College Mathematics Journal

Vol48-N4

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THE WORKS OF RAYMOND SMULLYAN Surveyed by Jason Rosenhouse (rosenhjd@jmu.edu), James Madison University, Harrisonburg, VA Aristotle is generally credited with inaugurating the study of formal logic in a work known as the Prior Analytics [1]. After a terse opening paragraph, Aristotle defines his terms in paragraph two: A premise then is a sentence affirming or denying one thing of another. This is either universal or particular or indefinite. By universal I mean the statement that something belongs to all or none of something else; by particular that it belongs to some or not to some or not to all; by indefinite that it does or does not belong, without any mark to show whether it is universal or particular, . . . It would seem that Aristotle also inaugurated the tradition that logical texts should be turgid and difficult. In the ensuing centuries, many scholars developed and enhanced Aristotle's basic ideas. By the late Middle Ages, the study of formal logic had reached an impressive state of refinement. Not everyone saw this as a positive development. Writing in 1644, John Milton lamented the pointlessness of it all [3]: [Universities] present their young unmatriculated Novices at first comming with the most intellective abstractions of Logick and Metaphysicks: So that they having but newly left those Grammatick flats and shallows where they stuck unreasonably to learn a few words with lamentable construction, and now on the sudden transported under another climate to be tost and turmoil'd with their unballasted wits in fadomless and unquiet deeps of controversie, do for the most part grow into hatred and contempt of Learning, mockt and deluded all this while with ragged Notions and Babblements, while they expected worthy and delight- ful knowledge. Mostly absent from these centuries of work was any sense that formal logic was a subject of particular interest to mathematicians. The mathematical turn in logic occurred largely in the nineteenth and twentieth centuries, pioneered by Frege and taken to a high art by Bertrand Russell. Russell believed that the grammatical structure of a sentence obscured its deeper, logical meaning [2]. Thus, a sentence as simple as, "My cat is furry," might be rendered symbolically as ( ∃x ) ( J x ∧ ( ∀y ) ( [ J y → ( y = x ) ] ∧ F x )) . Translated back into English, this says, "There exists an x such that x is Jason's cat, and if y is anything else that is Jason's cat then y is the same as x and x is furry." This is what awaits you upon undertaking the study of logic. Works on mathe- matical logic frequently contain simple thoughts expressed with daunting amounts of notation. Delve instead into works on philosophical logic, and you will find that fun- damental notions like "proposition" and "truth" are far murkier than what gets taught in introduction to proof courses. Regardless, you will come away with the impression that logic is for ivory tower aesthetes with no sense of fun or whimsy. Which is a pity, since working out the logical consequences of a set of premises can be surprisingly enjoyable. http://dx.doi.org/10.4169/college.math.j.48.4.302 302 © THE MATHEMATICAL ASSOCIATION OF AMERICA

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