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THE PROLEGOMENON: FAMILY HISTORY AND GENEALOGY (Collected Family Charts are found on on my personal page accessible from the home page by clicking on “Myron Evans”). Appendices are being added gradually with details (see end of this file). In the direct line, my paternal great great grandfather, Edward Evans, is recorded in the 1851 Census as having been born in Llanigon in Powys in 1812.
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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)
Available online here:•
www-unix.oit.umass.edu/~gmhwww/110/text.htm
Also available as hardcopy from bookstores like Amazon•
I will be referring to the online version•
Known typos are listed on Hardegree’s website•Course Website
www.rci.rutgers.edu/~amorgo/teaching/09s_201/
Provides downloads, including the syllabus and these course notes•
Provides news and information, including information about the •
homework and exams
Allows you to ask questions about the homework (see the site for •
instructions, or contact me)
Regularly updated throughout the semester, so check often!•
Assessment
Homework (20%)
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%)
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
This course is very different from most other courses in philosophy •
(and the humanities generally)
We’ll be learning how to use an artificial symbolic language, similar to •
mathematical ‘languages’ like algebra
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)What to Expect
If you enjoy programming, logic puzzles, Sudoku, etc., then you will •
probably take to this material quickly, and may even find it fun!
If not, you should be prepared to put in some extra work•
Either way, so long you put in the work, you’re almost guaranteed a •
good grade
However, some students have difficulty with the kind of abstract, rule-•
based 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
What to Expect
Please note that this is not the ‘easy logic course’ that you might’ve •
heard about! (that’s 730:101)
Here are some grade distributions from previous semesters:•
7 8
76
6
5
5
4
4
3
3
2
2
1 1
0 0
A B+ B C+ C D F A B+ B C+ C D F
Grade Grade
Advice
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
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
If you find yourself struggling with the course, please come see me •
after class or during office hours
# Students
# StudentsWhy Learn Logic?
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
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
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?
Logic is the study of the principles of ‘good’ or ‘correct’ reasoning•
Reasoning involves making inferences from one set of information •
to another set of information
Some inferences seem good, while others seem not so good•
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?
Systems of logic were studied in Ancient •
Greece, China and India
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)
Aristotle's system became the basis of •
Wester logic for almost 2,000 yearsWhat is Symbolic Logic?
In the late 1800s, logicians broke from the Aristotelian •
tradition and attempted bring the rigor and precision of
mathematics to bear on logic
They attempted to study logical inference using formal, •
axiomatic languages
This provided a more precise way of analyzing logical •
inferences by avoiding the ambiguity of natural languages
like English
The main figure in the development of symbolic logic •
was a German logician named Gottlob Frege
What is Logic?
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
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
What is an Argument?Statements
Recall that an argument is a set of statements•
A statement is a declarative sentence, i.e. a sentence that is •
capable of being true or false
We’re interested in these!
Different kinds of sentences:•
Declarative “The window is shut”‣
Interrogative “Is the window shut?”‣
Imperative “Shut the window!”‣
Statements
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
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
While a statement is something concrete (e.g. a symbol or a sound-•
wave), a proposition is abstractStatements vs. Propositions
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
The distinction is important, but won’t have much of an impact on •
what we do in this course
More on Arguments
Are these arguments good? Why?Examples of arguments:•
(1). If there is smoke, there is fire
PREMISES
There
Therefore, there is fire CONCLUSION
(2). If there is smoke, there is fire
PREMISES
There
Therefore, I am the King of France CONCLUSION
More on Arguments
(1). If there is smoke, there is fire This seems like a good
argument because the
There is smoke
conclusion in some sense follows
Therefore, there is fire from the premises
(2). If there is smoke, there is fire
This seems like a bad argument
There is smoke because the conclusion has
nothing to do with the premises!Therefore, I am the King of FranceValidity
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
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
An argument that is valid and has true premises is called sound•
Validity
Assume that the premises are true;More examples of arguments:• can the conclusion be false?
(3). All cats are dogs
NO!
All dogs are reptiles
The argument is valid
Therefore, all cats are reptiles
(4). All cats are vertebrates
YES!
All mammals are vertebrates
The argument is invalid
Therefore, all cats are mammals
Validity
If the premises were true, the (3). All cats are dogs T F •
conclusion would have to be
All dogs are reptiles T F true, so the argument is
valid.
TTherefore, all cats are reptiles F
However, the premises are in •
fact false, so the argument is
not sound
In terms of its form, the cats •
argument is ‘good’, but in
dogs terms of its content the
argument is notreptilesValidity
Even though the premises (4). All cats are vertebrates T T •
are true, the conclusion
All mammals are vertebrates T T could still be false, so the
argument is not valid
Therefore, all cats are mammals F T
Even though it has all true •
premises, it is not valid, so it
is automatically not sound
In terms of its content, the •cats mammals
argument is ‘good’, but in
terms of its form, the
argument is notvertebrates
Validity
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?‣
Example:•
(5). All cats are mammals (premise 1) T
Why is this
valid? Why All mammals are vertebrates (premise 2) T
sound?
Therefore, all cats are vertebrates (conclusion) T
Validity and Logical Form
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
All X are Y
Therefore, all cats are reptiles
All Y are Z
(5). All cats are mammals
Therefore, all X are Z
All mammals are vertebrates
Therefore, all cats are vertebratesValidity and Logical Form
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
If an argument is valid, then any argument with the same form is also •
validinvalid, then anorm is also •
invalid
Validity and Logical Form
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, ...and these are called sentence
statements like these are called forms. Sentence forms don’t express
concrete sentences... a particular proposition
Deductive vs. Inductive Logic
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 pre true, the conclusion has to be
true. This is called deductive logic
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