Logic

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)

In what sense, if any, do the laws of logic express necessary truths? The course will consider four fateful junctures in the history of philosophy at which this question received influential treatment: (1) Descartes on the creation of the eternal truths, (2) Kant's re-conception of the nature of logic and introduction of the distinction between pure general and transcendental logic, (3) Frege's rejection of the possibility of logical aliens, and (4) Wittgenstein's early and later responses to Frege. We will closely read short selections from Descartes, Kant, Frege, and Wittgenstein, and ponder their significance for contemporary philosophical reflection by studying some classic pieces of secondary literature on these figures, along with related pieces of philosophical writing by Jocelyn Benoist, Matt Boyle, Cora Diamond, Peter Geach, John MacFarlane, Adrian Moore, Hilary Putnam, Thomas Ricketts, Sebastian Rödl, Richard Rorty, Peter Sullivan, Barry Stroud, Clinton Tolley, and Charles Travis. (V)

An introduction to the concepts and principles of symbolic logic. We learn the syntax and semantics of truth-functional and first-order quantificational logic, and apply the resultant conceptual framework to the analysis of valid and invalid arguments, the structure of formal languages, and logical relations among sentences of ordinary discourse. Occasionally we will venture into topics in philosophy of language and philosophical logic, but our primary focus is on acquiring a facility with symbolic logic as such.

In this course, we will prove the soundness and completeness of deductive systems for both sentential and first-order predicate logic. We will also establish related results in elementary model theory, such as the compactness theorem for first-order logic, the Lӧwenheim-Skolem theorem and Lindstrӧm's theorem. (B) (II)

An introduction to the techniques of modern logic. These include the representation of arguments in symbolic notation, and the systematic manipulation of these representations in order to show the validity of arguments. Regular homework assignments, in class test, and final examination.

Key contemporary debates in the philosophical literature often rely on formal tools and techniques that go beyond the material taught in an introductory logic class. A robust understanding of these debates---and, accordingly, the ability to meaningfully engage with a good deal of contemporary philosophy---requires a basic grasp of extensions of standard logic such as modal logic, multi-valued logic, and supervaluations, as well as an appreciation of the key philosophical virtues and vices of these extensions. The goal of this course is to provide students with the required logic literacy. While some basic metalogical results will come into view as the quarter proceeds, the course will primarily focus on the scope (and, perhaps, the limits) of logic as an important tool for philosophical theorizing. (B)

In this course, we will prove the soundness and completeness of deductive systems for both sentential and first-order logic. We will also establish related results in elementary model theory, such as the compactness theorem for first-order logic, the Lowenheim-Skolem theorem and Lindstrom’s theorem. (B) (II)

Plato honors Parmenides with the title "father Parmenides", presumably for being the founder of philosophy as the "logical" study of being and thinking. In this course we shall discuss the struggle of ancient and modern philosophers to come to terms with this powerful heritage — in particular, we shall focus on the elaboration, reception and criticism of Parmenides' theses that being and thinking are the same, and that talk of negation or falsity is incoherent or empty. Among the philosophers whose work we shall discuss are Plato, Aristotle, Frege, Russell, and Wittgenstein.

Logic, traditionally conceived, aims to study the laws of thought. This makes it seem as though logicians share a concern with psychologists; but in fact, the proposal that logical laws can be studied empirically - also known as psychologism - came under attack by philosophers throughout the 19th and 20th centuries. The idea that logic is presupposed by all thinking was taken to disallow its empirical study, and to render the methods of psychology irrelevant to logic. For most of the 20th century, this philosophical position made sense to psychologists; at the very least, they did not seriously raise the question whether thinkers are actually rational in the sense prescribed by logic. This assumption has gradually been rejected; since the 70s, human rationality has become a central object of study for psychologists, with a focus on the defective logical patterns of thought that humans tend to exhibit. At the same time, in philosophy, the collapse of the analytic-synthetic distinction and the naturalistic turn gave way to a new conception of the relation between logic and psychology. Nowadays several fruitful research programs in the psychological study of reasoning and rationality exist side by side, and alongside them, many philosophers and logicians make room for psychological considerations. In reaction to the new sciences of rationality and to the new psychologism in logic, new forms of antipsychologism have also emerged; we will evaluate several such arguments and ask how psychologists and psychologically minded philosophers cope with them. We will conclude our inquiry with a look at the contemporary debate regarding the normative status of logic and its relation to thought.

Since Russell's discovery of the inconsistency of Frege's foundation for mathematics, much of logic has resolved around the question of to what extent we can or cannot prove the consistency of the basic principles with which we reason. This course will explore two main efforts in this direction. We will first look at proof-theoretic efforts towards demonstrating the consistency of various foundational systems, discussing the virtues and limitations of this approach. We will then closely examine Godel's theorems, which are famous for demonstrating limits on the extent to which we can formulate consistency proofs. Much has been written on the implications of Godel's theorems, and we will spend some time trying to carefully separate what they really entail from what they do not entail. Assessment will be by regular homework sets. (B) (II)