Today’s Lecture1/26/10

Logical Form

5 Famous Valid Forms

Announcements

Homework:

--Read Chapter 7 pp. 277-298 (doing the problems in partsA, B, and C pp. 298-300 are recommended but notrequired at this point).

Quiz this Thursday (the 28th):

--Be ready to state (i.e. write out) from memory the fivefamous valid forms.

The Text: (e-book option); library reserve

Adding the Course

Answers to HW

Ex 1.1 Part A (odds)

1. Statement 23. Neither

3. Sentence only 25. Sentence only

5. Statement 27. Sentence only

7. Neither 29. Neither

9. Neither

11. Statement

13. Statement

15. Sentence only

17. Statement

19. Statement

21. Neither

Answers to HW

Ex 1.1 Part C (odds)

1. Valid

3. Invalid

5. Valid

7. Invalid

9. Invalid

11. Invalid

13. Invalid

15. Valid

Answers

All birds are animals.

No tree is a bird.

So, no tree is an animal.

Key: don’t concern yourself

with the content of theargument i.e. the actualmeaning of the terms ‘bird’,‘animal’, and ‘tree’

Invalid.

It doesn't followabsolutely that no tree isan animal. I.e. it's possiblefor a tree to be an animalgiven the truth of thepremises. We grant thatany tree is not a bird(prem 2), and that any birdis an animal (prem 1). Butit's still possible that thereare other kinds of animals(besides birds) that a treecan be.

Answers

All animals are livingthings.

At least one cabbage is aliving thing.

So at least one cabbage isan animal.

Invalid.

Assume that if something isan animal, then it’s a livingthing (prem 1). It doesn’tfollow that if something is aliving thing, then it’s ananimal. So assuming that atleast one cabbage is a livingthing (prem 2), it doesn’tfollow that at least onecabbage is an animal.

Form and Content

The content of an argument is its subject matter(what the argument is about)

The form of an argument is its underlying patternof inference.

Identifying the Form

The form of an argument can be identified byuniformly replacing the component statementswith upper case letters while leaving the logicalwords intact.

The logical words that concern us for now are: 'it'snot the case' (or 'not'), 'if, then', 'or’, ‘therefore’ (or

‘so’).

Q: why do we want to be able to recognize anargument’s form, particularly if it is a famousvalid form?

-- such recognition is essential to translating English

arguments into logical symbols.

-- the five valid forms are also important inference rulesfor constructing logical proofs.

5 Famous Valid Forms

Modus ponens

Modus tollens

Hypothetical syllogism

Disjunctive syllogism

Constructive dilemma

Modus Ponens (MP)

If Bob is a philosopher,then he is neurotic

Bob is a philosopher

Therefore, he is neurotic

If today is Tuesday, thentomorrow is Wednesday

Today is Tuesday

Therefore, tomorrow isWednesday.

Though these argumentshave different content,they both share the sameargument form.

That form is:

If A, then B

A

Therefore, B

Modus Ponens Continued

If Bob is aphilosopher, then Bobis neurotic

Bob is a philosopher

Therefore, Bob isneurotic

The form is:

If A, then B

A

Therefore, B

A and B stand for the theargument’s componentstatements.

Modus Tollens (MT)

If Bob is a philosopher,then Bob is neurotic

It's not the case that Bob isneurotic

Therefore, Bob is not aphilosopher.

If today is Tuesday, thentomorrow is Wednesday

Tomorrow is notWednesday

Therefore, today is notTuesday.

Both arguments, thoughabout different things,have the same form:

If A, then B

Not B

Therefore, not A

Hypothetical Syllogism (HS)

If an apple is red, thensomething is red.

If something is red, thensomething is colored.

So, if an apple is red, thensomething is colored.

If the universe is 15 billionyears old, then the universe is afinite age.

If the universe is a finite age,then there was a first momentof time.

So, if the universe is 15 billionyears old, then there was a firstmoment of time.

Both arguments have thefollowing form:

If P, then Q

If Q, then R

So, if P then R

Disjunctive Syllogism (DS)

Either cats are canines or the ocean is green

It's not the case that cats are canines

So, the ocean is green.

Q: What is the form of this argument?

Disjunctive Syllogism (DS)

Either cats are canines orthe ocean is green

It's not the case that catsare canines

So, the ocean is green.

The form:

A or B

Not A

So, B

The following is also aversion of (DS):

A or B

Not B

So, A

Constructive Dilemma (CD)

1. Either cats are canines or the ocean is green

2. If cats are canines, then cats have four pointedteeth

3. If the ocean is green, then the ocean is colored

4. So, either cats have four pointed teeth or theocean is colored

Q: What is the form?

Constructive Dilemma (CD)

1. Either cats are caninesor the ocean is green

2. If cats are canines, thencats have four pointedteeth

3. If the ocean is green,then the ocean is colored

4. So, either cats have fourpointed teeth or the oceanis colored

The form:

A or B

If A, then C

If B, then D

So, C or D

Some Terminology

There is an indefinite number of substitution instances ofany argument form.

A substitution instance of an argument form is anargument that results from uniformly replacing thevariables in that form with statements.

A valid argument form is one in which every substitutioninstance is a valid argument.

Some Exercises

P. 29 Part B #s 1,2,4,6,8,10.

Assignment:

Identify the form; if it is a famous valid form,name which one it is.

# 1

If A, then B

A

Therefore, B

MP

# 2

If A, then B

Not B

So, Not A

MT

# 4

if A, then B

B

Therefore, A

(Note: this is an invalid argument form. Somesubstitution instances of this form are invalidarguments).

# 6

A or B

Not A

Therefore, B

DS

#8

A or B

If A, then C

If B, then D

So, C or D

CD

# 10

If A, then B

If B, then C

So, If A, then C

HS

A Few More Exercises

Part C: Pages 30-31. #s 2,7,11,15

#2

Not A

If B, then A

So, Not B

MT

#7

If A, then B

If B, then C

So, if A, then C

HS

#11

Not A

A or B

So, B

DS

#15

If A, then B

Not A

Not B

(Note: this is an invalid argument form. It’spossible to have a substitution instance that isan invalid argument)