Paradoxes

20 December 1999

Aren't paradoxes precisely something we cannot define?-)

For now, here is a piece of old news I posted two years ago. I'm sorry it's a bit pompous. It sounded better at the time, as I was adding my contribution to some people's attempts to define paradoxes thoroughly.

Subject: Paradox: a shocking new piece of information for one
Date: Wed, 17 Dec 1997 16:01:22 -0600
From Claude Chaunier
Newsgroups: sci.math, sci.logic, comp.theory, talk.philosophy.misc, alt.folklore.science, alt.philosophy.debate

Dr D.J. Marsay wrote:

> How about 'A statement in which what one considers to be a logic leads to a self-contradiction' as characterising an important class of paradox?

How about 'A statement in which what one considers to be the plain way to think of its matter leads to a self-contradiction because of well-admitted or hidden assumptions one should reject' ? I like your definition, although it took time for me to feel it clear, so I wonder if it's really clear. I can see up to five levels with a paradox:

A misunderstanding level, in which the observer's mind is blind to anything wrong while there is really a contradiction in it.

A contradiction level, in which the observer is confused. Everything looks obvious or unescapable and then at the same time everything looks wrong, though the observer considers to be using the only plain way to think.

A tautology level, in which the observer has clearly already rejected an assumption which would have led to a self-contradiction. Indeed the paradox looks like a foolish statement, a straight truth or a proof by reductio ad absurdum.

An enlightenment level, in which the observer sees the previous levels. The paradox is a funny and instructive piece of art or of the history of the human thought. The observer can speak of the apparent paradox and its hidden or well admitted assumptions in someone else's mind.

An epistemology level, in which the observer thinks of the very nature of a paradox in general.

You've managed to gather most of them in one sentence. For example, the 'one' you speak of is at the enlightenment level, because that 'one' considers 'a logic' among others instead of the only plain way to think. So that 'one' is indeed thinking of another 'one' who would be at the contradiction level. The misunderstanding level is suggested by the strong calling of 'logic'. Something was unassailable before the self- contradiction showed up. The epistemology level is of course implicit as it is a definition of a general class of paradox.

Still, I miss a little the tautology level, which I would like to be more explicit than the enlightenment level, and I feel uneasy with the use of 'a logic', because some paradox may only be known by mathematically oriented people who wouldn't word their conceptions of some rich structures as only 'a logic'. Alright if sociologists and Karl Popper use 'a logic' outside mathematics, but mathematicians would often like to distinguish at least vaguely the different reasoning levels and not always bother about the logical one. For example, I wouldn't call only 'a logic' the way I could feel confused with the Banach-Tarski Paradox or even the way some can with 0 = (1-1)+(1-1)+... = 1+(-1+1)+(-1+1)+... = 1 . Anyway, an observer at the contradiction level who even doesn't care about it is unable to point out what's wrong and could as well use any other strong calling like 'plain english' or 'empirical evidence' or 'universal conception' or 'I don't understand this is just life'. Perhaps you don't want to speak of an even broader class including graphical paradox, but it could help if we thought of them to find a better expression which wouldn't make one accuse an innocent logic.

But these definitions look so simple, I wonder if we don't see much more under the words than many would. I miss a more explicit presence of the levels.

Seemingly worse, we can deduce from our definitions that any statement is a paradox. Because we can always imagine someone special with a bad view on the matter of the statement making the statement painfully contradictory. Then how could our definitions help to distinguish a paradox from a non-paradox? So they are not definitions at all. What is wrong?

What is wrong is the beliefs that every new piece of information couldn't be a paradox and that we are trying to define 'a paradox' in an absolute meaning rather than to define 'a paradox for one'. It's sloppy indeed to say 'this is not a paradox' without telling for whom.

A more general definition should go round something like a 'shocking new piece of information for someone or something'.

I've heard Sartre once said he measured the value of a thought by the displeasure it gave him.

Claude Chaunier

[Up home]