Chap.6

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Which of the following propositions is a substitution instance of “p & q & q“?

1. The night is young, and you’re so beautiful, and my flight leaves in thirty minutes.

2. The night is young, and you’re so beautiful, and my flight leaves in thirty minutes, and my flight leaves in thirty minutes.

3. You’re so beautiful, and you’re so beautiful, and you’re so beautiful.

Exercise II

For each of the following propositions, give three different propositional forms of which that proposition is a substitution instance.

1. The night is young, and you’re so beautiful, and my flight leaves in thirty minutes.

2. The night is young, and you’re so beautiful, and you’re so beautiful.

Exercise III

Indicate whether each of the following sentences expresses a propositional con- junction or a nonpropositional conjunction—that is, whether or not it expresses a conjunction of two propositions. If the sentence could be either, then specify a context in which it would naturally be used to express a propositional conjunc- tion and a different context in which it would naturally be used to express a nonpropositional conjunction.

1. A monk married Devon and Nassier. 2. Fred had pie and ice cream for dessert. 3. The winning presidential candidate rarely loses both New York

and California.

Exercise IV

The proposition “The night is young, and you’re so beautiful” is a substitution instance of which of the following propositional forms?

1. p

2. q

3. p & q 4. p & r

Exercise I

5. p & q & r 6. p & p 7. p or q

General Notes on the Chapter

Note #1: there’s lots of room for extra practice on most of these exercises, and I heartily encourage you to do more! Learning logic is a matter mostly of practice and familiarity, and you can’t get that from cramming. The best policy is doing a little bit of logic every day – even 20-30 min every day will work wonders.

Note #2: the first major obstacle with learning formal logic is learning the procedure of how to go about doing these various tasks of analysis. If you’re getting stuck in that way where you don’t even know how to begin, contact me ASAP! 5 min on the phone with me can save you hours of frustration. I’m here to help prevent those sorts of tragedies!

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Now we can look at an argument involving conjunction. Here is one that is ridiculously simple:

Harry is short and John is tall. Harry is short.

This argument is obviously valid. But why is it valid? Why does the conclusion follow from the premise? The answer in this case seems obvious, but we will spell it out in detail as a guide for more difficult cases. Suppose we replace these particular propositions with propositional forms, using a different vari- able for each distinct proposition throughout the argument. This yields what we will call an argument form. For example:

p & q p

This is a pattern for endlessly many arguments, each of which is called a sub- stitution instance of this argument form. Every argument that has this general form will also be valid. It really does not matter which propositions we put into this schema; the resulting argument will be valid—so long as we are careful to substitute the same proposition for the same variable throughout.

Let’s pursue this matter further. If an argument has true premises and a false conclusion, then we know at once that it is invalid. But in saying that an argument is valid, we are not only saying that it does not have true premises and a false conclusion; we are also saying that the argument cannot have a false conclusion when the premises are true. Sometimes this is true because the argument has a structure or form that rules out the very possibility of true premises and a false conclusion. We can appeal to the notion of an argument form to make sense of this idea. A somewhat more complicated truth table will make this clear:

PREMISE CONCLUSION

p q p & q p

T T T T

T F F T

F T F F

F F F F

4. Susan got married and had a child. 5. Janet speaks both French and English. 6. Someone who speaks both French and English is bilingual. 7. Hei and Naomi are two of my best friends. 8. Miranda and Musa cooked dinner. 9. I doubt that John is poor and happy.

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form. There are, in fact, some obviously valid arguments that have yet to be shown to be valid in terms of their form. Explaining validity by means of logi- cal form has long been an ideal of logical theory, but there are arguments— many of them quite common—where this ideal has yet to be adequately fulfilled. Many arguments in mathematics fall into this category. At present, however, we will only consider arguments in which the strategy we used for analyzing conjunction continues to work.

Are the following arguments valid by virtue of their propositional form? Why or why not?

1. Donald owns a tower in New York and a palace in Atlantic City. Therefore, Donald owns a palace in Atlantic City.

2. Terrell owns a house. Therefore, Terrell owns a house and a piece of land. 3. Ilsa is tall. Therefore, Ilsa is tall, and Ilsa is tall. 4. Bernie has a son and a daughter. Bernie has a father and a mother.

