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not/negation/inversion

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Reconstructo
 
Reply Mon 19 Apr, 2010 10:37 pm
NOT is probably the most whittled down operator conceivable. I am thinking especially of the gate. One input. One output. As logic can be viewed as a modular binary arithmetic, there are not only 2 digits, but also only two numbers/values/states. 1 and 0. T and F. So we might as well call it Inversion, as it is sometimes called.

I heard it said that Nietzsche is an inversion of Plato, and of course addition is an inversion of subtraction, and so one. I feel that this not/inverter gate is a piece of art. Can we think of operators as something like constants? We have pi and e. Are operators/functions any less important? It seems that constants get their meaning from operators, and also the reverse/inverse.
 
Reconstructo
 
Reply Tue 20 Apr, 2010 09:11 pm
@Reconstructo,
We have the 1 and the 0 and the NOT. Let's exclude the other operators. Is this the minimal dynamic math/logic system? If so, what does that tell us about the way we humans think? Here's a serial logic gate.

0-->N-->1 -->N-->0 -->N-->1

---------- Post added 04-20-2010 at 10:16 PM ----------

Quote:

The question is not simply "how does one think non-being?" but also (and Parmenides also recognized this) "how does one name non-being?" The proper name, as Badiou points out in a passage immediately following the above, is not the transcendent God or the promise of the One or presence but the "un-presentation and the un-being of the one" (cf. Derrida's comments on the possibility of a negative theology).

I have only browsed B, but I like this quote. -1
 
Owen phil
 
Reply Wed 21 Apr, 2010 03:06 am
@Reconstructo,
Reconstructo;154712 wrote:
We have the 1 and the 0 and the NOT. Let's exclude the other operators. Is this the minimal dynamic math/logic system?


I'm not clear as to what you mean here but my impression is that you want to produce 'propositional logic' with the minimum of primitive terms and to use a minimal logical arithmetic to demonstate the theorems of this system.

We cannot produce the system of propositional logic with [1,0, ~], but we can produce the system with [1,0,|].

Let (p, q, r, ...) be propositional variables which take on the values 1 and 0 (truth and falsity).

(nor) primitive (|): 1|1=0, 1|0=0, 0|1=0, 0|0=1.

(not)
~p = p|p.
~1=1|1, ~0=0|0.

ie: ~1=0 and ~0=1.

(or)
(p v q) = ~(p|q).
(1 v 1)=~(1|1), (1 v 0)=~(1|0), (0 v 1)=~(0|1), (0 v 0)=~(0|0).

ie: (1 v 1)=1, (1 v 0)=1, (0 v 1)=1, (0 v 0)=0.

(if then)
(p -> q) = (~p v q).
(1 -> 1)=(~1 v 1), (1 -> 0)=(~1 v 0), (0 -> 1)=(~0 v 1), (0 -> 0)=(~0 v 0).

ie: (1 -> 1)=1, (1 -> 0)=0, (0 -> 1)=1, (0 -> 0)=1.

(and)
(p & q) = ~(~p v ~q).
(1 & 1)=~(~1 v ~1), (1 & 0)=~(~1 v ~0), (0 & 1)=~(~0 v ~1), (0 & 0)=~(~0 v ~0).

ie: (1 & 1)=1, (1 & 0)=0, (0 & 1)=0, (0 & 0)=0.

(equivalence)
(p <-> q) = ((p -> q) & (q -> p)).
(1 <-> 1)=((1 -> 1) & (1 -> 1)), (1 <-> 0)=((1 -> 0) & (0 -> 1)),
(0 <-> 1)=((0 -> 1) & (1 -> 0)), (0 <-> 0)=((0 -> 0) & (0 -> 0)).

ie: (1 <-> 1)=1, (1 <-> 0)=0, (0 <-> 1)=0, (0 <-> 0)=1.

With this simple arithmetic we can show the theorems (tautologies) of the system.

For example: (p & (p-> q)) -> q.

From the above we get:
(1 & (1 -> 1)) -> 1, (1 & 1) -> 1, 1 -> 1, 1.
(0 & (0 -> 1)) -> 1, (0 & 1) -> 1, 0 -> 1, 1.
(1 & (1 -> 0)) -> 0, (1 & 0) -> 0, 0 -> 0, 1.
(0 & (0 -> 0)) -> 0, (0 & 1) -> 0, 0 -> 0, 1.

