PHIL. 160
Deductive Logic II
Spring Semester 2019
Prof. Dowden
PHIL 160. Deductive Logic II. 3 Units
Prerequisite(s): Phil. 160.
Let's face it. There are three
kinds of people in the world: the kind who are good at a course like this,
and the kind who are not.
Catalog
description:
PHIL 160. Deductive Logic II. 3 Units
Prerequisites: CSC 28 or PHIL 60 or instructor permission.
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.
Textbooks:
The required textbooks and articles are free online in Canvas. The main textbook is the 2015 pdf revision of Logics by John Nolt.Grades:
Your grade will be determined by five required homework assignments (8% for the first, then 11% for each of the others), one in-class midterm exam (22%), and an in-class final exam (26%). Homework questions will be released at least a week in advance of the due date. Class attendance is optional, but you will be responsible in the above assignments for material covered in class that is not in the readings. All exams are open-book and open-notes. Homework is required, not optional. Late answers will receive 1/3-of-a-letter penalty for each 24 hours late, including weekends.
Due dates:
Homework 1: (wk 2) Jan. 31, 2019
Homework 2: (wk 4) Feb. 14
Homework 3: (wk 6) Feb. 28
Midterm: (wk 8) Mar. 14
Homework 4: (wk 11) Apr. 11
Homework 5: (wk 14) May 2
Final Exam: (wk 16)
Course Description:
This is a second course in symbolic logic. Our course presupposes you have had a first course in symbolic deductive logic, such as Sac State's PHIL 60, or CSC 28, or you have learned this material on your own. Initially, and also throughout the semester, we will review PHIL 60, but also will significantly enrich that material as we go along. Our course will include applications of formal logic to the fields of computer science, mathematics and philosophy.
Logic is the study of reasoning, especially of what has to be true if something else were to be true. Rhetoric, as opposed to logic, is the study of persuasion, of what is likely to be convincing to some particular person under certain circumstances. In a course on rhetoric you might study whether a particular piece of reasoning is more likely to be convincing to Democrats than to Republicans. Our course focuses on logic and not rhetoric.
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. This rule about logical form, called "modus
ponens," was discovered by Aristotle's pupil and successor Theophrastus of Eresus
(c. 371 – c. 287 B.C.E.) in Greece. By the word "logic"
in our course, 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.
In our course, we will develop logic symbolically by creating formal
systems, which are formal languages plus a system of making deductions of
some sentences from others. These formalisms don't have so many imperfections as ordinary English. In
exploring the formalisms we will use two
principal techniques: the method of proof and the method of finding a counterexample.
Intuitively and informally, a formal system is a system of symbols that
are manipulated by the logician in game-like fashion for the purpose of
more deeply understanding the properties of the structure that the formal
system is about.
Our long-term goal is to appreciate what can be done with symbolic
deductive logic and what can't. That is, we will explore the scope and limits of
symbolic deductive logic.
Deductive logic, as opposed to inductive logic, is a field of study that explores deductively valid reasoning, the most secure kind of reasoning. An argument is a set of premises together with a conclusion, and the most elemental piece of deductively valid reasoning is an argument that draws one conclusion from one premise (one assumption) such that the truth of the premise guarantees the truth of the conclusion. In any argument, no matter how complicated, 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 a deduction that the axe murderer had recently been in Sweden, he is really making what logicians call an "induction"—but readers of novels cannot be expected to know our 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 should be called invalid? Does it call every intuitively invalid argument "formally invalid"? Does it have the power to show that the conclusion "Someone's father is working" follows from the premise "Obama's father is working in his office? 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, that is, will never prove both some sentence A and its negation?
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.]
Regarding the issue of freeing our logic from contradictions, we will explore some of the interesting paradoxes. One of these is the dilemma of the crocodile, described by Patrick Suppes as follows: The crocodile has stolen a child. He says to the child's father, "I will return the child if you guess correctly whether or not I will return the child." The father replies, "You will not return the child." What should the crocodile do, assuming he wants to keep his promise? It seems to follow that the crocodile will if he won't, and he won't if he will.
During our course we will review Phil. 60 while providing a rigorous development of both sentential logic and predicate logic followed by an exploration of new logics beyond these two classical logics. 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 some truth trees 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.
