Logical Form 5 Famous Valid Forms Today s Lecture 1/26/10
Announcements Homework: --Read Chapter 7 pp. 277-298 (doing the problems in parts A, B, and C pp. 298-300 are recommended but not required at this point). Quiz this Thursday (the 28th): --Be ready to state (i.e. write out) from memory the five famous 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 the argument i.e. the actual meaning of the terms bird, animal, and tree Invalid. It doesn't follow absolutely that no tree is an animal. I.e. it's possible for a tree to be an animal given the truth of the premises. We grant that any tree is not a bird (prem 2), and that any bird is an animal (prem 1). But it's still possible that there are other kinds of animals (besides birds) that a tree can be.
Answers All animals are living things. At least one cabbage is a living thing. So at least one cabbage is an animal. Invalid. Assume that if something is an animal, then it s a living thing (prem 1). It doesn t follow that if something is a living thing, then it s an animal. So assuming that at least one cabbage is a living thing (prem 2), it doesn t follow that at least one cabbage 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 pattern of inference.
Identifying the Form The form of an argument can be identified by uniformly replacing the component statements with upper case letters while leaving the logical words intact. The logical words that concern us for now are: 'it's not the case' (or 'not'), 'if, then', 'or, therefore (or so ).
Q: why do we want to be able to recognize an argument s form, particularly if it is a famous valid form? -- such recognition is essential to translating English arguments into logical symbols. -- the five valid forms are also important inference rules for 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, then tomorrow is Wednesday Today is Tuesday Therefore, tomorrow is Wednesday. Though these arguments have different content, they both share the same argument form. That form is: If A, then B A Therefore, B
Modus Ponens Continued If Bob is a philosopher, then Bob is neurotic Bob is a philosopher Therefore, Bob is neurotic The form is: If A, then B A Therefore, B A and B stand for the the argument s component statements.
Modus Tollens (MT) If Bob is a philosopher, then Bob is neurotic It's not the case that Bob is neurotic Therefore, Bob is not a philosopher. If today is Tuesday, then tomorrow is Wednesday Tomorrow is not Wednesday Therefore, today is not Tuesday. Both arguments, though about different things, have the same form: If A, then B Not B Therefore, not A
Hypothetical Syllogism (HS) If an apple is red, then something is red. If something is red, then something is colored. So, if an apple is red, then something is colored. If the universe is 15 billion years old, then the universe is a finite age. If the universe is a finite age, then there was a first moment of time. So, if the universe is 15 billion years old, then there was a first moment of time. Both arguments have the following 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 or the ocean is green It's not the case that cats are canines So, the ocean is green. The form: A or B Not A So, B The following is also a version 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 pointed teeth 3. If the ocean is green, then the ocean is colored 4. So, either cats have four pointed teeth or the ocean is colored Q: What is the form?
Constructive Dilemma (CD) 1. Either cats are canines or the ocean is green 2. If cats are canines, then cats have four pointed teeth 3. If the ocean is green, then the ocean is colored 4. So, either cats have four pointed teeth or the ocean is 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 of any argument form. A substitution instance of an argument form is an argument that results from uniformly replacing the variables in that form with statements. A valid argument form is one in which every substitution instance 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. Some substitution instances of this form are invalid arguments).
# 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 s possible to have a substitution instance that is an invalid argument)