![]() Each historical source in the module introduces its own signs and symbols for logical thought, and the exercises examine how each new source contributes to the leitmotif. The organizing principle ( leitmotif) of the project is a discussion of the possible equivalence of Chrysippus’s verbal rules of deduction. Chrysippus stated verbally several rules of inference that are commonplace in modern discrete mathematics textbooks, while the symbolic logic needed to show their equivalence is essentially that of Russell-Whitehead. The most important selections are those from the ancient Greek philosopher Chrysippus and the modern writers Russell and Whitehead. The project offers excerpts from several sources, and not all sections need be covered in detail. Left to right: Logicians Gottlob Frege, Bertrand Russell, and Alfred North Whitehead (Sources: Convergence Portrait Gallery (Frege, Whitehead), Wikimedia Commons (Russell)) Covered in its entirety, this curricular module may take up to five weeks, although there are shorter paths through the material.įigure 2. Be sure to draw comparisons from one historical source to another, and follow the progress to what today appears rather abruptly in most textbooks on discrete mathematics, namely the truth table for an “if-then” statement. The instructor may wish to lead class discussion on certain excerpts from the original sources, and assign certain problems as out-of-class exercises. An examination of various rules of inference throughout history reveals a commonality of deductive thought, encapsulated today in the schemata of Ludwig Wittgenstein and truth tables of Emil Post. Particular emphasis is placed on the logic of deduction as expressed today in an “if-then” statement. Topics include propositional logic, truth tables, and the truth table of a logical implication. ![]() This project is designed for an introductory course in discrete or finite mathematics that covers elementary symbolic logic. For more projects, see Primary Historical Sources in the Classroom: Discrete Mathematics and Computer Science. Our primary source project module for students is part of a larger collection published in Convergence, and an entire introductory discrete mathematics or computer science course can be taught from various combinations selected from these projects. The comprehensive "Notes to the Instructor" presented next are also appended to the project itself. Our project, Deduction through the Ages: A History of Truth, is ready for students, and the Latex source is also available for instructors who may wish to modify the project for students. The reader is invited to embark on a journey from verbal rules of deduction stated by the ancient Greeks to a modern-day discussion of these rules via truth tables. ![]() Wittgenstein dubbed these tables “schemata,” while Post called them “truth tables,” the term by which they are known in present-day textbooks. A more streamlined notation to reduce questions in propositional logic to calculation is offered from the works of Ludwig Wittgenstein (1889–1951) and Emil Post (1897–1954), who independently introduced tabular formats to display the truth values of a (compound) proposition. The definitions of Russell and Whitehead can be used to settle the logical equivalence of Chrysippus’s rules. Whitehead (1861–1947) in Principia Mathematica, is in terms of a certain “or” statement of equivalent deductive power to Chrysippus’s first argument form, which he gave in terms of an “if-then” statement. The presently accepted definition of an implication, stated by Bertrand Russell (1872–1970) and Alfred N. Student exercises reveal that four of Chrysippus’s five rules can be written in terms of Frege’s condition stroke, which offers insight into the possible equivalence of these four rules. This is followed by a reading from Gottlob Frege’s (1848–1925) The Basic Laws of Arithmetic, in which a totally new notation, the concept-script, is introduced. An initial attempt to render verbal statements to a symbolic form is first studied from George Boole’s (1815–1864) treatise An Investigation of the Laws of Thought. The tools used to discuss these equivalences are the historical sources themselves. Beginning with five verbal argument forms attributed to the ancient Greek philosopher Chrysippus (280–206 BCE), the project examines the possible equivalence of these argument forms to a standard “if-then” statement (implication). ![]() This curricular module offers excerpts from selected primary historical sources from antiquity to the twentieth century that illuminate the thought process of deduction encapsulated today as an implication in propositional logic. (Source: Wikimedia Commons, courtesy of Eric Gaba) This bust of Chrysippus of Soli, owned by the Louvre Museum, is believed to be a Second Century Roman copy of a Greek original.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |