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:

• The economy will perform poorly again next year, and the Republicans will have a difficult time at the ballot box. (Use E and R to stand for the obvious propostional components of the sentence)
• This man must either be drunk or have a brain tumor. (D, T)

• If the economy performs poorly again next year, then the Republicans will have a difficult time at the ballot box. (E, R)

• If the economy does not perform poorly again next year, then the Republicans will not have a difficult time at the ballot box. (E, R)

• Neither the economy nor the Republican chances at the ballot box will improve next year. (E, R)

• Children will be promoted to the next grade only if they pass the essential competency test. (G, C)

• Passing the essential competency test is a necessary condition for promoting children to the next grade. (C, G)

• If you don't clean your room, then you don't go to the dance tonight. (R, D)

• You will not both study tonight and go to the dance. (S, D)

• The Iraq war situation and the Democratic chances at the ballot box will both not improve next year. (I, D)
• You have won $10 million only if you return the magazine subscription page. (M, R)

• You have won $10 million if only you return the magazine subscription page.

• If the security situation in Iraq does not improve, we can’t have credible elections. (S, C)

• Dividends will be tax-free under the new Bush tax plan if and only if the company that pays them has paid enough taxes. (D, T)

• Israeli soldier on the Israeli invasion of the West Bank, "We did not shoot civilians unless we had to. (S = We did shoot civilians; C = We had to shoot civilians.)

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:

• If not A, then neither B nor C.

• X only if not both P and Z.

• P and Z, if not D.

• Either D or not both P and Z.

• If Alice is going to the party, then Barbara and Carol will also go. (A, B, C)

• If Johnson gets the job, then neither Smith nor Kaneshiro will be hired. (J, S, K)

• Either Alice is not going to the party or Barbara and Carol are both not going. (A, B, C)

• The economy will not improve next year, but if the president decides to run again he still has his foreign affairs record to sustain his popularity with the voters.

  I = The economy will improve next year. R = The president decides to run again. F = The president still has his foreign affairs record to sustain his popularity with the voters.

• If the labor contract is renewed at the present salary levels, then there will either be an illegal strike or a work slowdown.

  R = The labor contract is renewed at the present salary levels. L = There will be a legal strike. W = There will be a work slowdown.

• If I find something in the laundry that shouldn't be there, someone is going to be in big trouble unless it is money.

  L = I find something in the laundry that should be there. T = Someone is going to be in big trouble. M = I find money in the laundry.

• Syria will recognize Israel only if Israel gives back all the Arab lands it captured in the 1967 Mideast war and accepts past U.N. resolutions.

  R = Syria will recognize Israel. G = Israel gives back all the Arab lands it captured in the 1967 Mideast war. A = Israel accepts past U.N. resolutions.

• If the basketball team has an exciting team and a winning team this year, then coach Little's contract will be renewed.

  E = The basketball team has an exciting team this year. W = The basketball team has a winning team this year. L = Coach Little's contract will be renewed.

• North Korea will not allow inspection of its nuclear research facility unless the U. S. nuclear involvement in the peninsula is stopped and international teams carry out simultaneous inspection of South Korean weapons sites.

  A = North Korea will allow inspection of its nuclear research facility. S = U.S. nuclear involvement in the peninsula is stopped. C = International teams carry out simultaneous inspection of South Korean weapons sites.

• If the U. S. and Israel both don't support Gemayal's Christian government in Lebanon, then Jumblatt will not remain in the Syrian camp for long.

  U = The U.S. supports Gemayal's Christian government in Lebanon. I = Israel supports Gemayal's Christian government in Lebanon. J = Jumblatt will remain in the Syrian camp for long.

• The State of Hawaii will preserve marriage between one man and one woman, provided that it does not deprive any person of civil rights on the basis of sex and the voters approve the Constitutional amendment this November.

  P = The State of Hawaii will preserve marriage between one man and one woman. D = It does deprive a person of civil rights on the basis of sex. V = Voters approve the Constitutional amendment this November.

• Enforcing the No Child Left Behind requirements is a necessary condition, but not a sufficient condition for us to meet the steep educational challenges of the twenty-first century.

  E = We will enforce the components of the No Child Left Behind requirements M = We will meet the steep educational challenges of the twenty-first century.)

• Military action should be taken against Iraq only if Iraq refuses to cooperate, and if the Security Council of the United Nations approves such action.

  M = Military action should be taken against Iraq. R = Iraq refuses to cooperate. A = The Security Council of the United Nations approves taking military action against Iraq.

• It is not true that having an argument with true premises is a sufficient condition for a valid argument. (T, V)

• From an article commenting on the little known Japanese nuclear bomb program in the 1940's. "If Japan considers American hands to have been soiled by the atomic bomb then Japanese hands are equally dirty, although most Japanese are unaware."

  S = Japan considers American hands to have been soiled by the atomic bomb. D = Japanese hands are equally dirty. A = Japanese are aware that Japanese hands are equally dirty.

• Political commentary on the 1992 elections: "The 1992 election choices will be serious if and only if political parties debate plans for escaping depression and militarism, and heed the people who have long been deprived of competent public services.

  C = The 1992 election choices will be serious. D = Political parties debate plans for escaping depression. M = Political parties debate plans for escaping militarism. H = Political parties heed the people who have long been deprived of competent public services.

• The United States has pledged to withdraw all of its landbased nuclear weapons from the Korean peninsula, although it would still be able to strike North Korean targets with nuclear missiles launched from submarines and will still have a substantial naval presence in Japan.

  P = The United States has pledged to withdraw all of its landbased nuclear weapons from the Korean peninsula. S = The United States would still be able to strike North Korean targets with nuclear missiles launched from submarines. N = The United States will still have a substantial naval presence in Japan.

