PHIL. 160
Deductive Logic II

Spring Semester 2013

Prof. Dowden







Catalog description:
Further study of deductive logic. Topics include: principles of inference for quantified predicate logic; connectives; quantifiers; relations; sets; modality; properties of formal logical systems, e.g. consistency and completeness; and interpretations of deductive systems in mathematics, science, and ordinary language. Prerequisite: MATH 031, or PHIL 060, or instructor permission. 3 units.

Textbooks: There are two required books. Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter; and Logics, by John Nolt. In addition to the books, there will be some webpages and class handouts that you should consider to be required reading.

The two books are available at the CSUS Hornet Bookstore, but here is a website that compares book prices at many online bookstores: You can get the book used at for $150: The book comes with a CD that we won't be using. We will be reading approximately half of the book Gödel, Escher, Bach; two copies of it are on reserve in the CSUS Main Library. If you understand a little about music theory, then this book will deeply enrich your understanding; but if you don't know about music theory, then you will be relieved to know that you won't be tested on any music knowledge, just knowledge of logic. The same goes for art and ancient Japanese poetry.

Grades: Your grade will be determined by four homework assignments (each 14%), a midterm exam (20%), and a comprehensive final exam (24%). Homework questions will be handed out a week in advance of the due date. Class attendance is optional, but you are responsible for material covered in class that isn't in the book.

Due dates:

Homework 1:  Feb. 14, 2013 (wk 3)

Homework 2:  Mar. 7 (wk 6)

Midterm: Mar. 21 (wk 8)

Homework 3:  Apr. 18 (wk 11)

Homework 4:  May 9 (wk 14)

Final Exam: May 23, 10:15 A.M.

For homeworks, you are responsible for any announced changes to questions that are made after the homework is handed out but before it is due, even if you didn't attend class the day the change was made.

Course Description: Our course presupposes you have had a first course in deductive logic, such as Phil. 60, or have learned this material on your own. The first month will contain a review of Phil. 60, but also will enrich that material. Our goal is to appreciate what can be done with deductive symbolic logic and what can't be done. That is, we will explore the scope and limits of deductive logic rather than its depth in one particular area.

Deductive logic explores deductively valid reasoning, the most secure kind of reasoning. A mathematical proof is deductively valid reasoning. Inductive reasoning, by contrast, is about less secure reasoning from the circumstantial evidence of the lawyer, the documentary evidence of the historian, the statistical evidence of the economist, and the experimental evidence of the scientist.

For a helpful metaphor, you might think of our symbolic deductive logic as a machine for detecting the presence of the most secure reasoning. In our course, we will not only use the machine but also study what it can and cannot do, and whether it can be revised to do other things. For example, does it have the power to show that "Obama's father is working in his office" logically implies "Someone's father is working"? Can we use the machine to demonstrate that no use of the machine will lead to a contradiction?

Our course will survey the deep results yielded by the developments in symbolic deductive logic. These results concern the surprising extent to which human knowledge can not be freed of contradictions, to what extent our knowledge can be expressed without loss of content inside of a formal language, and what our civilization has learned from the field of symbolic deductive logic about the limits to what people can know and about the limits of what computers can do, the major results here being the Unsolvability of the Halting Problem, the Church-Turing Undecidability Theorem, Tarki's Undefinability Theorem for Truth, and Gödel's Incompleteness Theorems.

We will begin our course with a review of Phil. 60 while providing a rigorous development of both propositional logic (also called statement logic and sentential logic and propositional calculus and statement calculus and the theory of truth functions) and elementary predicate logic (also called first-order logic, relational logic, quantificational logic and predicate calculus). Then we will learn about their applications, extensions, meta-theory, and non-classical variants. Regarding non-classical variants, this comment in 1970 by the American logician W.V.O. Quine is helpful:

Logic is in principle no less open to revision than quantum mechancis or the theory of relativity.

My role in our course will be to cut through the jargon and help you understand as quickly as possible.

A good analogy for our course is that learning symbolic logic is much like learning a computer language. The big difference is that in symbolic logic the focus is on using the formal language to assess argument correctness rather than on getting a computing machine to follow its intended program. To continue with the analogy, in our course we will not be focusing on doing actual programming so much as learning the capabilities of the computer.

Speaking about the second textbook Gödel, Escher Bach, the M.I.T. senior student Justin Curry, who gives the online lectures about it, says, "I advise everyone seriously interested to buy the book, grab on and get ready for a mind-expanding voyage into higher dimensions of recursive thinking."

Topics and reading assignments:
Click here.

Relevance of logic to other subjects: If you are curious about the relevance of deductive logic to other subjects such as philosophy, mathematics, and computer science, then click on the ticket below:

Student outcome goals: The hope is that by the end of the semester you will have achieved the following goals:

  • Be able to reason more effectively.

  • Be able to describe the scope of deductive logic, that is, what it can be used to do; and be able to describe the limits of logic, that is, what it cannot be used to do.

