Philosophy 135
Winter 2010
Introduction to Metalogic


Instructor

Time and Room

Office Hours

Office Hours for Exam Week

The location is Dodd 355.

Grading

Homework will be assigned weekly, and will be due by 5:00 PM on Wednesdays. Late homework will not be accepted.

Midterm Exam

The Midterm Exam will be an open-book, in-class exam.

Final Exam

The Final Exam will be a take-home exam.

Homework

  1. Exercises 1.1 and 1.2. Due Wednesday, January 13.
  2. Exercises 1.5, 1.6, and the "only if" part of Exercise 1.8.
  3. Exercises 1.10 and 1.11.
  4. Exercises 2.1 and 2.2. Due Monday, February 1.
  5. Exercises 3.1 and 3.3. Due Wednesday, February 17.
  6. Exercises 3.4 and 3.7. Due Wednesday, February 24.
  7. Exercises 3.8, 4.1, and 4.2. Due Wednesday, March 3.
  8. Exercises 4.3, 5.1, and 5.2. Due by class time, Wednesday, March 10.

Prerequisite

The official prerequisite is Philosophy 31. It is helpful to have also taken Philosophy 137. A different, but equally good, background would be provided by most upper division mathematics courses.

Text

The only text will be course notes, which will distributed a few pages at a time as the quarter progresses. Here are are the course notes that have been distributed so far:

Content of Course

In Philosophy 31 and 137, students are taught to use symbolic logic in various ways. In a metalogic course such this one, logic is studied rather than used.

Logic has two main aspects:

  1. It is concerned with logical truth and, more generally with the relation we will call logical implication. This latter is the relation that holds between premises and a conclusion if the truth of the former guarantees, as a matter of logic, the truth of the latter.

  2. It is concerned with logical reasoning and deduction. Formal systems like that used in Philosophy 31 and 137 belong to this aspect of logic. Such systems give rules that specify what counts as a derivation or deduction of a conclusion from premises.

These two aspects of logic are very different. In this course we will study both 1 and 2 for predicate logic (quantificational logic), and we will also study relations between 1 and 2. An example of such a relation is soundness of a formal deductive system. This relation holds if whenever a conclusion is derivable from some premises in the system then the premises logically imply the conclusion.

We will first introduce a formal symbolic language and study its syntax. Then we will introduce models for this language, defining semantic notions such as truth. We will prove the important Compactness Theorem. Next we will introduce a system of formal deduction (derivation) and prove theorems about its properties. Finally we will study the relation between models and deduction, proving the fundamental Soundness and Completeness Theorems.

As the outline just given indicates, content of the course is primarily mathematical. Moreover, it is mathematical in a different way from Philosophy 31 and 137. In those courses, the students have to learn techniques (for finding derivations, translations, etc.). In Philosophy 135, students have to master concepts and definitions, and they have to learn to produce informal proofs--as opposed to formal derivations.