Thread: converse & contrapositive

"Richmond will win the AFL premiership only if pigs can fly"

Here is how ive gone about doing this:

if p="pigs can fly" and r="Richmond will win the AFL premiership"
then the above statement can be written as p-->r

converse: (-p)-->(-r) If pigs can't fly, Richmond will not win the AFL premiership.

contrapositive: (-r)-->(-p) If Richmond do not win the AFL premiership then pigs cannot fly


Is what I have done correct? The answers given suggest otherwise, but I'm convinced that I haven't made a mistake. Can anyone clarify this?

Originally Posted by chrizzle

"Richmond will win the AFL premiership only if pigs can fly"

Here is how ive gone about doing this:

if p="pigs can fly" and r="Richmond will win the AFL premiership"
then the above statement can be written as p-->r

converse: (-p)-->(-r) If pigs can't fly, Richmond will not win the AFL premiership.

contrapositive: (-r)-->(-p) If Richmond do not win the AFL premiership then pigs cannot fly


Is what I have done correct? The answers given suggest otherwise, but I'm convinced that I haven't made a mistake. Can anyone clarify this?

You have fine understanding of converse and contrapositive, you merely made a mistake with the "only if". The sentence "A only if B" is actually A-->B, not B-->A. Think about it, you should see why. On the other hand "A if B" is B-->A.

I would like to add that AFAIK the converse of p -> r is r -> p, while ~p -> ~r is the contrapositive of the converse. Now, the contrapositive of any statement q is equivalent to q, so whether to consider r -> p or ~p -> ~r the converse of p -> r is a matter of nomenclature rather than logical equivalence.

Originally Posted by emakarov

I would like to add that AFAIK the converse of p -> r is r -> p, while ~p -> ~r is the contrapositive of the converse. Now, the contrapositive of any statement q is equivalent to q, so whether to consider r -> p or ~p -> ~r the converse of p -> r is a matter of nomenclature rather than logical equivalence.

I've really been making a lot of silly errors lately. From what I understand ~p -> ~r is generally called the inverse of p -> r, even though as you say it is logically equivalent to the converse r -> p. Thanks for the correction.