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