That is, (p & (p -> q)) resolves to 1 for all values of p and q, therfore,
(p & (p -> q)) -> q, is a theorem.
 
Reconstructo
 
Reply Wed 21 Apr, 2010 01:45 pm
@Owen phil,
Owen;154774 wrote:
I'm not clear as to what you mean here but my impression is that you want to produce 'propositional logic' with the minimum of primitive terms and to use a minimal logical arithmetic to demonstate the theorems of this system.

We cannot produce the system of propositional logic with [1,0, ~], but we can produce the system with [1,0,|].

Let (p, q, r, ...) be propositional variables which take on the values 1 and 0 (truth and falsity).

(nor) primitive (|): 1|1=0, 1|0=0, 0|1=0, 0|0=1.

(not)
~p = p|p.
~1=1|1, ~0=0|0.

ie: ~1=0 and ~0=1.

(or)
(p v q) = ~(p|q).
(1 v 1)=~(1|1), (1 v 0)=~(1|0), (0 v 1)=~(0|1), (0 v 0)=~(0|0).

ie: (1 v 1)=1, (1 v 0)=1, (0 v 1)=1, (0 v 0)=0.

(if then)
(p -> q) = (~p v q).
(1 -> 1)=(~1 v 1), (1 -> 0)=(~1 v 0), (0 -> 1)=(~0 v 1), (0 -> 0)=(~0 v 0).

ie: (1 -> 1)=1, (1 -> 0)=0, (0 -> 1)=1, (0 -> 0)=1.

(and)
(p & q) = ~(~p v ~q).
(1 & 1)=~(~1 v ~1), (1 & 0)=~(~1 v ~0), (0 & 1)=~(~0 v ~1), (0 & 0)=~(~0 v ~0).

ie: (1 & 1)=1, (1 & 0)=0, (0 & 1)=0, (0 & 0)=0.

(equivalence)
(p <-> q) = ((p -> q) & (q -> p)).
(1 <-> 1)=((1 -> 1) & (1 -> 1)), (1 <-> 0)=((1 -> 0) & (0 -> 1)),
(0 <-> 1)=((0 -> 1) & (1 -> 0)), (0 <-> 0)=((0 -> 0) & (0 -> 0)).

ie: (1 <-> 1)=1, (1 <-> 0)=0, (0 <-> 1)=0, (0 <-> 0)=1.

With this simple arithmetic we can show the theorems (tautologies) of the system.

For example: (p & (p-> q)) -> q.

From the above we get:
(1 & (1 -> 1)) -> 1, (1 & 1) -> 1, 1 -> 1, 1.
(0 & (0 -> 1)) -> 1, (0 & 1) -> 1, 0 -> 1, 1.
(1 & (1 -> 0)) -> 0, (1 & 0) -> 0, 0 -> 0, 1.
(0 & (0 -> 0)) -> 0, (0 & 1) -> 0, 0 -> 0, 1.

That is, (p & (p -> q)) resolves to 1 for all values of p and q, therfore,
(p & (p -> q)) -> q, is a theorem.


I appreciate your response. Actually I had something else in mind. And yes I agree that the not sign is not enough. A binary operator is a necessity.

What I'm looking at is not only the minimal ingredients for logic, but the minimal ingredients for any sort of dynamic symbolic system. Well, logic is pretty much the simplest, but this same logic is equivalent to a modular math. I'm looking at what amount of reduction is possible. And I'm doing so to examine the structure of human thought. So it's not about utility but about what such a minimal system can tell us about the way our minds work..
What is negation? Man can think the thing that is not. He can be in the spatial present and think "it is not raining." Can animals do this?

Yes, we need a binary operator for a working logic. But imagine a logic without falseness! Falseness is probably the source of not-operation. The core of logic is True and False, one and zero. Or such is my suggestion. I realize that fuzzy logic moves away from this, but that is not the point. They still have a spectrum that is defined by 1 and 0, true and false.
 
 

 
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