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Boolean Logic

  The whole idea of logic (that sentences have a form that differs from their content) originated with the ancient Greeks, whose investigation of the subject culminated in the work of Aristotle (384-322  BCE). Even though his work was severely limited, only insignificant additions were made to it for 2,000 years, despite the fact that inferences outside Aristotelian logic were used all the time.

Gottfried Leibniz (1646 - 1716) was the first great modern mathematician who was interested in logic. He made the first attempts to find a systematic notation for logic. He failed to follow his work through, and left it unfinished when he died. In the end, it was left to , George Boole (1815 - 1864) and , Augustus de Morgan (1806 - 1871) to come up with a fully worked-out notational system.

This notation (in its modern form) can perhaps best be demonstrated by breaking down a complex sentence into its symbolic form. Consider the sentence, ‘If tomatoes or beans mean that the food is nasty, then the fact that the food is nice means that there are no tomatoes and no beans.’ (The reason that this sentence reads rather fussily is that English is quite cavalier with logical pedantry, so one English sentence can express more than one logical form.) There are four concepts which give the content of this sentence: ‘tomatoes’, ‘beans’, ‘food’ and ‘nice’ (‘nasty’ is the same as ‘not nice’). Nice describes the food. It acts as a function from the food to the pair true/false. Thus we write N(F) for ‘the food is nice’. We put T for ‘tomatoes’, B for ‘beans’. Now, omitting words which serve no logical purpose, we have, ‘If T or B mean not N(F), then N(F) means that not T and not B.’ The sentence is of the form ‘If …, then …’, which is expressed by writing an arrow from the condition to the consequent, which are bracketed, to give ‘(T or B mean not N(F)) → (N(F) means not T and not B)’. Each bracketed clause is then analysed. ‘T or B mean …’ is the same as ‘If T or B then …’; the second clause is similar. The sentence is now ‘((T or B) → (not N(F)) → (N(F) → (not T and not B))’. The words ‘not’, ‘or’ and ‘and’ all have their own logical symbols (~, v and ^ respectively), so the final version of the sentence is ‘((TvB) → (~ N(F)) → (N(F) → ((~T)) ^ (~ B))’.

The point of all this is that the truth of such a sentence depends only on its logical form and not on its content; any sentence with the same logical form is true under the same circumstances (the one above is always true). The symbols make the logical form clearer, and its truth or fallacy can be more easily seen. Moreover, it can be determined in a purely mechanical manner whether the sentence is true or false, once the truth values of the symbols T, B and N(F) are known. Such logic is important, because it forms the basis of the way computers work. In a computer, the value true (or 1) is represented by a current flowing, and the value false (or 0) is represented by no current flowing. Circuits can be designed that transform the input values of A and B to an output value of A ^ B, AvB, A → B or ~A. The parts of a computer which perform calculations consist of very large numbers of such circuits. SMcL

See also symbolic logic.



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