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Question: Formal Logic Question!?
I just can't seem to figure this one out!. How do I translate this one question into sentence logic from English!?


John's getting a job is not a necessary and sufficient condition for Ravi's moving out!.Www@QuestionHome@Com


Best Answer - Chosen by Asker:
Well first start by translating your propositions:

John gets a job (J)
Ravi moves out (R)

If John getting a job were a necessary and sufficient condition for Ravi moving out, the logical relationship would be biconditional (if and only if)!. I don't have any biconditional or conditional symbols on my keyboard, so i am going to use = to symbolize a biconditional (it can be properly symbolized using three lines)!.

So J=R would translate as "Ravi moves out if and only if John gets a job!." (If John getting a job causes Ravi to move out, it is a sufficient condition; If Ravi moves out only if John gets a job, it is a necessary condition!.)

Since you are denying the biconditional relationship, you could translate your sentence as ~J=R!.

It could also be translated just using conditionals as ~ ((J>R) & (R>J))!. (A biconditional can also be written as (J>R) & (R>J)!.)Www@QuestionHome@Com

Let J = John got a job
R = Ravi moved out

J is not sufficient for R: ~(J->R)
(That is to say, J may be true while R is false!.)
J is not necessary for R: ~(R->J)
(In practical terms, R may be true while J is false!.)

So your formulation might be something like (~(J->R)) v (~(R->J))!.
(That is, J is neither necessary nor sufficient for R!.)

You can simplify that as follows:

(~(J->R)) v (~(R->J))
(~(~J v R)) v (~(~R v J))
(J ^ ~R) v (R ^ ~J)

Now that I look at it, I should have mentioned this: I used De Morgan's Law to change "not (necessary and sufficient)" to "not necessary or not sufficient"!. So you might come up with an additional step somewhere along the way!.

Actually, there's a shorter way to do this (the other answerer reminded me)!. "Necessary and sufficient" means there's an if and only if somewhere in the sentence!. <--> is the usual symbol for that!. So ~(J<-->R)!. You can simply write ~J<-->R, since they have the same truth table!.

Peace!Www@QuestionHome@Com

Instead of "condition" insert "reason!."Www@QuestionHome@Com