Philosophy 135
Spring 2012
Introduction to Metalogic


Instructor

Time and Room

Office Hours Exam Week

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 is due by Friday, June 15 at 5:00PM.

Homework

  1. Exercises 1.1, 1.2, and 1.3. Due Wednesday, April 11.
  2. Exercises 1.5, 1.6, and 1.8. Due Wednesday, April 18.
  3. Exercises 1.10, 1.11, 2.1, and 2.2. Due Wednesday, April 25. Solutions
  4. Exercises 3.1, 3.3, and 3.4. Due Wednesday, May 9.
  5. Exercises 3.6, 3.7, and 3.8. Due Wednesday, May 16.
  6. Exercises 3.10, 4.1, and 4.2. Due Wednesday, May 23.
  7. Exercises 4.3, 5.1, and 5.2. Due Wednesday, May 30.
  8. Exercises 5.5, 5.6, and 5.7. Optional: Exercise 5.4. Deduction Exercises Solutions Due Wednesday, June 6.

Solutions for Practice Problems

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.

Warning

The official prerequisite is Philosophy 31. But, as the outline just given indicates, 135 is very different from 31, and success in 31 does not guarantee success in 135. 31 mainly teaches techniques---most imporantly, techniques for doing formal derivations. 135 mainly teaches teaches concepts and their relations to one another, and it is much more conceptually sophisticated than 31. The student has to understand and apply concepts and definitions and produce informal proofs. Doing formal derivations is a relatively smal part of the course. 135 does not use computers, so there is no immediate feedback. Also course moves much more quickly than 21. Beacause Philosophy 137 is is the sequel to 31, 137 might be for some students a good alternative to 135.