Originally posted by 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, separate 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?
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 ∈ S}.
if S is a collection of any size of conjunctions of wffs, then
V(S)=inf{V(A): A ∈ S}.
being that {V(A): A ∈ 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 statement, "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.