  • Build on the abilities you learned in Phil. 60 to recognize when the quality of an English argument is capable of being analyzed with symbolic deductive techniques, to translate a symbolic deductive argument into English and vice versa, to determine if a symbolic deductive sentence is logically true, to determine if a set of symbolic sentences is consistent, to assess the logical correctness or incorrectness of arguments using the techniques of symbolic deductive logic, to create proofs in both predicate logic and propositional logic, and be capable of creating and analyzing rigorous proofs using the methods of classical symbolic deductive logic.

  • Understand Hilbert's program and the process of formally axiomatizing a theory.

  • Be familiar with the most important meta-theoretic results such as Gödel's Theorems, the Church-Turing Undecidability Theorem, Tarski's Undefinability Theorem, and the Löwenheim-Skolem Theorem. You will be able to appreciate why Gödel says all consistent axiomatic formulations of first-order number theory include undecidable propositions.

  • Be able to say how symbolic deductive logic has deepened our knowledge of some important philosophical issues, and how it has led to new issues of its own.

  • Know the extent to which human knowledge can be freed of contradictions.

  • Be able to say what our civilization has learned from the field of symbolic deductive logic about the limits to what people can know and about the limits of what computers can do.

  • Know that there are important extensions of classical first-order logic to non-standard logics such as modal logic, deontic logic, free logic, many-valued logic, second-order logic, many-sorted logic, fuzzy logic, and paraconsistent logic.

Laptops, cell phones: Photographing during class is not allowed without permission of the instructor. Audio recording is OK. During class, turn off your cellphone. Your computers may be used only for note taking, and not for browsing the web, reading emails, or other activities unrelated to the class. If you use a computer during class, then please sit in the back of the room or in a side row so that your monitor's screen won't distract other students.

Testing protocol: For in-class tests, you may use any books and notes but not your computer or cellphone.

Disabilities: If you have a documented disability and require accommodation or assistance with assignments, tests, attendance, note taking, and so forth, then please see me early in the semester so that appropriate arrangements can be made to ensure your full participation in class. Also, you are encouraged to contact the Services for Students with Disabilities (Lassen Hall) for additional information regarding services that might be available to you.

Plagiarism and Academic Honesty: A student tutorial on how not to plagiarize is available online from our library. The University's policy on academic honesty is at

Food: Except for water, please do not eat or drink during class. You are welcome to leave class (and return) any time you wish.

Late assignments, and make-up assignments:  I realize that during your college career you occasionally may be unable to complete an assignment on time. If this happens in our course, contact me as soon as you are able. If you promptly provide me with a good reason for missing a test or homework assignment (illness, accident, ...), then I'll use your grade on the final exam as your missing grade. There will be no make-up tests nor make-up homework. I do accept late homework with a grade penalty of one-third of a letter grade per 24-hour period beginning at the class time the assignment is due. Here are some examples of how this works. If you turn in the assignment a few hours after it is due, then your A becomes an A-.  Instead, if you turn in the same assignment 30 hours late, then your A becomes a B+.  Weekends count, so scan your late, but finished work on the weekend and email it as an attachment.  No late work will be accepted after the answer sheet has been handed out (normally this will be at the next class meeting), nor after the answers are discussed in class, even if you weren't in class that day.

Add-Drop: To add the course, try to do so by using the CMS system. If the course is full, then see me about signing up on the waiting list. To drop the course during the first two weeks, use the CMS system. No paperwork is required. After the first two weeks, it is harder to drop, and a departmental form is required, the "Petition to Add/Drop After Deadline." As with any university course, make sure you are dropped officially (by CMS or by the instructor or department secretary); don't simply walk away into the ozone or else you will get a "WU" grade for the course, which is counted as an "F" in computing your GPA (grade point average).


Professor: My office is in Mendocino Hall 3022, and my weekly office hours are TuTh 9:30-10:30 and 12:00-12:30. Feel free to stop by at any of those times, or to call. If those hours are inconvenient for you, then I can arrange an appointment for an alternative time. You may send me e-mail at or call my office at 278-7384 or the Philosophy Department Office at 278-6424. The fastest way to contact me is by email. My personal web page is at

photo of Dowden

Prof. Dowden

Study tips: As you read an assignment, it is helpful first to skim the assignment to get some sense of what’s ahead. Look at how it is organized and how the author signifies main ideas (section titles, bold face, italics, full capitals, and so forth). Make your own notes as you read. Stop every twenty minutes to look back over what you’ve read and try to summarize the key ideas for yourself. This periodic pausing and reviewing will help you maintain your concentration, process the information more deeply, and retain it longer. Notice connections between one section and another. You’ll be given sample questions now and then to help guide your studying for future assignments, but the homework and test questions in our course will usually require you to apply your knowledge to new questions not specifically discussed in class nor in the book. This ability to use your knowledge in new situations requires study activities different from memorizing. You goal is to improve your skills, rather than to memorize information. Think of the textbook more as a math book than a novel, so re-reading is important.

Contact me at if you'd like more information about our course.


Updated: May 4
, 2013