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Mazen February 27, 2020

Skylar explanation against Mehran's

Hello, In terms of answer-choice A, although I have been applying the logic, I have been doing so somewhat uneasily until I came across Skylar's explanation which justified to me my uneasiness with the blind application. Specifically, she stated that the sufficient condition must be present at least once in order for its separate necessary conditions to connect via the quantifier "some." Subsequently, I demonstrate my thinking for you to please correct it. GA---->O GA---->I Mehran's explanation is that from either of the principles diagrammed above we can infer as must be true that the necessary connects with the sufficient via "some." Accordingly, either O<SOME>GA, or I<SOME>GA. Next, we take either of the inferred quantifiers, for instance O<SOME>GA and combined it with the necessary-sufficient principle that involves the other necessary in this instance it would be GA------->I, and deduce O<SOME>I. I was honestly and humbly very nervous because we could have a necessary in the absence of the sufficient. As I understand the necessary-sufficient relationship, the sufficient variable guarantees the existence of the necessary variable. However, it is not the only trigger to the necessary variable. The necessary variable may be triggered by some variable other than that specific sufficient. In other words, we may very well have vegetables without any carrots. Just because all carrots are vegetables, it doesn't by necessity mean that some vegetables are carrots, UNLESS we know for a fact that at least one carrot exists (Here is where Skylar's explanation becomes very helpful for me). I was very uncomfortable with the language it "must be true" to infer from GA---->O that O<SOME>GA, because we don't know if "some" (at least one GA, possibly all GAs) "GA" exists/exist. Skylar in a reply, however, predicates the connection between the necessary variables "I" and "O" via a some on the "existence of a 'GA.'" In retrospect, am I safe to always infer as must be true a "some"- statement between the necessary and the sufficient without first being assured that some of the sufficient exists?

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BenMingov March 13, 2020

Hi Mazen, thanks for bringing this up.

Quite a bit to unpack here so let's try working through this.

You correctly demonstrated the approach to determining O <-some-> I.

I think something that might help you better understand necessary/sufficient is the following:

You correctly understand that sufficient condition guarantees the necessary. But I would like to think about the necessary condition a bit differently. Not so much about the fact that something else could trigger it, but rather that it is independent of the sufficient. It can simply occur without minding the sufficient.

As far as knowing whether at least one carrot exists..

Let's think of conditions simply as rules. If carrot then vegetable. This is the rule for if carrot. We know rule for not vegetable. If not vegetable, then not carrot. This is the contrapositive.

The rule doesn't apply unless the sufficient of either the original diagram or the contrapositive is met. However, it never changes the inferences we can make based on the rule. Such as carrots <-some-> vegetable. This is always true based on the condition.

Please let me know if this helps. I'd be happy to answer any other questions you have!