Saturday, April 25, 2015

Please Ignore This Post

Not-Quite Paradox

To do as the title of this post says, you must not have done as it says. Contradictions such as the "Please Ignore This Post" variety go round in a circle, but they don't go round and round. One might say that they are paradoxical, but that they are not paradoxes. 

Circular contradictions are not paradoxes as they lack circularity. Here are a few examples: 

All rules have exceptions

Never say never

It is forbidden to forbid 
(wall in Paris, May 1968)

An exchange from N. F. Simpson's play

Mr. Characterson: 

I wonder - if I might just put a supplementary question, Dr. Whinby - I wonder whether you would mind telling me - simply for the record - if there is any question you think might possibly come up in the foreseeable future to which you would reply unhesitatingly in the negative? 

James Whinby: (without hesitation)

Quite definitely not!  

A Narração de Filectas, 1887
Rodolpho Amoedo (1857-1941)
Rio de Janeiro, Museu Nacional de Belas Artes

Eubulides, the Megarian sixth century (the Megarian school of philosophy, which flourished in the 4th century BCE, was founded by Euclid of Megara, one of the pupils of Socrates) Greek philosopher and successor to Euclid, invented the paradox of the liar. 

In this paradox Epimenides, the Cretan, says, "All Cretans are liars." 

If he is telling the truth he is lying; and if he is lying, he is telling the truth. In the simpler form "I am lying" this paradox was known to the ancients as a pseudomenon. 

Relief-moulded black Megarian bowl
c. 225 - 175 BCE

Some variations of this form include: 

This sentence is false

Socrates: "What Plato is about to say is false."
Plato: "Socrates has just spoken truly."

Many more of these paradoxes and not-quite paradoxes can be found in Vicious Circles and Infinity: A panoply of paradoxes by Patrick Hughes and George Brecht. Of this book Isaac Asimov wrote: 

I enjoyed this enormously. It exercises the mind and, if one is not careful, puts one into a rat race consisting of 'Of course - and yet - and on the other hand - ' which leads, by definition of paradox, to no possible conclusion in a finite time and therefore gives infinite pleasure to those who like mental exercise. Of course, if they also like a triumphant solution, their pleasure will be infinitely delayed. And yet if they expect that eventually there may be a triumphant solution, the pleasure of anticipation will be infinite, and they won't know there won't be a triumphant solution till - 

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