PHIL. 160
Deductive Logic II


Spring Semester 2018
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.) 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. $222.95 for the paperback. $44.49 for a timed rental of the eBook through May 17, 2018. You can bring eBooks to all in-class tests. 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 as an eBook in various digital 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, the Zen Buddhis, 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.

Due dates:

Homework 1:  (wk 3) Feb. 7, 2018

Homework 2:  (wk 6) Feb. 28

Midterm:  (wk 8) Mar. 14

Homework 3:  (wk 11) Apr. 11

Homework 4:  (wk 14) May 2

Final Exam: (wk 16)

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

Extra credit: You have the option of earning extra credit by giving a 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 so far 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 from me 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.

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.

Rhetoric, on the other hand, is the study of persuasion, of what is or might be convincing to someone under certain circumstances. In a course on rhetoric you might study what reasoning is more likely to be convincing to Democrats than to Republicans. 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, a sense that includes being somewhat prescriptive.

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."

The rules of symbolic logic are rules that apply to reasoning regardless of what it is about. That is, the rules are about form and not content. For example, concluding sentence B from sentence A plus a sentence of the form 'If A, then B' is always good reasoning regardless of whether the content of sentences A and B is about basketball or the price of rice in China.

By "logic," we ordinarily will mean symbolic logic, also known as formal logic. This is logic using symbols, not words. A bad idea does not get any better by expressing it in symbols; but for many ideas, if you express them symbolically and are familiar with the symbolic techniques, then you can more easily see whether they are good ideas or bad ideas. The introduction of formal methods in the last one hundred years has led to enormous gains in clarity and conceptual power.

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 symbolic logic and what can't. That is, we will explore the scope and limits of deductive logic. We also will explore depth in one particular area, the Hilbert Program, which will be explained later in the course.

Deductive logic explores deductively valid reasoning, the most secure kind of reasoning. The most elemental piece of reasoning of this sort is an argument that draws one conclusion from one premise (also called an "assumption.") In any argument, if the conclusion is a logical consequence of the premises, that is, if the conclusion follows with certainty from the premises, then we call the argument deductively valid. A correct mathematical proof is an example of a (usually complicated) 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. Phil. 60 uses the machine. In our course, we will occasionally use the machine, but more importantly we will study what the machine 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 call every intuitively invalid argument "formally 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 really be confident that no use of the machine will lead us 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 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. Major results here are the completeness and consistency of classical predicate logic, Turing's analysis of a mechanical procedure and 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 construct a few truth tables and prove a few theorems within the logics, but our main focus will be about the logics. We will learn about their applications, extensions, meta-theory, and non-classical variants such as modal logic.

We will venture a little into non-classical logics, the philosophy of logic, and logic's connections to fields beyond philosophy such as mathematics and computer science. We follow the lead of C. I. Lewis who argued in 1922 that there are many logics for many purposes, and that it is a mistake to think that there is only a single, correct logic.

Regarding the certainty of logic,  the American logician W.V.O. Quine wrote:

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.

Grab on and get ready for a mind-expanding voyage into the higher dimensions of recursive thinking.

Topics and reading assignments:

Here is the detailed schedule of all the topics in our course, including when we will cover them, and their 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 deciding whether an attempted proof is incorrect. 

  • 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 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.


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.


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 2018 are Monday and Wednesday 10:45 A.M. - noon. Feel free to stop by at any of those times, or to call my office at 1-916-278-6424. If those hours are inconvenient for you, then I can arrange an appointment for an alternative time. I also have online office hours in SacCT every Wednesday evening from 9 - 10 P.M. Usually the fastest way to contact me is to send e-mail to You may expect an response within 24 hours, usually sooner. My personal web page is at

photo of Dowden

Prof. Dowden


Study tips:

If you'd like a gentler review of Sentential Logic and Predicate Logic than that offered by our Hausman textbook, then I recommend the very small book Schaum's Easy Outlines: Logic by John Nolt and others. For a more advanced book that is still more gentle than the Hausman textbook, I recommend the large book Schaum's Outlines: Logic Second Edition also by John Nolt, and others. Both the Schaum books call our Sentential Logic by the name "Propositional Logic," but for us these are the same thing. Logic Second Edition covers formal arithmetic (Robinson Arithmetic), but the easier Schaum book does not.

As you read an assignment, it is helpful first to skim the assignment to get some sense of what is 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 practice 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. Even-numbered exercises are answered in the back of the Hausman book. You are encouraged to practice by working some of these exercises.

Here are some helpful suggestions from Prof. McCormick.

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

Here are two fun items about logic that can be enlightening to think about:

If I call you a swine, then I call you an animal.
If I call you an animal, then I speak truly.
Therefore, if I call you a swine, then I speak truly.

There is an infinite line of people, and they all say, "Everybody behind me is saying something false." Is what the first person in the sequence says true, or is it false, or could it be either?


Updated: February 28, 2018