• An explanation for why the Kansas City Chiefs were out of the football playoffs. "The Chiefs beat the 49ers and Buffalo, however, the Raiders beat the 49ers and Buffalo both by more points."

  A = The Chiefs beat the 49ers. B = The Chiefs beat Buffalo. C = The Raiders beat the 49ers by more points. D = The Raiders beat Buffalo by more points.

• The Russian President Putin is not prepared to accept whatever the Bush administration offers in terms of verification of new nuclear weapons levels unless it is put on paper and accepted by other nuclear powers.

  A = The Russian president Putin is prepared to accept whatever the Bush administration offers in terms of verification of new nuclear weapons levels. P = Whatever the Bush administration offers in terms of verification of new nuclear weapons levels is put on paper. N = Whatever the Bush administration offers in terms of verification of new nuclear weapons levels is accepted by other nuclear powers.

• The Supreme Court has ruled that obscene material is illegal only if it is disseminated and not just possessed, unless there is an overwhelming societal interest in protecting children.

  L = Obscene material is legal. D = Obscene material is disseminated. P = Obscene material is just possessed. S = There is an overwhelming societal interest in protecting children.

Exercises IV:

• An adequate condition for being excused from the final is having a quiz average of over 90%. (F, Q)

• (Snoopy) "The commanding officer only offers me a root beer when there's a dangerous mission to be flown." (R, D)

• (Ecology poster) "Without you it won't get done." (H = you help, J = the job gets done)

• "It is not true there was not a threat in relation to weapons of mass destruction (in Iraq)." Toni Blair, 7/6/2004

  T = There was a threat in relation to weapons of mass destruction (in Iraq).

• There is only one possibility for a negotiated settlement: Konner must resign from the negotiation team. (S, K)

• Sports announcer Dan Derdorf at the Minnesota Metrodome, during a Bears and Minnesota football game. "It's noisy here even when it is quiet." (Q = quiet, ~Q = noisy)

• U. S. Rep. Charles Rangel, D-N.Y. commenting on the plight of Haitian refugee treatment: "There's no question, if they were not poor, if they were not black, that we would find some compassion to let these people in."

  P = If the refugees were poor. B = If the refugees were black. C = We could find some compassion to let these refugees in to our country.

• Australian Court decision ruling against a man charged with the rape of his wife. The man had argued that sexual intercourse with his wife was a marital right. "If it was ever the common law that by marriage (a women) gave irrevocable consent to sexual intercourse with her husband, it is no longer the common law." (C)

• Putting political issues aside and rescinding some of the adverse tax legislation produced in 1986 and 1989 is the only way we can avoid being economically killed by the Japanese and enslaved by the new European Common Market.

  P = We put political issues aside. R = We rescind some of the adverse tax legislation produced in 1986 and 1989. K = We avoid being economically killed by the Japanese. E = We avoid being economically enslaved by the new European Common Market.

• The CIA's assessment is that Iraq is unlikely to use biological or chemical weapons against the United States unless we attack Iraq and Saddam concludes he has nothing to lose.

  B = The CIA’s assessment is that Iraq is likely to use biological weapons. C = The CIA’s assessment is that Iraq is likely to use chemical weapons. A = We attack Iraq. S = Saddam concludes he has something to lose.

• Should the economy improve and consumer confidence return, the Republicans will do better at the ballot box provided that they get a handle on the embarrassment of the right wing.

  I = The economy improves. C = Consumer confidence returns. B = The Republicans do a better job at the ballot box. H = The Republicans get a handle on the embarrassment of the right wing.

• A doctor in the Netherlands will be allowed to perform an assisted suicide only when the following conditions are met: the doctor is convinced that the patient's request is well considered - the patient's suffering is unbearable - and finally, the doctor consults with another independent doctor who also examines the patient.

  A = A doctor in the Netherlands will be allowed to perform an assisted suicide. C = The doctor is convinced that the patient's request is well considered. B = The patient's suffering is bearable. I = The doctor consults with another independent doctor who also examines the patient.

• If Arafat cannot control his territory, it is in anarchy and Israel must subdue his territory - whereas, if Arafat can control his territory but refuses, then he has earned expulsion under the principle America cites in expelling the Taliban from power.

  C = Arafat can control his territory. A = Arafat's territory is in anarchy. S = Israel must subdue his territory. R = Arafat refuses to control his territory. E = Arafat has earned expulsion under the principle America cites in expelling the Taliban from power.

• If American Jews really care about Israel, if Arab leaders really care about Palestinians, and if Iraq hawks really want to get rid of Saddam, then these groups must lobby President Bush to station U.S. troops around Israel.

  J = American Jews really care about Israel. A = Arab leaders really care about Palestinians. I = Iraq hawks really want to get ride of Saddam. L = These groups must lobby President Bush to station U.S. troops around Israel.

• Rain and humidity are not uncommon in Hawaii.

  R = Rain is common in Hawaii. H = Humidity is common is Hawaii.

Exercises V: Writing Exercises.

• If W stands for "We have a winning team this year," and R stands for "The managers contract will be renewed," write out an English equivalent for each of the following. Then explain how each differs in meaning. What is (1) saying? How does what it differ from (2) in meaning? And so on.

1. W -> R

2. ~W -> ~R

3. ~(W -> R)

4. ~(~W -> ~R)

• Write out an explanation for the difference between

(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?

• Explain in writing why the statement, "It is not true that being female is a sufficient condition for being pregnant," ~(F -> P), is not equivalent to the statement, "If a person is not female, then that person is not pregnant," ~F -> ~P.
• If T stands for "An argument has true premises," and V stands for "An argument is valid," write out English equivalents for the following:

1. ~(T -> V)

2. ~T -> ~V

3. V -> T

4. ~(V -> T)

Essential Logic Ronald C. Pine