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.

Self-Defeating Sentences

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.

${\bullet}$ I enjoy your company most when I am by myself.

${\bullet}$ All simplifying assumptions are too complicated.

${\bullet}$ If you remember something too long, you might as well forget it.

${\bullet}$ Anyone who goes to a psychoanalyst should have his head examined — Samuel Goldwyn.

${\bullet}$ If I don’t see you again, auf wiedersehen!

${\bullet}$ I absolutely refuse to be assertive.

${\bullet}$ None of my close friends has a close acquaintance.

${\bullet}$ I am not contradicting you.

${\bullet}$ I won’t hesitate for a moment to avoid answering.

${\bullet}$ I will now predict an unanticipated result.

${\bullet}$ Ignore this sign.

${\bullet}$ You may steal this from me.

${\bullet}$ Henceforth, you will keep your communications to yourself.

${\bullet}$ Disregard any further announcements; disregard any further announcements.

${\bullet}$ Never say “never.”

${\bullet}$ Whether you mean it or not, be sincere!

${\bullet}$ If you want to think independently, you must imitate me.

${\bullet}$ Superstition brings bad luck.

${\bullet}$ Like everybody else, I’m different.

${\bullet}$ Statistics show that statistics can’t be trusted.

${\bullet}$ I saw him do it when no one was looking.

${\bullet}$ I used to be conceited, but now I’m perfect.

${\bullet}$ Words are incapable of describing what I am about to tell you.

${\bullet}$ No one goes there anymore; it’s too crowded. (Not ascribed by Gorn to Yogi Berra.)

${\bullet}$ Every once in a while it never stops raining.

${\bullet}$ We Scorpios don’t believe in astrology.

${\bullet}$ If your father was sterile, the same is probably true of you.

${\bullet}$ To distinguish the real from the unreal one must experience both.

${\bullet}$ 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:

${\bullet}$ Avoid clichés like the plague.

${\bullet}$ Don’t use no double negatives.

${\bullet}$ A preposition is something a sentence should never be ended with.

${\bullet}$ 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.

Anti-Self Sentences

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:

1. Do you understand each of these rights I have explained to you?

2. 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.

$\displaystyle \S$

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):

1. it is true by definition or follows logically from valid assertions; or

2. 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.

Open Problems

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]

63 Comments leave one →
1. July 23, 2011 10:19 pm

Groucho Marx:

“Those are my principles: if you don’t like them, I have others.”

Remarkably, if one replaces “principles” with “definitions”, the result makes perfect mathematical sense.

July 23, 2011 11:24 pm

Buddhism discourages insistence (執著). A common pitfall is to insist on being non-insist.

3. July 24, 2011 10:43 am

My favorite: “Can I ask you a question?”

This seems to fall into none of the categories but is somehow related.

s.

• July 24, 2011 12:12 pm

Actually last week I was forced into something similar at a bus stop. My school is at the end of the bus line, so late at night it’s common for a bus to disgorge passengers and then go out-of-service, or have the driver take a 20 minute stopover before the next run. So Monday night a bus pulls up and lets off a passenger; I’m the only one there:

Me: “Ready to board?”
Driver: “What?”
Me: “Ready to board?”
Driver: “I don’t understand the question.”
Me: (Stepping closer so he can hear me) “Can I board the bus?”
Driver: “I think you just did.”

• Serge Ganachaud permalink
July 25, 2011 2:56 pm

Once André Weil was asked: “May I ask you a silly question?” and he replied: “You just did!”

4. July 24, 2011 11:10 am

As far as i know, it was Georg Cantor who created the diagonal argument in his proof that there was more real numbers than integers.

http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

be well
-h-

• Rafee Kamouna permalink
July 24, 2011 11:55 am

Yes, you are right. But Dick is also correct as Turing version of Cantor’s diagonalization is the one used in computability and complexity theories.

5. July 24, 2011 3:34 pm

This sentence is a good example of a self-defeating one.

July 24, 2011 8:12 pm

“Do you subscribe to the principle of logical positivism?”

Doesn’t suscribing to PLP mean you believe one should only believe what one can deduce or learn from one’s senses? The key word is “should”, it’s not a contradiction to believe you should do something and then do something else. Maybe it’s hypocritical, but certainly not a contradiction.

July 24, 2011 10:15 pm

There is a joke that says that many people think (or act) like this: “Thanks God, I’m atheist”

8. July 24, 2011 11:38 pm

“This comment intentionally left blank” (cf http://www.this-page-intentionally-left-blank.org/ )

9. July 25, 2011 3:55 am

The comedian Arnold Brown:

“I like to do self-deprecating humour, even though I’m not very good at it.”

10. July 25, 2011 6:44 am

And here’s an idiotic paradox of my own.

• September 8, 2011 10:39 am

Paradoxes tend to disappear when one uses Fuzzy Logic rather than Boolean Logic. The sentence “Only idiots believe this sentence” is half right.

