LRB Chapter 6 page 161

[Stephanie Oswalt]

We recently received the following question from a student:

Hi there,

I have a question regarding Chapter 6 (Sufficient and Necessary Conditions).

On page 161 the paragraph that starts off with "Now, let's try that in English!" goes to explain that "if a sufficient condition occurs, you automatically know that the necessary condition also occurs. If a necessary condition occurs, then it is possible but NOT CERTAIN that the sufficient condition will occur" I understand this part however the next part further states "THUS they always occur TOGETHER"

This last part is misleading I think. Because if we're only CERTAIN that A --> B and B MAY cause A but not certain, then we cannot conclude that "they always occur together" can we?

Could you please help me make the connection between the first statement and the second confusing part?

Thank you in advance.

Regards,

Negar

An instructor will respond to this question below. Thanks!

[Dave Killoran]

Hi Negar,

Thanks for the question! Sure, I'm happy to talk about what that phrasing means! First, the context there is talking about the existence of sufficient and necessary conditions. Individually, the discussion shows that you can have one without the other at times in a "happens/not happens" sense, and at other times (such as when a sufficient condition occurs), if you have one then you have to have the other. I think that relationship aspect was clear to you.

Second, the comment about them both occurring together refers to the existence of the conditions themselves in an abstract sense. In other words, if you are going to say that a sufficient condition is present, then you automatically imply that somewhere else there is a necessary condition. You can't just have a sufficient condition alone because a sufficient condition is, by definition, related to a necessary condition. So, once there is a sufficient condition there must also be a necessary condition somewhere, and the reverse is true as well. A necessary condition has to be necessary for something, and that something is the sufficient condition. So, if you have a necessary condition, somewhere else there is a sufficient condition (which may or may not be occurring).

If you look at that relationship I'm discussing there, it reflects back on the formal relationships being defined above. Phrasing such as "A sufficient condition...indicates that a necessary condition must also occur" and "A necessary condition...is required in order for a sufficient condition to occur" is confusing unless you understand that these conditions automatically imply the existence of the other, and that's the point I was trying to get at here.

Please let me know if that helps. Thanks!