A trickier variant is “This sentence contains one error.” Of course it is in error, since it contains no errors. That is, it contains no spelling errors (“first-order errors”). Needless to say, there are such things as “second-order errors”—errors in the counting of first-order errors. So the sentence has no first-order errors and one second-order error. Had it talked about how many first-order errors it had, or how many second-order errors it had, that would be one thing—but it makes no such fine distinctions. The levels are mixed indiscriminately. In trying to act as its own objective observer, the sentence gets hopelessly muddled in a tangle of logical spaghetti.
C. H. Whitely invented a curious and more mentalistic version of the fundamental paradox, explicitly bringing in the system thinking about itself. His sentence was a barb directed at J. R. Lucas, a philosopher one of whose aims in life is to show that Gödel’s work is actually the most ineradicable uprooting of mechanism ever discovered—a philosophy, incidentally, that Gödel himself may have believed. Whitely’s sentence is this:
Lucas cannot consistently assert this sentence.
Is it true? Could Lucas assert it? If he did, that very act would undermine his consistency (nobody can say “I can’t say this” and remain consistent). So Lucas cannot consistently assert it—which is its claim, and so the sentence is true. Even Lucas can see it is true, but he can’t assert it. It must be frustrating for poor Lucas! None of us share his problem, of course! Worse yet, consider:
Lucas cannot consistently believe this sentence.
For the same reasons, it is true—but now Lucas cannot even believe it, let alone assert it, without becoming a self-contradictory belief system.
To be sure, no one would seriously maintain (we hope!) that people are even remotely close to being internally consistent systems, but if this kind of sentence is formalized in mathematical garb (which can be done), so that Lucas is replaced by a well-defined “believing system” L, then there arises serious trouble for the system, if it wishes to remain consistent. The formalized Whitely sentence for L is an example of a statement that the system itself could never believe! Any other believing system is immune to this particular sentence; but on the other hand is a formalized Whitely sentence for that system as well. Every “believing system” has its own tailor-made Whitely sentence—its Achilles’ heel.
These paradoxes all are consequences of a formalization of an observation as old as humanity: an object bears a very special and unique relationship to itself, which limits its ability to act upon itself in the way it can act on all other objects. A pencil cannot write on itself; a fly swatter cannot swat a fly sitting on its handle (this observation made by German philosopher-scientist Georg Lichtenberg); a snake cannot eat itself; and so on. People cannot see their own faces except via ext aids that present images—and an image is never quite the same as original thing. We can come close to seeing and understanding ours objectively, but each of us is trapped inside a powerful system with a unique point of view—and that power is also a guarantor of limitedness. And this vulnerability—this self-hook—may also be the source of ineradicable sense of “I.”
Malcolm Fowler’s hammer nailing itself is a new version of the ouroboros. (From Vicious Circles and Infinity: An Anthology of Paradoxes by Patrick Hughes and George Brecht.)
The “Short Circuit” serves to illustrate the short circuit of logical paradox. The negative invites the positive, and the inert circle is complete. (From Vicious Circles and Infinity.)
But let us go back to Cherniak’s story. As we have seen, the self-referential linguistic paradoxes are deliciously tantalizing, but hardly dangerous for a human mind. Cherniak’s Riddle, by contrast, must be far more sinister. Like a Venus’s-flytrap, it lures you, then snaps down, trapping you in a whirlpool of thought, sucking you ever deeper down into a vortex, a “black hole of the mind,” from which there is no escape back to reality. Yet who on the outside knows what charmed alternate reality the trapped mind has entered?
The suggestion that the mind-breaking Riddle thought would be based on self-reference provides a good excuse to discuss the role of looplike self-reference or interlevel feedback in creating a self—a soul—out of inanimate matter. The most vivid example of such a loop is that of a television on whose screen is being projected a picture of the television itself. This causes a whole cascade of ever-smaller screens to appear one within another. This is easy to set up if you have a television camera.
The results [see figure] are quite fascinating and often astonishing. The simplest shows the nested-boxes effect, in which one has the illusion of looking down a corridor. To achieve a heightened effect, if you rotate the camera clockwise around the axis of its lens, the first inner screen will appear to rotate counterclockwise. But then the screen one level farther down will be doubly rotated—and so on. The resulting pattern is a pretty spiral, and by using various amounts of tilt and zoom, one can create a wide variety of effects. There are also complicating effects due to such things as the graininess of the screen, the distortion caused by unequal horizontal and vertical scales, the time-delay of the circuit, and so on.
A variety of effects that can be achieved using a self-engulfing television system. (Photographs by Douglas R. Hofstadter.)
All these parameters of the self-referential mechanism imbue each pattern with unexpected richness. One of the striking facts about this kind of “self-image” pattern on a TV screen is that it can become so complex that its origin in videofeedback is entirely hidden. The contents of the screen may simply appear to be an elegant, complicated design—as is apparent in some shown in the figure.
Suppose we had set up two identical systems of this sort with identical parameters, so that their screens showed exactly the same design. Suppose we now made a tiny change in one—say by moving the camera a very small amount. This tiny perturbation will get picked up and will ripple down the many layers of screen after screen, and the overall effect on the visible “self-image” may be quite drastic. Yet the style of the interlevel feedback of the two systems is still in essence the same. Aside from this one small change we made deliberately, all the parameters are still the same. And by reversing the small perturbation, we can easily return to the original state, so in a fundamental sense we are still “close” to where we started. Would it then be more correct to say that we have two radically different systems, or two nearly identical systems?