11. July 25, 2011 10:07 am

I once gave a talk in honor of Saul Gorn (I was fortunate to have him as a professor),
and I opened with: “Before I begin, I’d like to make some remarks.”

July 25, 2011 10:12 am

The sentence “None of my close friends has a close acquaintance” is self-defeating only if you assume the relation “close friends” is symmetric.

August 1, 2011 9:27 pm

I don’t think so. What, after all, does a “close” acquaintance mean? The term “close” seems to make sense when applied to a friend, but it clearly makes no sense when applied to an acquaintance.

July 25, 2011 10:48 am

“strictly forbidden”, about 10,400,000 hits in google

• July 25, 2011 5:48 pm

From May 1968: “Il est interdit d’interdire” (it is forbidden to forbid).

14. July 25, 2011 10:59 am

“None of my close friends has a close acquaintance”
If I had no friends, the rest of the sentence would be trivially true. The defeat lies in the fact that I have no friends

July 25, 2011 11:02 am

Obi-Wan, criticizing the Dark Side, claimed “Only a Sith deals in absolutes.”

July 25, 2011 12:23 pm

“Cut off the end of the rope and it won’t have the end”.

17. Ross Snider permalink
July 25, 2011 12:53 pm

Yablo at the University of Michigan has crafted an infinite set of statements in which there are no self-references and yet a Liar’s-like paradox.

18. July 25, 2011 3:22 pm

The sentence, “The following joke is incredibly funny,” tends to defeat itself pretty comprehensively.

19. Rafee Kamouna permalink
July 25, 2011 9:39 pm

The empty string is the top self-defeating sentence (concept). Obviously, it destroys all computability and complexity theories. All complexity theory whom I know never want to discuss. It makes them run away. It will not only render a collapse of ZFC inconsistency, but also………they run away as they will realize that complexity theory is the biggest mistake in scientific history.

Try it and check whether you will run away or not, best of luck when you try to find the empty string on your keyboard.

20. Mark Betnel permalink
July 25, 2011 9:49 pm

Self-defeating performatives: “I promise [not] to keep this promise”, “I predict that this prediction will [not] come true” … via Lakoff (http://georgelakoff.files.wordpress.com/2011/01/performative-antimonies-in-foundations-of-language-8-lakoff-1972.pdf)

21. July 25, 2011 10:48 pm

Self-affirming sentences are fun too. I remember from my rural childhood a hat that read:

“If I hadn’t believed it with my own mind, I never would have seen it.”

Here’s one that many people seriously assert:

“God said it, I believe it, and that settles it.”

In Neil Stephenson’s Diamond Age we find a similar sentiment

“They have only one book in Sendaro, and it tells them to burn all other books.”

Scientists are not immune; in a 1944 letter from Oppenheimer to Groves we read:

“There are a few people here whose interests are exclusively ‘scientific’ in the sense that they will abandon any problem that appears to be soluble.”

An innocuous mathematical version might be:

“This sentence is self-affirming.”

• July 26, 2011 9:36 am

I don’t find “This sentence is self-affirming” innocuous. It seems to me that one can argue that it is just as likely to be false as true, since if it’s false, then it’s … er … false.

• Rafee Kamouna permalink
July 26, 2011 1:01 pm

This sentence is half-false; fuzzy logic.

July 27, 2011 10:20 pm

in a social network I used a personal description which can be translated to: “I’m concise”.

Other self-defeating sentences/texts:

“If you can’t understand this sentence, please ask a friend.”

August 11, 2011 12:33 pm

If I don’t arrive before midnight, don’t expect me today.

July 26, 2011 11:34 am

no comment

• July 26, 2011 11:47 am

.

• July 26, 2011 11:58 pm

July 26, 2011 3:44 pm

I told you a million times not to exaggerate.

24. July 27, 2011 3:50 am

In a discussion on Google+ on why it is superior to Twitter someone commented:

“I would go so far as to say that almost any idea that can be expressed in 140 characters is too trivial, and not worth expressing.”

25. William Gasarch permalink
July 27, 2011 10:28 am

I told you a million times not to exaggerate

July 27, 2011 1:09 pm

The smallest integer that cannot be described with at most a hundred words.

• July 27, 2011 5:46 pm

Not a sentence, so self-defeating but not in the intended manner. It gives me an idea though: “The presence of the main verb in this sentence.”

July 29, 2011 11:35 am

Since there is no clear definition of a sentence (no, the presence of a main verb is not needed), I have to disagree. I don’t care though.

• July 30, 2011 4:44 pm

I still think what you wrote wasn’t a sentence, but I’ll keep that to myself as I don’t want it to seem as though I’m trying to have the last word in this discussion.

August 11, 2011 12:32 pm

I hate people who always want to have the last word in an argument.

