First you had to work out which forms of reasoning made sense, and which didn't. If someone says that today's Prime Minister is named John Major, and then picks anybody from the street who happens to share that unfortunate name, and marches that person and all other Major name-alikes down to 10 Downing Street to take over, you would suspect that something had gone seriously awry. But if the reasoning is, say, that all future prime ministers must give the appearance of not being xenophobic maniacs, and then you see Michael Portillo give his Tebbit imitation, and so conclude, mercifully, that Mr Portillo cannot be a future Prime Minister, then you have a sound reasoning pattern of the sort that Aristotle labelled a syllogism.
Working through all the possible three-part syllogisms, Aristotle observed, there are 256 possibilities, only some of which are valid (leading from true premises to true conclusions). When you're arguing with someone, or just trying to work something out, you look at the list, see which syllogism matches the thoughts you're expressing, and then check if that syllogism is valid or not.
It sounds a roundabout way of working but it's an excellent way to pick up hidden assumptions. It's also convenient for highlighting new knowledge: you scoop through the universe, loading up your syllogisms with interesting data, and listening to see which ones clink accurately into place. When Warren Buffet, for example, says that you should invest in a company any idiot can run, because one day any idiot will, then you have a flight path directing you to hunt out a suitable "idiot-proof" company to invest in.
Unfortunately the syllogisms seemed such an impressive way of giving authority to an argument, that they largely ended up being used as a scaffolding for results that had already been worked out, most notably with Thomas Aquinas's 13th-century backing of Catholic theology. Only from the late 19th century was a fresh approach taken, with the development of a symbolic logic that could transcribe thoughts or mathematics with much finer detail.
The results seem odd, like a silly bunch of squiggles, at first glance. But the marks made by modern logicians are simply a compact way of notating such assertions as "For any x, if x exists, and if x is the king of France, and if x is bald, then the king of France is bald". (An example used by Bertrand Russell in his analysis of the logic of existence.) It's so pedantic that you can see why working scientists almost never use formal logic to develop new ideas.
But the pedantry reduces what's being said to a clean skeleton core, written in a form that even the lowest-IQ electronic circuits can follow. Computers positively thrive on such exact switching, which is why their early development depended, crucially, on the exploration of Peano, Russell and other logicians early in this century.
Since then logic has developed in many ways. One way of deciding if Ronald Reagan really was responsible for winning the Cold War by raising Pentagon budgets is to explore the logic of "possible worlds", where you run the historical tape forward several times. In one version you would look at the world where an unknown named Lebed led a coup against Gorbachev and matched Reagan's spending; in another, there would be the world where Reagan's nuclear policies led to the war which destroyed 99 per cent of all earth life in the Great Cataclysm of 1985, etc.
A different approach takes on the logic of ordinary reasoning. Don Norman has been especially good at showing how there's one sort of logic, call it engineering logic, where all the dials for a stove's burners should be laid out in a neat straight line, yet there's a second sort of logic, call it "familiarity" logic, where what we really want are for the dials to be arranged in a miniature copy of the burners' layout so we can remember which ones to turn.
None of this would have surprised Aristotle. Along with his writings on syllogisms he explored ordinary reasoning patterns at length: the egocentric world of young people, the prejudice-flatterings of politicans, the reflex for the familiar and comfortable, are all in there. Without realising it, he wasn't that far from the computer. The huge tableau of 256 possible syllogisms he outlined is a crude early precursor of a microprocessor with 256K units of memory. The number's the same, for they're both built up from the same multiplied sequences of rigid switchings and sub-switchings - which is what his logic, with its hunt for an ideal knowledge machine, automatically chugged out those 2,300 years ago.
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