Proof By Contradiction - Logic and Proof

Question

Consider the arguments below and decide whether they are valid. If they are, write down an inofrmal proof, phrased in complete, well.formed English sentences. When you use proof by case or proof by contradiction, say so. You do not have to be explicit about the use of simple proof steps like conjunction elimination. If the argument is invalid, constuct a counterexample.

1. Anna or Ben is not shopping
2. Ben is shopping or Ben and Anna are married
3. Anna and Ben are not married or Anna is shopping

          

Ben and Anna are married.

I am battling to get my head around the basics in this subject.

Firstly, I need to ascertain whether or not the statement is valid. I actually went as far as to construct a truth table and ultimately looked to see if there were instances where the premises were all true and the conclusion false. This was not the case so I have claimed that this is valid.

I understand the following concepts

1. To prove that $P$ is false, show that a contradiction ⊥ follows from $P$

2. To prove that $P$ is true, show that a contradiction ⊥ follows from $\neg P$

It seems to simple but each time I try a work around I somehow cannot contradict my assumption using the premises and the cases I try.

For example: Suppose Anna and Ben are not married. By premise 2 we cannot say they are married so then Ben is shopping. But premise 1 says Anna or Ben is not shopping but if Ben is not shopping this would be a contradiction so Anna is not shopping. By premise 3 Anna is not shopping which was determined by premise 1 so Anna and Ben are not married (which is not a contradiction at all)

I am also wondering if I should just be using proof by case here but as I see negation I thought perhaps proof by contradiction would be correct?

I know this is probably fairly simple but if I don't get the ground work here I'll miss the boat.

Please could I have some advice on where I am going wrong?

Thanks!

Answer 1

Your argument is fine: You show that the conclusion can be wrong even though the premises are all true. Once found, you can smoothen it for presentation someway like this:

Assume Anna and Ben are not married; Ben is shopping; Anna is not shopping. (This combination of facts is certainly consistent: By everyday experience, not being married does not prevent people from shopping or not shopping, and independently so). Then one readily verifies that each of the three premises is true, but the conclusion is false. Hence the argument is not valid.

In effect, this is a proof by counterexample that the argument is invalid, just as asked for by the problem statement.