29 Mar 2009

Deleuze's Wild Digital Computer Logic: A Formalization of Affirmative Synthetic Exclusive Disjunction's Deformities

by Corry Shores
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(I dedicate this entry to Deleuze scholar Rocky Clancy. Thanks for your firestorm lightning-bolt mind.)

[I obtain the following from the sixth paragraph of the 24th chapter in Deleuze's Logic of Sense / Logique du Sens.]

When something becomes, there is change.

Wood becomes fire.

Fire burns wood.

So wood cannot be fire. Thus the wood becomes not-wood. So being becomes nothing. And, the fire is not wood. Fire comes from not-fire. Thus nothing becomes being.

All of logic is built on the principle of identity: A is A. Closely related to this law is the law of non-contradiction: A cannot be not-A. But in becoming, A becomes not-A. This defies logic.

Yet someone might clarify. A is A at time 1, and B (or not-A) at time 2. But A is never both itself and not itself at the same time.

However, change is a movement. And physicists can measure "instantaneous velocity." It is an instantaneous change of sorts. Leibniz says that the moving object occupies more spacial points then are in its actual length. And at the instant when the pumpkin hits the ground, it is both moving and smashed at the same time. It is virtually smashed, because it is fully smashed as a tendency. It has intensively stretched into non-being, while still maintaining its fullest form, all in an instant. In other words, the world around us is one mess of logical contradiction.

Deleuze's logic provides a solution.

His correction to formal logic is "affirmative synthetic exclusive disjunction." But by its nature, this logic cannot be formalized like other logical operators. However, by using the metaphor of electrical logic gates, we will provide a formalization of identity's deformity in the event of becoming.

Let's first review two logical operators, conjunction and exclusive disjunction.

A and B are different things. B is not-A, and A is not-B.

Consider we say, "It is Sunday, and it's cloudy." We could reduce both clauses to symbols: "A and B." For this to be true, both A and B must be true. Otherwise, the whole conjunction is false.

Or instead we say, "Either it is cloudy, or the sky is clear." It cannot be both, and it must be one of the two. This is exclusive disjunction: "A or B."

Deleuze's new logical operation is affirmative synthetic exclusive disjunction. It first presumes that A and B logically exclude each other. So both cannot be true; for example, the two propositions "there is wood," and "the fire consumed it" supposedly cannot both be true. But, says Deleuze, they are contracted together.

We strike a match. At one point it is wood. At another point it is fire. But consider that instant when the match is right at the limit of its still being wood. In the next instant it will be fire. But between its limit as wood, and its initiation as fire, there extends no time at all. In other words, at the limit of the match being wood, nothing stands between the wood and fire. So the wood and fire contract together. That does not mean that they both assimilate into the same thing. They maintain their differences. For, one lies at the limit of the other: it contracts upon it, but it is not placed within it. So the two maintain their differences. Because they are different, they are not the same, and hence there is a sort of distance that remains between them. But as we saw, there is no extent of time between them. So there is not an extensive difference between them. Rather, there is an intensive difference. When the match sits in its box, it decomposes slowly. When we strike it, it changes rapidly. We can measure degrees of change from one instant to the next. So at the limit of wood being wood, it has a profound internal tendency to become fire. For this reason, it is intensely fire, even while it is wood. At that limit, the wood is virtually fire. But, its being fire is no less real than its being wood, just as the pumpkin's being smashed is no less real than its still being whole, right as it hits the ground.

So, this is not a logical conjunction, Deleuze claims. For, the wood and the fire maintain their individuality. They are differences that are forced together and are thereby contracted as differences. This contraction that maintains their distances is the 'synthesis.' Imagine that we force together the north ends of two extraordinarily powerful magnets, despite their enormous repulsive force. Their being together does not make the one end any less repulsive to the other. The extensive magnitudes between them are contracted. But there is an incredible intensive force that is trying to push them apart. The force is there, because the contraction causes them to communicate their differences to one another, without them homogenizing into one thing.

Let's compare the truth tables for conjunction (AND) with exclusive disjunction (XOR).

