A paradox is an assertion that is essentially self-contradictory, though based on a valid deduction from acceptable premises. What this means, more or less, is that there is some logical problem going on; either the deduction isn't really valid, or the premises aren't really acceptable. Alternately, the premises and the deduction are fine, and the universe really is self-contradictory.
Types of Paradox
Paradoxes come in different forms. Some are phrased as questions, for example, 'Which came first, the chicken or the egg?'. The answer seems paradoxical; the chicken must exist first to lay the egg, but the egg must exist first to hatch the chicken. Both answers seem to be backed up by solid logic, but only one can be the correct answer. Thus a paradox.
Some paradoxes take the form of proofs. Using apparently sound logic, one can prove that 2+2=5, that all motion is impossible, or that you are not, in fact, the one reading this entry.
Other paradoxes are presented as statements whose truth value is in question, for example, 'this statement is false'. Logic tells us that, if the statement is true, then it must be false. Likewise, if it is false, then it must be true. When logic sends us in circles like this, we have a paradox.
Apparent Paradoxes and Resolutions
It has been said that there are no true paradoxes, only apparent ones. It is certainly the case that many paradoxes come about when language is inappropriately applied to experience. These paradoxes are indications that our language must be looked at more carefully. Finding the 'resolution' of a paradox means finding a way of talking about it according to which its paradoxical nature vanishes.
The Chicken and the Egg
Which came first, the chicken or the egg? It seems that both answers (or neither) must be correct, for all chickens come from eggs, and all eggs come from chickens.
One resolution can be achieved by saying that it is incorrect to state that all eggs come from chickens. Many other creatures lay eggs, including the ancestor of the chicken. The paradox could however be restated, 'which came first, the chicken or the chicken egg?' In this case we can say that the chicken egg came first, and it was the mutant offspring of a not-quite-chicken.
A second answer could be that the boundary between 'chicken' and 'pre-chicken' is a fuzzy one, and so is the boundary between 'chicken egg' and 'pre-chicken egg,' therefore they (the chicken and its egg) emerged gradually and simultaneously, and at some point it became possible to say that both clearly existed.
Sometimes attributed to Zeno of Elea (on whom more later), but more often to Eubulides of Miletus, these are a family of paradoxes that take the following form:
10,000 grains of sand make a heap; removing one grain should not make much difference, so if 10,000 grains of sand make a heap, then 9,999 grains of sand make a heap. If 9,999 grains of sand make a heap, then 9,998 grains of sand make a heap... if two grains of sand make a heap, then one grain of sand makes a heap. Therefore, one grain of sand makes a heap.
It is certainly the case that 10,000 grains of sand make a heap and that one grain of sand does not, so where is the faulty step in this chain of inferences? The problem is that of vagueness. Proponents of so called 'Fuzzy Logic' say that this paradox can be solved by recognising that 'X is a heap' is not a statement that is necessarily 100% true or 100% false for every collection of objects.
The Irresistible Force and the Immovable Object
What happens when an irresistible force meets an immovable object? Goofy, yes, but it gives some people insomnia.
The general consensus is that the question is meaningless, because the force and the object in question are impossible.
Impossible for God?
These paradoxes illustrate problems with omnipotence (besides finding a decent game of chess). Can God create an object too heavy for Him to lift? Or, eschewing physical problems entirely, can God ask a question too difficult for Him to answer? And was that the question?
This paradox treads on theological grounds, which brings in special problems of heresy and damnation. Either God cannot in fact do everything, or He can, but He chooses not to, thus avoiding the problem.
Can Two Plus Two Equal Five?
Here is an example of a paradoxical mathematical proof. Several such proofs exist; all contain fallacies. The first line is taken as a given:
- A = B
- A+A = A+B (adding A to both sides)
- 2A = A+B (simplifying)
- 2A-2B = A+B-2B (subtracting 2B from both sides)
- 2A-2B = A-B (simplifying)
- 2(A-B) = 1(A-B) (factoring)
- 2 = 1 (cancelling)
Once it is established that 2=1, it is easy to prove that any other pair of numbers are equal, simply by multiplying and adding appropriate numbers to both sides of the equation.
'Proofs' like this always have a fallacy. The fallacy in this one is in the last step, when the (A-B) on both sides is cancelled. This is division by zero, which is illegal in mathematics, and indeed, this is why it's illegal.
Zeno of Elea (aka 'Xeno') was a student of Parmenides the pre-Socratic philosopher. He travelled with Parmenides to Athens where he interacted with Socrates, Plato and other notable thinkers, sages and luminaries. Parmenides used to claim that All was an indivisible, immutable One, and that change and motion were illusions. Zeno attempted to prove these claims mathematically by showing that motion implied certain contradictions. This gives rise to paradoxes, with his proofs showing that motion is impossible, while our experience cries out that motion happens all the time.
Zeno's most popular paradox runs something like this; suppose you want to cross the room, first you have to walk halfway across, then you have to cross half the remaining distance, then half the remaining distance, and so on. There are, eventually, an infinite number of distances you have to cross to get across the room, and this is impossible to do in a finite time, therefore, all motion is impossible.
Contrary to popular belief, the resolution of this paradox does not require calculus, or indeed anything unknown to the Greeks. It is merely required to point out that Zeno has assumed that the distance across the room is infinitely divisible, but not infinitely extended. Going with this assumption, the time required to cross it must also be infinitely divisible, and it is not required that the time be infinitely extended1. Whether space or time are actually infinitely divisible may be problematic in modern physics, but that's way over Zeno's head.
