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I have seen this example in a number of course material Show that the hypotheses “It is not sunny this afternoon and it is colder than yesterday”, “We will go swimming only if it is sunny”, “If we do not go swimming, then we will take a canoe trip”, and “If we take a canoe trip, then we will be home by sunset” lead to the conclusion “We will be home by sunset”.

and the propositions are identified as p - “It is sunny this afternoon” q- “It is colder than yesterday” r- “We will go swimming” s- “We will take a canoe trip” t- “We will be home by sunset”

All of them translate the second premise as r--> p. I am not able to understand why. Doesn't the second premise mean " if it is sunny then we will go swimming" and if yes then it should be p-->r or if the second premise is a biconditional then it should be r<-->p

user826717
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1 Answers1

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The second premise is an 'only if' statement, which is different from an 'if' statement (and also from an 'if and only if' statement). 'Only if' is equivalent to a necessary condition, whereas 'if' is equivalent to a sufficient condition, and 'if and only if' is both necessary and sufficient. So the second premise means that it must be sunny if we are to go swimming, but it doesn't require that we go swimming if it is sunny.

From the wikipedia page on 'if and only if':

Sufficiency is the converse of necessity. That is to say, given P→Q (i.e. if P then Q), P would be a sufficient condition for Q, and Q would be a necessary condition for P. Also, given P→Q, it is true that ¬Q→¬P (where ¬ is the negation operator, i.e. "not"). This means that the relationship between P and Q, established by P→Q, can be expressed in the following, all equivalent, ways:

P is sufficient for Q
Q is necessary for P
¬Q is sufficient for ¬P
¬P is necessary for ¬Q
Elenchus
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