Phil/Ling 455.001: Symbolic Logic

Spring 2022, MW 9:05-10:20 am in Caldwell 213, 3 credits

Professor Jim Pryor (he/him)

Sakai Site, Zoom, and Regular Updates

UNC students enrolled in the course (or otherwise authorized by the instructor) can access the Sakai webpages for this course at

Those pages include the Zoom links for the course meetings and for Professor Pryor’s office hours. These can also be retrieved from this restricted page.

Most of the information for the course will be published here, outside of the Sakai system, and can also be viewed by people not enrolled in the course.

This front web page won’t be updated frequently. Regular announcements, readings, and presentation notes will be posted at this page instead.

General Catalog Listing

I am obligated to copy this general information onto every syllabus. See below for a description of how this specific instance of the course will be run.

PHIL 455. Symbolic Logic. 3 Credits.

Introduction for graduates and advanced undergraduates.

Requisites: Prerequisite, PHIL 155; permission of the instructor for students lacking the prerequisite.

Gen Ed: QR.
Repeat rules: May be repeated for credit. 6 total credits. 2 total completions.
Grading status: Letter grade.
Same as: LING 455.

Description for Our Specific Instance of this Course

This course aims to equip advanced philosophy students with solid foundations to deal with logic in papers they read and write, and discussions they participate in. We will discuss some important metalogical results, but will tend more towards equipping you with a broad understanding of the field than towards refining your proof skills. We’ll discuss a smorgasbord of issues in logic and metalogic, and some in math and formal semantics.

The course will be divided into three roughly equal segments:

  1. Philosophically Useful Math, and Formal Grammars/Languages
  2. Logics: Their Syntax, Semantics, and Proof Systems
  3. Important Metalogical Results, Numbers, and Arithmetic

This course is required for Philosophy Graduate Students.

Other students require permission from the instructor to enroll in the course.

Prerequisites: a solid grasp of elementary first-order logic (as covered for example in PHIL 155).

Target Audience and Course Goals

This course is aimed at grads and undergrads in Philosophy, Linguistics, and related fields like Math and Computer Science.

As stated in the catalog listing, it’s expected that you have taken Phil 155 or its equivalent. Understanding that material well will be essential for being able to keep up with this course.

Goals for the course include:


The course is offered by Professor Jim Pryor (he/him).

Professor Pryor’s office is Caldwell 108A. He can best be reached by email, at

Professor Pryor’s office hours are on Mondays starting at 3 pm, and Wednesdays starting at 1 pm. (If you have a quick question, you can also ask just after class.) If you’re unable to meet in person, we can also arrange to meet by Zoom. The Zoom link for office hours can be found on this restricted page.

Feel free to drop into office hours to discuss anything you like about our course. I’m happy to talk about your homeworks, continue discussion, and so on. If you do come to my office and I’m already speaking with someone, make sure that we know that you’re waiting for us to finish.


All readings for the course will be provided by web links. Some of these are in a restricted section of the course website. The username and password for these were emailed to you, and will also be announced in class.

Course Requirements

It is essential that you attend the class meetings regularly. Material not in the readings will often be presented there, and useful background and framing for some of the readings will also be provided. The University’s Policy on Class Attendance can be found here. In brief, they authorize absences only for some University activities, religious observances, disabilities, significant health conditions including pregnancy, and personal or family emergencies. If these include your situation, then consult the link about how to get your absence approved. (The University Approved Absence Office (UAAO) also has useful information.) If you need to miss class meetings for other kinds of reasons (like a job interview or to attend your mother’s wedding), ask me about it well in advance. In any case, you will be responsible for catching up with missed course content; and permission to miss a class doesn’t excuse you from deadlines for work due before or after the class.

Though this is an in-person course, attending a class meeting doesn’t necessarily mean being bodily present in the room. As the carolinatogether website says:

Each time prior to coming to campus, all members of the Carolina community should self-assess whether you are experiencing any symptoms.

If you have symptoms, you should stay home. You should not enter any campus building, attend any class or report to work.

Information about Covid testing is available here.

Information about quarantining and isolation is available here.

If you need to stay home during any of our class meetings, try to attend the meeting by Zoom instead.

See the Policies section below about wearing masks and using laptops or other devices in class.

When you join the class meetings, you are expected to have read any material assigned for that day, and to be ready to discuss it and/or ask questions about it.

It is essential that you ask questions when things in the readings or my presentations are unclear, and be ready to participate when I invite class discussion. I encourage you also to actively engage with each other outside the classroom, for example in study groups or to work together on homeworks.

There will be reading assignments for some class meetings, and I’ll often post summaries of material I presented in class. You should read these materials carefully as soon as they’re available, and expect that you’ll have to read many of them more than once.

There will be weekly homeworks due throughout the semester. These must be turned in (in-person or by email) before the start of most Wednesday class meetings. (No homeworks are due on Jan 12, Mar 23, Apr 13, or Apr 27.) There will also be a final homework due on Thursday May 5 at 8 am, which is the time scheduled for our final exam. Instead of the exam, we will meet at that time to review the final homewok and discuss outstanding philosophical issues that emerged during the semester.

