PHIL. 160
Deductive Logic II

Spring Semester 2016
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: PHIL 060 (or CSC 28 or instructor permission). 3 units.


There are two required books.

We will begin by reading selected chapters from the book Logic and Philosophy: A Modern Introduction by Alan Hausman, Howard Kahane, and Paul Tidman, 12th edition. (You must use the 12th edition, not an earlier edition.) You will be required to read only chapters 7, 8, 10, and 13, although the earlier chapters provide a helpful review of Phil. 60 in the notation needed for our course.

The book is available at Their pricing is complicated. $212.95 for the paperback. $33.99 for a timed rental of the paperback for a semester. $42.49 for the ebook (no time limit). $8.99 for electronic versions of any chapter (no time limit).

Even-numbered exercises are answered in the back of the book. Do not buy their LogicCoach computer program that can also be purchased with the book.

The second book is Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter, 1979. Any edition and printing is fine. We will eventually read chapters 1, 2, 3, 4, 5, 7, 8, 9, and 14. The book is available free in various formats such as pdf at

picture of book cover of Godel, Escher, Bach

In addition to these two books, there will be some webpages and class handouts that you should consider to be required reading.

If you understand a little about music theory, then the Gödel, Escher, Bach 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 the art and the ancient Japanese poetry that is covered in the book.

A note about page numbers: The actual content on the paper pages is available in the pdf pages, but the pdf has odd page numbering. All the various paper editions have the same page numbering, but the pdf version has a table of contents page that agrees with the paper edition but not with the pdf's own later pages. The page numbering in the pdf somewhere shifts all page numbers by 8. For example the paper page 71 is displayed as page 79 in the pdf. The paper page 433 is page 441 in the pdf.


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 is not in the readings.

Extra credit: You have the option of earning extra credit by giving a five to ten minute presentation in class one day on something new, but relevant. Then respond for a few minutes to student questions, and the professor's questions, about your presentation. The presentation can raise your lowest homework grade by a full letter. For this extra credit project you must describe your project in a sentence or two in an email sent to the professor asking for approval to give the presentation. This needs to be sent no later than 7:00 P.M. on the night before you wish to make your presentation. Sending a notification further in advance is preferable so you can profit from early feedback on what is covered. Here are some suggestions for kinds of presentations; remember to keep the duration to under ten minutes even if you could speak for an hour on the topic.

Due dates:

Homework 1:  (wk 3) Feb. 11, 2016

Homework 2:  (wk 6) Mar. 3

Midterm: (wk 8) Mar. 17

Homework 3:  (wk 11) Apr. 14

Homework 4:  (wk 14) May 5

Final Exam: (wk 16)

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 did not attend class the day the change was made. There will be no class meeting on Thursday, 3\31/16 nor on Thursday 4/21/16.

Course Description:

The field of logic is more prescriptive than descriptive. That is, it is not the study of how persons actually do reason, but rather the study of how they ought ideally to reason insofar as they are being "logical," that is, reasoning correctly. Not only does the field of logic study correct reasoning, but also it explores the errors to which thinking can be prone when it drifts away from the ideals of correct reasoning.

Rhetoric, on the other hand, is the study of persuasion, of what is convincing to whom under what circumstances. Our course focuses on logic and not rhetoric.

Logic in the most narrow sense of the term is not prescriptive and is only about logical consequence, that is, about what has to be true if something else is true. Our course will use a broader and less specific sense of what the term "logic" means. And we will venture into deviant logics, the philosophy of logic, and logic's connections to fields beyond philosophy.

The field of logic explores the structural properties of reasoning. The field isn't interested in building bigger piles of good arguments, but in understanding their structural features. The structure is called "logical form."

Our course presupposes you have had a first course in symbolic deductive logic, such as Sac State's Phil. 60, or you have learned this material on your own. Initially, and also throughout the semester, we will review Phil. 60, but also will enrich that material as we go along.

Our long-term 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. The most elemental piece of reasoning is a simple argument that draws one conclusion from one premise or assumption. In any argument, simple or not simple, if the conclusion follows with certainty from the premise we call the argument deductively valid. A mathematical proof is an example of a deductively valid argument.

Inductive reasoning, by contrast, is about less secure reasoning. Its conclusion follows from its premises with probability but not with certainty. When Sherlock Holmes says he is making deductions, he is really making inductions--but readers of novels usually cannot be expected to know the technical term "induction."

For a helpful metaphor, you might think of our symbolic deductive logic as a machine for detecting the presence of deductively valid reasoning. In our course, we will occasionally use the machine, but more importantly we will study what it can and cannot do, and whether it can be revised to do other things. Here are some interesting questions about the machine. Does it call any arguments valid that we should call invalid? Does it have the power to show that the premise "Obama's father is working in his office" logically implies the conclusion "Someone's father is working"? In classical logic we assume that contradictions are impermissible, but can we be confident that no use of the machine will lead us to a contradiction even if none has appeared so far?

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 necessarily 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 are the completeness and consistency of classical predicate logic, the Unsolvability of the Halting Problem, the Church-Turing Undecidability Theorem, Tarki's Undefinability Theorem for Truth, and Gödel's Incompleteness Theorems. [You are not yet supposed to know what any of these results are.] Of these, we will spend the most time on Gödel's Incompleteness Theorems

During our course we will review Phil. 60 while providing a rigorous development of both sentential logic and predicate logic. Sentential logic is also called statement logic and propositional logic and propositional calculus and statement calculus and the theory of truth functions. Elementary predicate logic is also called first-order logic, relational logic, quantificational logic and predicate calculus. We will learn about their applications, extensions, meta-theory, and non-classical variants such as modal logic. Regarding non-classical variants, the following controversial but helpful comment was made by the American logician W.V.O. Quine:

Logic is in principle no less open to revision than quantum mechanics 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.

As the student Justin Curry says, "grab on and get ready for a mind-expanding voyage into higher dimensions of recursive thinking."

Topics and reading assignments:

Here is the detailed schedule of all the topics in our course, when we will cover them, and the accompanying reading assignments.

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.

  • Know about, without actually having proved, the most important meta-theoretic results such as Gödel's Theorems, the Church-Turing Undecidability Theorem, and Tarski's Undefinability 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.

  • 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 revisions 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. Audio recording is OK. During class, turn off your cellphone's ringer. 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 on the end of a row so that your monitor's screen won't distract other students. Educational research shows that students learn more when they take their own notes by using a pencil or pen rather than by typing.

Testing protocol:

For in-class tests, you may use any books and notes and laptop computer, but not a cellphone.


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. See also

Plagiarism and Academic Honesty:

Browse the University's policy on academic honesty. A student tutorial on how not to plagiarize is available online from our library.


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


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 and email your answers on the weekend if that is when you finish it. No late work will be accepted after the answer sheet has been handed out (normally this will be at the next class meeting after it is turned in), nor after the answers are discussed in class, even if you weren't in class that day.


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).


My office is in Mendocino Hall 3022, and my weekly office hours for Spring Semester 2016 are there on Tuesday and Thursday 10:30-11:45. Feel free to stop by at any of those times, or to call. I also have online office hours in SacCT every Wednesday evening from 8-10 P.M. 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. Usually 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. Educational research shows that, for the same amount of study hours, it is better to study regularly over a long time and not try to "cram" all at once.

Here are some helpful suggestions from Prof. McCormick.

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


Updated: April 18, 2016