PHI 201 Introductory Logic

Hans Halvorson

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Logic is the oldest and most foundational of the university disciplines — having been taught every year for nearly 1000 years. The goal is to equip you with most general possible framework for sound and rigorous reasoning — one that will support you in any intellectual activity you pursue (e.g. mathematics, politics, chemistry, English literature, law, religion, etc.). What’s more, logic aspires to be independent of the quirks of a particular natural language, programming language, or mathematical system. In this sense, logic aims to to capture that which is universally rational. (We might argue later about the extent to which logic is universal, but we will have to use logic to do that!)

Instructors

Rebecca Mullen, Cosim Sayid (head TA), Lingzhi Shi

Prof Halvorson has office hours Thu 2:30-3:20pm.

Course format

Late policy

We are unable to accept late homework, except in case of a documented medical or family emergency.

Recommended work protocol

  1. Weekend before lecture

    1. Look over next problem set

    2. Skim relevant section of text

  2. Monday: Attend lecture and take notes

  3. Between lecture and precept

    1. Read text more closely

    2. Work on pset

    3. Look at feedback on previous week’s pset

  4. Attend precept

    1. Review previous week’s pset

    2. Work on this week’s pset

  5. Optionally: visit preceptor’s office hours

  6. Friday: submit pset

Academic integrity

The policy for this course is that collaboration between students is encouraged, but in no case should one student copy another’s work. That is hardly a precise criterion, but the spirit of the law is that you need to understand what you write down on your paper. If you fail to do that, then your lack of understanding will most likely reveal itself on the exam. As a general rule, we would suggest that you put away any notes from joint brainstorming sessions before writing your final answers.

It would be pointless for us to try to prevent you from using ChatGPT and the like. But please note: we have tested ChatGPT on the problem sets for this class, and it is not generally reliable in giving correct solutions to the kinds of problems we pose. It gave us, e.g., a “proof” of Pierce’s law that did not use the DN rule (which is provably impossible).

Resources

Subject matter

While this subject has been taught for many hundreds of years, the content got an overhaul in the mid 1900s, when modern symbolic logic was consolidated. (The content has been mostly stable now since the 1950s.) Since then, the standard format for an introductory logic class has been to learn the “predicate calculus” in three steps: propositional logic, monadic predicate logic, polyadic predicate logic. We follow this same outline, but with a few twists.

  1. We emphasize reasoning techniques (natural deduction) over calculational techniques (truth tables or trees).

  2. We teach proofs before truth tables. It’s harder, but makes your brain stronger.

  3. We incorporate more analysis of arguments in natural language (e.g. argument mapping).

  4. We illustrate logical concepts, when helpful, by displaying them concretely in the general framework of functional programming languages. However, this course presupposes no background in programming, nor does it presuppose that students are interested in programming.

Schedule

Week 1

pset1 is due on the first Friday of classes. It covers basic concepts, some translation, and some simple proofs.

Reading: Chapters 1 and 2

Week 2

pset2 covers proofs with dependency numbers (conditional proof and elimination)

Reading: Chapter 3

Week 3

pset3 covers more proofs ( elimination and RAA)

Reading: Chapter 3

Here’s a draft of the lecture notes: lecture3

Week 4

pset4 covers truth-table problems

Reading: Chapter 5 (note that we are skipping over Ch 4 for now)

Week 5

pset5 asks you to synthesize the topics we have covered so far, and it covers meta-rules (substitution, cut, and replacement)

The lecture will have two parts.

  1. Jillian Roberts will tell us about the paradoxes of material implication.

  2. A more practical part that covers (a) using meta-rules for proofs, and derived rules more generally, (b) using truth-tables to judge whether a proof is correct (soundness and completeness), and (c) reasoning about semantic relations between sentences. (This stuff is all fair game on the exam.) lecture5 notes

Reading: Chapter 4

Week 6

Midterm exam

The exam is administered at the normal time and place for the Monday lecture. You have 80 minutes to complete it. You are not permitted to use any computational devices, especially not any that are connected to the internet. You may bring a a single letter-sized sheet of paper (which may have text on front and back) for reference during the exam. On this sheet you may write whatever information you think would be helpful for you during the exam.

Types of questions you can expect:

  1. Translation

  2. Truth-table problems (classifying sentences, arguments, relations between sentences)

  3. Proofs

  4. Conceptual questions, e.g. about whether certain line fragments could be part of a correctly written proof, or about whether a certain new rule could be derived from those we already have.

You also need to be clear on the definitions of key concepts such as: counterexample, inconsistency, main connective, completeness, etc.

Week 7

pset7 covers translation into predicate logic, and proofs with UE and UI

Reading: Chapter 6 (pp 84-99)

Week 8

pset8 covers more translation, and proofs with EI and EE.

Reading: Chapter 6 (pp 99–115)

Week 9

We skip over Chapter 7 for now and move straight to “semantics” (Chapter 8). In pset9, you will construct “interpretations” of symbols that make predicate logic sentences either true or false. You will use interpretations classify sentences as either tautologies, contingencies, or inconsistencies, and to classify arguments as either valid or invalid.

Week 10

pset10 covers theories about equality, ordering, and functional relations

The equality relation greatly increases our expressive power. In particular, we are now able to formulate (finite) numerical claims. We can also formulate “definition descriptions”, such as the infamous “The present king of France is bald”.

Reading: Chapter 7 (pp 116-131)

Week 11

pset11 will review some of the more challenging things in the previous few weeks – especially interpretations and translation.

Week 12

We will run through a full-length practice final exam.