# Thread: Deriving rules of inference

1. ## Deriving rules of inference

I'm going over old test results, and for whatever reason I just can't get this one into my head.

Given the proposition

$(p \rightarrow q)\rightarrow(p \vee r \rightarrow q \vee r)$

determine whether it is a tautology
[it is, of course], and, if it is a tautology, write the valid argument determined by it.

In other words, figure out what rule of inference can be derived from this argument. I believe it is modus tollens, but am not sure, and just can't see what method I would use to go about this. Any ideas?

Thanks to all the good people out there!

2. Originally Posted by epicurean
I'm going over old test results, and for whatever reason I just can't get this one into my head.

Given the proposition

$(p \rightarrow q)\rightarrow(p \vee r \rightarrow q \vee r)$

determine whether it is a tautology
[it is, of course], and, if it is a tautology, write the valid argument determined by it.

In other words, figure out what rule of inference can be derived from this argument. I believe it is modus tollens, but am not sure, and just can't see what method I would use to go about this. Any ideas?

Thanks to all the good people out there!
Hi epicurean.

Modus tollens is the tautology $((p\rightarrow q)\wedge\neg\,q)\rightarrow\neg\,p.$ I have no idea what your tautology is called.

3. Abstractionist,

Thanks for taking a look at it. Now that I look at it again, it appears to be much closer to Resolution, but still isn't quite right. Methinks it could bear more looking at from my side. Thanks again!