These are old fond paradoxes to make fools laugh i' the alehouse.
- Othello, Act 1, Scene 1

Paradoxes are as old as humankind. The ancient Greeks studied them intensely which eventually helped lead to the discovery of irrational numbers, and paradoxes are mentioned in the Bible: "It was one of their own prophets who said 'Cretans were never anything but liars, dangerous animals, all greed and laziness;' and that is a true statement." (Titus 1:12-13)

Even today, we are surrounded by paradoxes such as Blackwood's "the more terrible the prospect of thermonuclear war becomes the less likely it is to happen," or the Moebius Strip - a topological paradox.

For this article, we define a paradox as a statement or sentiment that is seemingly contradictory or opposed to common sense and yet is perhaps true in fact. Another way of thinking of a paradox is a statement that is actually self-contradictory and hence false even though its true character is not immediately apparent.

One of the oldest paradoxes is the one cited by the Apostle Paul in his letter to Titus (see above.) The following famous Liar Paradox

is interesting because it cannot be true because it would make the speaker a liar and therefore what he says is false. Neither can it be true because that would imply that Cretans are truth-tellers, and consequently what the speaker says would be true. (For classic Star Trek fans... "Norman, coordinate." {I, Mudd})
Self-reference would appear to be the problem with the Liar Paradox, but upon further study, eliminating reference to oneself does not eliminate the paradox. Try this one on for size:

The Jourdain Truth-Value Paradox

If sentence A is true, then B is false, and if B is false then A must also be false. But if sentence A is false, then B must be true, and if B is true, then A must be true. Neither sentence talks about itself, but taken together they keep changing the truth-value of the other, so that eventually, we are unable to say whether either sentence is true or false.

Closely related to the Liar Paradox and the Jourdain Truth-Value Paradox are prediction paradoxes such as the Unexpected Exam.

Most people admit that Mike's reasoning is correct, that the exam can't be offered on Friday, because it would not be "unexpected." Once this is admitted as sound reasoning, the rest of Mike's reasoning seems to follow. However, even the first step in Mike's reasoning is faulty. Suppose he has attended class every day. As he walks to class on Friday, can he deduce correctly that there will be no test that day? No, because if he makes such a deduction, he might walk into class and see the unexpected examination.

The consensus among logicians is that although the professor knows he can keep his word, there is no way that Mike can know it. Therefore, there is no way he can make a valid deduction about the test on any day, including Friday.

The Ancient Greeks were confounded by many paradoxical situations, which, as noted above, helped lead to the discovery of irrational numbers (numbers such as the square root of 2 or pi). Without irrational numbers we would not have progressed beyond elementary arithmatic and such things as geometry or calculus would be unknown. Zeno, a 5th Century B.C. Greek philosopher, used a paradox to argue that a person could never cross a room because to do so would require an infinite amount of time. Zeno's Paradox, where the sum of a set of infinite numbers can be derived to a finite total reads thus.

But obviously, it is possible to cross a room and touch the other wall and obviously it does not take an infinite amount of time to do this. Zeno's Paradox was not "resolved" until Newton and Liebnitz discovered the concept of the limit. "An irresistible inference is in conflict with an inescapable fact." A Tour of the Calculus, David Berlinski, Pantheon Books, 1995.

An irresistible inference in conflict with an inescapable fact: A marvelous modern definition of a paradox.

Bertrand Russell, a philosopher/mathematician/political activist, changed the direction of mathematics in the early 20th Century when he reported his famous Barber Paradox. Russell's Paradox arises within set theory by considering the set of all sets which are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself.

The significance of Russell's paradox can be seen once it is realized that, using classical logic, all sentences follow from a contradiction. In the eyes of many, it therefore appeared that no mathematical proof could be trusted once it was discovered that the logic and set theory apparently underlying all of mathematics was contradictory.

Based on Russell's Paradox, metaphysicians, mathematicians and philosophers have introduced the concept of Metalanguages to help describe such sets. Russell's basic idea is that we can avoid reference to S (the set of all sets which are not members of themselves) by arranging all sentences into a hierarchy. This hierarchy will consist of sentencesindividuals at the lowest level, sentences about sets of individuals at the next lowest level, sentencessets of sets of individuals at the next lowest level, etc. This hierarchy has helped bridge the gap between mathematics, logic and philosophy in an important way.

Another of Zeno's paradoxes, the Arrow Paradox, also illustrates the impossibility of motion or change. This paradox is part of the work of Werner Heisenberg, who was awarded the 1932 Nobel Prize for his work on the eponymous Heisenberg Uncertainty Principle. Heisenberg observed that the closer one gets to measuring the velocity of a particle of energy, the further one gets from observing its position. The Heisenberg Uncertainty Principle prompted the famous remark from Einstein: "I shall never believe that God plays dice with the universe."
One of the latest and most profound prediction paradoxes is Newcomb's Paradox. This paradox is closely related to game theory (See my other page on politics for more information about game and decision theory).

The paradox reveals a lot about whether or not a person believes in free will. Reactions are almost equally divided between believers in free will, who favor taking both boxes, and believers in determinism who favor taking only Box B. Others argue that conditions demanded by the paradox are contradictory regardless of whether or not the future is or isn't completely determined.

Aside from their amusing nature, paradoxes have practical applications as well. Based on the Omnipotence Paradox , legal scholars have debated whether or not a constitution can be non-paradoxically amended. While some may argue that this is an angels dancing on the head of a pin question, there are important implications for the rights of people. For instance, can a constitution contain a clause that prevents certain sections of the constitution from being amended in the future? What about amending the section that deals with amendments? For more information about self-amendment paradox, see Peter Suber, "The Paradox of Self-Amendment in American Constitutional Law," Stanford Literature Review, 7, 1-2 (Spring-Fall 1990) 53-78.
Political scientists have been looking at the Voter Paradox as an example of a "government failure." Just as individual choice sometimes fails to promote social values in desired and predictable ways, so to does collective choice. Collective choice exercised through governmental structures offeres at least the possibility for correcting the perceived deficiencies of individual choice (See The Prisoners Dilemma as an example). However, at times even government cannot overcome these deficiencies.

For more information about the Voter Paradox and Kenneth Arrow's General Possibility Theorem which states that any fair rule for choice will fail to ensure a transitive social ordering of policy alternatives (in other words, cyclical social preferences like those appearing in the Voter Paradox can arise from any fair voting system), see William H. Riker, Liberalism Against Populism (San Francisco: Freeman, 1982). and Kenneth Arrow, Social Choice and Individual Values 2nd ed. (New Haven, Conn.:Yale University Press, 1963)

The St. Petersburg Paradox, played an important role in the development of decision theory and the concept of the utility function of money. While much too complex to get into in great detail here, decision theory is the development of systemic rules for decision-making. The utility function of money was first developed by the 18th century mathematician Daniel Bernoulli, who proposed that the true worth of an individual's wealth is the logarithm of the amount of money possessed.

Imagine that there is a scientific law that says "All crows are black." If only three or four crows are observed, then the law is weakly confirmed. If millions of crows are seen to be black, then it is strongly confirmed.

Professor Carl Hempel, who invented this paradox, believes that a purple cow actually does slightly increase the probability that all crows are black.

Hempel's Paradox and Nelson Goodman's "grue" paradox are examples of confirmation paradoxes. For more information about confirmation, see Wesley C. Salmon, "Confirmation" Scientific American May, 1973.

Copyright 1997 Mark C. Gribben

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