# Seminar in Semantics / Philosophy of Language

or: What Philosophers and Linguists Can Learn From Theoretical Computer Science But Didn't Know To Ask

This course is co-taught by Chris Barker and Jim Pryor. Linguistics calls it "G61.3340-002" and Philosophy calls it "G83.2296-001." The seminar meets on Mondays from 4-6, in the Linguistics building at 10 Washington Place, in room 104 (back of the first floor). One student session will be held every Wednesday from 3-4 on the fourth floor at 10 Washington Place.

## Announcements

Older Announcements

## Lambda Evaluator

There is also now a library of lambda-calculus arithmetical and list operations, some relatively advanced.

## Lecture Notes and Assignments

(13 Sept) Lecture notes for Week1; Assignment1.

Topics: Applications, including Damn; Basics of Lambda Calculus; Comparing Different Languages

(20 Sept) Lecture notes for Week2; Assignment2.

Topics: Reduction and Convertibility; Combinators; Evaluation Strategies and Normalization; Decidability; Lists and Numbers

(27 Sept) Lecture notes for Week3; Assignment3; an evaluator with the definitions used for homework 3 preloaded is available at assignment 3 evaluator.

Topics: Evaluation Order; Recursion with Fixed Point Combinators

(4 Oct) Lecture notes for Week4; Assignment4.

Topics: More on Fixed Points; Sets; Aborting List Traversals; Implementing Trees

(18 Oct, 25 Oct) Lecture notes for Week5 and Week6; Assignment5.

Topics: Types, Polymorphism, Unit and Bottom

(1 Nov) Lecture notes for Week7; Assignment6.

(8 Nov) Lecture notes for Week8.

(15 Nov) Lecture notes for Week9; Assignment7. Everyone auditing in the class is encouraged to do this assignment, or at least work through the substantial "hints".

(22 Nov) Lecture notes for Week10

Topics: Calculator Improvements, including mutation

(30 Nov) Lecture notes for Week11; Assignment8.

(6 Dec) Lecture notes for Week12; Assignment9.

(13 Dec) Lecture notes for Week13; Assignment10.

Topics: CPS and Continuation Operators; Curry-Howard

Topics: Version 4 lists, Monads in Category Theory

## Scheme and OCaml

See below for how to get the programming languages running on your computer.

There's lots of links here already to tutorials and encyclopedia entries about many of the notions we'll be dealing with.

## Course Overview

The goal of this seminar is to introduce concepts and techniques from theoretical computer science and show how they can provide insight into established philosophical and linguistic problems.

This is not a seminar about any particular technology or software. Rather, it's about a variety of conceptual/logical ideas that have been developed in computer science and that linguists and philosophers ought to know, or may already be unknowingly trying to reinvent.

Philosphers and linguists tend to reuse the same familiar tools in ever more (sometime spectacularly) creative ways. But when your only hammer is classical logic, every problem looks like modus ponens. In contrast, computer scientists have invested considerable ingenuity in studying tool design, and have made remarkable progress.

"Why shouldn't I reinvent some idea X for myself? It's intellectually rewarding!" Yes it is, but it also takes time you might have better spent elsewhere. After all, you can get anywhere you want to go by walking, but you can accomplish more with a combination of walking and strategic subway rides.

More importantly, the idiosyncrasies of your particular implementation may obscure what's fundamental to the idea you're working with. Your implementation may be buggy in corner cases you didn't think of; it may be incomplete and not trivial to generalize; its connection to existing literature and neighboring issues may go unnoticed. For all these reasons you're better off understanding the state of the art.

The theoretical tools we'll be introducing aren't very familiar to everyday programmers, but they are prominent in academic computer science, especially in the fields of functional programming and type theory.

Of necessity, this course will lay a lot of logical groundwork. But throughout we'll be aiming to mix that groundwork with real cases in our home subjects where these tools play central roles. Our aim for the course is to enable you to make these tools your own; to have enough understanding of them to recognize them in use, use them yourself at least in simple ways, and to be able to read more about them when appropriate.

Once we get up and running, the central focii of the course will be continuations, types, and monads. One of the on-going themes will concern evaluation order and issues about how computations (inferences, derivations) unfold in (for instance) time. The key analytic technique is to form a static, order-independent model of a dynamic process. We'll be discussing this in much more detail as the course proceeds.

