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Ncontrer18 on November 13, 2019

Missing Premise with 3 Premises

Watching the video on youtube was very helpful! Thank you for taking the time to explain the missing premise assignment. I feel like I've got a good handle on most of the missing premises, but I continue to struggle a bit when there are three missing premises. Specifically, when the conclusion and its contrapositive share sufficients with the other two premises. For example P: A-->B NotB-->NotA P: Not X-->NotD D-->X P: C:Not X-->NotA A-->X I get confused because I could go either way. I could connect the conclusion to premise 2 and it would look as such: NotX-->NotD-->NotA and I would conclude that my missing premise is the ending: NotD-->NotA. Or I could connect the conclusion to the first premise and get the following answer: A-->B-->X Where I would conclude the missing premise is B-->X. Neither is the correct answer and now I feel like I'm way overthinking it. Please help!

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Irina on November 13, 2019


I think the point of these exercises is to use all the premises to reach your conclusion. Both of your examples only use two out of three premises, if we assume that the missing premise is B ->X, the conclusion is reached by applying rules of inference only to premises 1 and 3. We could take transposition of premise 2 but we could not connect it to premises 1 and 3 otherwise:

(1) A-> B
(2) ~X -> ~ D
(3) B -> X
(4) A -> X 1,3
(5) D -> X 2 transposition
(6) A v D ->X 4,5

In your first example, if the missing premise is ~D -> ~A, then we again only use premises 2 and 3 to reach the conclusion:
(1) A -> B
(2) ~X -> ~D
(3) ~D -> ~A
(4) ~X -> ~A 2,3

So the missing premise needs to connect premises 1 and 2 to infer the conclusion:
(1) ~B -> ~A
(2) ~X -> ~ D
(3) ?
(4) ~X -> ~A

We can see that to get from ~X to ~A using all the premises, the logical chain would have to be ~ X -> ~D -> ? -> ~B -> ~A . The missing connection is therefore ~D -> ~B or B -> D .

Let me know if this helps and if you have any other questions.

Ncontrer18 on November 20, 2019

This was very helpful! It made it much simpler, thank you.