Course Descriptions

'Going Higher-Order'

This seminar will take careful look at higher-order logics and their applications in philosophy of mathematics, philosophy of language, and metaphysics.  We will start with traditional questions about second-order logic, such as whether it deserves the status of logic at all, or whether it is a variant of set theory. We will then turn our attention to more general higher-order logics, starting with the simple theory of types.  We will explore a range of modern type theories, including higher-order typed lambda-calculi, constructive type theory, and a little bit of modern homotopy type theory.  You will see from a few examples that the world of higher-order logics is rich, complex, and somewhat unruly.  There are a lot of things that can count as higher-order logic. Applications can be chosen according to participants’ interests. One likely application is the role of second or higher-order logic in philosophy of mathematics, where a range of structuralist positions rely on higher-order logic. We will also ask whether weaker logics can establish adequate uniqueness results for the natural numbers, or for sets.  In philosophy of language, we can examine the role of types in formal semantics, the role of type shifting, and what background logic is required for type-shifting systems.  Finally, we can examine some of the many recent applications of higher-order logic in metaphysics, such as work of Bacon, Dorr, Fritz, Goodman, Williamson, and others.

A sample of potential readings includes:

1.Bacon, "A Case for Higher-Order Metaphysics," A Philosophical Introduction to

Higher-Order Logics

2. Barendregt, Dekkers, and Statman, Lambda Calculus with Types

3. Barker, "A Gentle Introduction to Type Logical Grammar, the Curry-Howard

Correspondence, and Cut Elimination"

4. van Benthem, Language in Action

5. Boolos, “To Be is to Be a Value of a Variable (or to Be Some Values of Some

Variable)”

6. Button and Walsh, Philosophy and Model Theory

7. Dorr, "To Be F Is To Be G"

8.Gamut, Logic, Language, and Meaning

9. Jané, “Higher-Order Logic Reconsidered”

10. Linnebo and Rayo, "Hierarchies Ontological and Ideological"

11. Maddy and Väänänen, Philosophical Uses of Categoriticy Arguments

12. Meadows, “What Can a Categoricity Theorem Tell Us?”

13. Parsons, "The Uniqueness of the Natura Numbers"

14. Partee and Rooth, "Generalized Conjunction and Type Ambiguity"

15. Quine, Philosophy of Logic

16. Shapiro, “Higher-Order Logic,” Foundations without Foundationalism: The Case

for Second-Order Logic

17. The Univalent Foundations Program, Homotopy Type Theory

18. Väänänen, “Second Order Logic and Set Theory”

19. Williamson, “Everything,” Modal Logic as Metaphysics

20. Winter, Flexibility Principles in Boolean Semantics