The Lying Cretan
This and the following several paradoxes rely on a fun and headache-inducing phenomenon known as self-reference. The Lying Cretan, like Zeno's Paradoxes, comes to us from the Ancient Greeks. One version of the Lying Cretan paradox occurs, interestingly, in the Bible:
One of themselves, even a prophet of their own, said, The Cretans are always liars, evil beasts, slow bellies. This witness is true.Titus 1:12-13
The 'prophet' in question was Epimenides, who said 'all Cretans are liars' when he was being interrogated by Athenians. The problem here is that if what he says is true, then he, a Cretan, is always a liar, so what he says is therefore false. There must, in fact, be at least one Cretan who is not a liar. This is not quite a paradox, it is just an odd way to lie. The writer's claim that 'this witness is true', cannot be correct. Lying Cretans: 1 - the Apostle Paul: 0.
A more paradoxical version is due to Eubulides of Miletus, who lived a couple of centuries after Epimenides. Eubulides said 'I am lying'. If he was lying, then what he said was false, and he was in fact telling the truth. On the other hand, if he was telling the truth, then his statement was true, so he was lying after all. This paradox has been further refined to 'this statement is false'.
It must not be the case that every statement is either true or false, but there must be a third option: paradoxical or indeterminate.
The Barber Paradox
In a certain town lives a barber who has a peculiar rule by which he operates. The barber cuts the hair of everyone who doesn't cut their own hair, and he does not cut the hair of anyone who does cut their own hair. Who cuts the barber's hair?
The resolution to this is that the barber is bald. Alternately, no barber can follow such a rule.
The Librarian Paradox
This one is a little tricky to follow; there is a chief librarian who is in charge of several libraries. He asks the librarians of each library to prepare for him, in book form, a catalogue of all of the books in their respective libraries. Each librarian is faced with a dilemma. Since the catalogue is a book, should it be included in the catalogue of books? Some decide yes; some decide no. When the chief librarian receives all of the catalogues, he divides them into two groups: those which list themselves, and those which don't. There are many of each kind, and the chief librarian sets out to catalogue the catalogues. He prepares one catalogue which lists all catalogues which list themselves. Another lists all catalogues which do not list themselves. Where is this last catalogue to be listed? If it does not list itself, then it really should, but if it does, then it really shouldn't.
The librarian paradox is more succinctly stated as a problem in set theory. Let X be a set, defined as follows: X is the set of all sets which do not contain themselves as members. Any set not containing itself is a member of X, and any set containing itself is not a member of X. For example the set (1,2,3,4,5) does not contain itself as a member, for its members are numbers and not sets. It, therefore, is a member of X. On the other hand, the set containing everything that is not a teacup does contain itself as a member, for it is not a teacup. This set is, therefore, not a member of X. So, the paradox is easily expressed; is X a member of itself? If it is, then it violates its own definition by containing a set that is a member of itself. If it is not, then it violates its own definition by not containing a set that is not a member of itself.
In the library version of the paradox, the resolution is simple; the librarian cannot make a catalogue according to the stated rule. In the purely mathematical version, it is more difficult, because mathematicians assume (until they've considered this paradox) that every set exists if it can be defined. This assumption is known as the 'Axiom of Unbounded Comprehension'. Therefore, the paradox is resolved by rejecting the Axiom of Unbounded Comprehension, at which point it becomes possible to say that a set with such problems simply does not exist.
Speaking of mathematics, there is a rather famous result which is typically brought up in discussions about paradox, although it is not strictly about paradoxes. Kurt Godel (1906-1978) proved in 1931 his famous First Incompleteness Theorem2. This theorem states that in any formal system complicated enough to be able to talk about basic arithmetic, there will exist propositions which are true, yet cannot be proved nor disproved within that formal system. This is sort of the converse of a paradox, in which a statement can be proved, and yet is not true.
Since Godel's Theorem is not a paradox, it needs no resolution. It can be neatly illustrated by looking at the statement, made in system X, 'This statement cannot be proved true in system X'. This statement is clearly true, for if it weren't, then it could be proved true, which would be absurd. There is, however, no contradiction in supposing that it is true. It cannot be proved within system X. We know that it is true, but that knowledge comes from thinking outside of system X3.
You are taking part in an experiment. In front of you on a table are two boxes, box A and box B. Box A is made of glass so you can see inside, and it contains £1000. Box B is opaque and contains either nothing or £1,000,000. You are allowed to take either box B alone, or both box A and box B.
Here's the catch, the people running the experiment claim to be able to predict with astounding accuracy which of the two options you will go for, and will have put the £1,000,000 in box B only if they predict that you will only take box B. If they think you will take both then they've left box B empty.
In the previous thousand experiments like this, their predictions about what the subject will do have been correct... so, what do you do?
One argument is that you're just another subject, so if you take both boxes you get £1000, if you take just box B then you get £1,000,000. So take just box B.
On the other hand, either the money is sitting there in box B or it's not, so whatever is in box B, taking both means you get £1,000 more than if you take just box B.
The resolution of this paradox is left as an exercise for the reader. Good luck!
One Last Paradox
This paradox concerns language and numbers. Numbers can be described by sequences of words, for example, 'the number of states in the USA', 'the largest number of people that have fit into a phone box at the same time', 'the smallest number of colours you need to colour a map of the world such that no two neighbouring countries have the same colour', '42'.
Some sequences don't specify a number, such as 'some sequences don't specify a number'. But lots of them do. The paradox is contained in the following sentence:
The smallest number not expressible in fewer than eleven words.
Now, there is a finite set of word sequences of fewer than eleven words (supposing 100,000 possible words, that's only 100,00010 possible sequences - that's a one with 50 noughts after it - a large, but finite number) so there's a finite set of numbers they can express. So we can find the smallest number not in that set. Then that must be the number expressed by the above phrase. Except of course, that the above sentence has only ten words in it...