Arguments that succeed in revealing why they cannot succeed
|By permission of Pierangelo Boog, artist: source 1, source 2.|
Alan Turing created the basic form of the diagonal argument that we use in computation theory. This drew inspiration, however, from the idea of a diagonal sentence, such as in Gödel’s diagonal lemma in logic. A typical diagonal sentence is, “This sentence is false.” We covered diagonal sentences thematically here.
Today Ken and I want to talk about a different kind of special sentence, one that is self-defeating. A self-defeating sentence is one that ensures it cannot achieve its desired end, which in this instance is to illustrate a self-defeating sentence.
We can also think of self-defeating arguments and conversations. Here again we see Turing as a pioneer, though he was not the first. This leads in to the defining principle of postmodernism, which is that no principle can be definitive.
Some people speculate that the several known “barriers” to strong complexity lower bounds combine to impute that straightforward attempts to prove circuit lower bounds are self-defeating. This suggests the need for new ideas. In a fight-fire-with-fire mood, we hope that exploring the arena of self-defeat will stimulate some.
Our first examples are drawn from a larger collection of “Self-Annihilating Sentences” compiled by Saul Gorn of the University of Pennsylvania’s Moore School of Electrical Engineering. Gorn helped standardize the view of Computer Science as a separate discipline, and received the ACM Distinguished Service Award in 1974 for his work on standardizing computer languages.
Gorn’s sentences are any with hitches that nullify them. We are attempting a finer distinction by which the process of falling short is ingrained into the plan of the sentence or argument. Hence we’ve selected only those examples that rise to our standard of being lower. We considered making Gorn the featured person of this column, but since we are cutting away from his broader idea this would have been self-defeating. Here is a picture of him anyway:
The following are taken from what Gorn also called his compendium of “Rarely Used Clichés.” More could be included; these represent a wide variety of self-defeating mechanisms.
I enjoy your company most when I am by myself.
All simplifying assumptions are too complicated.
If you remember something too long, you might as well forget it.
Anyone who goes to a psychoanalyst should have his head examined — Samuel Goldwyn.
If I don’t see you again, auf wiedersehen!
I absolutely refuse to be assertive.
None of my close friends has a close acquaintance.
I am not contradicting you.
I won’t hesitate for a moment to avoid answering.
I will now predict an unanticipated result.
Ignore this sign.
You may steal this from me.
Henceforth, you will keep your communications to yourself.
Disregard any further announcements; disregard any further announcements.
Never say “never.”
Whether you mean it or not, be sincere!
If you want to think independently, you must imitate me.
Superstition brings bad luck.
Like everybody else, I’m different.
Statistics show that statistics can’t be trusted.
I saw him do it when no one was looking.
I used to be conceited, but now I’m perfect.
Words are incapable of describing what I am about to tell you.
No one goes there anymore; it’s too crowded. (Not ascribed by Gorn to Yogi Berra.)
Every once in a while it never stops raining.
We Scorpios don’t believe in astrology.
If your father was sterile, the same is probably true of you.
To distinguish the real from the unreal one must experience both.
We need to learn how determinism forces us to act in a certain way, in order to be able to act the way we want to. (revised by us)
Related Ideas That Our Concept is Independent Of
One is simply a self-denying sentence.
“The Treachery Of Images” (1928-29) by René Magritte depicts a pipe along with text stating “This is not a pipe.” Once the onlooker grasps the difference between an object and its representation, however, the sentence resolves to be true and not self-defeating.
When Shakespeare’s Macbeth soliloquizes, “nothing is real but what is not,” we think his logic is merely chasing itself in circles.
Fumblerules define rules of good writing via sentences that violate those very rules. Examples are:
Avoid clichés like the plague.
Don’t use no double negatives.
A preposition is something a sentence should never be ended with.
The passive voice should never be employed.
More general kinds of self-reference can lead to paradoxical situations. Miguel de Cervantes in Don Quixote references a fake version of Don Quixote in a way that readers who follow his logic will be led to conclude that Don Quixote and Sancho Panza are real, while the readers themselves are characters in the novel.
A anti-self sentence is one with the following property: It is usually given in response to a question, and the answer is self-defeating with regard to the question. That is, the answer is not in your best interests, but of a kind that people make routinely anyway.
This specialization of self-defeating sentences may seem rare, but they arise every day. Yet they hurt those who make the anti-self sentence. One of the reasons they occur, I believe, is that the answer is “forced,” yet seems to be not forced.
One great example is a story about Alan Turing that I discussed long ago here. Here is a shortened version:
In 1940, Turing wanted to be ready to fight an invasion, so he signed up for the Home Guard. When he joined, the form that he had to sign asked the question:
You understand by signing this form that you could be drafted into the regular army at any moment.
He signed the form, but he answered the question “No.” An official apparently filed it, without bothering to read it, and Turing duly joined the Guard.
The Home Guard officer for his locale later decided to call up Turing into the regular army, little knowing that since his work was critical to the war effort, Prime Minister Winston Churchill would have stopped it personally.
Turing went to see the officer anyway. At their meeting, the officer pointed out that Turing had signed a form that standardly allowed Turing to be put directly into the army. Turing smiled and said, “Take a look at my form.” There was Turing’s answer “No.” Apparently, Turing was thrown out of the meeting.
The question: “You understand by signing this form that you could be drafted into the regular army at any moment.” is an example of an anti-self sentence. The usual answer is “yes,” which is of course not in your best interest, as seen by Turing’s situation.
More Anti-Self Situations
A second example concerns the approved protocol for the Miranda warning, which in the U.S. must be read before interrogating a criminal suspect. Several states require the following additional questions, after the standard formula by which an officer reminds an arrestee of the U.S. Constitution’s rights not to be forced to give self-incriminating evidence:
- Do you understand each of these rights I have explained to you?
- Having these rights in mind, do you wish to talk to us now?
An affirmative answer to both of the above questions waives the rights. If the suspect responds “no” to the first question, the officer is required to re-read the Miranda warning. Saying “no” to the second question invokes the constitutional rights themselves. In either case the officer is stopped from further questioning.
Thus the expected double-yes answers are self-defeating for a suspect who happens to be guilty. However, guilty suspects often answer yes because of other perceived advantages of co-operation.
A third example comes from the attempt to justify the principle of logical positivism (PLP). This states that an assertion should be subscribed to only if (and if):
- it is true by definition or follows logically from valid assertions; or
- it is apprehendable from sense experience.
The anti-self situation results from the question,
Do you subscribe to the principle of logical positivism?
A “no” answer is fine, but the expected “yes” answer runs into the issue that PLP is neither deducible logically nor apprehendable from sense experience.
Other examples of purportedly self-defeating arguments can be found on this Wikipedia page.
Do you have pithy examples of self-defeating sentences besides those in Gorn’s compendium? Just as we go to press, we note this reader comment on “Mathematical creativity can be automated.”
Have you encountered any anti-self situations?
Can we build any interesting theorems on such examples?
[changed Turing picture, revised intro]