Contrapositives and Logical Opposite

[atirvine88]

Hey everyone,

I have been reading the PS Logical Reasoning Bible and just came to logical opposites and I am confused about the relationship between contrapositive and the logical opposite of a statement.

Example:
Original Statement: A B
Contrapositive: not B not A

So I would assume that the logical opposite of both these statements would yield similar/same outcome. Instead, when I negate the second element, I get two different statements.

Logical opposite-original statement: A not B
Logical opposite- contrapositive: not B A

The logical opposite of the contrapositive looks like a mistaken reversal!

Now I know that the logical opposite looks to negate the necessary condition, and therefore I could just ignore the the contrapositive's logical opposite. Still, I do not understand why the two are not the same - why can't I treat the contrapositive's necessary condition (not A) and negate it to make the logical opposite?

[Dave Killoran]

Hi AT,

Thanks for the excellent question. When you look at a statement like A B, the proper logical negation is actually A not nec B. That is, instead of B being required, it's not necessary. That means it might occur, or may not occur; it just doesn't have to occur all the time. However, in many instances, we use a the fast and dirty notation of A B because that typically works to solve the problem as efficiently as possible.

Keep in mind also that a contrapositive expresses the same relationship, but in different terms. So, when you negate a statement and its contrapositive, it will come out "looking" different. Is that a problem? No, it's just a reflection of the fact that a contrapositive necessarily transforms the original statement into something different, even if the underlying meaning reflects the same truth. Regardless, good insight on your part to see that relationship

Please let me know if that helps. Thanks!

[atirvine88]

Thanks- That makes a whole lot more sense. Would it be accurate to say that for a statement A B and not B A (the contrapositive) essentially both disallow A and not B? Another way, is the possibility of A and not B sufficient to disprove this conditional relationship?

[Dave Killoran]

Hi AT,

Yes, you have it

The relationship A B by itself would say that A and not B is impossible. If A and not B occurred, you'd have a contradiction (I could see something like that in a two-speaker stimulus where the first person argues that a conditional is in force and the second speaker disputes it by citing A and not B).

Another way of thinking about this is to examine what is possible if A B is known to be true, focusing only on the A/B outcomes (meaning, order isn't important below):

Possible

1. A, B (repeat of the original)
2. not A, not B (the CP)
3. not A, B


Impossible

4. A, not B


Please let me know if that helps. Thanks!

[alphonse92m@gmail.com]

How do you know when to diagram a stimulus with the contrapositive vs logical opposite?

[Dave Killoran]

Typically it's in the stimulus language itself that they use the contrapositive. They'll say A B, then introduce a B term, triggering the contrapositive. The "not nec" idea appears infrequently.

Thanks!