Introduction to Logic



Introduction to Logic Sample Final Exam

The rules of Replacement and Quantifier use are at the end.

For #1-3, write at least two paragraphs—thorough discussion that shows genuine knowledge.

1. What is an argument? Choose two non-argument forms of speech that might be confused with arguments, and explicate how arguments are really quite different from them. Explicate the difference between the two families of arguments at length, using and clarifying all the basic terminology needed to make this clear. (6 pts)

2. Explain what a counter-example is, what it does, and how it does it. Take care to correctly differentiate between the use of “valid” and “true.” Give an example of a counter-example; first give an example of an invalid argument (either in propositional or categorical logic). (6 pts)

3. Thoroughly explicate what truth tables can tell us about a) individual statements, b) sets of statements, and c) arguments. Then put the following argument into propositional notation, and fill in the eight-line truth table. Report what you can about a) the individual statements (each one by itself) b) the total set of statements (i.e., what can be said about them in relation to each other, and c) the argument. Make sure you are clear and your answer is complete. Either you’re with us or you’re against us. If you’re with us, you’re a patriot, but if you’re against us, you’re a terrorist, so you’re either a patriot or a terrorist. (12 pts)

4. List four conclusion-indicating and four premise-indicating words. From each set, identify a word that is not always an argument indicator, and explicate (write a few sentences) why it is not always a logical indicator. (3 pts)

5. Identify the fallacy in each passage. You can get credit for only one fallacy per passage --that means do not provide more than one fallacy or you will get no credit. Support your answer in a complete sentence or two, which includes a definition of the fallacy, and write out the conclusion of each passage. (15 points)

A. The last three times we sat down for a fancy dinner, my mother called. We haven’t heard from her in a while, and I wonder how she’s doing. Why don’t we have a fancy dinner tonight and I’m sure we’ll hear from her?

B. When a car breaks down so often that repairs become pointless, it is thrown on the junk heap. Likewise, when a person becomes too old and sick, he should just be put to death.

C. The universe is spherical in form because all the constituent parts of the universe, that is the sun, the moon, and the planets, appear in this form. --Copernicus, "The New Idea of the Universe"

D. We know that what the Bible says is true, because it is the word of God, and we know it’s the word of God because it says so right there in the Bible.

E. Mr. Larme, a noted defense lawyer, wins his cases by crying in court while pleading his clients' cases to the jury. This sort of thing --swaying people by such blatant appeals to their sense of pity-- is unfair and despicably low. You just can't trust lawyers.

6. Analyze these in Categorical Logic. Be thorough, cover every step on the way to showing whether each is valid or not: a) translation to standard form, b) correct order, c) mood and figure, d) Venn diagram, e) rule broken if invalid, f) counterexample (12 pts)

A. “Of course every club member can vote. Here are the rules: everyone who can vote subscribes to the newsletter, and you’re automatically a subscriber if you’re a member of the club.”

B. No one who misses weeks of class expects to get a good grade, so Adam doesn’t expect a good grade. (What is this kind of argument called?)

7. Write a proof for this using Indirect Proof. (6 pts)

1. ~ (T . ~P) ( ~G

2. (S v T) ( J

3. G / J

8. Use CP to write a proof for this one. (6 pts)

1. P ( [(L v M) ( (O . N)]

2. (T v O) ( W / P ( (M ( W)

9. Translate these into the notation of Predicate Logic. (9 pts)

A. Only registered voters will be given a ballot, and Joe’s not registered.

B. A categorical syllogism is invalid if it breaks a rule, and if it doesn’t break a rule, it’s valid.

C. If there are no legumes in the garden, and beans and peas are legumes, then there are no beans or peas in the garden.

10. Translate this and write a proof in Predicate Logic. (10 pts)

All ambassadors are diplomats. Furthermore, any experienced ambassador is cautious, and all cautious diplomats have foresight. Therefore all experienced ambassadors have foresight.

11. Using the predicates indicated, write this out as a word problem in colloquial English. (5 pts)

A = is alright

W= is worried

H = is happy

P = is a person

1. (x) Ax > (x) (Px > Hx)

2. (x) [(Px . Wx) > ~Hx]

3. (Pa . Pb) . (Wa . Wb) / (Зx) ~Ax

Write a proof for it (in symbols, not in English), using CQ (not CP!). (5 pts)

12. (5 pts)

Either (a) Put each of these pairs into English:

(x) Fx :: ~(Зx) ~Fx

(x)~Fx :: ~(Зx) Fx

(Зx) Fx :: ~(x)~ Fx

(Зx)~Fx :: ~(x) Fx

Or (b) Write a proof for this one :

1. (Зx) (Ax v Bx) ( ~(Зx) Ax / (x) ~Ax

Rules:

Removing or adding quantifiers.

UI --Universal Instantiation

(x) (Ax > Bx)

---------------------

Ax > Bx or Aa > Ba

UG --Universal Generalization

Ax > Bx not allowed: Aa > Ba

-------------- ------------

(x) (Ax > Bx) (x) (Ax > Bx)

EI --Existential Instantiation

Not allowed: (Зx) (Ax . Bx)

(Зx) (Ax . Bx) -------------------

------------------ Ax . Bx

Aa . Ba

Only instantiate to a name. Restriction: The name must be a new name

EG --Existential Generalization

Aa . Ba or Ax . Bx

---------------------

(Зx) (Ax . Bx)

Rules of Equivalence/ Replacement

DN p :: ~~p

CM (p . q) :: (q . p) (p v q) :: (q v p)

AS ((p . q) . r) :: (p . ( q . r)) ((p v q) v r) :: (p v (q v r))

DM ~(p v q) :: (~p . ~q) ~(p .q) :: (~p v ~q)

DIST (p v (q . r)) :: ((p v q) . (p v r)) (p . (q v r)) :: ((p . q) v (p . r))

TRAN (p > q) :: (~q > ~p)

IMP (p v q) :: (~p > q)

EQ (p ≡ q) :: ((p > q) . (q > p)) (p ≡ q) :: ((p . q) v (~p . ~q))

EXP (p > (q > r)) :: ((p . q) > r)

TAUT (p v p) :: p (p . p) :: p

-----------------------

CQ: Change of Quantifier

(x) Fx :: ~(Зx) ~Fx (x)~Fx :: ~(Зx) Fx

(Зx) Fx :: ~(x)~ Fx (Зx)~Fx :: ~(x) Fx

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