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i. If it rains next Sunday, then Andy will only go to the football game if Bob does.

ii. If Andy will only go to the foot ball game on Sunday if it rains, then Bob will definitely go to the Sunday football game regardless of the weather.

My gut feeling is they are not equivalent. I chose the following three predicates: r: It rains next Sunday, b: Bob goes to the game, a: Andy goes to the game. And then it seems like the statements would then be:

i. $r \rightarrow (b \leftrightarrow a)$ ii. $(a \leftrightarrow r) \rightarrow b$

Then if I choose the case where it rains on Sunday, Bob goes to the game, but Andy doesn't, statement i is false, but statement ii is true. Have I translated these correctly?

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3 Answers

up vote 2 down vote accepted

I don't think you've interpreted the end of (i) correctly: "Andy will only go to the football game if Bob does." This is not an if-and-only-if statement; it is only an only-if statement.

You have the same issue in your interpretation of (ii).

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Thank you for pointing this out. I had never come across an only-if statement until now. –  yunone Aug 31 '10 at 8:33
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if we set R = "Saturday it will rain", A = "Andy goes to football match", B = "Bob goes to football match", then your statement (i) says for triples (R,A,B) that (Y,Y,Y) and (Y,N,N) are true, while (Y,Y,N) and (Y,N,Y) are false. Nothing can be said about the other four triplets.

Your statement (ii) is indeed different: the part "Andy will only go to the foot ball game on Sunday if it rains" says for couples (R,A) that (Y,Y) and (N,N) are true while (Y,N) and (N,Y) are false. But the statement "If S then T" is always true when S is false (Latins said "ex falso quodlibet", from falsehood comes everything); therefore the triple (Y,N,Y) is true.

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I'm curious as to why (Y,N,Y) would be false for statement (i). I see this would be a case there it rains, and Bobs goes but Andy doesn't. Doesn't 'Andy will go to the football game only if Bob does' mean that if it is the case that Andy goes, then Bob must have gone as well. However, if Bob goes, Andy does not necessarily have to go. So would statement (i) be of the form $r \rightarrow (a \rightarrow b)$, and putting in the truth values gives a true statement? –  yunone Aug 31 '10 at 8:32
    
maybe I misread the semantic of the sentence, but to allow (Y,N,Y) you should have written something like "then Andy could go to the football game only if Bob does". –  mau Aug 31 '10 at 9:32
    
Ok, so (Y,N,Y) would be true for statement (i) then. I see that p only if q is the same as if p then q, so the two statements now would be of the form i) $ r \rightarrow (a \rightarrow b)$ and (ii) $ (a \rightarrow r) \rightarrow b$. So the two triples (Y,N,N) and (N,N,N) would give different truth values for the two statements. Thanks. –  yunone Aug 31 '10 at 10:02
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I will take your translation:

  • $r$ : It rains next Sunday
  • $b$ : Bob goes to the game
  • $a$ : Andy goes to the game

The first assertion $P_1$ can be translate by $r \Rightarrow (a \Rightarrow b)$ since "$\phi$ occurs only if $\psi$ occurs" is equivalent to "if $\phi$ occurs then $\psi$ occurs". The second assertion $P_2$ can be translation by $(a \Rightarrow r) \Rightarrow b$.

Following this, we can use the definition of $\Rightarrow$ to find that $$P_1 = \neg r \vee (a \Rightarrow b) = \neg r \vee \neg a \vee b$$ and $$P_2 = \neg (a \Rightarrow r) \vee b = a \vee \neg r \vee b$$

Using this, it is now easy to see that when $r$ is true and $b$ is false, $P_1$ and $P_2$ differ. Indeed, when "I will rain next Sunday" and if "Bob will no go to the game", then if "Andy goes to the game", $P_1$ is false and $P_2$ is true.

Writing your propositions in a normal form (here a disjunction of positive and negative atoms) helps you, to find a clear counter example ;)

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