So what is paradox? It is a string of symbols in a human language which we interpret as expressing an unresolvable logical contradiction. Put more simply, a paradox is when we have two statements - A and B - both of which are true, yet they contradict each other. So they can't both be true. Simple, isn't it? More formally, a paradox contradicts the first law of logic, viz A exclusive or ~A. In English: either A is true or its opposite is true, but not both. This is called the law of the excluded middle. You're either pregnant or you're not.
There are paradoxes in every field of human endeavour:
Religion - the problem of evil
History - the only thing we learn from history is that we learn nothing from history
Counselling - the only way to help someone is to resist the impulse to help
Genetics - insanity is hereditary, you get it from your kids
Psychology - we only really want those things we can't have (and if we got them we wouldn't be happy)
Art - "Form liberates" (Goethe)
Fashion - whoever follows fashion wants to stand out by conforming
Cosmogony - why is the universe 5 billion years old? Because it took that long for a being to develop who would ask the question
Lucid dreaming - being simultaneously aware and asleep
Eroticism - that nudism isn't sexy in the slightest
Criminology - prescription of the death penalty for rape did not deter criminals but caused them to kill as well as rape their victims.
Language - "You've got the morals of an alley-cat," turns out to mean exactly the same as "You haven't got the morals of an alley-cat."
Pharmacology - coffee makes you sleepy (on the day after).
Scheduling - Hofstadter's Law: "It always takes longer than you expect, even when you take Hofstadter's Law into account".
Paradox causes stress in rationally inclined people, me included. So whenever a paradox is encountered, attempts are made to mend the torn fabric of rationality. The usual way is to show that one of the mutually contradictory statements is actually false, or that they don't in truth contradict each other. This latter possibility is the one explored in this essay; but first let's look at some other approaches.
A more radical way out is to reject logic. This solution appeals to only a minority of people. Rejection of logic isn't an illogical notion, since one can't logically use logical arguments in favour of adopting a logical approach. But this is itself a logical argument, and we seem to he headed for infinite regress (whatever that is). Personally, I like to stay with logic, so let's keep with it and hope it'll be fun.
Another way out is to identify the paradox in question as an example of self-reference, called a 'strange loop'. The simplest example of which is the statement: "This statement is false". There are many intriguing and complicated versions of "strange loops" (see my puzzles page). D. Hofstadter has written an impressive book called, "Godel, Escher, Bach" devoted to this topic. Bertrand Russell constructed the theory of types (simply a fancy way of forbidding self-reference) explicitly to abolish just this kind of paradox. Philosophers can be killjoys.
In the human world, far from philosophic discourse, statements of fact are frequently neither entirely true nor entirely false. Quite often the most appropriate answer to a question is "Yes and no". If we are asked whether we have read a book, do we answer yes or no, if we are 90% through it? This is a trivial example, showing how imprecise is our everyday use of language. At a deeper level, our experience of life is richer and more complex than our verbal statements about it can ever be. Or take something seemingly unequivocal: death. Has a person actually died if they are declared clinically dead but revive?
That's enough preamble. The method of paradox resolution that I favour is to recognise that two apparently contradictory statements are at different levels of description, and hence are incommensurate. I think I picked up this idea from Hofstadter. Of course this method does not resolve all paradoxes. I confidently predict that no general method of paradox elimination will ever be devised.
Confusion of levels helps us fall into the trap of false analogies. For that matter, an argument by analogy can never be more than an indication of plausibility. Someone said that Germany had behaved like a major criminal in the war and should therefore be punished. The flaw is that you can punish a person but not a nation. Almost anything can be "proved" by selecting the appropriate analogy. Many paradoxes arise in this way.
Let's look at a few examples. There is the notorious wave/particle duality in physics. Under certain circumstances, an electron will behave like a wave. Under different circumstances it will behave like a particle. The same seems to apply to all sub-atomic objects, including the photon, the smallest unit of light. So is an electron a wave or a particle? It's neither. The terms 'wave' and 'particle' are drawn from our ordinary experience on the human scale. There is really no reason to expect that familiar terms like 'particle' will have any equivalent at the microscopic scale of being. It would be like expecting grains of sand to be composed of microscopic bricks and mortar.
That was fairly straightforward. Let's take another paradox from physics. Quantum theory asserts that sub-atomic events are 'indeterminate', ie that the concept of causality is only partially applicable to the sub-atomic scene, where we can only talk about probabilities. For example, the half-life (time to decay) of uranium is precisely known, but this is a statistical average only. Any particular uranium atom will decay when it damn well pleases, apparently at random. Yet when billions of atoms are averaged out, the behaviour of a lump of uranium is perfectly predictable. But isn't science based on the assumption that the universe is ruled by causality? Quantum theory seems to reject this basic principle.
There are various ways of dealing with this paradox. The way that I adopt is this: that causality is indeed a universally valid principle or law. What quantum physics tells us is not a statement about reality (= what is) but about our knowledge of reality. My resolution of the paradox is that (assuming quantum physics is right) our knowledge of the physical world will never reach the point where we'll be able to predict when an individual uranium atom will decay. That doesn't mean there isn't some reason or cause that determines its individual timetable, only that we can't ever know it.
