1.
a. Valid - A valid argument is one whose logical form is good. In a valid argument, the premises entail the conclusion and it there is no case in which all the premises are true and the conclusion false.
b. Sound - A sound argument is one in which the argument form is valid and all the premises are true. Sound arguments thus lead us to real, true conclusions about the world.
c. Statement Connective - An expression with one or more 'blanks' that takes one or more propositions/statements to form a new 'complex' statement.
d. Truth-functional - A statement connective is truth-functional when its operators consider only the truth values of the propositions and generate truth values as the values of complex formulas. The truth value of any compound formula should be determinable by the definition of the operator and the truth values of the propositions it takes on.
e. Tautology - A tautology is a complex truth-functional
formula whose truth value is "true" in every possible
assignment of truth values to its variables.
2.
a. (PvQ) -> (~Q -> (P v R))
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b. ~(P -> (P v Q))
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c. ((PvR)->Q) & ~(~Q ->~R)
| ((P | v | R) | -> | Q) | & | ~ | (~ | Q | -> | ~ | R) |
| T | T | T | T | T | F | F | F | T | T | F | T |
| T | T | F | T | T | F | F | F | T | T | T | F |
| T | T | T | F | F | F | T | T | F | F | F | T |
| T | T | F | F | F | F | F | T | F | T | T | F |
| F | T | T | T | T | F | F | F | T | T | F | T |
| F | F | F | T | T | F | T | F | T | T | T | F |
| F | T | T | F | F | F | T | T | F | F | F | T |
| F | F | F | T | F | F | F | T | F | T | T | F |
d. ((P v Q) & ~R) <-> (P v (Q & ~R))
| ((P | v | Q) | & | ~ | R) | <-> | (P | v | (Q | & | ~ | R)) |
| T | T | T | F | F | T | F | T | T | T | F | F | T |
| T | T | T | T | T | F | T | T | T | T | T | T | F |
| T | T | F | F | F | T | F | T | T | F | F | F | T |
| T | T | F | T | T | F | T | T | T | F | F | T | F |
| F | T | T | F | F | T | T | F | F | T | F | F | T |
| F | T | T | T | T | F | T | F | T | T | T | T | F |
| F | F | F | F | F | T | T | F | F | F | F | F | T |
| F | F | F | F | T | F | T | F | F | F | F | T | F |
e. (P <-> (Q -> (R <-> (P & (Q v R)))))
| P | <-> | (Q | -> | (R | <-> | (P | & | (Q | v | R))))) |
| T | T | T | T | T | T | T | T | T | T | T |
| T | F | T | F | F | F | T | T | T | T | F |
| T | T | F | T | T | T | T | T | F | T | T |
| T | T | F | T | F | T | T | F | F | F | F |
| F | T | T | F | T | F | F | F | T | T | T |
| F | F | T | T | F | T | F | F | T | T | F |
| F | F | F | T | T | F | F | F | F | T | T |
| F | F | F | T | F | T | F | F | F | F | F |
4.
a. (P<->Q) -> R
~(R v P)
-------------
Q
| (P | <-> | Q) | -> | R | ~ | (R | v | P) | Q | |
| T | T | T | T | T | F | T | T | T | T | |
| T | T | T | F | F | F | F | T | T | T | |
| T | F | F | T | T | F | T | T | T | F | |
| T | F | F | T | F | F | F | T | T | F | |
| F | F | T | T | T | F | T | T | F | T | |
| F | F | T | T | F | T | F | F | F | T | |
| F | T | F | T | T | F | T | T | F | F | |
| F | T | F | F | F | T | F | F | F | F |
b. P ->Q
P -> (R v P)
-------------
P -> (Q->R)
| (P | -> | Q) | P | -> | (R | v | P) | P | -> | (Q | -> | R) | |
| T | T | T | T | T | T | T | T | T | T | T | T | T | |
| T | T | T | T | T | F | T | T | T | F | T | F | F | <== |
| T | F | F | T | T | T | T | T | T | T | F | T | T | |
| T | F | F | T | T | F | T | T | T | T | F | T | F | |
| F | T | T | F | T | T | T | F | F | T | T | T | T | |
| F | T | T | F | T | F | F | F | F | T | T | F | F | |
| F | T | F | F | T | T | T | F | F | T | F | T | T | |
| F | T | F | F | T | F | F | F | F | T | F | T | F |
c. (P v R)
((P v Q) -> Q) -> R
(Q & R)
------------------
P v ~Q
| (P | v | R) | ((P | v | Q) | -> | Q) | -> | R | (Q | & | R) | P | v | ~ | Q) | |
| T | T | T | T | T | T | T | T | T | T | T | T | T | T | T | F | T | |
| T | T | F | T | T | T | T | T | F | F | T | F | F | T | T | F | T | |
| T | T | T | T | T | F | F | F | T | T | F | F | T | T | T | T | F | |
| T | T | F | T | T | F | F | F | T | F | F | F | F | T | T | T | F | |
| F | T | T | F | T | T | T | T | T | T | T | T | T | F | F | F | T | <== |
| F | F | F | F | T | T | T | T | F | F | T | F | F | F | F | F | T | |
| F | T | T | F | F | F | T | F | T | T | F | F | T | F | T | T | F | |
| F | F | F | F | F | F | T | F | F | F | F | F | F | F | T | T | F |
(30 points total)
5. Translations (2 points each)
a. College graduates make 70 percent more than high school
graduates do.
C: You gradute from college.
H: You make 70% more than high school graduates on average.
C -> H
b. What you invest in a college education is likely to pay
you back many times over.
I: You invest in a college education.
P: You are paid back many times over.
