Nesting Conditionals

[Foti]

I have two questions concerning nesting conditionals.

I am referencing the soil depletion example that was used in the past forums that you suggested I look into (43-2-7)

Is it safe to say that to negate this relationship: A --> B it is the same thing as A + B?
Example:
"If nutrients are completely depleted additional crops cannot be grown unless fertilizer is applied to the soil"
To negate this first relationship would mean that “if nutrients are completely depleted AND additional crops can be grown"?

Therefore... D + AC --> F
~F --> ~D or ~AC

According to past forums however, the correct way to diagram this would be:
D and ~F --> ~AC
AC --> ~D or F

Also according to the shortcut you gave me: "If A, then B, unless C" can automatically be translated as: A --> B or C
Why does this example not translate to:

A: Depleted
B: No additional crops
C: Fertilizer

Depleted --> No additional crops OR Fertilizer

Thank you for your help and time. It is truly appreciated.

[Jeremy Press]

Great question, Foti!

That’s an example of the “If not B, then not A, unless C” form of statement I was referring to, where you have to add an additional step to the process. Notice how both of the concepts in the original A/B portion of the statement have a “negative” flavor to them (“deplete” means “not maintain”; and “cannot be grown” is explicitly negative). When we confront such a statement, to utilize the handy rule I mentioned (that Paul originally referred to in the older thread), first convert the A/B portion of the statement to its contrapositive, i.e. “If crops can be grown, then nutrients must be maintained (“not depleted”), unless fertilizer is applied. Then you can apply the simple rule, to get:

Grow Maintain Nutrients or Apply Fertilizer

Make sense? Keep up the great work—you’re doing the hard work of deep analysis you’ll need to be very successful on this test!

Jeremy

[Foti]

I am referencing the soil depletion example that was used in the past forums regarding this same topic (43-2-7)

"If nutrients are completely depleted additional crops cannot be grown unless fertilizer is applied to the soil"

1) Using this above example, if I were to keep depleted, without adjusting for its negative tone, could I manipulate it to use this trick:
A --> B unless C = A --> B or C ? For instance, if the relationship were to read A --> ~B Unless C can I use the trick or does it have to be a relationship where a positive sufficient condition leads to a positive necessary condition in order for it to work? I understand that if it were ~A --> ~B then I could just use the contrapositive and continue on from there. I just want to know if it can work when dealing with a mutually exclusive relationship?

In this specific example it would be: depleted --> ~additional crops UNLESS fertilizer

2) Is it safe to say that to negate this relationship: A --> ~B it is always the same thing as A + B?
Using the same example as above:
"If nutrients are completely depleted additional crops cannot be grown unless fertilizer is applied to the soil"
To negate this first relationship would mean that “if nutrients are completely depleted AND additional crops can be grown"?

Therefore... D + AC --> F
~F --> ~D or ~AC

3 ) Is the correct way to negate the relationship A --> B just “If A not necessarily B”? I know it’s negating the entire relationship but I just want to put it into words so I can better understand it conceptually.

Thank you very much for all your time and help.

[Jeremy Press]

Hi Foti,

You can use that simple trick for an "If A, then not B, unless C" statement. Consider this example: "If you are a lawyer, then you are unhappy unless you are working for the public interest." That equates validly with L ~H or WPI. The valid contrapositive of that relationship would be: H and ~WPI ~L.

Let's talk about the negation of a conditional statement next. The negation of a conditional statement is the possibility that the sufficient condition can occur without the necessary condition. Be careful with your understanding of the A/B relationship there. The negation of A B is simply that there can be some A's without a B, or, as you correctly stated: A but not necessarily B. That shouldn't be understood, though, as "All A without B."

In the depletion example, both the first and the second relationship you note (D ~AC or F; D + AC F) are generally valid, so long as you understand precisely what those conditions mean in context. I'd restate the first one as follows: "If soil has been depleted, then it must be the case that either additional crops cannot be grown in that soil or fertilizer must added to grow such crops." I'd restate the second one as follows: "If soil has been depleted but it turns out additional crops can be grown in that soil, then it must be the case that fertilizer was added to the soil." As this shows, there are multiple ways of looking at the nested conditional relationship. Focusing, however, on context and the purpose of the stated relationship is helpful. What are we interested in with that particular stimulus and relationship? Growing additional crops. Find the necessary conditions for that first (and diagram the relationship and contrapositive if necessary). Then evaluate answer choices from there.

I hope this helps!

Jeremy

[Foti]

Thank you very much. I truly appreciate all of your help.