The logical systems we'll be looking at include:

• the pure/untyped lambda calculus
• combinatorial logic
• the simply-typed lambda calculus
• polymorphic types with System F
• some discussion of dependent types
• if time permits, "indeterministic" or "preemptively parallel" computation and linear logic

## Who Can Participate?

The course will not presume previous experience with programming. We will, however, discuss concepts embodied in specific programming languages, and we will encourage experimentation with running, modifying, and writing computer programs.

The course will not presume lots of mathematical or logical background, either. However, it will demand a certain amount of comfort working with such material; as a result, it will not be especially well-suited to be a first graduate-level course in formal semantics or philosophy of language. If you have concerns about your background, come discuss them with us.

This class will count as satisfying the logic requirement for Philosophy PhD students; however if this would be your first or only serious engagement with graduate-level formal work you should consider carefully, and must discuss with us, (1) whether you'll be adequately prepared for this course, and (2) whether you'd be better served by taking a logic course (at a neighboring department, or at NYU next year) with a more canonical syllabus.

Faculty and students from outside of NYU Linguistics and Philosophy are welcome to audit, to the extent that this coheres well with the needs of our local students.

During the course, we'll be encouraging you to try out various things in Scheme and Caml, which are prominent functional programming languages. We'll explain what that means during the course.

• Scheme is one of two major dialects of Lisp, which is a large family of programming languages. Scheme is the more clean and minimalistic dialect, and is what's mostly used in academic circles. Scheme itself has umpteen different "implementations", which share most of their fundamentals, but have slightly different extensions and interact with the operating system differently. One major implementation used to be called PLT Scheme, and has just in the past few weeks changed their name to Racket. This is what we recommend you use. (If you're already using or comfortable with another Scheme implementation, though, there's no compelling reason to switch.)

Racket stands to Scheme in something like the relation Firefox stands to HTML.

• Caml is one of two major dialects of ML, which is another large family of programming languages. Caml has only one active implementation, OCaml, developed by the INRIA academic group in France.

• Those of you with some programming background may have encountered a third prominent functional programming language, Haskell. This is also used a lot in the academic contexts we'll be working through. Its surface syntax differs from Caml, and there are various important things one can do in each of Haskell and Caml that one can't (or can't as easily) do in the other. But these languages also have a lot in common, and if you're familiar with one of them, it's not difficult to move between it and the other.

Family tree of functional programming languages

Translating between OCaml Scheme and Haskell

## What is Functional Programming?

Here's a survey conducted at Microsoft asking programmers what they understand "functional programming" to be. Don't take their responses to be authoritative... this is a just a "man in the street" (seat?) poll.

• An Introduction to Lambda Calculi for Computer Scientists, by Chris Hankin, currently $17 on Amazon. • (Another good book covering the same ground as the Hankin book, but more thoroughly, and in a more mathematical style, is Lambda-Calculus and Combinators: an Introduction, by J. Roger Hindley and Jonathan P. Seldin, currently$52 on Amazon. If you choose to read both the Hankin book and this book, you'll notice the authors made some different terminological/notational choices. At first, this makes comprehension slightly slower, but in the long run it's helpful because it makes the arbitrariness of those choices more salient.)
• (Another good book, covering some of the same ground as the previous two, but also delving much deeper into typed lambda calculi, is Types and Programming Languages, by Benjamin Pierce, currently $61 on Amazon. This book has many examples in OCaml.) • The Little Schemer, Fourth Edition, by Daniel P. Friedman and Matthias Felleisen, currently$23 on Amazon. This is a classic text introducing the gentle art of programming, using the functional programming language Scheme. Many people love this book, but it has an unusual dialog format that is not to everybody's taste. Of particular interest for this course is the explanation of the Y combinator, available as a free sample chapter at the MIT Press web page for the book.
• The Seasoned Schemer, also by Daniel P. Friedman and Matthias Felleisen, currently $28 on Amazon • The Little MLer, by Matthias Felleisen and Daniel P. Friedman, currently$27 on Amazon. This covers some of the same introductory ground as The Little Schemer, but this time in ML. It uses another dialect of ML (called SML), instead of OCaml, but there are only superficial syntactic differences between these languages. Here's a translation manual between them.