I -> P
c. Should interest rates go up or your stock price falls, sell
immediately so that you can diversify your portfolio.
I: Interest rates go up
F: Your stock price falls
S: You should sell immediately
D: You should diversify your portfolio
(I v F) -> (S & D)
d. It snows one more time in this godforsaken town and
Iím out of here.
S: It snows one more time in this godforsaken town.
I: I leave
S -> I
e. When you lose your job, you need to apply for unemployment,
unless you have enough savings to hold you over till a new job
comes.
L: You lose your job
A: You apply for unemployment
S: You have enough savings to hold you over.
L -> (~S -> A)
I also would have accepted ~S -> (L -> A) or (L & ~S) -> A
f. I can only go to the convention on the condition that I
can find a cheap ticket.
G: I can go to the convention
C: I find a cheap ticket
~C -> ~G
g. Although coffee keeps me up at night, I need it when a paper
is due.
C: Coffee keeps me up at night
D: There is a paper due
N: I need coffee
C & (D ->N)
I also would have accepted (with the addition of I: I drink coffee): (I -> C) & (D -> N)
h. Kalamazoo is West of Detroit, which makes it west of Cleveland.
K: Kalamazoo is west of Detroit
W: Kalamazoo is West of Cleveland.
K & (K -> W)
I also would have accepted (with the addition of D: Detroit is west of Cleveland): (K & D) -> W
i. Unless Flynn and Liz show up, weíll leave in an hour,
but only if I can get the car started.
F: Flynn shows up
L: Liz shows up
H: We leave in an hour
S: I get the car started
(~(F & L) -> H) & (~S -> ~H))
I also would have accepted:
~S -> ~(~(F & L) -> H)
j. You have to pass logic to be a philosophy major, but that
doesn't make you one automatically.
L: You pass logic.
P: You are a philosophy major.
(~L -> ~P) & ~(L -> P)
k. You robbed the bank or you organized the robbery.
Either way, you're guilty.
R: You robbed the bank
O: You organized the robbery
G: You're guilty
(R v O) & ((RvO)->G)
I also would have accepted just (RvO)->G.
l. Kares will paint your house if you want unless she has other
houses to paint, but only if you give her a week's notice.
K: Kares paints your house
W: You want Kares to paint your house
O: Kares has other houses to paint
N: You give Kares a week's notice
(~O -> (W -> K)) & (~N -> ~K)
I also would have accepted:
~N -> ~ (~O -> (W -> K))
((~O -> (W ->K)) <-> N)
m. Paul wants the supervisor position and a new truck (but
only if it's cheap), unless he invents something and makes a million.
I: Paul invents something
M: Paul makes a million dollars
S: Paul wants the supervisor position
T: Paul wants a new truck
C: The truck is cheap
~(I & M) -> ((S & T) & (~C-> ~T))
I also would have accepted:
~(I & M) -> (S & (~C-> ~T))
~(I & M) -> (S & (C <-> T))
n. Cats are great pets whether they have fleas or not.
G: Cats are great pets
F: Cats have fleas
(F v ~F) -> G
I also would have accepted:
(F -> G) & (~F -> G)
o. Things usually work out fine, unless it's one of those times
where they don't.
T: Things work out fine.
~~T -> T
Some people also did this with two sentences, one to the effect of W: this is one of those times when things work out fine, giving us
~W -> ~T
Others included a sentence like D: Things don't work out fine, which I was less happy about since it clearly contains a negation that should not be part of the simple proposition.
(30 points total)
Extra Credit (5 Points)
'One of these things is not like the others, one of these things just doesn't belong...'
Two of these formulas are truth-functionally equivalent to one another. Which is the odd one out, the one that is not truth-functionally equivalent to any of the others listed here?
(~A v ~(B & C)) -> ((~B v ~C) v A) <=====
(A & ((B v ~C) v (~B v C)))
~(A -> ((B v C) & ~A))
| (~ | A | v | ~ | (B | & | C)) | -> | ((~ | B | v | ~ | C) | v | A) |
| F | T | F | F | T | T | T | T | F | T | F | F | T | T | T |
| F | T | T | T | T | F | F | T | F | T | T | T | F | T | T |
| F | T | T | T | F | F | T | T | T | F | T | F | T | T | T |
| F | T | T | T | F | F | F | T | T | F | T | T | F | T | T |
| T | F | T | F | T | T | T | F | F | T | F | F | T | F | F |
| T | F | T | T | T | F | F | T | F | T | T | T | F | T | F |
| T | F | T | T | F | F | T | T | T | F | T | F | T | T | F |
| T | F | T | T | F | F | F | T | T | F | T | T | F | T | F |
| (A | & | ((B | v | ~ | C) | v | (~ | B | v | C))) |
| T | T | T | T | F | T | T | F | T | T | T |
| T | T | T | T | T | F | T | F | T | F | F |
| T | T | F | F | F | T | T | T | F | T | T |
| T | T | F | T | T | F | T | T | F | T | F |
| F | F | T | T | F | T | T | F | T | T | T |
| F | F | T | T | T | F | T | F | T | F | F |
| F | F | F | F | F | T | T | T | F | T | T |
| F | F | F | T | T | F | T | T | F | T | F |
| ~ | (A | -> | ((B | v | C) | & | ~ | A)) |
| T | T | F | T | T | T | F | F | T |
| T | T | F | T | T | F | F | F | T |
| T | T | F | F | T | T | F | F | T |
| T | T | F | F | F | F | F | F | T |
| F | F | T | T | T | T | T | T | F |
| F | F | T | T | T | F | T | T | F |
| F | F | T | F | T | T | T | T | F |
| F | F | T | F | F | F | F | T | F |