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January 26th, 2018, 03:02 AM   #1
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Axioms logic problem

Hello everyone,

I am currently stuck on one problem for an assignment for a university logic course. I will provide a picture in the attachment.

I have to prove the blue formula at the bottom of the note in the picture by only using the axioms written above in pencil. I would start by using axiom 7 twice (getting rid of the universal quantifiers, so I can limit myself to axioms 2-4), and end with axiom 6 to re-introduce the universal quantifier.

The steps in-between, however, are a mystery to me, even after -literally- hours of looking for a solution. I do know that ((Ax->Rxx)->~Ax) should be equivalent to (Rxx->~Ax), so a deduction should be possible. But doing that with axioms seems to be impossible for me.

Can someone give me some more insight in these axioms or put me on the right tracks, please?

Thanks in advance!
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January 26th, 2018, 03:10 AM   #2
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Here's a solution, I'll let you formalize it in your language:

Take $x$ arbitrary.
Assume $A_x$.
If $R_{xx}$, then $A_x\rightarrow R_{xx}$, hence by hypothesis $\neq A_x$.
Contradiction.
Hence $\neg R_{xx}$.

Hence $A_x\rightarrow \neg R_{xx}$.
Hence $R_{xx}\rightarrow \neg A_x$.

So you'll want to do some proof by contradiction. The formal way of doing this is (3).
Thanks from Atrend
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January 26th, 2018, 04:45 AM   #3
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Thank you for your quick reply, Micrm@ss! Your help is greatly appreciated!
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