Wednesday, 14 January 2009

Paradoxes 1

I've been dipping my toe in the world of logical paradoxes, and I’m disappointed. Since the first instinct when confronted with a paradox (this, of course, is Logic 101) is to assume that the contradiction is actually a way of disaffirming one or other premises underlying the paradox, I’m disappointed (reading R M Sainsbury’s book, Paradoxes) how little trouble the framers of mathematical paradoxes take with their premises. They shape them in such as way as to make the paradox that derives from them watertight, but that involves them in starting from positions of idiocy. For example, from Sainsbury’s book:

Newcomb’s paradox. You are confronted with a choice. There are two boxes before you, A and B. You may either open both boxes, or else just open B. You may keep what is inside any box you open, but you may not keep what is inside any box you do not open. The background is this. A very powerful being, who has been invariably accurate in his predictions about your behaviour in the past, has already acted in the following way: He has put $1000 in box A. If he has predicted that you will open just box B, he has in addition put $1,000,000 in box B. If he has predicted that you will open both boxes, he has put nothing in box B. The paradox consists in the fact that there appears to be a decisive argument for the view that the most rational thing to do is to open both boxes; and also a decisive argument for the view that the most rational thing to do is to open just box B. [53]
But the ‘decisive argument’ invoked here depends upon accepting that there is such a powerful entity (Sainsbury calls his ‘the Predictor’). There isn’t. In fact, it’s better to read this as a thought experiment about the sort of thing that cold happen if God exists, as a way of disproving that God exists (or at least disproving that a god with the power to predict what you will do with invariable accuracy).

Sorities paradoxes. Suppose two people differ in height by one-tenth of an inch. We are inclined to believe that either or both of them are tall. If one were 6’ 6” and the other 0.1” shorter than this, then both are tall. If one is 4’ 6” tall and the other 0.1” taller then neither is tall. This apparently obvious and uncontroversial statement appears to lead to the paradoxical and uncontroversial supposition that everyone is tall. Consider a series of heights starting with and descending by steps of 0.1”. A person of 6’ 6” is tall. By our supposition, so must be a person of 6’ 5.9”. However if a person of this height is tall, so must be a person of one-tenth of an inch smaller, and so on without limit until we find ourselves forced to say, absurdly, that a 4’ 6” person is tall. Indeed that everyone is tall. [22]
No we don’t. The logic here depends upon the unspoken assumption (contained in the two sentence beginning ‘Consider a series…’) that tallness is measured in relation to 6’ 6”; or even more absurdly tallness is measured top-down from a notional upper height. It’s not. It’s measured in terms of standard deviation from a rough sense of the average heights of a whole population. Nor is tall an absolute, but is rather a relative marker (quite tall, very tall and so on) moving upwards from that perceived mean.

Sainsbury ends with an appendix ‘some more paradoxes’. They don’t seem to me to be very paradoxical.

The Gallows. The law of a certain land is that all who wish to enter the city are asked to state their business there. Those who reply truly are allowed to enter and depart in peace. Those who reply falsely are hanged. What should happen to the traveller who, when asked his business, replies ‘I have come to be hanged’?
If we assume that this certain land has other laws, and that there are other reasons why people can be hanged (murder, say), then there’s nothing paradoxical in this. Probably they’d send him away; conceivably they’d hang him; there’s not enough information in the premises to enable us to determine which. If, though, the paradox is supposed to inhere in the supposition that in this land they only hang people who come and state their business, always hang liars, regardless of how trivial or unintentional the lie and so on … well then the paradox is being generated by a set of impossible-to-believe premises, and in fact functions as a way of demonstrating that no such state could exist.

The Lawyer. Protagoras, teacher of lawyers, had this contract with pupils: “Pay me a fee if and only if you win your first case”. One of his pupils, Euathlus, sues him for free tuition, arguing as follows: ‘if I win the case, then I will win free tuition as is what I am suing for. If I lose, then my tuition is free anyway, since this is my first case. Protagoras, in court, responds as follows: ‘If you give the judgment for Euathlus, then he will owe me a fee, since it is his first case and that was our agreement; if you give judgment to me, then he will owe me a fee, since that is the content of the judgment.’
But this isn’t a paradox. Protagoras just has a weaker legal argument than his pupil (his pupil is canny and has found a nice loophole in P.’s offer). P’s argument disingenuously slides from the nature of the agreement (in his first part) to the content of the judgment (in the second). These two things are not the same. Specifically, the judgment is a judgment about the nature of the agreement. That's a crucial distinction.

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