I am currently writing a paper with some colleagues that has made me realize I must obtain a more solid grounding in the art of mathematical proofs. This involves venturing into the fun world of formal logic, which I am currently doing with the help of Velleman’s “How to prove it: A structured approach.” In that book, there is an interesting argument, the validity of which I am supposed to test with a truth table. It goes like this:

Either sales or expenses will go up. If sales go up, then the boss will be happy. If expenses go up, then the boss will be unhappy. Therefore, sales and expenses will not both go up.

Let \(S\) be “sales go up”. Let \(E\) be “expenses go up.” And let \(H\) be “boss will be happy.”

The first premise is \(S\lor E\). At first I translated the “either … or …” as “either … or … but not both”, i.e., \((S\land \lnot E) \lor (\lnot S\land E)\), but this automatically gives the conclusion \(\lnot(S \land E)\) without the boss.

The second and third premises are \(S \to H\) and \(E \to \lnot H\).

And finally the conclusion is \(\lnot(S \land E)\).

It was surprising to me to find that this is a valid argument.

The boss cannot become happy and unhappy (a contradiction),

and this keeps the sales and expenses from both increasing.

However, consider the following slightly altered argument:

Either sales or expenses have gone up. If the boss is happy, then sales have gone up. If the boss is unhappy, then expenses have gone up. Therefore, sales and expenses have not both gone up.

It turns out that this argument is invalid! Now, why?

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The counterexample is when both sales and expenses have gone up and the boss is happy. In this case, all the premises are true, but the conclusion is false :)

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