This document provides information about an introductory symbolic logic course. It includes: - Instructor and contact details for Alex Morgan - Information about the required textbook and online course materials - An outline of assessments including homework and exams - An overview of what students can expect in the course, including that it will emphasize skills over facts and require rigorous precise thinking - Advice that students should practice exercises from the textbook extensively and seek help if struggling - An introduction to logic including what it is, symbolic logic, arguments, statements, propositions, validity, and deductive vs inductive reasoning - Examples of valid and invalid arguments are provided and the importance of logical form is discussed - An overview of

1. PHIL 201:
Introduction to Symbolic Logic
Spring 2009
Instructor Information
Instructor:
Alex Morgan
Office:
Room 011, Davison Hall,
Douglass Campus
Office Hours:
M 6.00-7.30pm, Scott Hall (locn. TBA)
Email:
amorgan@philosophy.rutgers.edu
Phone:
(732) 932 9861, ext.172
Internet:
http://eden.rutgers.edu/~amorgo/
Textbook
Hardegree, G. ‘Symbolic Logic, A First Course’ (2nd Edition)
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Available online here:
www-unix.oit.umass.edu/~gmhwww/110/text.htm
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Also available as hardcopy from bookstores like Amazon
I will be referring to the online version
Known typos are listed on Hardegree’s website

2. Course Website
www.rci.rutgers.edu/~amorgo/teaching/09s_201/
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Provides downloads, including the syllabus and these course notes
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Allows you to ask questions about the homework (see the site for
instructions, or contact me)
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Regularly updated throughout the semester, so check often!
Provides news and information, including information about the
homework and exams
Assessment
Homework (20%)
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A total of 10 bi-weekly homework assignments based on the exercises
in the textbook, each worth 2%. Collected at the end of the Monday
class. The main point of the homework is to demonstrate that you’re
actively working through the material.
Exams (80%)
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Two exams, a mid-term and a final, each worth 40%. They’ll be held
around March 4 and May 4, respectively. I’ll provide more information
about the exams later.
What to Expect
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This course is very different from most other courses in philosophy
(and the humanities generally)
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We’ll be learning how to use an artificial symbolic language, similar to
mathematical ‘languages’ like algebra
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The emphasis will be on...
‣ skills rather than facts and ideas,
‣ rigor and precision rather than creativity and interpretation (at least in
these early stages)

3. What to Expect
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If you enjoy programming, logic puzzles, Sudoku, etc., then you will
probably take to this material quickly, and may even find it fun!
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If not, you should be prepared to put in some extra work
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However, some students have difficulty with the kind of abstract, rulebased thinking required in this course. If this sounds like you (e.g. if
you have difficulty with algebra or computer programming), please
come talk to me after class
Either way, so long you put in the work, you’re almost guaranteed a
good grade
What to Expect
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Please note that this is not the ‘easy logic course’ that you might’ve
heard about! (that’s 730:101)
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Here are some grade distributions from previous semesters:
7
5
6
# Students
8
6
# Students
7
4
3
2
5
4
3
2
1
1
0
0
A
B+
B
C+
C
D
F
A
B+
B
C+
C
D
F
Grade
Grade
Advice
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The material we’re covering might seem easy to begin with, but it
quickly gets much harder. If you get behind it will be very difficult for
you to catch up
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The course is more about learning skills than learning facts, so it is
crucial that you do lots and LOTS of practice using the exercises in
the textbook
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If you find yourself struggling with the course, please come see me
after class or during office hours

