Catch-22 (logic)

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Catch-22 is a term coined by Joseph Heller in his novel Catch-22, describing a false dilemma, where no real choice exists. In probability theory, it refers to a situation in which multiple probabilistic events exist, and the desirable outcome results from the confluence of these events, but there is zero probability of this happening, as they are mutually exclusive.

1 Logic
2 Other uses from the novel
3 Popular culture example
4 See also

[edit] Logic

The prototypical Catch-22, as formulated by Heller, involves the case of John Yossarian, a U.S. Army Air Forces bombardier, who wishes to be excused from combat flight duty. In order to be excused, he must submit an official medical diagnosis from his squadron's flight surgeon, demonstrating that he is unfit to fly because he is insane. In fact, according to regulations, any sane person would naturally not want to fly combat missions because they are so dangerous. However, by requesting to have one's sanity evaluated, the airman demonstrates that he is in fact sane and therefore is fit to fly because one has to be sane to recognize one's own insanity.

Conversely, any flyer who wished to fly on combat runs implicitly demonstrates that he is insane and unfit to fly and ought to be excused. To be excused the airman would only need to submit a request. Of course, if they did, the "Catch" would assert itself, automatically short-circuiting attempts to be excused from flying.

A logical formulation of this situation is:

If S (you're sane, the default), then F (you have to fly).

If not S (you're not sane), then not F (you don't have to fly).

In fact, flying is so dangerous that if F, then not S (you have to be crazy to fly).

Officially, the only way to prove not S to be true is for D to be true (an evaluation from the Doctor, which must be true because doctors understand that only a crazy man would want to fly).

If not D (you're not evaluated), then S (by default you're assumed sane)

You can't get D without A (asking for it).

However, if not S, then not A (an insane person would not have enough presence of mind or fear of death to ask for an evaluation)

Therefore A automatically proves S (you're sane enough to not want to fly by USAAF guidelines, so you're sane).

Therefore if A, then F.

The only way for D and not S to be true (which is the only official way for not F) is if not A, but A is required for D so that can never happen.

So:

S → F

~S → ~F

F → ~S (known to be true, but not acceptable as proof without D)

D → ~S (acceptable as proof)

~D → S

~A → ~D

~S → ~A

A → S

Therefore:

A → F

~A → F (Catch 22)

In formal Propositional logic symbolism, Catch-22 can be expressed as " $((C \Rightarrow (A \land B)) \land (A \Rightarrow \neg B) \land (B \Rightarrow \neg A))$ ", which reads as "if C, then A and B, and if A then not B, and if B then not A", thus C is impossible. In this example, in order to not be required to fly (outcome $\!\,C$ ), one must be both crazy (proposition $\!\,A$ ) and request to have a sanity evaluation (proposition $\!\,B$ ). If one is crazy ( $\!\,A$ ) then they would never (according to airforce regulations) request an evaluation i.e.( $\neg B$ ). But someone who does request an evaluation ( $\!\,B$ ) is self aware enough that they are considered not crazy! i.e.( $\neg A$ ). Thus one can never leave ( $\!\,C$ cannot occur).

[edit] Other uses from the novel

Another Catch-22 is depicted in a scene with Luciana and Yossarian. Luciana is distraught because no man will marry her because she is not a virgin. Yossarian offers to marry her, but she claims he is crazy for wanting to marry a non-virgin like herself and says she can't marry him.

Yet another Catch-22 is described in a scene in which Orr tells Yossarian that Appleby has flies in his eyes. Appleby supposedly cannot realize that he has flies in his eyes because the very fact means that his vision is clouded, making it unable for him to see anything clearly, including whether or not he has flies in his eyes. Appleby doesn't realize that he has clouded vision either, and therefore has no way of knowing he has flies in his eyes.

However, in Heller's text, two meanings for Catch-22 exist beyond that of an unsolvable logical dilemma. In the first chapter, officers who censor the privates' letters must sign their own name according to Catch-22, and in the final chapters it is restated simply as "anything can be done to you that you can not prevent," the logical conclusion being eventual destruction or assimilation ...unless one breaks out of the system entirely. The solution for Heller is, to borrow a Prussian expression, die Flucht nach vorne antreten ("to take flight (flee) forward") decisively freeing oneself from a situation in which it is not possible to withdraw. In the case of Orr, a friend of Yossarian (Heller's main character), this was done by deserting and fleeing to Sweden. Note that the act of fleeing is not the primary issue: The central point is a decision to be free of a constraint combined with a strong course of action; such decisions are probably more typically found in the philosophy of existentialism than other systems of philosophy.

[edit] Popular culture example

In a famous retort to U.S. country clubs of the period not accepting minorities, including Jews, as members, Groucho Marx, in claiming a club had made him an exception, quipped: "I sent the club a wire stating: Please accept my resignation. I don't want to belong to any club that will accept me as a member."

Here, Marx is either not a member of a club, or if a club would accept him, he would immediately resign on the grounds that it would accept him. This leaves Marx completely unaffiliated no matter what – while he might wish to join a club, his acceptance would immediately cause him to leave.

The children's song "There's a hole in the bucket" takes a very circuitous route to a Catch-22. There can be any number of interjections, but the simplest form is that there's a hole in the bucket, which must be fixed by plugging with wood, but the only hatchet is too dull to use, the sharpening stone is too dry to hone the blade, and the water to wet the stone can only be carried by the bucket which has the hole. This uses circular probabilistic givens, of the form

$((A \Rightarrow B) \land (B \Rightarrow C) \land (C \Rightarrow D) \land ... \land (X \Rightarrow A))$ ,

thus because all related events are each only possible if another has happened, none of them are possible.

[edit] See also

Similar to a Catch-22:

Circular logic
False dilemma
Irony
No-win situation - real choices exist, but no choice leads to success.
Reductio ad absurdum
The Lady and the Tiger - a short story involving a princess who must make a decision in a false dilemma situation.

Sometimes confused with a Catch-22:

Chicken or the egg - a seemingly unbreakable cycle of causation, which has an unknown origin.
Cornelian dilemma - a choice between actions which will all have a detrimental effect on the chooser or on someone they care for.
Deadlock - in computing, when two commands reach a standstill or impasse; paradoxically waiting for the other to finish.
Double bind - a forced choice between two logically conflicting demands.
Hobson's choice - the choice between taking an option or not taking it.
Lesser of two evils principle - a choice between two undesirable outcomes.
Morton's Fork - a choice between two equally unpleasant alternatives.
Paradox - often mistakenly used to describe situations that are ironic.