translation

Natural language and (propositional) logic

Adapted from the page http://home.honolulu.hawaii.edu/~pine/Book2/chap7EL-2.html by Ronald C. Pine

Essentially, what I have done here is pull out the exercises from the above page. There is copious advice and discussion of strategy there: you might have a look.

Examples

  1. John passed the final exam and the course. (F, C)

    F & C

  2. Either John passed the final exam or he passed the course. (F, C)

    F v C

  3. If John passes the final exam, then he will pass the course. (F, C)

    F -> C

  4. John will pass the course if and only if he passes the final exam. (C, F)

    C <-> F

  5. John passed the course but not the final exam. (F, C)

    C & ~F

  6. John did not pass the course but he did pass the final exam. (C, F)

    ~C & F

  7. John passed the course even though he did not pass the final exam. (C, F)

    C & ~F

  8. Even though he did not pass the final exam, John passed the course (F, C)

    ~F & C

  9. John did not pass both the final exam and the course. (F, C)

    ~(F & C)

  10. John did not pass the final and he did not pass the course. (F, C)

    ~F & ~C

  11. Johnson and Kaneshiro will not both be hired. (J, K)

    ~(J & K)

  12. Johnson and Kaneshiro will both not be hired. (J, K)

    ~J & ~K

  13. John either passes the final exam or he does not pass the course. (F, C)

    F v ~C

  14. Either John did not pass the final exam or he did not pass the course. (F, C)

    ~F v ~C

  15. John passed neither the final exam nor the course. (F, C)

    ~(F v C) or ~F & ~C

  16. John will take the bus to school unless his girl friend drives him in her car. (B, D)

    B v D or ~D -> B

  17. John will pass the course, if he passes the final exam. (C, F)

    F -> C

  18. John will pass the course only if he passes the final exam. (C, F)

    C -> F

  19. If only John passes the final exam, he will pass the course. (F, C)

    F -> C

  20. Provided that John passes the final exam, he will pass the course. (F, C)

    F -> C

  21. John will pass the course, provided that he passes the final exam. (C, F)

    F -> C

  22. Passing the final exam is a necessary condition for passing the course. (F, C)

    C -> F

  23. Passing the final exam is a sufficient condition for passing the course. (F, C)

    F -> C

  24. If John does not pass the final exam, then he will not pass the course. (F, C)

    ~F -> ~C

  25. It is not true that if John passes the final exam, then he will pass the course. (F, C)

    ~(F -> C)

Exercises I:

  1. A but not B.
  2. If not A, then B.
  3. Z only if not B.
  4. Not both A and G.
  5. P, if not G.
  6. Either not P or not D.
  7. Neither R nor H.
  8. S unless not P.
  9. Not Z, if and only if Y.
  10. A necessary condition for P is not Y.
  11. A sufficient condition for P is not Y.
  12. If only K, then B.
  13. Z if and only if not J.
  14. A, provided that not Z.
  15. Not P, even though A.

Exercises II:

Complex Translations, the Use of Parentheses, and Arguments

In translating complex statements, we use parentheses in much the same way that we use punctuation in ordinary language. Consider the difference between the following two statements, noting the position of the commas in the originals and the use of parentheses in the translations.

If the president implements his tax program, then the deficits will continue to increase and the economy will not improve. (T, D, E)

T -> (D & ~E)

If the president implements his tax program then deficits will continue to increase, and the economy will not improve. (T, D, E)</p>

(T -> D) & ~E

These say very different things!

Exercises III:

Exercises IV:

Exercises V: Writing Exercises.

1. W -> R

2. ~W -> ~R

3. ~(W -> R)

4. ~(~W -> ~R)

(1) F -> (C & G) and (2) (F -> C) & G.

F = "John passes the final exam." C = "John passes the course." G = "John's GPA is high enough for eligibility for the Dean's list."

How would (2) best be expressed in English?

1. ~(T -> V)

2. ~T -> ~V

3. V -> T

4. ~(V -> T)

Essential Logic Ronald C. Pine