** 2.** Consider the following axioms:

- Every child loves Santa.

*∀ x (CHILD(x) → LOVES(x,Santa))* - Everyone who loves Santa loves any reindeer.

*∀ x (LOVES(x,Santa) → ∀ y (REINDEER(y) → LOVES(x,y)))* - Rudolph is a reindeer, and Rudolph has a red nose.

*REINDEER(Rudolph) ∧ REDNOSE(Rudolph)* - Anything which has a red nose is weird or is a clown.

*∀ x (REDNOSE(x) → WEIRD(x) ∨ CLOWN(x))* - No reindeer is a clown.

*¬ ∃ x (REINDEER(x) ∧ CLOWN(x))* - Scrooge does not love anything which is weird.

*∀ x (WEIRD(x) → ¬ LOVES(Scrooge,x))* - (Conclusion) Scrooge is not a child.

*¬ CHILD(Scrooge)*

** 3.** Consider the following axioms:

- Anyone who buys carrots by the bushel owns either a rabbit or a
grocery store.

*∀ x (BUY(x) → ∃ y (OWNS(x,y) ∧ (RABBIT(y) ∨ GROCERY(y))))* - Every dog chases some rabbit.

*∀ x (DOG(x) → ∃ y (RABBIT(y) ∧ CHASE(x,y)))* - Mary buys carrots by the bushel.

*BUY(Mary)* - Anyone who owns a rabbit hates anything that chases any rabbit.

*∀ x ∀ y (OWNS(x,y) ∧ RABBIT(y) → ∀ z ∀ w (RABBIT(w) ∧ CHASE(z,w) → HATES(x,z)))* - John owns a dog.

*∃ x (DOG(x) ∧ OWNS(John,x))* - Someone who hates something owned by another person will not date
that person.

*∀ x ∀ y ∀ z (OWNS(y,z) ∧ HATES(x,z) → ¬ DATE(x,y))* - (Conclusion) If Mary does not own a grocery store, she will not date
John.

*(( ¬ ∃ x (GROCERY(x) ∧ OWN(Mary,x))) → ¬ DATE(Mary,John))*

** 4.** Consider the following axioms:

- Every Austinite who is not conservative loves some armadillo.

*∀ x (AUSTINITE(x) ∧ ¬ CONSERVATIVE(x) → ∃ y (ARMADILLO(y) ∧ LOVES(x,y)))* - Anyone who wears maroon-and-white shirts is an Aggie.

*∀ x (WEARS(x) → AGGIE(x))* - Every Aggie loves every dog.

*∀ x (AGGIE(x) → ∀ y (DOG(y) → LOVES(x,y)))* - Nobody who loves every dog loves any armadillo.

*¬ ∃ x ((∀ y (DOG(y) → LOVES(x,y))) ∧ ∃ z (ARMADILLO(z) ∧ LOVES(x,z)))* - Clem is an Austinite, and Clem wears maroon-and-white shirts.

*AUSTINITE(Clem) ∧ WEARS(Clem)* - (Conclusion) Is there a conservative Austinite?

*∃ x (AUSTINITE(x) ∧ CONSERVATIVE(x))*

( ( (not (Austinite x)) (Conservative x) (Armadillo (f x)) ) ( (not (Austinite x)) (Conservative x) (Loves x (f x)) ) ( (not (Wears x)) (Aggie x) ) ( (not (Aggie x)) (not (Dog y)) (Loves x y) ) ( (Dog (g x)) (not (Armadillo z)) (not (Loves x z)) ) ( (not (Loves x (g x))) (not (Armadillo z)) (not (Loves x z)) ) ( (Austinite (Clem)) ) ( (Wears (Clem)) ) ( (not (Conservative x)) (not (Austinite x)) ) )

** 5.** Consider the following axioms:

- Anyone whom Mary loves is a football star.

*∀ x (LOVES(Mary,x) → STAR(x))* - Any student who does not pass does not play.

*∀ x (STUDENT(x) ∧ ¬ PASS(x) → ¬ PLAY(x))* - John is a student.

*STUDENT(John)* - Any student who does not study does not pass.

*∀ x (STUDENT(x) ∧ ¬ STUDY(x) → ¬ PASS(x))* - Anyone who does not play is not a football star.

*∀ x (¬ PLAY(x) → ¬ STAR(x))* - (Conclusion) If John does not study, then Mary does not love John.

*¬ STUDY(John) → ¬ LOVES(Mary,John)*

** 6.** Consider the following axioms:

- Every coyote chases some roadrunner.

*∀ x (COYOTE(x) → ∃ y (RR(y) ∧ CHASE(x,y)))* - Every roadrunner who says ``beep-beep'' is smart.

*∀ x (RR(x) ∧ BEEP(x) → SMART(x))* - No coyote catches any smart roadrunner.

*¬ ∃ x ∃ y (COYOTE(x) ∧ RR(y) ∧ SMART(y) ∧ CATCH(x,y))* - Any coyote who chases some roadrunner but does not
catch it is frustrated.

