Philosophy 3349
Topics in Logic


Curtis Brown
Spring, 2002

This is an intermediate-level course in formal logic.  The course presupposes that you have taken a course which provides a good background in classical first-order logic, such as PHIL 2340, Symbolic Logic; MATH 2326, Introduction to Abstract Mathematics; or CSCI 1323, Discrete Structures.  The course will fall into two main parts. In the first part of the course, we will cover a number of metalogical results culminating in Gödel's incompleteness theorems. We will also consider relations between these metalogical results and results in computability theory, and will discuss arguments that these results have significant philosophical implications, notably Penrose's argument that the incompleteness theorems show that artificial intelligence is impossible and Hilary Putnam's argument that the Löwenheim-Skolem theorem shows that metaphysical realism is untenable. In the second part of the course, we will consider some proposed extensions and modifications of first-order logic.  The principal extension we will consider is modal logic, which adds resources for dealing with the logic of possibility and necessity. Other possible extensions include deontic logic, doxastic logic, etc. Modifications we will consider may include  many-valued logics, intuitionistic logic, and fuzzy logic. Other possible topics include nonmonotonic logics, paraconsistent logics, etc. We will also consider actual and possible computational applications of these extensions and revisions.


Christopher C. Leary, A Friendly Introduction to Mathematical Logic (Prentice-Hall, 2000). Don't be misled by the title: this is not an introduction to symbolic logic per se, but rather an introduction to metalogic. It includes proofs of soundness and completeness theorems for first-order logic, the compactness theorem, the upward and downward Löwenheim-Skolem theorems, and culminates in proofs of Gödel's incompleteness theorems. However, compared with other texts in the area, this one is fairly friendly.

We will also make use of a variety of other materials, including handouts and online materials on the philosophical implications of Gödel's results, and material on alternatives to, and extensions of, classical first-order logic. Especially valuable resources are the Routledge Encyclopedia of Philosophy (available in the library either in hard copy or in a CD-ROM version) and the Stanford Encyclopedia of Philosophy.

Office Hours

MW, 10:30 - 11:30, 2:00 -3:00; TR, 2:30 - 3:30. 

I am usually in my office during office hours, but occasionally a meeting or another commitment prevents this. If you just drop by during office hours, you will probably find me in; if you want to see me at another time, or if you want to be certain I'll be in, we can set up an appointment.


There will be a mid-term examination and a final examination.  There will be regular shorter assignments -- sometimes homework problems, sometimes short papers.  Every member of the class will be expected to give at least two in-class presentations on topics germane to the course:  an extension to, or revision of, first-order logic or a metalogical result.  There will also be a final project for the course.  Percentages:  Mid-term, 20%; Final, 20%; presentations and class participation, 20%; Short Assignments, 20%; Final Project, 20%.  (Easy to remember!)

Very Rough Schedule

Preliminary: Some Set Theory (Barwise and Etchemendy)

Metalogic (main text: Leary, Friendly Introduction)

structures and languages


proof of soundness
proof of the deduction theorem

completeness and compactness

proof of completeness theorem
proof of compactness theorem
proofs of upward and downward Skolem-Lowenheim theorems

incompleteness - groundwork

recursive sets and functions
Gödel numbering
NUM and SUB functions

incompleteness - proofs

proofs of Gödel's first and second incompleteness theorems

significance of the incompleteness results

relation between incompleteness and uncomputability 
Penrose: incompleteness shows AI impossible (handout from Penrose)
criticisms of Penrose (Chalmers essay)
Putnam on realism (handout)
criticisms of Putnam (recent Journal of Philosophy article)

Alternative Logics

equivalent systems

axiomatic systems, natural deduction systems, sequent introduction systems
equivalent sets of axioms, rules, etc.


modal logic (handout from Forbes)
deontic logic


intuitionistic logic (handout from Forbes)
nonmonotonic logics?
dialethic logic?


Last update: January 13, 2002
Curtis Brown | Topics in Logic | Philosophy Department | Trinity University