Paradox

Cogito ergo sum Logic and rhetoric

Key articles

Logical fallacy Syllogism Argument

General logic

Appeal to consequences Argument Bulverism Ipse dixit Non sequitur Ultimate end

Bad logic

Appeal to fiction Common sense Genetic fallacy Logical fallacy Slippery slope Sunk cost v - t - e

“”A paradox, a most ingenious paradox! We've quips and quibbles heard in flocks, but none to beat this paradox!

—W. S. Gilbert, The Pirates of Penzance

A paradox is:

• A statement or argument which leads to a contradiction in its own foundation or premise, such as "The next sentence is false. The previous sentence is true."
• A statement that, while true, defies common knowledge by leaving out important details. For example "Alex was born February 29^th, 1984. He has had 8 birthdays."
• Sometimes used to refer to a linguistic fallacy.

Types of paradox

• Veridical paradoxes: a paradox whose conclusion is true despite its air of absurdity e.g., the Monty Hall paradox
• Falsidical paradoxes: a paradox whose conclusion not only appears absurd, but is also false e.g., invalid mathematical proofs that 1 = 2.
• Antinomy paradoxes: a paradox that is neither veridical nor falsidical e.g., Russell's paradox. There is currently no way to deal with antinomies, either by convincing us that there conclusion is true or by finding the fallacious step in the argument.[1][note 1]

Examples

• The grandfather paradox — If you invent a time machine (let's assume you could), go back in time and kill your grandfather (particularly before your own father was manufactured), your father would never have been born, and thus you would never have been born. Consequently, you weren't around to invent any time machine with which to go back in time to kill your grandfather — thus, your father actually lived, you were born, and you would in fact be free to head back in time and kill your grandfather before your father was manufactured. And so on. A generalised version of this is: any situation whereby going back in time prevents you from going back in time originally, but as you were prevented from doing so, you never actually did, and thus are not prevented from doing so. Ad infinitum.[note 2]
• The omnipotence paradox — Asserts that a truly omnipotent being (e.g. God) cannot exist, since some of the actions it could perform would challenge its own omnipotence — e.g. creating a stone so heavy that it could not lift it or a more powerful being than itself, self-destructing (it would also have to be able rise back out of its own "ashes" or else not truly be omnipotent), believing a contradiction or disbelieving a tautology, etc. One way of putting it is to ask what would happen if an immovable object collided with an unstoppable force. But the answer to this is that nothing really happens when they do because the equation F=ma asserts that they are physically sort of the same thing: either way you want it, you have to make F equal infinity.
• Zeno's paradox — Zeno of Elea made three bizarre paradoxes that could be best described as taking mathematics far too literally. In his most famous one, a faster object (such as a hare) cannot overtake a slower object (such as a tortoise) providing the slower one has a head start. This is because by the time the faster object catches up, the slower one will have moved on to a new location. When it has caught up again, the slower object would have again moved (although less than before). This continues to infinity, proving that motion is impossible. Zeno, however, didn't account for the fact that the time interval between successive catchings-up gets shorter and shorter. The two rates cancel each other out, and as a result, the world functions as observed. This is also an example of a 'false' description. In the paradox, static measurements between the hare and tortoise are taken at times. However the time and distance measurements don't describe the system; both the hare and tortoise are moving.
• The missing square puzzle in any form in which it may occur is the result of using actual geometrical figures and not only textual descriptions and the axioms of geometry to reason about geometrical figures. In all forms of this puzzle, the area of the shape made by all the pieces put together appears not to be conserved under an arbitrary permutation of the pieces. The solution to this apparent paradox is two of the pieces of the puzzle are dissimilar shapes with diagonal sides or that a diagonal side of one of the pieces is not actually a straight line, and if the puzzle is made from 3D blocks, it may even have a piece whose obverse and reverse sides have different areas[2].
• The liar paradox — Exists in two forms : the first, said by a Cretan is "All the Cretans (here) are liars" and is usually solved by saying that liars tell the truth sometimes, or more easily by saying that the speaker is a liar (the ancient Greeks viewed this statement as paradoxical, but it is just false). The other form is: "this phrase is false" (or "Next phrase is true. Last phrase is false."), which is trickier to manage.
• The unexpected fire drill — A fire drill that by definition can never happen, because it is announced as "A surprise fire drill will be called next week, but we aren't saying on what day." It obviously cannot happen on the Friday, because if Friday dawns and there's been no fire drill, it won't be a surprise. So that's Friday out. If Thursday dawns and there's been no fire drill, it lacks any element of surprise because we already know it can't be on Friday. And so on.[note 3]
• The St. Petersburg paradox — Imagine that I am running a betting game where I toss a fair coin. If it comes down heads, you receive triple your stake and I invite you to play again. If it comes down tails, you lose. Simple probability says that you get back, on average, $1.50 for each $1 you bet. A naive strategy to maximise your winnings would mean that you would never stop playing, betting everything you have at each step. The paradox arises because this behaviour is certain to lose you all the money you bet — the probability of getting tails at some point in the infinite game is 1. That this is one of the simpler paradoxical results in probability theory says a lot about statistics. And statisticians.
• Russell's paradox (a.k.a. The Barber's paradox) — Imagine the set of "all sets which are not members of themselves". Is this set a member of itself? If it is, then it is not. If it is not, then it is. Another famous way of putting it is: suppose there is a town with just one barber. The barber is a man in town who shaves all those, and only those, men in town who do not shave themselves. In this town, every other man keeps himself clean-shaven, and he does so by doing one of two things: either by shaving himself, or by being shaved by the barber (read: by the man who only shaves other people). The question that results in the paradox is: who shaves the barber? Answering this question results in apparent contradiction. The barber cannot shave himself (as he only shaves those who do not shave themselves). As such, if he were to shave himself, he would cease to be the barber (since we've already defined "being the barber" as only shaving people who do not shave themselves). Now, if the barber follows the above reasoning and does not shave himself, one could then argue that he clearly qualifies among his customer demographic — the men who don't shave themselves — that is, among the group of regular men who would be shaved by the barber. And so, as the barber, it's his job to shave all members of that group, which now includes himself. Which is a contradiction. The paradox is thus that the barber can either shave himself, or go to the barber (which happens to be himself). However, neither of these possibilities are valid: they both result in the barber shaving himself, but he cannot do this (or else he'd no longer be the barber), because the barber only shaves those men who do not shave themselves.
• The Paradox of Tolerance - A perfectly tolerant society would be tolerant of intolerance, therefore would create a paradox onto itself (see Karl Popper).
• The principle of totality for material implication - Two mutually contradictory statements must imply each other either way.
• The principle of sufficient reason - Either of two mutually contradictory conditions might have the same reason for obtaining.

