p = q for P

q = l - p for Q.

These make no sense if p and q can only be 0 or 1. But there's a unique solution in fuzzy logic: p = q = 0.5. So each side of my card is a half-truth, and there's no paradox anymore."
"Precisely," Lukasiewicz said. "But it goes further than that. What we've been discussing is the beginnings of a whole new theory of dynamic logic, invented by Gary Mar and Patrick Grim in the department of philosophy of the State University of New York at Stony Brook. It provides a link between semantic paradoxes and chaos theory."
It was Socrates' turn to look puzzled.
"Oh, wake up. You know what chaos is. Simple deterministic dynamics leading to irregular, random-looking behavior. Butterfly effect. That stuff."
"Of course, I know that," Socrates said in irritation. "No, it was the idea of dynamic logic that was puzzling me. How can logic be dynamic?"
 Lukasiewicz looked surprised. "How can it be anything else when discussing self-referential statements? The statement itself forces you to revise your estimate of its truth value. That revised value has to be revised again and again. Consider the Paradox of the Liar , your statement P: 'this statement is false'. Earlier I wrote an equation for its truth value: p = 1 - p. But what I should really have written was a process that forces constant revision of your assessment of its truth value, p ← 1 - p. If you assume that P has a particular truth-value p, then P itself tells you to replace that truth value by 1 - p. For example, if you started out thinking that P was 30 percent true, so that p = 0.3, the revision rule implies that p = 0.7, which in turn implies that p = 0.3 again...and you get an infinite sequence of truth values, oscillating periodically between the two values 0.3 and 0.7. The classical paradox, with p = 0 or 1 , leads to the sequence 0, 1, 0, 1,...which faithfully reflects the logical argument if P is false, then P is true, so P is false, so P is true, so.... the logical oscillations of the paradox are captured by the dynamics of the truth value."
"And p = 0.5 is the only value that doesn't lead to an oscillation," Plato mused.
"Precisely. Now, the Dualist Paradox on your business card is really a logical dynamic:

p ← q

q ← l - p

Suppose you start out by estimating p = 0.3, q = 0.8. Then your first revision is to p = 0.8, q = 0.7. A further revision leads to p = 0.7, q = 0.2, a third to p = 0.2, q = 0.3. A fourth revision gives p = 0.3, q = 0.8, and you're back where you started. It cycles with period four—unless you start at p = 0.5, q = 0.5, when everything stays unchanged."
"Okay, I'll buy that," I said. But what about the chaos?"
Lukasiewicz's face went very serious.
"Before I can explain that, I must be more precise," he said. "If you want to play around with these ideas for yourself, I'd better tell you how to calculate fuzzy truth values for combinations of logical statements [see box below]. Although all you really need to know at the moment is that not-P has truth value 1 - p if P has truth value p. Second, you must know how to assess the truth value of statements about statements.

This statement is as true as it is assessed to be false.

If its truth value is c, then it instructs you to assess a truth value of 1 - c. So by the assessment formula, its truth value is 1 - |c - ( 1 - c)| = 1 - |1 - 2c|.
In short, there is a dynamic process

c ← l - | 1 - 2c |

of reassessment of the truth value c. Choose any starting value for c, say c = 0.12345, and calculate successive values. You'll find they are chaotic. Actually, I should warn you that because of round-off errors in your calculator or computer, the process may appear to settle down to either 0 or 1. It may help to replace the dynamic by c <- 1 - | 0.999999 - 2c|. You can even observe the famous butterfly effect of chaos theory-if a butterfly flaps its wings, it can cause a hurricane a month later. More prosaically, small changes in initial conditions make big changes to the subsequent dynamics. If you use a start value of 0.12346 instead, you get a different image."
Lukasiewicz paused. "Next there is the Chaotic Dualist, which involves two statements:

X: X is as true as Y is true

Y: Y is as true as X is false

"It's rather like your business card, Epimenides. The dynamics are

x ← l - |x - y|

y ← l - |y - (1 - x)|

To see what it does, you choose an initial pair of values, say, (x,y) = (0.2, 0.9), and calculate successive pairs of values. Think of them as coordinates and plot them in the plane. You get a geometric shape, called the attractor of the dynamic system. In this case, you get a triangle, densely filled with points [see right illustration below]. This representation can be transformed into a beautiful and intricate image known as an escape-time diagram. To create it, temporarily relax the conditions that x and Y lie between 0 and 1. The idea is to watch how far (x,y) moves from the origin (0,0) and to count how many calculation steps are needed before it goes beyond some threshold value. Then the point (x,y) is plotted in a color that depends on the number of steps required. To start, you should try a threshold value just larger than 1 [see left illustration below].

The assessed falsehood of this statement is not different from its assessed truth.

In fuzzy logic, it is standard to interpret the adjective 'very' as forming the square of a truth value. So think about the rather woollier statement

The assessed falsehood of this statement is not very different from its assessed truth.

The statement leads to the dynamic

p ← l - (p - (l - p))²,

which converts into the form

p ← 4p(1 - p).

Chaos theorists call this the logistic dynamic system—so my statement is the Logistic Liar. It's chaotic, too—try it."

At midnight, the Paradox Club closed, and Lukasiewicz and I walked out into the street. I realized I had been so absorbed working out examples of fuzzy-logical chaos that I had forgotten to ask one very important question. "Luke, it's all very pretty, but how significant is it?"
"Well," he said. Mar and Grim point out that it gives a geometric approach to semantic complexity, letting you distinguish between different systems of self-referential statements. They also say it can be used to prove there is no decision procedure that will tell you whether or not a given system is chaotic. That's a result in the same general line as Kurt Gödel's famous theorem on the undecidability of arithmetic. It's potentially rather deep stuff, Epimenides."
"So I see. Connections between logic and chaos— Amazing! But wait a second. How can I be sure everything you just told me is true?"
"If I have ever lied to you, I ask the gods to strike me down with two lightning bolts."
Just then, thunderclouds formed in the sky, and a single lightning bolt zapped Lukasiewicz into oblivion. I looked up, shaking my fist at the clouds: "So was he telling me the whole truth, or only half?"