There are several opinions on what constitutes a paradox, and you can find several differing definitions, but, in general a paradox is either:
- a statement that appears to be correct but also appears to contradict itself, or
- a statement that appears to be correct but also appears to contradict commonly accepted opinions, or
- a statement that appears to be correct but also appears to be counter intuitive.
The apparent contradiction can almost invariably be attributed to one or more of the following:
- The ambiguity and context dependency of natural languages such as English.
- The incorrect application of a logic to a statement of natural languages such as English.
- An unproven assumption.
- Faulty intuition.
Some Paradoxes Discussed
Some paradoxes raise interesting issues and some of the more interesting ones will be discussed on this site. More may be added at a later date. Currently there are the following:
- Balls in the Urn Paradox
- Berry Paradox
- Bertrand’s Paradox
- Church’s Paradox by Gödel
- Curry’s Paradox
- Drinker Paradox
- Goodman’s Paradox
- Grelling-Nelson Paradox
- Liar Paradox
- Platonist Rod Paradox
- Quine’s Paradox
- Richard’s Paradox
- Smullyan’s Paradoxes
- Wang’s Paradox
- Yablo’s Paradox
Note that this is not intended to be in any way comprehensive, since there are plenty of descriptions of paradoxes available elsewhere.
A contradictory delight in paradoxes
The response of mathematicians and logicians to paradoxes has shifted considerably since the beginning of the twentieth century.
At that time, when Bertrand Russell discovered (in 1901) his famous paradox, (Footnote: See, for example Russell’s paradox entry in Stanford Encyclopedia. ) such paradoxes were viewed as intolerable contradictions that were totally unacceptable in the world of logic. Russell’s paradox marked a crisis point in logic and mathematics, and logicians and mathematicians tried, without success, to come up with a foundation of logic and mathematics that was simple and appeared intuitively to be correct.
As time progressed, however, particularly after Gödel published his incompleteness proof, the way in which paradoxes such as Russell’s paradox were viewed underwent a subtle shift. Logicians began to believe that contradictory paradoxes were an inevitable part of any comprehensive system of logic, and that one might as well learn to live with them.
So, instead of the viewpoint that was prevalent at the time of the discovery of Russell’s paradox, when the common opinion was that the source of such contradictions should be rooted out and dealt with, we now have a situation where many people (for example, Gregory Chaitin and Raymond Smullyan) take a perverse delight in such contradictions, and embrace them as desirable notions, rather than as problems to be solved. And today there seems to be an increase in the willingness of certain academics to follow this line, and to ignore any need for critical logical analysis.
But it is not only possible to eliminate such contradictions from logical language and logical analysis, it is imperative that this should be done. In fact, when it is seen that the root of most of these contradictions arise from confusion of levels of language, it is seen that the elimination of such contradictions is quite straightforward.
As site owner I reserve the right to keep my comments sections as I deem appropriate. I do not use that right to unfairly censor valid criticism. My reasons for deleting or editing comments do not include deleting a comment because it disagrees with what is on my website. Reasons for exclusion include:
Frivolous, irrelevant comments.
Comments devoid of logical basis.
Comments with excessive number of different points.
Questions about matters that do not relate to the page they post on. Such posts are not comments.
Comments with a substantial amount of mathematical terms not properly formatted will not be published unless a file (such as doc, tex, pdf) is simultaneously emailed to me, and where the mathematical terms are correctly formatted.
Reasons for deleting comments of certain users:
Bulk posting of comments in a short space of time, often on several different pages, and which are not simply part of an ongoing discussion. Multiple anonymous usernames for one person.
Users, who, when shown their point is wrong, immediately claim that they just wrote it wrong and rewrite it again - still erroneously, or else attack something else on my site - erroneously. After the first few instances, further posts are deleted.
Users who make persistent erroneous attacks in a scatter-gun attempt to try to find some error in what I write on this site. After the first few instances, further posts are deleted.
Difficulties in understanding the site content are usually best addressed by contacting me by e-mail.
Note: a password enables editing of comments, an email enables notification of replies