According to xkcd, mathematics is the über-science (or unterscience, if you prefer) (http://www.xkcd.com/435/):

Logic is a branch of mathematics (and philosophy) with applications in fields as wide-ranging as physics, computer science, digital software and hardware engineering, and in getting rid of that pesky evil robot in your life.

Mathematical logic, as a distinct field, dates from about the mid 19th Century, when George Boole developed Boolean algebra – a system for representing logical propositions symbolically. Boolean algebra is binary: logical propositions can either be true or false (and must be one or the other).

Logic is about the most fun type of mathematics ever. One of the particularly ace aspects of logic is the application of the paradox.

Consider a town in which the men are all clean-shaven. The town has one barber, Mr Jones. Each of the men in the town either shaves himself, or he is shaved by Mr Jones. Now consider Mr Jones: does he shave himself? Well, if he shaves himself, Mr Jones must be one of the men in the town who doesn’t shave himself. But if he doesn’t shave himself, then he is one of the men who does not shave himself and so is therefore shaved by Mr Jones – so he shaves himself. You see the paradox here.

This barber paradox is a paradox of the type that Bertrand Russell conceived in 1901 to illustrate a contradiction in Gottlob Frege’s “naïve set theory”, a critique which led to further development of the concept of sets as mathematical objects.

As part of Frege’s set theory, for any criterion or rule, a set could exist for which the test of whether or not something is a member of that set is whether or not it meets the criterion or rule – if it does, it’s in the set, if it doesn’t, it’s not. So for example, we could have the set of living things. The test for membership of this test is whether or not the thing in question is a living thing (so people, plants, and zombies are members of the set; rocks and brooms are not).

In essence, Russell’s paradox is as follows: consider a set (let’s call it A) that contains all the sets, and only the sets, that are not members of themselves. Is A a member of itself? Let’s assume that A is not a member of itself. Well, then, because all the sets that are not members of themselves are members of A, A must be a member of itself. But, if A is a member of itself, then, by definition, it is not a member of itself. A is like Mr Jones: just as Mr Jones doesn’t shave himself so therefore shaves himself, A must be a member of itself, but it cannot be a member of itself.

If you’re thinking “well that’s great Anna, but that’s not going to help me in my day-to-day life,” consider the liar paradox. The liar paradox is a statement to which a Boolean truth value cannot be applied (it cannot be said to be true, and it cannot be said to be false). The liar paradox is as follows: “this statement is false.” If the statement is true, then it is false, but if it false, then it must be true.

If that sounds strange, a more concrete example of the liar paradox which may make it clearer is illustrated in the following example. Imagine the ultimate lie-detector test: one that is 100% accurate in detecting lies, and that sends a fatal zap of electricity to the person strapped to it if it that person utters a lie. So, do you get fried if you’re strapped up to the machine and say “this statement is a lie”?

The liar paradox is one of the problems that led to the development of logical systems with more than the two truthvalues of Boolean algebra (for example, the addition of a third value that means, in some systems “both true and false” and in others “neither true nor false”).

The liar paradox can also destroy evil robots – perhaps. In a 1960s Star Trek episode ‘I, Mudd’, Captain Kirk and another character, Harry Mudd, destroy an android who has turned on his masters by utilising the liar paradox. Mudd says that he is lying, and Kirk says that everything that Mudd says is a lie, which eventually short-circuits the attacking android. But be warned: when Lisa tried this approach on crazed robots 30 years later in ‘Itchy and Scratchy Land’ it failed miserably. So, pick your evil robot, and pick your logical paradox.

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