Course Outline—Logic 3003, Spring 2007
Instructor: Bryson Brown, TH 222, 329-2506, brown@uleth.ca
Office Hours: Tuesday, Thursday from 2:00-3:00 and by appt.
Logic is often described as a normative study of reasoning. By this we mean a study of how to evaluate
reasoning as good or bad. Reasoning is something done (or at least expressed) in language. However,
ordinary language isn’t very well suited to producing systematic theories of reasoning. Though we can
often arrive at a pretty wide consensus that a particular argument is persuasive, or that it’s completely
unconvincing, it’s very hard to articulate clear and straightforward criteria for these judgments. The result
is that there turn out to be many cases where people just plain disagree about whether an argument is good
or not. Aside from bias and prejudice and sheer stubborn cussedness, there are perfectly honest reasons for
this kind of disagreement. Ordinary language is fuzzy, irregular in its usage, and sometimes outright
ambiguous—the sentences used in an argument may well mean something different to different audiences.
Further, each of us brings background beliefs to our evaluation of arguments. Since these are rarely made
explicit, and sometimes we may not even be aware of them, they can lead us to differ on whether an
argument succeeds without being able to see clearly why we differ.
This was a particularly acute problem in mathematics, where apparently convincing proofs were
occasionally overturned by clever people who found ways to construct odd and unanticipated counterexamples. Math isn’t supposed to be like that—results convincingly established in mathematics are
supposed to be settled once and for all! So mathematicians responded by trying to systematize the process
of reasoning, in ways that would ensure (so long as the rules were followed) that anything that was proven
would stay proven. Today’s techniques in propositional logic and quantificational logic grew out of this
effort. They also have deep roots in philosophy, reaching back to Aristotle, early mathematicians like
Pythagoras, and figures like the Stoics, who had interesting ideas about some of the connectives we treat
using truth functions today. But their systematic nature, as well as some of the things we will be proving
about them, emerged from ideas and ambitions that were articulated in the late nineteenth and through the
twentieth centuries.
The result of this work, begun in the 19th Century by Gotlob Frege and continued by Russell and
Whitehead, Wittgenstein, Post, Carnap, Quine, Lewis, Gödel, Kleene and others, is modern formal logic.
Our focus will be on classical formal logic, which remains the heart of the discipline and the best place to
learn the ropes. In formal logic we use artificial languages with simple, fully-specified structures. This
allows us to pass over the ambiguities and fuzziness of ordinary language, though of course they re-emerge
when we try to connect the elegant results of formal proofs back to arguments in our natural languages
(which are the ones we were interested in, in the first place).
This is the second run of Logic 3003, following a re-working of our logic program. 3003 is intended as a
second course in classical formal logic. We will be reviewing quite a lot of familiar material as we go
along, including proof theory (natural deduction) for propositional and quantificational logic. But we will
go beyond this practical approach to prove some important facts (called metatheorems) about these systems
and their relation to classical semantics for these logics. This formal attention to the formal properties of
these systems marks the step from (merely) doing logic, to systematically studying logic.
Some of the course work will involve using software packages called Simon and Simon Says. The
software has been developed at Simon Fraser University, and has been in use for some time, so I’m
confident that it will work well. The details of the assignments that I will be using Simon for are still being
worked out. This is partly because I’m working with this version of the software for the first time, and
partly because I’m still trying to work out the balance we need to strike between review/improvement of
your basic logical skills, and studying the systems formally, including proving a few things about them.
Grades will be based on two tests (25% each), two written assignments (15% each), and five smallish
Simon-based assignments (4% each). My usual grading scale is:
A+: 95 and higher; A: 85-94; A-: 80-84; B+: 78-79; B 73-77; B-: 70-72, and so on.
Course Schedule:
Week 1: January 4. Introductory remarks. Get Simon up & running! Proof and Consequence, Chapter 1.
Week 2: January 9, 11. Chapter 2. Propositional proof theory: logic as a search for general kinds or
structures of good arguments (not topic or context specific); telling good arguments from bad; English
connectives; sequents and ; documenting proofs (some discussion here of alternative ways to approach
proof construction & documentation); L and the rules of L; working with Simon’s proof editor.
Week 3: January 16, 18. Chapters 2,3. Brief review of translation (working with Simon’s formulation
editor); structural properties & the general idea of a consequence relation (remarks re. multiple-conclusion
systems & duality); compactness. Formal definition of L (our propositional language) R (our rules of
deduction), and proofs. Theorems and uniform substitution. First metatheorems (3.1, 3.2). Theorem and
Sequent introduction rules.
Week 4: January 23, 25. Chapter 3. Effectiveness. Arithmetizing our language & proof theory;
searching the proof space. The standard interpretation of L. Tautologies, inconsistencies and substitution
instances. N-ary truth functions and expressive completeness. Conjunctive normal forms, disjunctive
normal forms. Sheffer strokes, dagger strokes.
Major Assignment 1: Distributed January 25, due February 3.
Week 5: January 30, February 1. Chapter 3. Strong Soundness of L: A proof by strong mathematical
induction. (Remarks on mathematical induction.)
Week 6: February 6, 8. Chapter 3. Weak & Strong completeness of L: Kalmar’s method. Sequents &
truth-table lines: finding a proof for every such sequent. Completeness demonstrated constructively by
defining canonical proofs of the tautologies. Strong completeness & conditionals corresponding to
sequents. The significance of soundness and completeness; maximality.
Week 7: February 13, 15. Why stay with just one proof when you can have two? A Henkin-style weak
completeness proof for propositional logic: The Strategy—show that every consistent wff is satisfiable (i.e.
has a ‘1’ somewhere in its truth table).
Test 1: February 15.
Week 8: February 27, March 1. Introduction to quantifiers & quantifier rules. Simple translations &
reading Lq sentences back into English; UI and EE rules, the importance of arbitrary names, and methods to
ensure that arbitrary names are correctly used.
Week 9: March 6, 8. Practice with some more quantifier proofs; some discussion of proof technique.
Introduction to the formal account of Lq.
Week 10: March 13, 15. Interpretation of Lq. Satisfaction of wffs and quantificational models:
Domains, interpretations of proper names, and predicate letters, and the treatment of variables and arbitrary
names by assignments.
Second assignment distributed March 15, due March 29.
Week 11: March 20, 22. Soundness of Lq. I will be in for extended office hours & watching email
closely to provide feedback and advice as you work on the second assignment.
Week 12: March 27, 29. Completeness of Lq. The Henkin strategy reviewed, construction of canonical
models for sets of sentences,
Week 13: April 3, 5. Remarks on the strength and limits of Lq, second-order logic and induction, and
incompleteness issues.
Final exam (takehome) distributed April 5.
Week 14: April 10, 12: Ideas about truth and the trouble with having truth in the object language.
Tarski’s hierarchy of languages.
Exam Period: I will be in during the first week of exams—I’d like to schedule a couple of sessions for
discussion/ advice on completing the exam, review of any issues that remain puzzling for you all.