27. William Gasarch permalink
July 28, 2011 9:13 am

(From Commercials on 45, a sequence of commercial satires, by Stevens and Grdnic.
Its on you-tube and also on mp3)

This offer will not be repeated, I repeat this offer will not be repeated!

28. July 29, 2011 12:23 pm

I once bought a second hand copy of Hofstadter’s Metamagical Themas in which there was a bookmark that read ‘This Book Is Not Marked’.

July 30, 2011 9:17 am

A few days late, but I thought I’d share this self-defeating sentence based on it’s location 🙂
http://imgur.com/OyXeO

30. July 30, 2011 10:29 am

Thanks for all these great entries. My wife Debbie contributes one, quoting a late neighbor and her true story. As a girl she had been fidgety on outings to Sage Chapel and other events in Ithaca, NY. So her mother told her, “The next time I take you somewhere, I’m leaving you at home!” Seventy-five years later, she was still puzzled by that…

July 30, 2011 2:49 pm

That’s great 🙂 I have family from out that way, as well, and it seems a number of odd statements originate there.

31. July 30, 2011 3:44 pm

Two from Hungary, via physicist Valentine Telegdi:

I. Singly self-defeating (attributed by Telegdi to Dirac)

“A Golden Era arrives when the contributions of ordinary researchers are great.”

II. Doubly self-defeating (said to be a Hungarian proverb?)

“It is not enough to be rude; one must also be wrong.”

32. July 31, 2011 5:18 am

One more from physics — a self-defeating sentence that is hundreds of years old, yet only in the 20th century was it recognized as being self-defeating:

“We will observe the system’s dynamical trajectory without disturbing it.”

It is self-defeating because (from the modern QIT point-of-view) a device is a measurement apparatus if-and-only-if it alters the dynamics of the trajectory to agree with the recorded measurement.

Does this mean that when we watch a bird flying in the sky, that our eye optics, retinas and cerebral cortex jointly act upon the bird so that its flight path agrees with our cognition of that path? The answer is “yes” …the mechanism of action has subtle interferometric aspects …and this is why eyes, retinas, and cortices are richly structured devices. Obviously, it took nature many hundreds of millions of years to evolve a visual/cognitive apparatus of sufficient sophistication to achieve this effect.

The self-defeating aspect of this sentence explains why, as 21st century engineers press against fundamental limits to size, speed, efficiency, accuracy, prediction, and verification, the appreciation grows that systems engineering broadly conceived is quantum systems engineering.

• July 31, 2011 9:09 am

A natural followup question, which just now was asked by my bird-watching wife, is this:

“When we see a bird, and simultaneously the bird sees us, does it happen that the bird’s eyes, retinas, and visual cortex back-act upon *us*, such that our human actions accord with avian cognition?”

The answer from quantum information theory is “yes” … the mathematical details are set forth in chapters 2 and 8 of Nielsen and Chuang … the physics details in the simplest case are associated with trapped ions and detector diodes linked by optical cavities … and the generalization from the idealized laboratory case of ions, diodes, and optical cavities, to the natural case of birds, bird-watchers, and binoculars, is mainly a matter of increasing the dimensionality of the state-spaces, without materially altering the physical dynamics of the information exchange between ions-and-diodes, and similarly, between birds-and-bird-watchers.

Evidently in both mathematics and science the contradiction inherent in a self-defeating sentence need not be obvious … indeed every proof-by-contradiction begins with a postulate that is shown to be self-defeating.

• August 1, 2011 1:18 am

I see the point—and it may simply be what’s meant by the expression “seeing eye-to-eye”. Which is also at least a little bit mis-directed, if seeing “eye-to-eye” really requires cognition in the brain apart from the eye itself.

Dick perhaps-unconsciously put a self-defeating sentence into the intro of our new post on guessing: “We guess that we two guess pretty poorly.”

August 2, 2011 4:49 pm

‘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.’

In that event, no sentence in natural language and outside formal language can be truly self-defeating, since no sentence outside formal language can ensure anything.

34. August 9, 2011 6:25 am

Papere audi

• August 9, 2011 10:23 am

The non-Google part of my brain did recognize the play on Sapere aude, “boldly seek to know,” but I can’t grok or Web-trace your changes. The closest I can come is the artist Christo Javacheff wrapping an Audi in paper, which is self-defeating since you can’t then drive it…!?

35. Nathan Ramella permalink
August 11, 2011 3:00 pm

You can trust me.

36. Nathan Ramella permalink
August 11, 2011 3:03 pm

“It was never a problem until today.”

37. August 12, 2011 6:59 am

As I told you before, I never repeat anything.

38. Anonymous Human permalink
August 16, 2011 5:08 pm

Here’s an example from ethics. The imperative sentence “Be evil!” is self-defeating, since any imperative is, implicitly, a claim of the goodness of the action. (“Comb your hair!” implies that combing your hair is good and, therefore, not evil.) By the same reasoning, Google’s “Don’t be evil!” corporate motto is meaningless tautology.