is built in to calling this a function.

in this system, if A and B are two wffs, using 0 as true and 1 as false,

V(AvB)=min{ V(A), V(B) }
and V(~A) = 1 - V(A).

all other connectives come from these two "generators" of connectives.

9. Dec 19, 2003

StarThrower

I don't understand this at all.

There is only one logic which is non contradictory, and that is binary logic.

10. Dec 19, 2003

Sikz

That's because you're looking at it from a binary perspective. Even "non contradictory" is a negation of "contradictory"- but in a non-binary system that negation wouldn't work the same way. As long as you try to describe a non-binary system by using binary you will come out with the result that it is impossible. Binary cannot describe higher systems, but higher systems can describe binary :)

Something I've been thinking about though:
True (T) is represented by a 1.
False (F) is represented by a 0.
This is because the math works out:

T * T = T
T * F = F
F * F = F

1 * 1 = 1
1 * 0 = 0
0 * 0 = 0

You can see that F can be replaced with 0 and T replaced with 1 and the equations still come out fine (you can also see that F is actually dominant to T). In case you don't understand where I am getting the T and F equations, they are like this:

For our example, "a chair is next to me" and "I am standing up" are true. "a llama is next to me" and "I am sitting down" are false.

T * T = T
A chair is next to me (True) * I am standing up (True) = A chair is next to me and I am standing up (True)
T * F = F
A chair is next to me (True) * a llama is next to me (False)= A chair and a llama are next to me (False)
F * F = F
A llama is next to me (False) * I am sitting down (False) = A llama is next to me and I am sitting down (False)

Now... That is binary, using zero and one. Isn't negative one just as valid as positive one though? Why not have another option, seperate from T and F, that can be represented by -1? The equations would then be:

1 * 1 = 1
1 * 0 = 0
0 * 0 = 0
0 * -1 = 0
-1 * -1 = 1
-1 * 1 = -1

Obviously I havn't developed the entire idea... I just thought I should post the idea about negative one. Any thoughts?

11. Dec 19, 2003

phoenixthoth

no, there are other logics though in the light of what hurkyl is suggesting, which i disagree with, you may be right. check out fuzzy logic. what follows is a relatively advanced treatment of fuzzy logic:

http://plato.stanford.edu/entries/logic-fuzzy/

i believe that other logics can actually resolve all paradoxes in binary logic. can you not see that paradoxes neccessitate enlarging the number of truth values to at least 3?

however, i wonder if there are archparadoxes that no logic can resolve.

12. Dec 19, 2003

phoenixthoth

it may be confusing to others that on two places in this thread, F has meant 0 and false has meant 1.

your * corresponds to the ^ operation. conjunction. AND.

to do v, or, you can say that AvB=A+B-A*B which looks like what you get when you do set union and intersection.

another equally valid way to do it is this:
V(AvB)=max{V(A),V(B)} (like join in lattices)
V(A^B)=min{V(A),V(B)} (like meet)

this can be generalized to infinitary fuzzy logic in which S is a collection of any size of disjunctions of wffs. then
V(S)=sup{V(A): A &isin; S}.

if S is a collection of any size of conjunctions of wffs, then
V(S)=inf{V(A): A &isin; S}.

being that {V(A): A &isin; S} is bounded above by 1 and below by 0, the sup and inf exist.

the incorperation of infinitary logic as well as the lattice theoretical notions is why i prefer max and min to * and + - *. technically, as long as V is any function having the domain of all wffs and range [0,1] and as long as V is an extention of any function f such that for all A and B,
f(AvB)=f(BvA)
f(AvB)=1 if f(A)=1 and f(B)=1
f(AvB)=1 if f(A)=1 and f(B)=0
f(AvB)=0 if f(A)=f(B)=0
f(~~A))=f(A).

all other logical connectives can be built from these:
A^B := ~( (~A) v (~B) )
A->B := ~AvB
A<->B := (A->B) ^ (B->A).

this form of logic captures the statment, "the universe is not just black and white" although one can just view it as white and various degredations of white:
T
~T
~(T v ~T)
etc.,
so in a way, all of it can be reduced to unitary logic.

in that sense, absolute black cannot be achieved, only approximated. there is only white.

anyways, i think veracity functions since they are not uniquely determined could be called perspectives. from one perspective, "phoenix is beautiful" is true and from another, "phoenix is beautiful" is not true.

in order for perspectives to make sense, they must be generalizations of binary logic. this is what's encapsulated in that they must be extentions of any function f with the properties above.

Last edited: Dec 19, 2003
13. Dec 19, 2003

Sikz

Yes, but I'm not trying to come up with a system that uses shades of grey (fuzzy logic does that perfectly fine). I'm trying to come up with one that uses more independent colours than black and white in the first place. Instead of things being black and white (or in fuzzy logic shades of grey), I'm trying to put things in, say, yellow, red, and blue.

0 1
That's current logic.
0 0.25 0.5 0.75 1
That's fuzzy logic (sort of, heh).
-1 0 1
That's an example of the kind of thing I'm trying to get... Something with more than two "polarities".

14. Dec 19, 2003

phoenixthoth

fuzzy logic could certainly act like [0,255]^3 with (255,255,255) being true and (0,0,0) being false. however, defining logic on this would be a bit complicated. this is a way to view 256^3 colors on computers. similarly, [0,1]^[0,1] would be increasingly difficult to define logic on but it would represent infinitely many colors. then again, [0,1] already does.

if you stick to [0,1] it's fine to say 0=black and 0.25=blue and 1=white or change it to shades of grey. i just hope your perspectives will be consistent.

Last edited: Dec 20, 2003
15. Feb 14, 2004

classical and quantum logic

Classical logic, is based on the way we think of "things by using logos, and Quantum logic, is the description of the "experience of things" by experience. For example to describe a "quanta" using classical logic yöu can look at a Feynman drawing. Using quantum logic a "quanta" might be the sound of a one hand clapping or simple the potentiality of possibilities. Quantum logic being the more "real representation" of the matrix. It seems to me that it is not mathematically possible to describe experience or the subatomic world with classical logic, because the "our real world" follows different rules. Mathematics is another language like English to describe how we apply classical and quantum logic. Somehow there seems to be a paradox here. Help

16. Feb 17, 2004

agnostictheist

re: phoenixthoth

This could be rubbish but lets play around with something here:

I amssume you mean the following...unless you meant the thing before.

"this form of logic captures the statment, "the universe is not just black and white" although one can just view it as white and various degredations of white: "

T
~T
~(T v ~T)
etc.,
so in a way, all of it can be reduced to unitary logic.

in that sense, absolute black cannot be achieved, only approximated. there is only white."

ok so one never gets to the other side has it were, but here is a thought, what if this works on the level of laws, princples, defintions, methods..that its very applying:

for example: you feed in white and you assume there is also black - its opposite?

Now lets apply this on scale of laws:

there is the same and there is opposoites:

now you get levels of the sames, but never complete opposoites, but if that is true, then one never gets compelte black has you say, but what happens when we ask whats the complete opposites of sames...opposites? (by defintion) but according to the method just described we never quite get that. on the level of defintion however we do!!)

Last edited: Feb 17, 2004
17. Feb 17, 2004

Loren Booda

18. Feb 24, 2004

elibol

anyone know of any pages/books that explains binary logic?

19. Feb 24, 2004

phoenixthoth

20. Feb 24, 2004

thanks.