andrew-bacon

PHIL 451: Higher-order Logic

Time and place

Thursdays, 12:30-1:50. TTH 108.

Contact

Andrew Bacon

office: 224 STO

email: abacon@usc.edu

in person office hours: Thursdays 2-3 (by appointment)

zoom office hours: Wednesday 9-10 by appointment

Textbook

A Philosophical Introduction to Higher-order Logics. Andrew Bacon. Routledge.

Links

dropbox

signup sheet

guidelines for writing a paper proposal

exercises

Course description

Higher-order languages expand the language of first-order logic by allowing generalizations into grammatical positions other than that of singular terms. They have found numerous applications in philosophy, computer science, mathematics and linguistics. This course introduces students to higher-order languages, their proof and model theory, and their applications.

Objectives: In this course students will learn to be proficient with the $\lambda$ notation; to formulate ideas in higher-order languages and to reason from them; how to build theories stated in higher-order languages and to use models to probe their consequences; about the philosophical applications of higher-order languages.

Prerequisites

PHIL 350, PHIL 222. The prerequisites may be waived if the student can demonstrate some relevant background, which may include relevant courses in mathematics, linguistics or computer science.

Grading

• Participation: 20%
  • Contribute to class discussion. Be prepared to answer questions if called upon, and to raise your hand if you do not understand something. You are collectively responsible for setting the pace of the class, and clarificatory questions. 5%
  • Every week, three students will present a solution to an exercise of their choice on the board. The signup sheet for these presentations can be found here. Once you have signed up for an exercise please do not sign up again until we have made a full cycle of the class. 15%.

• Exercises 50%

  • Weekly exercises will be graded on a pass fail basis, and must be submitted by Tuesdays 9:30, before we discuss them in class.

    A pass does not necessarily mean your answer is correct: it means that you have engaged seriously with the question, and that you can explain your strategy and any difficulties you encountered. Correct answers will be discussed in class.

  • Exercises must be uploaded to course management software. Submissions normally consist of photos of handwritten solutions; LaTeX or other software for typesetting logical symbols can be used but is not necessary.
  • If you can provide a reason that you will not complete the exercises by the deadline, let me know in advance, and we can discuss an extension.
• Final Project: 5% + 25%

  • Proposal: 5%. Due April 14.

    Submit a detailed proposal of what you intend to do your final project on.

  • Final project: 25%. Due May 1st.

Schedule

Theory

• Week 1. Typed and higher-order languages: what are they, and why are they useful?

  • Informal introduction to types, first and higher-order generalizations, and surrounding philosophical issues.
  • Reading: Chapter 0
• Week 2. Types and typed languages.
  • Types, typed languages, the “concept horse” problem.
  • Reading: Chapter 1.
• Week 3. An informal introduction to abstraction.
  • Definitions by abstraction, $\lambda$-notation, Currying.
• Weeks 4, 5. $\lambda$-languages.
  • $\lambda$-languages, combinators, $\alpha$, $\beta$, and $\eta$-equivalence, translations between $\lambda$-languages and combinatory languages.
  • Reading: Chapter 3.
• Weeks 6. Higher-order languages and higher-order logics.
  • Higher-order languages, reasoning in higher-order theories and logics, inductive definitions in higher-order languages.
  • Reading: Chapters 4 and 5.
• Week 7. Higher-order theories of granularity.
  • Extensionalism, the Boolean theory of propositional individuation, weakenings thereof and philosophical motivations.
  • Reading: Chapter 6.
• Week 8. Models.
  • Models of H, soundness and completeness, philosophical issues surrounding incompleteness.
  • Reading: Chapter 15.

Applications

This part of the course is focuses of applications of higher-order logic in philosophy. The schedule here is subject to change, and may be influenced by students interests.

• Weeks 9, 10. Modal metaphysics.
  • Higher-order logic and its relation to modal logic, discussion of topics in the philosophy of modality.
  • Reading: Chapter 7 and 8. Selection from Williamson Modal Logic as Metaphysics.
• Week 11. General $\lambda$-languages.
  • General $\lambda$-languages and motivations for studying them, Curry typing.
  • Reading: Chapter 9 and 10.
• Weeks 12. The metaphysics and propositions and relations.
  • Philosophical applications to structured theories, the Russell-Myhill paradox, converse relations.
  • Reading: Chapters 11-13.
  • Background reading: Fine, “Neutral Relations”, selections from Dorr, “To be F is to be G”.
• Week 13. Type-theory in natural language semantics.
  • Applications of type-theory in natural language semantics: categorial grammars, type logical semantics, variable free approaches to semantics.
  • Background readings: Selections from Bob Carpenter’s “Type Logical Semantics”, Lambek “The Mathematics of Sentence Structure”, Jacobson “Towards a variable free syntax”,
• Week 14. Higher-order logic and the philosophy of mathematics.
  • We will look at uses of higher-order logic in contemporary philosophy of mathematics, and its historical origins in mathematical foundations.
  • Background readings: We may look at some early work by Frege, Russell and Wittgenstein, as well as contemporary uses in the literature on structuralism, modal approaches to the philosophy of set theory, and more.
• Week 15. Metaethics and philosophy of mind.
  • Higher-order language can be used to clarify questions to do with supervenience, reduction, and identifications in various areas of philosophy. We will look at some applications in metaethics and the philosophy of mind.

Classroom norms

Please come to class ready to learn, and prepared to engage in class discussions. Clarificatory questions are always appreciated, and can be helpful to your classmates. Be constructive in your interactions with other students, and try when possible to help each other out.

Laptops, phones and other distractions must be out of sight during class. I encourage students to take handwritten notes; if you have a special reason to use an electronic device contact me about it.

Statement on academic conduct and support services

Academic Conduct

The University of Southern California is foremost a learning community committed to fostering successful scholars and researchers dedicated to the pursuit of knowledge and the transmission of ideas. Academic misconduct is in contrast to the university’s mission to educate students through a broad array of first-rank academic, professional, and extracurricular programs and includes any act of dishonesty in the submission of academic work (either in draft or final form).

This course will follow the expectations for academic integrity as stated in the USC Student Handbook. All students are expected to submit assignments that are original work and prepared specifically for the course/section in this academic term. You may not submit work written by others or “recycle” work prepared for other courses without obtaining written permission from the instructor(s). Students suspected of engaging in academic misconduct will be reported to the Office of Academic Integrity.

Other violations of academic misconduct include, but are not limited to, cheating, plagiarism, fabrication (e.g., falsifying data), knowingly assisting others in acts of academic dishonesty, and any act that gains or is intended to gain an unfair academic advantage.

The impact of academic dishonesty is far-reaching and is considered a serious offense against the university and could result in outcomes such as failure on the assignment, failure in the course, suspension, or even expulsion from the university.

For more information about academic integrity see the student handbook or the Office of Academic Integrity’s website, and university policies on Research and Scholarship Misconduct.

Support Systems

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