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This course covers the development of analytic philosophy in its early stage: from Frege's logical revolution to logical positivism at its highest point, before critiques by Quine and Wittgenstein fundamentally transformed the analytic tradition in the early 1950s. We begin with Frege's attempt to address traditional metaphysical and epistemological problems about numbers through the logical analysis of language and then trace how this project - the analysis of language with formal logic to resolve philosophical questions - develops through Russell, Wittgenstein, and the Vienna Circle, particularly Carnap. We then consider how Gödel and Tarski's logical work forces a reinterpretation of the project and conclude with Carnap and Quine's debate "on what there is".

(Graduate students register for 241F.)

Undergrad lecture.

An introduction to the concepts and techniques used in mathematical logic, focusing on propositional, modal, and predicate logic. Highlights connections with philosophy, mathematics, computer science, linguistics, and neighboring fields.

In this course we will go through some of the seminal ideas, constructions, and results from modern logic, focusing especially on classical first-order ("predicate") logic. After introducing general ideas of induction and recursion, we will study a bit of elementary (axiomatic) set theory before then covering basic definability theory, viz. assessing the theoretical limits of what can and cannot be expressed in a first-order language. The centerpiece result of the class is the completeness - and closely related compactness - of first-order logic, a result with a number of momentous consequences, some useful, some philosophically puzzling. We will then study a connection with game theory, whereby a certain type of game characterizes precisely the expressive power of first-order logic. Further topics may include: the 0-1 law in finite model theory, second-order logic, and the algebraic approach to logic. Prerequisite: 150 or consent of instructor.

Graduate students enroll in 251D.

Kurt Goedel's ground-breaking Incompleteness Theorems demonstrate fundamental limits on formal mathematical reasoning. In particular, the First Incompleteness Theorem says, roughly, that for any reasonable theory of the natural numbers there are statements in the language that are neither provable nor refutable in that theory. In this course, we will explore the expressive power of different axiomatizations of number theory, on our path to proving the Incompleteness Theorems. This study entails an exploration of models of computation, and the power and limitations of what is computable, leading to an introduction to elementary recursion theory. At the conclusion of the course, we will discuss technical and philosophical repercussions of these results. Prerequisite: 151/251.

(Graduate students register for 254.) Syntax and semantics of modal logic and its basic theory: including expressive power, axiomatic completeness, correspondence, and complexity. Applications to classical and recent topics in philosophy, computer science, mathematics, linguistics, and game theory. Prerequisite: 150 or preferably 151.

This year's topic is Non-Classical Logic. May be repeated for credit.