PHIL 22401/32401 Modern Logic and the Structure of Knowledge
In this course, we will examine the various ways in which the concepts and techniques of modern mathematical logic can be utilized to investigate the structure of knowledge. Many of the most well-known results of mathematical logic, such as the incompleteness theorems of Gödel and the Löwenheim-Skolem theorem, illustrate the fundamental limitations of formal systems of logic to fully capture the structure of the semantic models in which truth and validity are assessed. Some philosophers have argued that these results have profound epistemological implications, for instance, that they can be used to ground skeptical claims to the effect that there must be truths that logic and mathematics are powerless to prove. One of the aims of this course is to assess the legitimacy of these epistemological claims. In addition, we will explore the extent to which the central results of mathematical logic can be extended so as to apply to systems of inductive logic, and examine what forms of inductive skepticism may emerge as a result. We will, for example, discuss the epistemological implications of Putnam's diagonalization argument, which shows that, for any Bayesian theory of confirmation based on a definable prior, there must exist hypotheses which, if true, can never be confirmed. (B) (II)