Solutions to predicate translation exercises

1. Paris is beautiful. Bp

2. Tokyo is overcrowded. Ot

3. If Paris is beautiful then it’s popular.

Bp ⊃ Pp

4. If Gonzales is tortured, then he’ll talk.

Tg ⊃ Ag

5. All lawyers are members of the Bar Association. (x) (Lx ⊃ Mx)

6. Some flowers are not pretty. (∃x)(Fx ∙~Px)

7. All laptop computers have batteries. (x)((Lx ∙ Cx) ⊃ Bx)

8. No students carry cellphones, but Mary is not a student. (x) (Sx ⊃ ~Cx) ∙ ~Sm

9. Not all Senators are communists. ~(x)(Sx ⊃ Cx) or (∃x)(Sx ∙ ~Cx)

10. Obama is running for President and so is Clinton. Ro ∙ Rc

11. Obama’s not a communist and neither is Clinton. ~(Co v Cc)

12. Either Clinton will be the candidate or there will be no woman candidate. Cc v ~(∃x)(Cx ∙ Wx)

13. Horses exist, but not unicorns. (∃x)Hx ∙ ~(∃x) Ux or (x)~Ux

14. Sea lions are mammals. (x) (Sx ⊃ Mx)

15. Squirrels live on this campus. (∃x)(Sx ∙ Lx)

16. Only snakes and lizards thrive in the desert. (x) (Tx ⊃ (Sx v Lx))

17. Peaches are delicious unless they are rotten. (x) (Px ⊃ (Dx v Rx))

18. Dogs bite if they are frightened or harassed. (x) (Dx ⊃ (Bx ⊃ (Fx v Hx)))

19. Bears and eagles are talked about alot on The Colbert Report. (∃x) (Bx ∙ Tx) ∙ (∃x) (Ex ∙ Tx)

20. Sean Penn and Steven Colbert love metaphors. Lp ∙ Lc





21. Only arguments can be valid.

(x) (Vx > Ax)


22. Arguments are sound if and only if they are deductive, valid, and have all true premises.

(x) {Ax > {Sx ≡ [Dx . (Vx . Tx)]}}


Also (x) {(Ax . Sx) ≡ [Dx . (Vx . Tx)]}

Notice that the difference between these two versions is a matter of Exportation.



23. All sentences that are statements are either true or false.


(x) [(Ex . Ax) > (Tx v ~Tx)]


24. If all sentences are statements, then they are all true or false.

(x) (Ex > Ax) > (x) (Ex > (Tx v ~Tx))


25. Only inductive arguments are weak or uncogent.

(y)((Wy v ~Cy) > Iy)


Why “y” in this case? Just for a change. “x,” “y,” and “z” are all available in the role of variables; there is no difference in meaning when we use “y” as in this case, from if we used “x,” which most people use most of the time as long as only one variable is called for.

You’d use a second variable if you were representing a predicate that links two subjects together, like “likes.” “Likes” is “…likes…,” what logicians call a two-place predicate.

“Tony likes Christopher” would be Ltc. “Everbody likes something” would be (x) (∃y) Lxy (“for every x, there is a y such that x likes y” or “everyone is such that there is a thing that s/he likes”)


26. A categorical syllogism is invalid if it has an undistributed middle term.

(x) [(Sx . Cx) > (~Dx > ~Vx)]


27. A good violin is rare and expensive.

(y) [(Vy . Gy) > (Ry . Ey)]


28. Violins and violas are stringed instruments.

(x) [(Ix v Ox) > (Sx . Tx)]


29. A room with a view is available.

(∃x) ((Rx . Vx) . Ax)


30. A room with a view is expensive.

(x) ((Rx . Vx) > Ex)


31. Hurricanes and tornados are violent and destructive.

(y) ((Hy v Ty) > (Vy . Dy))


32. Scooter is guilty if all the witnesses told the truth.

(x) (Wx > Tx) > Gs


Use the same predicate letters for 32 and 33.


33. If any witnesses lied, then Scooter’s innocent.

(∃x) (Wx . ~Tx) > ~Gs



34. If all journalists are interesting people, then Bob Novak is an interesting person.

(x) (Jx > (Ix . Px)) > (In . Pn)


35. Not all jazz fans like Monk.

(∃x) (Jx . ~ Lxm) or you could stipulate that “…likes Monk” is a one place predicate for our level of work, and say (∃x) (Jx . ~Lx)


36. Balcony seats are never chosen unless all the orchestra seats are taken.

(x) ((Sx . Bx) > ~Cx) v (x) ((S x . Ox) > Tx)


This is a compound statement, linking an “E” with an “A.”



37. Some employees will get raises if and only if some managers are overly generous.

(∃x)(Ex . Rx) ≡ (∃z) (Mz . Oz)


38. Some local artists ask too much for their work, but not Bill Harris.

(∃x) ((Ax . Lx) . Tx) . ~Th


39. If the scientists and technicians are conscientious and exacting, then some of the mission directors will be either pleased or delighted.

(x) [(Sx v Tx) > (Cx . Ex)] > (∃x) (Dx . (Px v Lx))



40. The physicists and astronomers at the symposium are listed in the program if they either chair a session or read a paper.

(x) [((Px v Ax) . Sx) > ((Cx v Rx) > Lx)]


41. Some words are predicates, but not “existence.”

(∃x) (Wx ∙ Px) ∙ ~Pe

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