vir'tu.al (vur'tu.al),adj. 1. Archaic. Of or relating to a virtue or efficacious power; energizing. 2. Being in essence or in effect, but not in fact; as the virtual rulers of a country. -- vir'tu.al'i.ty(-al'i.ti),n.--vir'tu.al.ly,adv.
I take the meaning of the word virtual in the archaic sense. Virtual logic is not logic, it is that which energizes reason and so brings the forms of logic and mathematics into being. Virtual logic is not logic, nor is it the actual subject matter of the mathematics, physics or cybernetics in which it may appear to be embedded. Virtual logic lives in the boundary between syntax and semantics. It is the pivot that allows us to move from one world of ideas to another.
This is a column on virtual logic. That which empowers reason is not necessarily itself reasonable or logical. There are many ways in which we encounter this sort of virtuality. One way is to proceed from within an apparently logical system and push its boundaries, find its limits. Another is to arrive from without in a leap, a bound, a jump into something new.
A word about the use of the word "I" in this column.
I am the one who says I
That statement applies equally well to you as to me. So you are invited to take the I-saying in this essay as your own I-saying. You are invited to try on the points of view to which the self-referential pronoun points. The author him/her self tends to exfoliate into dialoguing parts, each saying I. Please join the fugue.
There will be time to explore many themes in this column. I want to begin with some remarks about the relationship of paradoxes, self reference, imaginary values and fixed points. This sounds like a mouthful, but lets take it step by step.
This statement is false
Here is the famous paradox of Epimenides the Liar. The statement asserts that it is false. Are we to believe it? If we believe this statement, then we must disbelieve it. If we disbelieve it, then we must believe it. What are we to do?
If we must either believe or disbelieve that statement, then our only recourse is to oscillate between the possibilities of its true and false values. Oscillation is not a bad solution. The statement is a buzzer, a feedback loop that turns itself off when it is on and turns itself on when it is off. The paradox creates time. (I take the creation of time to be the creation of temporal action.) The paradox demands time for its resolution and time is what it will get.
Must we either believe or disbelieve?
I would rather just look.
The statement can be schematized:
S = NOT(S).
The statement is equivalent to its own negation!
Well, why not?
I like to solve equations. Lets solve this one. Lets see,
S = NOT(S)
= NOT(NOT(S))
= NOT(NOT(NOT(S)))
...
Are we getting anywhere? The key to solving math equations is to get the unknown (S) on only one side of the equation. Here the S on the right hand side keeps getting buried deeper and deeper. I know!
Lets send it to infinity! That will fix its wagon. It will never get back from there:
S = NOT(NOT(NOT(NOT(NOT(....))))).
So this "paradox" is no paradox at all. It is just an infinite hall of NOT mirrors. Paradox does not generate time. Paradox generates INFINITY! Perfect, eternal timeless infinity. How satisfying. Not an oscillation in sight. And as for the value of S? It is neither true nor false. It just is. It NOTS itself.
Are you sure you mean that it nots itself? Don't you mean that it knots itself? I have a vision of this statement all entangled with its own assertions? Well that would seem to be the contents of another column. Lets go back to the equation solving.
All right, I have another problem with your method of solving equations. Is it legitimate to let the S disappear to infinity?
Well, you can do this in mathematics. For example, consider the equation X = 1 + aX. I will solve it in the same way.
X= 1 + aX
= 1+ a(1+aX)
= 1 + a + a2X
= 1 + a +a2(1+aX)
=1 + a + a2 + a3X
...
So you see, the solution by letting X slip off to infinity on the right hand side is the infinite sum
X = 1 + a + a2 + a3 + a4 + ...
This infinite sum certainly solves our equation:
1 + a(1 + a + a2 + a3 + a4 + ... ) = 1 + a + a2 + a3 + a4 + ...
and it does this for specific numbers! For example, if a = 1/2 then X= 2 since 2 = 1 + (1/2)2. And we all understand that
2 = 1 + (1/2) + (1/2)2 + (1/2)3 + (1/2)4 + ...
Wait! Do we understand this? Let me see. First I step halfway towards two from one. Then I step half the remaining distance. Then I step half the still remaining distance. Hey! This goes on forever. I can never reach two! You mathematicians. I suppose you want me to imagine taking an infinite number of steps. Anyway, how did we get here? Isn't this Zeno's Paradox? I thought we were discussing the paradox of the Liar. You started us down a slippery slope.
Indeed! In fact, lets look at another value for a. Let a=-1. Then our equation is X= 1 - X or X+X = 1. Thus X=1/2. (You cannot complain about this method of solution. I did it in a finite number of allowed steps.) Now lets apply our infinity to this. We get
1/2 = 1 + (-1) + (-1)2 + (-1)3 + (-1)4 + ...
Thus
1/2 = 1 -1 + 1 -1 +1 -1 +1 -1 + ...
Oh no! This is even worse than Zeno. In fact now I will show you the error of your ways. Let Z = 1 -1 + 1 -1 +1 -1 +1 -1 + .... Then clearly Z = 1 - Z hence Z=1/2. but we also have
Z = (1 -1) + (1 -1) +(1 -1) +(1 -1) + ....
= 0 + 0 + 0 + 0 + ...
= 0
So 1/2 =0. This is as bad as the Liar paradox.
Well, I think we have done well today. We have seen that Zeno's paradox and the Liar Paradox are really part of one structure. This structure can even leap to correct answers.
I will leave you with something to ponder for the next time.
I can solve any equation!
For example, lets try to solve
Z= F(Z).
Let G(X) = F(X(X)).
Then G(G) = F(G(G)).
So Z=G(G).
We have done it.
Any self referential equation can be solved.
This G is a lovely gremlin. She duplicates anyone you give her and inserts the copies neatly inside F. Why if you let F(X) =NOT(X) then G(G) = NOT(G(G)), and we did not even need infinity! View Figure 1.
| The Incredible Duplicating Gremlin
G(#) = F(#(#)) Apply G to herself and she replicates herself inside F. G(G) = F(G(G)) |
Figure 1
How did that happen without using infinity?
I thought you were going to solve it with
Z = F(F(F(...))).
Is G(G) = F(F(F(F(...))))?
What does G(G) mean anyway?
Stay tuned for the next episode when we follow the adventures of the self replicating gremlin G from paradox to imaginary values, Gödel's theorem and the structure of DNA.