Therefore, Bernie has a son and a mother. 5. Elena got married and had a child. Therefore, Elena had a child and got

married. 6. Bess and Katie tied for MVP. Therefore, Bess tied for MVP.

Exercise V

For each of the following claims, determine whether it is true or false. Defend your answers.

1. An argument that is a substitution instance of a valid argument form is always valid.

2. An argument that is a substitution instance of an invalid argument form is always invalid.

3. An invalid argument is always a substitution instance of an invalid argu- ment form.

Exercise VI

Is a valid argument always a substitution instance of a valid argument form? Why or why not?

Discussion Question

By “virtue of their propositional form” the instructions mean that if the argument is valid, it is true just because of the structure of the argument and not because of the content of what the claims are talking about. Remember how we could swap out elements from the “Clue” example we did in class and the argument would stay valid? That is an example of an argument that is valid because of the form (and not the content).

You only need to do the even problems officially, but feel free to do more for extra practice!

This exercise is like a “training wheels” version of truth-tables. Here you are making single calculations instead of the table of calculations you’d do with a full truth-table. Your only job here is to state whether the claim is true or false under the conditions where the simple propositions A, B, and C are true and X, Y, and Z are false.

Here just do the odd numbered problems. Again you can do more for extra practice (and extra practice is really REALLY helpful for learning logic!)

Do problems 5, 7, 13, and 18 minimally, but there’s lots of room for extra practice!

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13. Unless there is a panic, stock prices will continue to rise. (P, R) 14. I won’t scratch your back unless you scratch mine. (I, Y) 15. You will get a good bargain provided you get there early. (B, E) 16. You cannot lead a happy life without friends. (Let H = You can lead a

happy life, and let F = You have friends.) 17. The only way that horse will win the race is if every other horse drops

dead. (Let W = That horse will win the race, and let D = Every other horse drops dead.)

18. You should take prescription drugs if, but only if, they are prescribed for you. (T, P)

19. The grass will die without rain. (D, R = It rains.) 20. Given rain, the grass won’t die. (R, D = The grass will die.) 21. Unless it doesn’t rain, the grass won’t die. (R, D = The grass will die.)

Exercise XXVIII

(a) Translate each of the following arguments into symbolic notation. Then (b)" test each argument for truth-functional validity using truth table techniques.

Now we’re putting it all together! Many of the skills we’ve been learning are linked together here.

6. If Bobby moves his queen there, he will lose her. Bobby will not lose his queen. Therefore, Bobby will not move his queen there. (M, L)

7. John will play only if the situation is hopeless. But the situation is hopeless. So John will play. (P, H)

8. Although Brown will pitch, the Rams will lose. If the Rams lose, their manager will get fired. So their manager will get fired. (B, L, F)

9. America will win the Olympics unless China does. China will win the Olympics unless Germany does. So America will win the Olympics unless Germany does. (A, R, E)

10. If you dial 0, you will get the operator. So, if you dial 0 and do not get the operator, then there is something wrong with the telephone. (D, O, W)

11. The Democrats will run either Jones or Borg. If Borg runs, they will lose the South. If Jones runs, they will lose the North. So the Democrats will lose either the North or the South. (J, B, S, N)

12. I am going to order either the fish special or the meat special. Either way, I will get soup. So I’ll get soup. (F, M, S)

13. The grass will die if it rains too much or it does not rain enough. If it does not rain enough, it won’t rain too much. If it rains too much, then it won’t not rain enough. So the grass will die. (D = The grass will die, M = It rains too much, E = It rains enough.)

14. If you flip the switch, then the light will go on. But if the light goes on, then the generator is working. So if you flip the switch, then the generator is working. (F, L, G) (This example comes from Charles L. Stevenson.)

Note that for these problems you'll need to provide your own "universe of discourse". In other words, you need to define the simple propositions that each propositional letter will represent in your translation.

I will make it easier for you on the exam by providing you with a universe of discourse for each problem.