12/15/06

You might try Wikipedia:History of Logic (but just read the sections on Traditional Logic and The Advent of Modern Logic)

- Aristotle's Logic
- George Boole
- Gottlob Frege,
- Gottlob Frege
- Kurt Godel
- Gottfried Liebniz
- Augustus de Morgan
- C. S. Peirce
- Bertrand Russell
- Alan Turing

In brief...

Aristotle identified and classified certain kinds of arguments ('syllogisms') as being valid or not, and he used variables to pick out kinds of arguments. For example, Aristotle went beyond merely identifying 'All swans are white; all white things are colored; therefore, all swans are colored' as being a particular valid argument, to the wider assertion that all arguments with the form 'All As are B; all Bs are C; therefore, all As are C' are valid.

Leibniz sought to reduce all reasoning and dispute to calculation (via the theory that complex ideas were compounds of simple ideas, and that relations among ideas could be established by techniques similar to arithmetic (such as combining and evaluating)).

Boole and de Morgan extended this composition-decomposition and calculation to what we would now call propositional logic (just what propositional logic is you will learn in the present course).

Frege founded modern predicate logic (perhaps with notational assistance from the independent work of Peirce). However the young Bertrand Russell destroyed Frege's system by finding a paradox in it. Then, in turn, Godel, with some of the most profound mathematical results of the 20th century showed that the axiomatic systems in the style of Russell would not quite do what was initially expected of them.

Many and various others produced viable (smaller scale) logical systems. In particular we favor the use here of the 'Natural Deduction' of Gerhardt Gentzen's Sequent Calculus

The formalization of logical deduction, especially as it has been developed by Frege, Russell, and Hilbert, is rather far removed from the forms of deduction used in practice in mathematical proofs. . . . In contrast I intended first to set up a formal system which comes as close as possible to actual reasoning. The result was a calculus of natural deduction (NJ for intuitionist, NK for classical predicate logic). [Gentzen:

Investigations into logical deduction]