The History of Mathematical Logic

(vastly abbreviated and horribly simpliied) Michaa l Walicki 1997 The term \logic" may be, very roughly and vaguely, associated with something like \correct thinking". Aristotle deened a syllogism as \discourse in which, certain things being stated something other than what is stated follows of necessity from their being so." And, in fact, this intuition not only lies at its origin, ca. 500 BC, but has been the main force motivating its development since that time until the last century. There was a medieval tradition according to which the Greek philosopher Parmenides (5th century BC) invented logic while living on a rock in Egypt. The story is pure legend, but it does reeect the fact that Parmenides was the rst philosopher to use an extended argument for his views, rather than merely proposing a vision of reality. But using arguments is not the same as studying them, and Parmenides never systematically formulated or studied principles of argumentation in their own right. Indeed, there is no evidence that he was even aware of the implicit rules of inference used in presenting his doctrine. Perhaps Parmenides' use of argument was inspired by the practice of early Greek mathematics among the Pythagoreans. Thus it is signiicant that Parmenides is reported to have had a Pythagorean teacher. But the history of Pythagoreanism in this early period is shrouded in mystery, and it is hard to separate fact from legend. We will sketch the development of logic along the three axes which reeect the three main domains of the eld. 1. The foremost is the interest in correctness of reasoning which involves study of correct arguments, their form or pattern and he possibilities of manipulating such forms in order to arrive at new correct arguments. The other two aspects are very intimately connected with this one. 2. In order to construct valid forms of arguments one has to know what such forms can be built from, that is, determine the ultimate \building blocks". In particular, one has to ask the questions about the meaning of such building blocks, of various terms and categories of terms and, furthermore, of their combinations. 3. Finally, there is the question of how to represent these patterns. Although apparently of secondary importance, it is the answer to this question which can be, to a high degree, considered the beginning of modern mathematical logic. The rst three sections sketch the development along …