# odd logical structures

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

• See this post : So $A$ only if $B$ means $A \rightarrow B$. Commented Jan 24, 2015 at 15:21
• See also Converse and Contrapositive. Commented Jan 24, 2015 at 15:26
• I did read those so this is the conclusion I came to for these 1. If john is Lumberjack, then he cut down the tree contrapositive If john is not a lumberjack, then he did not cut down the tree. 2. Converse If you don't fight, then you can't win. Contrapositive If you do fight, then you can win. 3. Converse If person is from Oregon then they root for the Ducks. If person is not from Oregon, then they dont root for the Ducks. are these correct? Commented Jan 24, 2015 at 15:52

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.

For 2), we simply have :

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

Thus, converse and contrapositive must be straightforward.

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.

• How would we rewrite the other two? Commented Jan 24, 2015 at 15:57
• oh okay so this is what I came up with for 2 and 3 2. Converse If you can't win, then you didnt fight Contrapositive If you did win, then you did fight 3. If person is from Oregon, then he roots for the Ducks Contrapositive If person is not from Oregon, then person does not root for the Ducks Commented Jan 24, 2015 at 16:04