Negation in Conditional Reasoning

[ghan0510]

Greetings,

I've learned that if the original conditional statement is A -> B, then:

~B -> ~A would be its contrapositive,
~A -> ~B would be its Mistaken Negation, and
A -> ~B would be its Logical Negation (from the Power Score LR Bible book page #357), which is different from Mistaken Negation because it negates just the necessary condition in an attempt to produce the opposite of the original statement.

Then my question is, what is ~A -> B? Would this also be considered a Logical Negation and work as the opposite of the original statement?

Thanks!

[Dave Killoran]

Hi G,

Great question! This would actually not be considered anything at all In the original statement, the sufficient condition is A, where here it is ~A. In other words, it's the case when the sufficient condition isn't met, and when that occurs, anything is possible. For example, consider the following example:

• Speaker: If you attend Harvard Law School, you will be successful.
  
  Student: Well, I didn't attend HLS, so what does that mean?
  
  Speaker: Umm, I don't know. Anything is still possible. I just know what happens if you attend Harvard...

Obviously that's a bit of a humorous example, but the point is that when the sufficient condition goes unmet, anything can happen, so it's neither a flaw nor a logical cousin of the original statement.

Please let me know if that helps. Thanks!