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.

please...

Re: prove by rules of inference

Which exactly system of inference rules are you using? See this sticky thread.

Re: prove by rules of inference

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 $\displaystyle \left( {p \vee q} \right) \to r$.

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).

Re: prove by rules of inference

So, premise (1) and premise (2) are not really necessary here?

Re: prove by rules of inference