Proofs

From Wolf:

1. A statement Q is said to be a propositional consequence of statements iff the single statement is a tautology.
2. Theorem: Suppose the statement R is a consequence of premises , and another statement Q is a consequence of and R. Then Q is a consequence of just .

Example: Prove that the following argument is correct: If Al shows up, Betty won't. If Al and Cathy show up, then so will Dave. Betty or Cathy (or both) will show up. But Al and Dave won't both show up. Therefore, Al won't show up.

1. Solution Method 1: Show that the following statement is a tautology:

   

2. Solution Method 2:

   

3. Solution Method 3: We are given that Al and Dave won't both show up. Therefore, if Al shows up, Dave won't (using tautology 19).

   Now, let's assume Al shows up. Then we are told that Betty will not show up. But we also know that Betty or Cathy will show up. Therefore, Cathy must show up. But that means Al and Cathy show up, and we are told that if they both show up, then Dave must show up. so we have shown that if Al shows up, then Dave shows up.

   Putting both previous paragraphs together, we have shown that if Al shows up, then Dave will show up and Dave won't show up. That is, if Al shows up, something impossible occurs. Therefore, Al cannot show up (tautology 28).