4. Why Learn Logic?
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Symbolic logic will help you to be a better reasoner; it will provide you with a
set of tools for analyzing arguments and determining whether they’re any good
‣ Note that the emphasis of the course is not on practical reasoning; if that’s
your main interest, take 730:101
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Some understanding of logic is presupposed in virtually all areas of
contemporary philosophy. Logic is used to analyze complex arguments, and
underlies philosophical theories of meaning, truth and thought
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Logic is used in linguistics to understand syntax and semantics
Logic provides the conceptual foundations of computer science, and is studied in
its own right as a branch of pure math (heard of Goedel’s incompleteness
theorems?)
What is Logic?
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Logic is the study of the principles of ‘good’ or ‘correct’ reasoning
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Some inferences seem good, while others seem not so good
Reasoning involves making inferences from one set of information
to another set of information
‣ If I see smoke and infer that there is fire, this seems like a good
inference
‣ If I see smoke and infer that the moon is made of cheese, this
doesn’t seem like a good inference
What is Logic?
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Systems of logic were studied in Ancient
Greece, China and India
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In Ancient Greece, Aristotle developed a
system of logic that was based on the
analysis of certain kinds of inferences called
syllogisms (more on these later)
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Aristotle's system became the basis of
Wester logic for almost 2,000 years

5. What is Symbolic Logic?
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In the late 1800s, logicians broke from the Aristotelian
tradition and attempted bring the rigor and precision of
mathematics to bear on logic
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They attempted to study logical inference using formal,
axiomatic languages
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This provided a more precise way of analyzing logical
inferences by avoiding the ambiguity of natural languages
like English
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The main figure in the development of symbolic logic
was a German logician named Gottlob Frege
What is Logic?
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Recall that logic in general is the study of good inferences. In formal
logic, we focus on a particular kind of inference, called an argument
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An argument means many things in ordinary language, but for us it will
mean something quite specific:
‣ An argument is a collection of statements, one of which is the
conclusion, and the remainder of which are the premises,
where the premises are intended to ‘support’ or justify the
conclusion
What is an Argument?

6. Statements
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Recall that an argument is a set of statements
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Different kinds of sentences:
A statement is a declarative sentence, i.e. a sentence that is
capable of being true or false
We’re interested in these!
‣ Declarative
“The window is shut”
‣ Interrogative
“Is the window shut?”
‣ Imperative
“Shut the window!”
Statements
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Which of the following are declarative sentences?
‣ Shut the door
‣ It is raining
‣ Are you hungry?
‣ 2+2=4
‣ I am the King of France
Note that whether or not a sentence is declarative doesn’t depend on whether
the sentence is in fact true, but whether it expresses something that could be true
Statements vs. Propositions
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A statement (i.e. a declarative sentence) is said to express a
proposition. You can think of a proposition as (roughly) the
meaning of a statement
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While a statement is something concrete (e.g. a symbol or a soundwave), a proposition is abstract

7. Statements vs. Propositions
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The distinction is similar to the distinction between mathematical
expressions and the numbers they stand for:
‣ ‘4’ and ‘2+2’ and are different mathematical expressions for the
same number, namely 4
‣ Similarly, ‘snow is white’ and ‘der Schnee ist weiss’ are different
statements that express the same proposition, namely that snow is
white
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The distinction is important, but won’t have much of an impact on
what we do in this course
More on Arguments
• Examples of arguments:
Are these arguments good? Why?
(1). If there is smoke, there is fire
There is smoke
Therefore, there is fire
(2). If there is smoke, there is fire
There is smoke
Therefore, I am the King of France
PREMISES
CONCLUSION
PREMISES
CONCLUSION
More on Arguments
(1). If there is smoke, there is fire
There is smoke
Therefore, there is fire
(2). If there is smoke, there is fire
This seems like a good
argument because the
conclusion in some sense follows
from the premises
This seems like a bad argument
There is smoke
because the conclusion has
Therefore, I am the King of France
nothing to do with the premises!

8. Validity
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How can we make this notion of ‘following from’ more precise?
With the notion of validity:
‣ To say that an argument is valid means that it is impossible for the
conclusion of the argument to be false if the premises are true
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Validity has to do with the structure, or form, of the argument, and is
independent of whether the premises of the argument are in fact true
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An argument that is valid and has true premises is called sound
Validity
• More examples of arguments:
Assume that the premises are true;
can the conclusion be false?
(3). All cats are dogs
NO!
The argument is valid
All dogs are reptiles
Therefore, all cats are reptiles
(4). All cats are vertebrates
YES!
The argument is invalid
All mammals are vertebrates
Therefore, all cats are mammals
Validity
T
F
All dogs are reptiles
T
Therefore, all cats are reptiles
T
F
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If the premises were true, the
conclusion would have to be
true, so the argument is
valid.
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However, the premises are in
fact false, so the argument is
not sound
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In terms of its form, the
argument is ‘good’, but in
terms of its content the
argument is not
F
(3). All cats are dogs
cats
dogs
reptiles

