Michael Clark’s *Paradoxes from A to Z* is already recognised as one of the best books on the topic. It has recently gone into a second edition. In this interview for Virtual Philosopher he explains what a paradox is and why philosophers should be interested in them. And if you want to know which came first, the chicken or the egg, you can download his discussion of the Chicken and Egg Paradox: Download Chicken and Egg Paradox.pdf.

**What is a paradox?**

Etymologically the paradoxical is what is contrary to received opinion or general belief.

A frequently cited characterization of paradox is Mark Sainsbury’s, in his excellent book, Paradoxes (2nd. edn. 1995, CUP). This can be illustrated with the sorites paradox, often known as ‘the heap’:

(1) A pile of 10,000 grains is a heap.

(2) Take a single grain away from a heap and it will not cease to be a
heap. And another. And so on. Until you have only one grain left.

(3) So one grain makes a heap.

Here the two premisses, (1) and (2), are apparently acceptable but the conclusion, (3), which seems to follow from them is not – it is contrary to received opinion.

But not all paradoxes naturally exemplify this same pattern. Take the Ship of Theseus, where the ship has its planks gradually replaced during the course of maintenance and the old planks are kept and reconstituted into a ship: which is Theseus’ ship now? Here we have a conflict between competing criteria. A number of paradoxes in my book are dilemmas of this sort: the Lawyer, the two-envelope paradox, the numbered balls, to mention just three. What is distinctive about such paradoxes is that the arguments for competing conclusions mirror one another, making the problem seem especially puzzling.

**Why should philosophers be interested in them?**

Philosophy thrives on paradoxes, and many have borne abundant fruit. As Quine says, ‘More than once in history the discovery of paradox has been the occasion for major reconstruction at the foundation of thought’. The development of nineteenth-century mathematical analysis (Zeno’s paradoxes), twentieth-century set theory (the paradoxes of set theory), the limitative theorems of Gödel, and Tarski’s theory of truth (the liar group) are dramatic illustrations.

And paradoxes are strongly motivating. There is a strong desire to resolve a paradox – generally stronger than with ordinary philosophy problems.

**How did you first get interested in paradoxes?**

At school a maths master lent me Lancelot Hogben’s Mathematics for the Millions, and encouraged me to read about Zeno’s paradoxes of motion. In any case, an interest in paradoxes comes with an interest in philosophy itself, since so many philosophical questions involve them.

Do paradoxes ever get solved?

Yes, I think they do. A good example would be Zeno’s paradoxes, which have been resolved by the mathematical analysis developed in the nineteenth century.

Less deep ones also tend to have agreed resolutions. These would include statistical illusions like the Monty Hall paradox. (Indeed, some purists don’t even count these as paradoxes.)

In this paradox, there are three doors, with a prize behind just one of them. The contestant picks a door but does not open it yet. She knows that the game host, Monty Hall, knows where the prize is and, when he opens another door to reveal nothing behind it, that he has used his knowledge to do so. The host then offers the opportunity of changing doors. She will double her chances of winning by accepting this offer.

When the contestant ﬁrst picks a door, the chance that it has the prize is 1⁄3. She knows that the host will be able to open a door concealing no prize, since at least one of the other doors must be a loser. Hence she learns nothing new which is relevant to the probability that she has already chosen the winning door: that remains at 1⁄3. Since if she swaps she will not choose the door the host has just revealed to be a loser, the opportunity to swap is equivalent to the opportunity of opening both the other doors instead of the one she has picked, which clearly doubles her chances of winning.

**Which is your favourite paradox and why? **

My favourite paradox varies depending on what I’m working on or thinking about. Once, it was the two-envelope paradox, in which I got very involved and for which I had to learn some mathematical analysis. Then I got very interested in Newcomb’s problem, because a book I was reviewing offered a clearly perverse resolution and I wanted to diagnose just what had gone wrong. At present, it’s Moore’s paradox. (For example, I cannot intelligibly say ‘Marilyn committed suicide but I don’t believe it’, though it could be true). I examined a very good PhD on it recently, and I have just received an interesting Analysis submission which offers a novel approach. Moore’s paradox is a deep one, having significance not only for the nature of belief and assertion but also for the mind-body problem.

