Translating “only if”

Mr. Nance,

Question 6 of Intermediate Logic Quiz 2 asks to symbolize this proposition: “The knight attacks the dragon only if the dragon devours the damsel.” The answer key says K ⊃ D (“If K then D”). I would have thought the answer was D ⊃ K (“If D then K”). Am I wrong?

Yes, you are wrong. The answer would be D ⊃ K had the statement not included the word “only”. But “p only if q” is equivalent to “If p then q”; it is not equivalent to “If q then p.” This is the difference between “p if q” and “p only if q.”

To demonstrate this, consider this statement: “A person is a mother only if a person is a woman.” This is a true statement. It is equivalent to this true statement: “If a person is a mother then a person is a woman.” It is not equivalent to this false statement: “If a person is a woman then a person is a mother.”

The only before the if makes all the difference.


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