Cornell Math - MATH 788, Fall 1999

MATH 788 — Fall 1999
Topics in Applied Logic: Nonclassical Logics

Instructor: Richard Platek

Time: M 4:30-6:30

Room: Malott 206

By "classical logic" we mean customary two valued logic which uses truth functions to model propositional connectives. There are a number of alternative logics (e.g., modal logic, relevance logic, paraconsistent logic, quantum logic, temporal logic, situational logic, etc.) which have been introduced to cover cases which are inadequately handled by classical logic. In this course we will survey some of these. Our interest is to show how logics introduced for basically philosophical reasons have been found to be extremely useful as practical tools. For example, paraconsistent logics, which allow inconsistencies but contain them and donít let them contaminate other statements have been used to analyze Zeno's paradoxes of motion as well as large data bases which might contain conflicting information. Quite a sweep!

For another example consider the formula:

[p & q –> r] –>[(p –> r) v (q –> r)]

which says that if r is a consequence of p and q then it must be a consequence of p or a consequence of q alone. This is a tautology. But it is patently false for any reasonable understanding of "implies" (this is one of C. I. Lewis' infamous paradoxes of material implication introduced almost a century ago.) It is doubtful if any reasoner (e.g., a mathematician) would ever make us of this "Law of Logic" in his/her deductions. If that is the case what is it doing in formalized mathematics and what happens when it and fellow illegal aliens are removed? Come find out.