next up previous
Next: Some Problems Up: Brief Logic Previous: Some Useful Tautologies

Proofs

From Wolf:

  1. A statement Q is said to be a propositional consequence of statements tex2html_wrap_inline241 iff the single statement tex2html_wrap_inline243 is a tautology.
  2. Theorem: Suppose the statement R is a consequence of premises tex2html_wrap_inline241 , and another statement Q is a consequence of tex2html_wrap_inline241 and R. Then Q is a consequence of just tex2html_wrap_inline241 .

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:

    displaymath237

  2. Solution Method 2:

    tabular88

  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).



Carl Lee
Wed Nov 18 12:16:44 EST 1998