9. Validity
All mammals are vertebrates
F
Even though the premises
are true, the conclusion
could still be false, so the
argument is not valid
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Even though it has all true
premises, it is not valid, so it
is automatically not sound
In terms of its content, the
argument is ‘good’, but in
terms of its form, the
argument is not
T T
Therefore, all cats are mammals
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•
T T
(4). All cats are vertebrates
cats
mammals
T
vertebrates
Validity
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Comprehension questions:
‣ Can a valid argument have a false conclusion? Yes
‣ Can a valid argument with true premises have a false conclusion? No
‣ Can anyone give an example of a valid argument with true premises?
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Example:
(5). All cats are mammals
All mammals are vertebrates
(premise 1)
T
(premise 2)
T
Therefore, all cats are vertebrates (conclusion)
T
Why is this
valid? Why
sound?
Validity and Logical Form
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We saw that arguments (3) and (5) are both valid, and that validity has to
do with form. In fact, (3) and (5) have the same form:
(3). All cats are dogs
All dogs are reptiles
Therefore, all cats are reptiles
All X are Y
All Y are Z
(5). All cats are mammals
All mammals are vertebrates
Therefore, all cats are vertebrates
Therefore, all X are Z

10. Validity and Logical Form
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On the other hand, (4) has a different form:
All X are Y
(4). All cats are vertebrates
All mammals are vertebrates
All Z are Y
Therefore, all cats are mammals
Therefore, all X are Z
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If an argument is valid, then any argument with the same form is also
valid
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If an argument is invalid, then any argument with the same form is also
invalid
Validity and Logical Form
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On the other hand, (4) has a different form:
(4). All cats are vertebrates
All X are Y
All mammals are vertebrates
All Z are Y
Therefore, all cats are mammals
Therefore, all X are Z
Note that in the textbook,
statements like these are called
concrete sentences...
...and these are called sentence
forms. Sentence forms don’t express
a particular proposition
Deductive vs. Inductive Logic
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The kind of logic that we study in this class is concerned with
arguments in which the premises are supposed to logically guarantee
the conclusion -- if the premises are true, the conclusion has to be
true. This is called deductive logic
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There is another kind of logic that is concerned with arguments in
which the premises are supposed to make the conclusion more likely,
but not necessarily certain. This is called inductive logic, and is a
much more complicated subject than deductive logic

11. Deductive vs. Inductive Logic
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Recall argument (1):
Now consider argument (7):
‣ If there is smoke, there is fire
‣ There is smoke
‣ There is smoke
‣ Therefore, there is fire
‣ Therefore, there is fire
This is a deductive argument
because the truth of the premises
logically guarantees the truth of the
conclusion
This is an inductive argument
because the truth of the premise
makes the conclusion more likely,
but doesn’t guarantee it
Syllogisms
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A syllogism has two premises and a
conclusion
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The statements that make up a syllogism
contain descriptive terms that refer
to sets of things (e.g. ‘cat’, ‘dog’)
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The statements also contain logical
terms like ‘all’, ‘some’, ‘none’, which
describe relations between sets of things
(7). Some cats are dogs
All dogs are reptiles
Therefore, all cats are reptiles
dogs
cats
reptiles
Syllogisms
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For example, the first premise in (7)
says that some cats are dogs - in other
words, that some of the things in the
‘cat set’ are in the ‘dog set’
(7). Some cats are dogs
All dogs are reptiles
Therefore, all cats are reptiles
Questions:
‣ Is (7) valid? Sound?
‣ What is the logical form of (7)?
dogs
cats
reptiles

12. Next Time...
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Please finish Ch. 1 and make a start on Ch. 2