# why do we use contrapositive?

#### calvinexpress

• 111
##### why do we use contrapositive?
« on: January 16, 2009, 08:10:14 PM »
Ok, I know how to contrapositive, but I don't understand why we have to do it or how it helps us find the correct answer. Anybody explain this to me? Thanks

#### Dilettante

• 239
##### Re: why do we use contrapositive?
« Reply #1 on: January 16, 2009, 09:15:12 PM »
You can get pregnant without one.

#### Jamie Stringer

• 1571
##### Re: why do we use contrapositive?
« Reply #2 on: January 16, 2009, 09:16:06 PM »
You can get pregnant without one.

#### 'blueskies

• 461
• your changing skyline is twisting me up inside
##### Re: why do we use contrapositive?
« Reply #3 on: January 16, 2009, 09:23:53 PM »
You can get pregnant without one.

me too.  that was a good one

#### Pop Up Video

• 1818
##### Re: why do we use contrapositive?
« Reply #4 on: January 16, 2009, 09:25:22 PM »
I'm going to delete my serious post so the funny post is even funnier.

#### noumena

##### Re: why do we use contrapositive?
« Reply #5 on: January 17, 2009, 01:55:20 AM »
It's one of the basic principles of logic, synonymous in purpose and meaning with the more impressive sounding modus tollens. Knowing how and why it works will get you through many problems utilizing much more complex variations on proposition forms. This will help especially in arguments composed of chains of such statements (syllogisms).

T

#### calvinexpress

• 111
##### Re: why do we use contrapositive?
« Reply #6 on: January 17, 2009, 08:33:21 AM »
So nobody knows the answer either?

#### noumena

##### Re: why do we use contrapositive?
« Reply #7 on: January 17, 2009, 08:59:43 AM »
So nobody knows the answer either?

I attempted an answer. It's difficult to see its inherent importance if you don't understand the principle underlying it. But here's one example of reasoning with the contrapositive for you.

Let's say all Catholics are Christians. What can you infer from this? The contrapositive tells us that all non-Christians are non-Catholics (and on the individual level, if someone's not a Christian, then he can't be Catholic), and that makes sense, because were there a non-Christian that was a Catholic, then there would be a Catholic that wasn't a Christian, thereby violating the conditional. It may seem less than intuitive at first, but note that what the principle implies is that the subject and predicate are merely placeholders that with any subject/predicate combination will lead to the same result. This is because the contrapositive applies universally to statement forms that have the same structure, which is why we can reason without referring to the subject and predicate without error all the time.

What you cannot draw from the conditional, however, is that just because someone isn't Catholic, he isn't Christian, because he might belong to some other denomination just as Christian, like Mormonism, for instance. This is a common logical fallacy known as "denying the antecedent." Thus, besides its internal consistency and universal applicability, remembering that the contrapositive isn't bidirectional will help guard against such errors. Symbolically,

A-->B;
Therefore, ~B-->~A
and NOT ~A-->~B

where "~" means "not" and "-->" means roughly "if-then" or "therefore."

Better?

T

#### zippyandzap

• 274
• Clint Dempsey FTW
##### Re: why do we use contrapositive?
« Reply #8 on: January 17, 2009, 10:23:52 AM »
So nobody knows the answer either?

I attempted an answer. It's difficult to see its inherent importance if you don't understand the principle underlying it. But here's one example of reasoning with the contrapositive for you.

Let's say all Catholics are Christians. What can you infer from this? The contrapositive tells us that all non-Christians are non-Catholics (and on the individual level, if someone's not a Christian, then he can't be Catholic), and that makes sense, because were there a non-Christian that was a Catholic, then there would be a Catholic that wasn't a Christian, thereby violating the conditional. It may seem less than intuitive at first, but note that what the principle implies is that the subject and predicate are merely placeholders that with any subject/predicate combination will lead to the same result. This is because the contrapositive applies universally to statement forms that have the same structure, which is why we can reason without referring to the subject and predicate without error all the time.

What you cannot draw from the conditional, however, is that just because someone isn't Catholic, he isn't Christian, because he might belong to some other denomination just as Christian, like Mormonism, for instance. This is a common logical fallacy known as "denying the antecedent." Thus, besides its internal consistency and universal applicability, remembering that the contrapositive isn't bidirectional will help guard against such errors. Symbolically,

A-->B;
Therefore, ~B-->~A
and NOT ~A-->~B

where "~" means "not" and "-->" means roughly "if-then" or "therefore."

Better?

T

Word.

If we're given P then Q, without contrapositive our only possible conclusion (on its face) is
Q, given P

Using the contrapositive, we can conclude ~P. given ~Q.

You might say that this second conclusion is obvious, but in my opinion in games with a lot of different things going on it's easiest to have everything written down because P then Q and ~Q, therefore ~P seems less obvious then ~Q then ~P and ~Q, therefore ~P.

#### noumena

##### Re: why do we use contrapositive?
« Reply #9 on: January 18, 2009, 09:30:10 AM »
Also, one more thing of note to supplement what zippyandzap said:

Thinking of the contrapositive and plain jane modus ponens, i.e., a simple "If A then B" statement, as restating the same conditional under the different but related terms of necessary and sufficient conditions is very helpful. When we have, as a rule, that if A, then B (A-->B), we know in addition to A being sufficient for B, that B is necessary for A. Thus, since the property of B is necessary for A, if A obtains, B MUST obtain. Hence the contrapositive--if B does not subsist as a property of something x, then x cannot be A.

Hope this helps.

T