The same approach can be used to interpret the famous uncertainty principle, which states that the more precisely we know the position of a sub-atomic particle, the less we can say about its velocity. I interpret this to mean not that the world is indeterminate but that our knowledge of it is necessarily limited.
It is also paradoxical that the uncertainty principle, ie a formalised and precise statement of our necessary ignorance, leads to a rich theory that generates many valid predictions.
The dichotomy between knowledge and reality, or between 'what is' and belief, is well illustrated by the statement of a New Guinea native regarding a local form of witchcraft:
"I'm a Christian so I don't believe it, but it's true."
A long-standing conundrum in philosophy is the so-called mind-body problem. If body and mind are of entirely different natures then how can they interact? To put it another way, are the thoughts in my mind merely the result of chemical and electrical events in my brain, or is it the other way around? Various answers have been proposed, eg dualism - that body and mind are fundamentally different in nature, so that neither can be reduced to the other. This begs the question: how can they interact? Another approach is to regard the universe as being made up of 'mind-stuff' and that matter is like a precipitation of mind, or else just an illusion. Behaviourist psychology goes the other way, effectively denying the existence of mind.
My proposed solution to this hoary paradox is that it is again a result of confusion of levels - the mental and physical. It's like asking whether hardware or software is the basic nature of a computer, whether one is actually a form of the other, and how can two such different entities interact. Clearly, the question itself is faulty. Computers can be, indeed must be, understood in terms of both hardware and software. Yet neither level is more "basic". Without hardware software cannot run, without software there is no point in having hardware. I am not saying that the computer analogy solves the mind-body problem, only that it hints that its nature is illusory, ie that there is no "problem".
The situation with body and mind is similar. Both the physical and mental descriptions of a person are important, yet neither can be reduced to the other. How do body and mind interact? They don't. There is the activity of the body and there is the activity of the mind, but neither causes or affects the other. The two sets of phenomena are at different levels of description, though they refer to the same entity. Each mental state has an associated physical state and vice versa, but one cannot be reduced to (ie fully explained by) the other.
Perhaps this is clearer in a simpler case. A book can be seen at many different levels of description:
1) As a lump of inanimate matter, much like a rock or broken branch.
2) As a pattern of black marks on sheets of paper, ie a purely visual approach that ignores the meaning or purpose of the book.
3) As a long permutation of the 26 letters of the alphabet (again with no interpretation as to meaning).
4) In terms of sentences taken in isolation from each other. This is the level of syntax and context-less semantics.
5) As a collection of meanings, plot, characters, world view and so forth.
Some of these levels are echoed in how differently a book is perceived by:
1) An animal, to whom it is just an inedible object.
2) A pre-literate savage, who may realise that it is man-made.
3) A foreigner, who knows it has meaning but can't read it.
4) A native speaker, for whom it holds heaps of meaning.
5) Its author, for whom it will contain meanings not apparent to anyone else.
Yet none of these levels of description is more correct or more basic than any of the others. All are valid and each is the most appropriate under certain circumstances.
Turing devised an ingenious test to determine whether a machine can think. If after asking written questions of the machine we are convinced that it is actually a human being then the machine can think. This seems plausible, for we know that humans can think, so if a machine can mimic a human then it too can think. There is a flaw in Turing's test. In practice we would need to limit the length of the test to an hour, a day, a year, or some other finite period. It is in principle possible to write down the answers that a human being would make to all the questions that one could ask in that length of time (a year, say). A machine could match up any question it were given with the previously prepared answer. Yet such a machine would have no intelligence whatever, apart form the ability to compare the question it was presented with its store of prepared possible questions and answers.
The theoretical flaw in Turing's test indicates that thinking cannot be operationally or objectively defined. It can only be experienced (or not) on its own level, ie within the mind. Consciousness is also subjective. Likewise with intelligence. You can record a clever person talking but all you have captured is the results of their cleverness of mind.
Another philosophical paradox is the problem of free will and determinism. If my physical body (including my brain) strictly obeys deterministic physical laws then does it make sense to say that I have free will? If we accept the argument of the preceding paragraphs, then the paradox vanishes, since free will is a mental concept, determinism a physical one. The two are as much on different levels of description as grammar and the plot of a novel.
One of the most profound paradoxes is that of unity and the separate self. All the mystics tell us that there is only the One (or maybe there is only one mystic). Yet each of us is heartily convinced of being a separate being. The paradox arises from mixing the psychological and metaphysical levels of description. Likewise, it is fruitless to argue whether the physical world is an illusion or whether it is 'less real' than Brahman, the ground of being. The mystical insight cannot be expressed in words, if only because language is fundamentally dualistic.
"What is immortal and what is mortal are harmoniously blended, for they are not one, nor are they separate," wrote Ashvaghosha. It echoes, "The self and the world are not two". This mother of all paradoxes can be interpreted in terms of our levels of description. Reality in itself, or Brahman, or the Void, cannot be described. In so far as we attempt to talk about it, we fall into paradox because reality is not commensurate with any of our levels of description, since these all derive from the world of sense phenomena and are filtered through language.
My partner, Carla, put it better: "Since all boundaries are false, each time we come up against a boundary we will hit a paradox."