For one example, a famous meta-theorem of predicate logic is Gödel's First Incompleteness Theorem. In addition to studying its proof, we will ask this controversial question: Does the theorem imply the existence of facts that must be true, but that our minds can never prove? This is a controversial because some experts answer with "yes," and some with "no."
We will venture a little into
non-classical logics, into the philosophy of logic, and into 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.
As we explore the various systems of formal logic, "each system is
introduced," says the author of your textbook John Nolt, "first, by way of
concrete problems that motivate it and then by an account of its semantics
[of how truths require other truths]. Proof theory [what can be proved
from what], though usually historically prior, is relegated to third
place, since much that is puzzling about proofs can be elucidated
semantically, whereas relatively little that is puzzling about semantics
can be illuminated by proofs. The ultimate step for each system is an
ascent to the vantage point of metatheory, where the deepest understanding
may be achieved."
Nolt and Dowden are proponents of logical
pluralism, which is the claim that there is not one privileged or uniquely
true logical system; rather, different logics are appropriate for
different applications.
This sentence cannot be shown to be true using any kind of sound reasoning.If this sentence is false, then it can be shown to be true using sound reasoning. So, we've now learned that the sentence is true if it is false. But, if it is true, then we have found an odd restriction on the human mind because we can't use sound reasoning to establish that it is.
Grab on and get ready for a mind-expanding voyage into the higher dimensions of recursive thinking.
Be able to reason more effectively.
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 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 to be capable of deciding whether an attempted
proof is incorrect.
Understand Hilbert's program and the process of formally axiomatizing a
theory.
Know about important meta-theoretic
results such as the Löwenheim-Skolem Theorem, Gödel's Theorems,
the Unsolvability of the Halting Problem for computers, the
Church-Turing Undecidability Theorem, and Tarski's Undefinability
Theorem.
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.
For example, be able to explain why, no matter what algorithms we create,
no matter how sophisticated or fast we make our computers, there will
always be logic problems that we cannot solve; and we know that this
limitation on logic can
be proved logically.
Know that classical logic can be augmented to produce richer logical
systems. 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 and videotaping during class are 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.
Computers and cell phones can be used during in-class exams (you might be
asked to work a problem in the textbook).
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. See also http://www.csus.edu/sswd/index.html.
Plagiarism and Academic Honesty:
Browse the University's policy on academic honesty.
I
realize that during your college career you occasionally may be unable
to complete an assignment on time. 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 whenever you finish it. No late work will be accepted
after the answer sheet has been handed out or posted (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. Here are some helpful suggestions from Prof. McCormick. Contact me at dowden@csus.edu if you'd like more information about our course. Food:
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:
Add-Drop:
To add
the course, try to do so by using “My Sac State” (https://www.my.csus.edu) . 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
My Sac State. 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; 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 for Spring Semester 2019 are
Tuesday and Thursday 10:45 A.M. - noon.
Feel free to stop by at any of those times. If
those hours are inconvenient for you, then I can arrange an appointment
for an alternative time. I also have online office hours in Canvas
every Wednesday evening from 9 - 10 P.M. Usually the fastest way to
contact me is to send e-mail to dowden@csus.edu.
You may expect a response within 24 hours, usually sooner. My personal web page is at
http://www.csus.edu/indiv/d/dowdenb/index.htm 
Prof. DowdenStudy tips:
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.
Nolt wrote two other books that are simplified versions of his Logics textbook.
One is Schaum's
Outlines: Logic ($13.00 on amazon.com). A second, even
simpler book, is called Schaum's
Easy Outlines: Logic ($9.63 on amazon.com), but it doesn't provide as
many worked problems and it covers fewer topics. Both books provide a wonderful way to get the main ideas of his
Logics more easily. You will find the treatment of the semantics
of predicate logic (which is chapter 7 in Logics) to be
especially helpful. Our Logics book is free in Canvas and covers
other interesting topics beyond those in the two simpler books.
The Logics book
contains hundreds of typographical errors, so you often face the problem of
deciding, for example, whether that apostrophe is there on purpose or by accident.
Updated:
February 17, 2019