# prove by rules of inference

• Sep 30th 2011, 05:27 AM
deniselim17
prove by rules of inference
I have premises
(1) $\displaystyle p$
(2) $\displaystyle p\rightarrow q$
(3) $\displaystyle r$
and conclusion
($\displaystyle p\vee q$) $\displaystyle \rightarrow$ r

I've checked that the argument is valid.
But I still cannot find a way to show it by using rules of inference.
• Sep 30th 2011, 05:33 AM
emakarov
Re: prove by rules of inference
Which exactly system of inference rules are you using? See this sticky thread.
• Sep 30th 2011, 04:51 PM
deniselim17
Re: prove by rules of inference
• Sep 30th 2011, 05:23 PM
Plato
Re: prove by rules of inference
Quote:

Originally Posted by deniselim17
I have premises
(1) $\displaystyle p$
(2) $\displaystyle p\rightarrow q$
(3) $\displaystyle r$
and conclusion
($\displaystyle p\vee q$) $\displaystyle \rightarrow$

From (3) we get $\displaystyle \neg \left( {p \vee q} \right) \vee r$ by addition.

Then by material implication we get $\displaystyle \left( {p \vee q} \right) \to r$.
• Oct 1st 2011, 02:10 AM
emakarov
Re: prove by rules of inference
Another way is to have $\displaystyle p\lor q\vdash r$ just because r is a premise and use the Deduction theorem (or Conditional Introduction).
• Oct 1st 2011, 02:37 AM
deniselim17
Re: prove by rules of inference
So, premise (1) and premise (2) are not really necessary here?
• Oct 1st 2011, 03:17 AM
emakarov
Re: prove by rules of inference
That's right.