Impossible objects

One kind of paradoxical concept is something called an "impossible object". Impossible objects are drawings which, due to their construction, could not possibly exist in the Euclidean 3-dimensional world without an amount of cheating. Impossible objects typically make sense when only a part of them is viewed, but together, its impossibility is apparent. Examples of the impossible object include:

• The Penrose triangle, or tribar which includes three rectangular bars arranged in a triangle, each at right angles to each other. Any single vertex when viewed looks like a right angle, but viewing all three together presents the most well-known of all the impossible figures.
• The poiuyt or blivet. If you only look at the top half, it appears to be a fork with two prongs. If you only look at the bottom half, it appears to be three columns. Viewed as a whole, it can't really exist.

• 
  A Penrose triangle
• 
  A poiuyt

Obligatory trivia section

Paradox was also the name of a database programme for Windows which was released by Borland. Icons used for the programme were often visual puns on the name, such as two ducks ('pair of ducks') or two wooden jetties ('pair of docks').

See also

• Logical fallacy
• Mathematical paradoxes
• Proof
• Rhetoric

Notes

1. This section is currently stubby, and as a result will be expanded in due time.
2. That, or whatever you do while you travelled to the past has no consequences to whatever the place you came from. Science fiction writers typically handle this by arguing that at every moment, timestreams split infinitely according to, say, a given butterfly flapped its wings once, twice, or just glided for a few seconds longer. Then, the timestream in which you arrived in the past would instantly no longer be the timestream which led to your birth, the development of time travel, and your decision to become a time traveller rather than, say, a Taoist monk.
3. This is a lighter version of the unexpected hanging paradox.

References

1. Quine, W.V., 1966. The ways of paradox.
2. Xiaoke Ding - 3D visual explanation of Winston Freer Tile Puzzle (https://youtu.be/QbpfjM0NP7Q)

External links

• Wikipedia has a list of paradoxes; some are intriguing but many are only described as paradoxes because they disobey common sense rather than actual reality, mathematics, or probability.