Some of the homework exercises will ask you to define a specified function or predicate (as you might do in a computer programming course). Many will ask you to argue for some conclusion, in a mathematically rigorous way but not in a formal proof system. (Only occasionally will you be asked to prove things in a formal system like first-order logic or extensions of it; instead most of this course will involve reasoning about formal systems.) Some homework exercises will ask you to explain why some mathematical fact does/doesn’t obtain (as an educator or philosopher might, to a student who is stuck or confused). Taken together, the homeworks will constitute (substantially more than) the intellectual equivalent of ten pages of writing.

Homeworks cannot be turned in late, as I’ll be posting sample solutions after you submit them. Your grade for the course will be based on the quality of your homeworks. The weakest two homeworks you submit (or fail to submit) will be ignored. (The final homework cannot be omitted.)

The University Honor Code applies to all course assignments, and petitions for absences or rescheduling. In brief, this means students are expected to refrain from “lying, cheating, or stealing” in the academic context. For more information or to clarify which actions violate the honor code, consult with me,, and/or The Instrument of Student Judicial Governance.

What constitutes “lying, cheating, or stealing” depends on the academic activity.

Grade Appeals: If you feel you have been given an incorrect grade for the course, we can review together how I evaluated your work. If this doesn’t resolve the issue, you have the right to discuss with our department’s Director of Undergraduate Studies (currently Professor Markus Kohl), or to appeal through a formal University process. You’ll be expected to make a case that the grade reflects an arithmetic/clerical error, arbitrariness, discrimination, harassment, or personal malice. To learn more, consult the Academic Advising Program website, or this summary of University policies.

Most requests that I and other professors hear for changing grades are based on how good/bad it would be for a student to get a given grade; but it would be unfair and inappropriate for justifications like that to succeed.


This schedule lists the rough order of our topics. See this other page for course announcements, specific readings, presentation notes, and any minor tweaks to the schedule. Check that page frequently.

Meetings 1-2 / Mon Jan 10, Wed Jan 12
Strings (Finite Sequences)
Mon Jan 17
MLK Day, No classes
Meeting 3 / Wed Jan 19
More on Strings
Meeting 4 / Mon Jan 24
Sets, Tuples, Functions
Meeting 5 / Wed Jan 26
Functions and Relations
Meeting 6 / Mon Jan 31
Equivalence Relations and Orders
Meeting 7 / Wed Feb 2
Graphs and Trees
Meeting 8 / Mon Feb 7
Meeting 9 / Wed Feb 9
Algebras and Relations between Them
Meeting 10 / Mon Feb 14
Cardinality and Powersets
Meeting 11 / Wed Feb 16
Automata and Effective Decidability
Formal grammars and languages
Meeting 12 / Mon Feb 21
Meeting 13 / Wed Feb 23
Meeting 14 / Mon Feb 28
Syntax, Formula Complexity
Normal forms
Bound/free variables, Substitution
Meeting 15 / Wed Mar 2
Semantics for sentential and predicate logic
Logical truth and consequence
Meeting 16 / Mon Mar 7
Proof systems
Meeting 17 / Wed Mar 9
Logical extensions: Multisorted logic
Logical extensions: Restricted quantification
Logical extensions: Multivalues logic, presuppositions, descriptions
Logical extensions: Free logic
Mon Mar 14, Wed Mar 16
No classes (Spring break)
Meetings 18-19 / Mon Mar 21, Wed Mar 23
Normal modal logics
Uses/extensions of modal sentential logic
Meeting 20 / Mon Mar 28
Meetings 21-22 / Wed Mar 30, Mon Apr 4
Completeness and Compactness
Meeting 23 / Wed Apr 6
Models and Theories
Meeting 24 / Mon Apr 11
Ordinals and Cardinals
Meetings 25-26 / Wed Apr 13, Mon Apr 18
Peano Arithmetic, Standard/Nonstandard models
Meetings 27-28 / Wed Apr 20, Mon Apr 25
Gödel’s Incompleteness Theorems
Meeting 29 / Wed Apr 27
Last class
Thu May 5
Scheduled for Final Exam at 8-11 am
Instead of an exam, you’ll turn in your final homeworks; and we’ll have a final review/discussion meeting of the course

Other Information

If you wish to be in the course, but aren’t yet enrolled

(Whether or not you’re on the waitlist, the procedure is the same.) Come to the first week of classes and be in touch with me asap about your interest in the course, how it fits into your larger educational plans, and what your background in logic and other philosophy and/or formal courses might be. I’ll accommodate you if it seems that you’ll be adequately prepared for this course; but you should also figure out a backup plan.

Rescheduling/missing deadlines

Policies and Resources

The first few of these are specific to our course; the rest are information that the University mandates I include on every syllabus. (So you will see a lot of overlap with your syllabi for other courses.)


I reserve the right to make changes to the syllabus, including assignment due dates. These changes will be announced as early as possible so that students can adjust their schedules.


I welcome your input about the course at any time. You are welcome to approach me directly. I’ll also provide opportunities for anonymous evaluation and feedback during the term.