# Introduction to Logic

## Exercises for Section 7:Predicate Calculus

Answers To see an answer to any odd-numbered exercise, just click on the exercise number.

Translate each of the sentences in Exercises 1-26 into a statement in the predicate calculus. (Red letters are to be used for the relevant predicates or terms where appropriate.)

 1. Every good girl deserves fruit. 2. Good boys deserve fruit always. 3. All cows eat grass. 4. No cows eat grass. 5. Some cows eat grass. 6. Some birds are fishes. 7. Some cows are not birds and some are. 8. Some cows are birds but no cows are fishes. 9. Although some city drivers are insane, Dorothy is a very sane city driver. 10. Even though all mathematicians are nerds, Waner and Costenoble are not nerds. 11. If one or more lives are lost, then all lives are lost. 12. If every creature evolved from lower forms, then you and I did as well. 13. Some numbers are larger than two; others are not.. 14. Every number smaller than 6 is also smaller than 600.

In Exercises 15-26, you can use the convention that the letters i through n represent positive integers.

 15. 12 is divisible by 6. 16. 13 is not divisible by 6. 17. For any positive integer m, if 12 is divisible by m, then so is 24. 18. If 13 is not divisible by m, then neither is 17. 19. 15 is divisible by some positive integer. 20. 15 is divisible by a positive integer other than 15 or 1. 21. 17 is prime (that is, not divisible by any positive integer except itself and 1). 22. 15 is not prime. (See (21).) 23. There is no smallest positive real number. (Use the convention that the letters x through z represent real numbers.) 24. There is no largest positive integer. 25. If 1 has property P, and if (n+1) has property P whenever n does, then every positive integer has property P. (This statement is called the Principle of Mathematical Induction.) 26. If 2 has property P, and if (n+2) has property P whenever n does, then every even positive integer has property P.

Translate the statements in Exercises 27-34 into words.

 27. ∀x[RxSx]; R = "is a raindrop," S = "makes a splash." 28. ∀y[CyMy]; C = "is a cowboy," M = "is macho." 29. ∃z[DzWz]; D = "is a dog," W = "whimpers." 30. ∃z[Dz~Wz]; D = "is a dog," W = "whimpers." 31. ∀x[Dx~Wx]; D = "is a dog," W = "whimpers." 32. ~∀x[DxWx]; D = "is a dog," W = "whimpers." 33. ∃z,y[CzCyWz~Wy]; C = "is a cat," W = "whimpers" 34. ∀x[Px ∃y[PyL(x,y)]], P = "is a person," L(x,y) = "y is older than x."

Communication and Reasoning Exercises

 35. The claim that every athlete drinks ThirstPro is false. In other words, no athletes drink ThirstPro, right? 36. Give one advantage that predicate calculus has over propositional calculus. 37. Your friend claims that the quantifiers ∀ and ∃ are insufficient for her purposes; she requires new quantifiers to express the phrases "for some" and "there does not exist". How would you respond? 38. Consider a new quantifier, "∇" meaning "for no" (as in "for no x can x be larger than itself") Express ∇ in terms of the quantifiers you already have.

Last Updated: December, 2001
Copyright © 1996 StefanWaner and Steven R. Costenoble

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