This is a course on basic metalogic (with a bit on computability theory if we have the time). Metalogic is the study of facts about and properties of logical systems as a whole (as opposed to learning to use a particular logic system, e.g., to construct proofs within that system). One of the central issues we will focus on is the distinction between the syntactic features of a formal system and the system's semantic features. We will learn how in a good formal system these features line up exactly, and we will prove that they do for a particular formal system. Topics we will cover along the way will include Truth-Functional Completeness, Models and Interpretations, the Soundness, Completeness, Compactness, and the Undecidability of First-Order Logic. Through the study of these topics we will consider the scope and limits of formal theorizing. Computability theory investigates what it is for a function to be computable, that is, when there is a mechanical procedure or algorithm for solving a particular mathematical problem. This study includes such topics as Turing computability, uncomputability, and the Halting Problem.
The Learning Goals for this course are an extension of the Philosophy Department's general goal to have students exhibit facility in the theory and practice of argumentation, reasoning, and critical thinking. These include the following.
1. Master the practice of reasoning well, including
A. The ability to construct clear and concise summarizations and assessments of the reasoning in complex passages by
i) Extracting their conclusions
ii) Distilling the lines of reasoning in support of those conclusions
iii) Evaluating how well such reasoning supports those conclusions.
B. The ability to construct cogent arguments for their own conclusions and to express their reasoning in a coherent and convincing manner.
2. Demonstrate knowledge of, and competence with, the theory of argumentation and logic through their abilities to:
A. Describe different approaches to logical theory, and to articulate their aims and scope.
B. Define and apply central concepts and techniques of logical theory.
C. Describe major results of logical theory.
D. Sketch how to arrive at those results.
Participation—This requirement is designed to take into account contributions during class (e.g., asking questions, suggesting moves for proofs done in class, etc.) and improvement throughout the term. To do well on this requirement it is vital that you keep up with the reading assignments.
Homework—This requirement covers completion of and performance on the homework assignments. The homework provides practice with the techniques presented in class, so it is crucial that you keep up with the assignments. There will be assignment due every week. No late assignments accepted.
The First Test—There will be a timed, in-class test in mid September. The test questions will cover concepts and definitions and include problems like those on the homework assignments and in the readings, and the proofs done in class.
The Second Test—There will be a second timed, in-class test in mid October. Again, the test questions will consist of problems like those on the homework assignments and proofs done in class.
The Third Test—There will be a third timed, in-class test in early November. Again, the test questions will consist of problems like those on the homework assignments and proofs done in class.
The Final Exam—There will be a timed, in-class final exam given on Thursday, Dec. 12, 2013 at 3:10pm in our regular classroom. The final will essentially be cumulative, but it will emphasize the material since the Third Test. The exam questions will include problems similar to the homework and classroom proofs, as well as some pertaining to concepts.Note: All course requirements must be satisfactorily completed in order to pass the course. More than 3 unexcused absences reduces your final grade by 1/3 of a letter grade, more than 5 is a full letter grade deduction, more than 8 is automatic failure of the course.
The class will consist mostly of lectures, demonstrations of
problem-solving techniques, and sample exercises. However, I want to
encourage student participation throughout the class--both in the form
of questions and suggestions about how to approach problems we are
considering. Class meetings will typically consist of two different
(not necessarily equal) parts: one in which I will lecture on the
material you have read about for the day and work some sample problems,
and one in which I will answer questions about problems from homework
assignments that students would like to go over.
V. COURSE SCHEDULE
The course units and topic covered in them are as follows.
1. Artificial Formal Language (FOL) Basics
Review of basic syntax (Lexicon and Formation Rules) and semantics (e.g., intensional interpretations, intuitive models, truth-tables) for the language of First-Order Logic. (3 weeks)2. General Semantics and Model Theory
Extensional interpretations, models, contingency, consistency, semantic proofs, logical truth/falsity, entailments/enfailments, inconsistency, semantic equivalents (4 weeks)3. Deductive Apparatus
Tree system rules, theorems/anti-theorems/neutrals, compatibility/incompatibility, establishment/non-establishment, coupled/uncoupled sentences, mathematical induction (3 weeks)4. Metalogic
Soundness, completeness, Koenig's Lemma, compactness, undecidability (incompleteness?) (4 weeks)*The instructor of this course reserves the right to change any aspect of the syllabus, with the understanding that any such changes will be announced in class.