Deleuze writes that synthetic disjunction cannot be reduced to conjunction. So we may not use the conjunction truth table. But he also says that the disjunction is affirmative. When A and B are contracted, they maintain their distance, which affirms their difference. Thus because it is affirmative, we cannot use the exclusive disjunction (XOR) truth table. For, we see it lists a negative truth value for the combination of true disjuncts.

Deleuze also says that the exclusive disjunction disperses its ramifications everywhere. Imagine that the magnets we force together have such incredible repulsive forces that they both explode and fly off away from each other in all directions. So the contraction that happens in this operation causes a multitude of divergent heterogeneous series to be disjunctively "coordinated" at once. And at the disjunction is the "aleatory point." How, where, and into what forms the series disperse is decided by both the competition of forces and pure chance.

Because we will be using a digital logic formalization, I need first to distinguish two types of chance. To do so, we imagine an electrical circuit. The circuit designer includes a random variable generator. At mathematically random times, the generator causes the current to randomly jump to another part of the circuit. Now consider instead that we overload the circuit. This causes it to "short circuit." The current jumps wildly around the circuitry (and maybe also outside the circuit system). I want to be very clear about these two types of chance: mathematically random vs wild. Random means that it goes somewhere that we cannot predict. But the places and the ways that it gets there we have predetermined. And the rest is left to chance.

When instead the circuit jumps wildly, it could go anywhere in any way. It could jump out of the circuit. It could jump to places that are impossible using artificial means. No one can predict where the lighting-bolt strikes. It's wild. It's the wilderness of chance. We will keep this sort of variable in mind when we design our Deleuzean Digital Logic Gate.

Let's first just look at how the conjunction and exclusive disjunction logic gates are symbolized.

[Thanks you image source:
Tony R. Kuphaldt at cybermike.net]

I propose the following shape for Deleuze's Affirmative Synthetic Exclusive Disjunction Gate. Instead of XOR for exclusive disjunction, I name it SOR for synthetic exclusive disjunction.

I choose this shape because it bears the golden ratio, which is a ubiquitous proportion in nature and the cosmos. As an irrational number, it cannot be represented using digital numerals. It is undeniably a real number. But it will always defy digital logic.

Let's imagine now that there are two channels of electrical current that enter the SOR gate: A and B. Let's also imagine that the SOR gate can handle the electricity from either channel, just one at a time. But when both channels feed charge into the gate, then it overfills its carrying capacity. Still the logic of the gate forces the charges through. This contracts them together even though they are too large to both merge together in the gate.

Because the SOR gate is essentially an exclusive disjunction, we will follow the logic of exclusive disjunction up to the point where synthesis happens.

Consider first when no electricity flows through the channels. None should flow out of the gate.

But if it flows through just A, then the gate can handle the current, and it lets the charge through.

Likewise if only B's current is moving. Again, the gate can carry the charge and let it pass through.

However, we noted that the SOR gate cannot carry both charges at once. When they force themselves into the gate, they do not then become one same current flowing out the other end. Rather, they ramify wildly (at this aleatory point.) So they never converge. However, they contract, because they both entered the gate. Nonetheless, they shoot out wildly to any place and in any way whatsoever.

I have chosen a "w" to represent the logical value of this combination. It is not a 1 or a 0. We previously saw that it cannot be either of those. But it is still something. It is a variable. Yet not a mathematically random variable, but rather a wild variable. It can become anything anywhere depending on the battle of intensive forces that deform it in the instant of contraction.

I present the SOR gate as a possible formalization for Deleuze's logic. It has the advantage of metaphorically making use of real forces like electricity. A plain truth table will not suffice. I also think that even though it is a formalization, it is loyal to the way that intensities deform whatever they embody on account of their pushing-and-pulling it in many different directions at once. This we display as the wild dispersion of ramified current "series" and symbolize with the random variable w for its wild logical value.

Another advantage of this formalization will be when discussing a problem in Artificial Intelligence theory. Some are trying to replicate the human mind. And even a few think we can replicate someone's mind so well that the computer simulation is no different from the actual person herself. I do not argue that this is impossible. I will leave it for experiments to demonstrate. Rather, I am trying to use Deleuze's philosophy to suggest a way to replicate or at least explain human creativity. My hypothesis is that it is wild.

AND and XOR gate images from:
Tony R. Kuphaldt

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