*∀ x (COYOTE(x) ∧ ∃ y (RR(y) ∧ CHASE(x,y) ∧ ¬ CATCH(x,y)) → FRUSTRATED(x))* - (Conclusion) If all roadrunners say ``beep-beep'', then all coyotes
are frustrated.

*(∀ x (RR(x) → BEEP(x)) → (∀ y (COYOTE(y) → FRUSTRATED(y)))*

( ( (not (Coyote x)) (RR (f x)) ) ( (not (Coyote x)) (Chase x (f x)) ) ( (not (RR x)) (not (Beep x)) (Smart x) ) ( (not (Coyote x)) (not (RR y)) (not (Smart y)) (not (Catch x y)) ) ( (not (Coyote x)) (not (RR y)) (not (Chase x y)) (Catch x y) (Frustrated x) ) ( (not (RR x)) (Beep x) ) ( (Coyote (a)) ) ( (not (Frustrated (a))) ) )

** 7.** Consider the following axioms:

- Anyone who rides any Harley is a rough character.

*∀ x ((∃ y (HARLEY(y) ∧ RIDES(x,y))) → ROUGH(x))* - Every biker rides [something that is] either a Harley or a BMW.

*∀ x (BIKER(x) → ∃ y ((HARLEY(y) ∨ BMW(y)) ∧ RIDES(x,y)))* - Anyone who rides any BMW is a yuppie.

*∀ x ∀ y (RIDES(x,y) ∧ BMW(y) → YUPPIE(x))* - Every yuppie is a lawyer.

*∀ x (YUPPIE(x) → LAWYER(x))* - Any nice girl does not date anyone who is a rough character.

*∀ x ∀ y (NICE(x) ∧ ROUGH(y) → ¬ DATE(x,y))* - Mary is a nice girl, and John is a biker.

*NICE(Mary) ∧ BIKER(John)* - (Conclusion) If John is not a lawyer, then Mary does not date John.

*¬ LAWYER(John) → ¬ DATE(Mary,John)*

** 8.** Consider the following axioms:

- Every child loves anyone who gives the child any present.

*∀ x ∀ y ∀ z (CHILD(x) ∧ PRESENT(y) ∧ GIVE(z,y,x) → LOVES(x,z)* - Every child will be given some present by Santa if Santa
can travel on Christmas eve.

*TRAVEL(Santa,Christmas) → ∀ x (CHILD(x) → ∃ y (PRESENT(y) ∧ GIVE(Santa,y,x)))* - It is foggy on Christmas eve.

*FOGGY(Christmas)* - Anytime it is foggy, anyone can travel if he has some
source of light.

*∀ x ∀ t (FOGGY(t) → ( ∃ y (LIGHT(y) ∧ HAS(x,y)) → TRAVEL(x,t)))* - Any reindeer with a red nose is a source of light.

*∀ x (RNR(x) → LIGHT(x))* - (Conclusion) If Santa has some reindeer with a red nose, then
every child loves Santa.

*( ∃ x (RNR(x) ∧ HAS(Santa,x))) → ∀ y (CHILD(y) → LOVES(y,Santa))*

** 9.** Consider the following axioms:

- Every investor bought [something that is] stocks or bonds.

*∀ x (INVESTOR(x) → ∃ y ((STOCK(y) ∨ BOND(y)) ∧ BUY(x,y)))* - If the Dow-Jones Average crashes, then all stocks that are
not gold stocks fall.

*DJCRASH → ∀ x ((STOCK(x) ∧ ¬ GOLD(x)) → FALL(x))* - If the T-Bill interest rate rises, then all bonds fall.

*TBRISE → ∀ x (BOND(x) → FALL(x))* - Every investor who bought something that falls is not happy.

*∀ x ∀ y (INVESTOR(x) ∧ BUY(x,y) ∧ FALL(y) &rarrm; ¬ HAPPY(x))* - (Conclusion) If the Dow-Jones Average crashes and the T-Bill
interest rate rises, then any investor who is happy bought some gold stock.

*( DJCRASH ∧ TBRISE ) → ∀ x (INVESTOR(x) ∧ HAPPY(x) → ∃ y (GOLD(y) ∧ BUY(x,y)))*

** 10.** Consider the following axioms:

- Every child loves every candy.

*∀ x ∀ y (CHILD(x) ∧ CANDY(y) → LOVES(x,y))* - Anyone who loves some candy is not a nutrition fanatic.

*∀ x ( (∃ y (CANDY(y) ∧ LOVES(x,y))) → ¬ FANATIC(x))* - Anyone who eats any pumpkin is a nutrition fanatic.

*∀ x ((∃ y (PUMPKIN(y) ∧ EAT(x,y))) → FANATIC(x))* - Anyone who buys any pumpkin either carves it or eats it.

*∀ x ∀ y (PUMPKIN(y) ∧ BUY(x,y) → CARVE(x,y) ∨ EAT(x,y))* - John buys a pumpkin.

*∃ x (PUMPKIN(x) ∧ BUY(John,x))* - Lifesavers is a candy.

*CANDY(Lifesavers)* - (Conclusion) If John is a child, then John carves some pumpkin.

*CHILD(John) → ∃ x (PUMPKIN(x) ∧ CARVE(John,x))*

Gordon S. Novak Jr.