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Entry updated 9 May 2022. Tagged: Theme.

A logical contradiction or seeming contradiction. The most famous in sf are the many forms of Time Paradox and the so-called Fermi Paradox concerning our continuing lack of expected (according to certain arguments) First Contact with Aliens. Also frequently cited is the "twins paradox" of Relativity.

Logical paradoxes – like the traditional statement of Epimenides the Cretan that "All Cretans are liars." – are generally regarded as trifling verbal games. They are routinely deployed in sf to immobilize or destroy insufficiently resilient Computers, as in Gordon R Dickson's "The Monkey Wrench" (August 1951 Astounding) – which cites Epimenides – and many other stories, to the point of Cliché. They are also useful to bewilder not-so-bright Aliens, as in Eric Frank Russell's "Diabologic" (March 1955 Astounding). However, seeming quibbles may have profound implications. The impossibility of excluding self-referential statements from formal mathematical logic (despite efforts by Bertrand Russell and others) was a major crux of twentieth-century Mathematics, encapsulated in Gödel's Theorem and most lucidly explained in Gödel, Escher, Bach: An Eternal Golden Braid (1979) by Douglas Hofstadter. This book also incorporates and expands on Lewis Carroll's paralogical fantasia "What the Tortoise Said to Achilles" (April 1895 Mind #4), which rings the changes on the paradoxes of infinity and motion proposed by Zeno of Elea (circa 490 BCE-circa 430 BCE) – the race between Achilles and the Tortoise being perhaps the most famous – to argue the impossibility of logical deduction. Zeno had pretended to argue the impossibility of an arrow's flight from A to B because first half the distance must be covered, then half the remaining distance, and so on in an infinite sequence of steps that apparently never reaches the goal; the problem was resolved by the mathematical insight that the infinite series in question has a finite sum and may be traversed in finite time.

The racing rivals in the Achilles and Tortoise Thought Experiment are often replaced by the more familiar figures of the Tortoise and Hare from Aesop's fable, as in Tom Stoppard's Jumpers (first performed 2 February 1972; 1972 chap), where Zeno's paradoxes are tragicomically illustrated with an actual arrow, hare and tortoise. Terry Pratchett's Discworld analogues of Greek philosophers adopt a similarly practical attitude in Pyramids (1989), setting up an Axiom Testing Station where Zeno's sophistical argument that an arrow in flight can never reach its destination is checked by experiment, with unwilling tortoises as the targets.

In Jorge Luis Borges's Tlön, Uqbar, Orbis Tertius (May 1940 Sur; trans James E Irby 1983 chap), the alien philosophy of the imagined world Tlön is highlighted by its equivalent of a Zeno sophistry, which to us seems a trivial anecdote. Several coins are lost and later found: surely these are the same coins? This proves to be a shockingly heretical paradox in a culture that denies materialism and sees blasphemy in "attributing the divine category of being to some ordinary coins" (from Alastair Reid's translation).

Gödel's Theorem itself, a highly technical reductio ad absurdum demonstration of the incompleteness or inconsistency of any formal mathematical system sufficiently rich to support ordinary arithmetic, is often invoked in an incantatory fashion as "proving" the unknowability of the universe or the future – for example, in Samuel R Delany's The Einstein Intersection (1967; 1 chapter restored 1968). [DRL]

see also: Information Theory.

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