odd logical structures

Question

How you find contrapositive and converse of these sentences.

1. Only if John chops down the tree, will he be a lumberjack.

2. You can't win if you don't fight.

3. All people that root for the Ducks are from Oregon.

The logical has me thrown at where the if is happening so I can switch it around from p implies q to q implies p. Please help

Answer 1

HINT

We have to rewrite 1) as :

"Only if John chops down the tree, will he be a lumberjack"

as :

if John will be a lumberjack, then he chops down the tree.

This one has the "logical form" : $p \rightarrow q$; thus, its converse ($q \rightarrow p$) will be :

if John chops down the tree, then he will be a lumberjack,

while its contrapositive ($\lnot q \rightarrow \lnot p$)will be :

if John does not chop down the tree, then he will not be a lumberjack.

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For 2), we simply have :

if you don't fight, then you can't win.

Thus, converse and contrapositive must be straightforward.

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For 3), assuming that we have to "analyze" it without predicate logic, I agree with you :

if a person roots for the Ducks, then he is from Oregon.

Again, having reduced it to the standard "logical form" : $p \rightarrow q$, we have only to apply the above formulae to get converse and contrapositive.