## Sunday, July 29, 2012

### Self-Reference

Edit: Begging a question says that "there is no fallacy if the missing premise is acknowledged, and if not, there is no circle". That is, A in A→A may be true or false, as I've said in what fallows.

I recently watched a lecture on Russel Paradox and got to know that self-referential definition is not "well founded". So that you define A based on B, A→B, then it is ok but A ↺A is not. I have also got to know that A ↺A is the simplest self-reference known as "Quine Atom". And, it seems to me that you cannot scratch your back with 0-length hand. This is a recurrence in space domain. But it is also mandatory in time -- you cannot read a value as soon as you produced it because infinite frequency implies uncertainty.

I was taught to implement the logic in digital circuits and optimize them. If you represent the Liar paradox by a circuit of logic gates, you'll get a basic oscillator. Start by optimizing

b = not a

a = b

is identical to

b = not b

What is b? If that would be a function b(x) and x was a known input value, i.e. ground voltage 0 or 1, then we could say what b(x) value is. If x would be a function itself, we could unroll it and reach the inputs. But, b refers to itself rather than staying on constants.

The quine atom, b = b

is the basic memory element, it feeds the input to its own output. This is known as positive feedback in saturation and just keeps its value rather than growing.

In the case b = not b, the inverse of output is submitted to the input. It means that output value  is inverted at every step.

If you do not lie, you lie and if you do lie, you do not. At which frequency? The feedback reaction speed is dependent of the physical length of the feedback. But, it also can be delayed and synchronized with something external. So what self-reference does is memory and evolution of the state in addition to simple combinatorial evolution of the function, given the arguments. It is a dynamical system. Self-reference also needs initial state specification and synchronization circuits. Otherwise it will have unpredictable result, oscillating at max speed. The shorter is the feedback the faster it will switch.

You cannot refer to yourself immediately. The paradox appears when you try. It is like asking what is position of pendulum oscillating at infinite frequency or much faster than you pick the answer. If you don't ask such stupid thing, you get (a slowly) evolving system. To heel yourself your hand must be should have quite powerful "service part" that can operate autonomously while your first part is in service. Self-repairing circuits are redundant. You cannot make the self-reference infinitely simple. You must have some matter for it. The snake bitying its tail can be shorter and shorter and shorter but you cannot make the loop of zero length infinitely short (to grasp yourself). Oh dear, what I'm talking here? Just wanted you to know about the logic and paradox modelling with logic gates.

Update I have got to know that Liar's paradox is also known as Liar's circle or just Circuit. So, my idea to draw circuits was right and the most insightful. See also Circular definition and infinite regress.

Moreover, It is cohesitivist approach to truth so state that a thing is true if it is not self-contradictory. Therefore, inverter-based generator is liar but trush-teller is true.

Somebody refers Norbert Weiner "A machine to answer this question would give the successive temporary answers: 'yes,' 'no,' 'yes,' 'no,' and so on, and would never come to equilibrium"