geography lecture

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: [email protected]

Phone: (732) 932 9861, ext.172

Internet: http://eden.rutgers.edu/~amorgo/

Textbook

• 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

Hardegree, G. ‘Symbolic Logic, A First Course’ (2nd Edition)

Course Website

• 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!

www.rci.rutgers.edu/~amorgo/teaching/09s_201/

AssessmentHomework (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:

0

1

2

3

4

5

6

7

A B+ B C+ C D F

Grade

# S

tude

nts

0

1

2

3

4

5

6

7

8

A B+ B C+ C D F

Grade

# S

tude

nts

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

Why 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 years

What 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 conclusion

What is an Argument?

‣ Declarative “The window is shut”

‣ Interrogative “Is the window shut?”

‣ Imperative “Shut the window!”

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

• Different kinds of sentences:We’re interested in these!

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 abstract

Statements 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

• Examples of arguments:

(1). If there is smoke, there is fire

There is smoke

Therefore, there is fire


There is smoke

Therefore, I am the King of France

PREMISES

CONCLUSION

PREMISES

CONCLUSION

Are these arguments good? Why?

More on Arguments(1). If there is smoke, there is fire

There is smoke

Therefore, there is fire


There is smoke

Therefore, I am the King of France

This seems like a good

argument because the

conclusion in some sense follows

from the premises

This seems like a bad argument

because the conclusion has

nothing to do with the premises!

Validity

• 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

• More examples of arguments:

(3). All cats are dogs

All dogs are reptiles

Therefore, all cats are reptiles

(4). All cats are vertebrates

All mammals are vertebrates

Therefore, all cats are mammals

Assume that the premises are true;can the conclusion be false?

YES!The argument is invalid

NO!The argument is valid

Validity(3). All cats are dogs



reptiles

dogs

cats

T

T

• If the premises were true, the conclusion would have to be true, so the argument is valid.

• However, the premises are in fact false, so the argument is not sound

• In terms of its form, the argument is ‘good’, but in terms of its content the argument is not

F

F

FT

ValidityT

T

• Even though the premises are true, the conclusion could still be false, so the argument is not valid

• 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

TF

vertebrates

mammalscats




Validity• Comprehension questions:

‣ Can a valid argument have a false conclusion?

‣ Can a valid argument with true premises have a false conclusion?

‣ Can anyone give an example of a valid argument with true premises?

• Example:

(5). All cats are mammals (premise 1) T

All mammals are vertebrates (premise 2) T

Therefore, all cats are vertebrates (conclusion) T

Why is this valid? Why sound?

Yes

No

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 X are Y

All Y are Z

Therefore, all X are Z(5). All cats are mammals


Therefore, all cats are vertebrates

Validity and Logical Form• On the other hand, (4) has a different form:




All X are Y

All Z are Y

Therefore, all X are Z

• If an argument is valid, then any argument with the same form is also valid

• If an argument is invalid, then any argument with the same form is also invalid

Validity and Logical Form• On the other hand, (4) has a different form:




All X are Y

All Z are Y

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

• 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

• 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

Deductive vs. Inductive Logic

‣ If there is smoke, there is fire

‣ There is smoke

‣ Therefore, there is fire

‣ There is smoke

‣ Therefore, there is fire

• Recall argument (1): • Now consider argument (7):

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• A syllogism has two premises and a

conclusion

• The statements that make up a syllogism contain descriptive terms that refer to sets of things (e.g. ‘cat’, ‘dog’)

• The statements also contain logical terms like ‘all’, ‘some’, ‘none’, which describe relations between sets of things

(7). Some cats are dogs



reptiles

dogscats

Syllogisms• 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’

• Questions:

‣ Is (7) valid? Sound?

‣ What is the logical form of (7)?

(7). Some cats are dogs



reptiles

dogscats

Next Time...

• Please finish Ch. 1 and make a start on Ch. 2

geography lecture

Technology

system of logic

formal logic

semantics logic

logic puzzles

systems of logic

easy logic course

development of symbolic

basis of wester logic