**Has your work as editor of Analysis had an influence on your writing in this book?**

Yes. I have been introduced to new paradoxes like the Sleeping Beauty and variants of old ones, and published a number of papers on them. The concise analytical style of Analysis papers is especially amenable to the treatment of paradoxes.

**How does the second edition of your book differ from the first?**

As well as making corrections and revisions, I have added ten new paradoxes, including perhaps the oldest one ever, the Chicken and the Egg. That makes a total of 84 altogether, plus the entry on paradox itself. A pretty recent one I have added is the Parrondo paradox: there are pairs of losing games which if played one after the other become a winning combination. This won’t make you a fortune in the casinos, but there are possible applications in engineering, biology, economics and population genetics.

**Who is your book for?**

Academics can use it as a reference work. (I was delighted to hear from one distinguished philosopher that his copy was much thumbed), general readers can amuse and enlighten themselves, philosophy students can use it in courses or for reference. I know of a number of cases where my book has been used as the basis for a course. And paradoxes are a good way of being introduced to philosophy.

**Thank you very much**.

I sometimes wonder if Zeno's Arrow Paradox has really been solved. Or even if it has, after some fashion, whether it is raising an interesting metaphysical point. If you look at the world at a specific instant in time, what is it about a moving arrow that distinguishes it from one at rest?

Posted by: Heraklites | May 30, 2007 at 07:44 PM

Thanks! I never realized the depths of the subject before reading this post. - TL

Posted by: Tim Lacy | May 30, 2007 at 09:43 PM

I think you need to amend the explanation of the sorites paradox a bit; I had to look it up on Wikipedia to be sure what you meant. (2) needs to say something like "Take a single grain away from a heap and it will not cease to be a heap. And another. And so on. Until you have only one grain left".

Posted by: potentilla | May 30, 2007 at 11:43 PM

To Heraklites

See the entry on The Arrow in my book.

Posted by: Michael Clark | June 06, 2007 at 12:19 PM

To potentilla

Thanks. A good idea. I'll get it changed to your suggestion.

Posted by: Michael Clark | June 06, 2007 at 12:20 PM

Done!

Posted by: nigel warburton | June 06, 2007 at 12:48 PM

The paradox of The Body, and the paradox of The Ship of Theseus (Plutarch, Hobbes):

Over a period of years, in the course of maintenance a ship (Ship A) has its planks replaced one by one. However, the old planks are retained and themselves reconstituted into a ship (Ship B). At the end of this process there are 2 ships. Which one is the original ship of Theseus?

The human body comprises up to 100 trillion total cells (reports vary). Only a proportion of them are human cells. The rest are parasites and assorted microbes, much smaller than our human cells. Over 1000 different species of bacteria live in the human body. Each of your cells contains about 10 billion protein molecules of approximately 10,000 different varieties. There may be 200,000 different proteins. The average cell makes 2,000 proteins/sec, and 173 million /day, ==> if 74 trillion cells in the body, = 1.28x10²¹/body/day. Each cell is continually involved in billions of chemical reactions.

100,000 or so cells are dying off every second and replaced. In a human body a trillion atoms may be replaced every millionth of a second. Every atom in the body get turned over at least once if not many times every few years. They are now in plants, other animals including humans, or the soil, air and oceans.

A person (A) may have no atoms in common with his former body (B) from a previous period of his life, and the tiny number of possible atoms which may again be present will be in a different cell and location. The existence of a soul involves a hypothesis without proof. If a person doesn't have one, then all that A and B have in common is part of their history and memories, plus a genetic code, changing and older cellular structural blueprints including some common neuronal structures/pathways, and personality traits. The bundle that comprises self A now is different to his former B self. So is he the same person, or different?

Posted by: jim rogers, gosford, australia | May 03, 2008 at 01:10 PM