How do you use the rules of inference to construct formal proofs?

Answer 1

The point of a formal proof of validity is to get back to the conclusion of a syllogism in as few steps as possible. Let's say we have the syllogism:

1. P>Q (that's supposed to be a conditional...)

2.P

3.Q>R /.'.R

What you want to do is keep going with the syllogism. You can use steps 1, 2,and 3, but you cannot use the conclusion. How you use them is try to find which rules of inference start with any of your premises. For instance, step #1, P>Q and step #3, Q>R are the first two premises in the Hypothetical syllogism. So you could make step #4 P>R. Next to this step you will put what is called the 'justification', which would look something like this: 1,3 H.S. (which means: I used steps 1 and 2 and a hypothetical syllogism to make this step). Now we can use the step we just made in a Modus Ponens. This would use steps 4 and 2, and would look like this: R. Do you recognize that? That was our conclusion. We have now finished this formal proof of validity. Here's what the whole thing looks like:

1. P>Q

2.P

3.Q>R /.'. R

4.P>R 1,3 H.S.

5.R. 4,2 M.P.

(If you want to look like you really know what you're doing, you will want to put Q.E.D. at the end of a formal proof. That's what the real logicians do).

Hope this helps!!

(By the way, I'm 13.) :D