Course Outline—Logic 3003, Spring 2005
Instructor: Bryson Brown
Office: TH 222
Phone: 329-2506
Email: brown@uleth.ca
Office Hours: Tuesday, Thursday from 11:00-12:00 and by Appt.
Logic is often described as a normative study of reasoning, by which we mean a study of
how to evaluate reasoning as good, or bad. And reasoning is done (or at least expressed)
in language. But ordinary language isn’t very well suited to producing systematic
theories of how to reason in it. 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 a lot of 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—so 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 aren’t often
made explicit (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 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,
unanticipated counter-examples. Math isn’t supposed to be like that, so mathematicians
responded by trying to systematize the process of reasoning. Today’s techniques in
propositional logic and quantificational logic grew out of this effort, though they 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.
The result of this work, well begun in the 19th Century by 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 logic, which remains the heart of the
discipline. 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 they re-emerge when we try to connect the elegant results of formal
proofs back to arguments in our natural languages (which are, of course, the ones we
were interested in in the first place).
This is the first 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 some
familiar material as we go along, including proof theory (natural deduction) for
propositional and quantificational logic. But we will go beyond this practical approach &
prove some important facts (called metatheorems) about these systems and their relation
to classical semantics for these logics.
Some of the course work will involve using a software package called Simon and Simon
Says. The software is under development at Simon Fraser University, but this is not a
first run, so I’m optimistic 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 software for the first time, and partly because I’m still trying to work out the
balance 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.
I’ll let you know if there are any adjustments to this.
Initial Course Schedule (tentative):
Week 1: January 6, 2005. Introductory remarks. Get Simon up & running! Proof and
Consequence, Chapter 1.
Week 2: January 11, 13. 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 18, 20. 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 25, 27. 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 27, due February 3.
Week 5: February 1, 3. Chapter 3. Strong Soundness of L: A proof by strong
mathematical induction. (Remarks on mathematical induction.)
Week 6: February 8, 10. 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 15, 17. 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 17.
Week 8: March 1, 3. 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 8, 10. Practice with some more quantifier proofs; some discussion of
proof technique. Introduction to the formal account of Lq.
Week 10: March 15, 17. 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 17.
Week 11: March 22, 24. Soundness of Lq. I will be away March 24. But I will be in for
extended office hours later that afternoon & watching email closely to provide feedback
and advice as you work on the second assignment.
Week 12: March 29, 31. Second assignment due March 29. Completeness of Lq. The
Henkin strategy reviewed, construction of canonical models for sets of sentences,
Week 13: April 5, 7. Remarks on the strength and limits of Lq, second-order logic and
induction, and incompleteness issues. Final exam (takehome) distributed April 7.
Week 14: April 12, 14: No Classes: I will be at a conference in Dubrovnik, Croatia. I
will be watching my email and responding to questions while there as best I can.
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.