Propositional logic problem: Sales, expenses and happiness of the boss

Question

Either sales will go up and the boss will be happy, or expenses will go up and the boss won’t be happy. Therefore, sales and expenses will not both go up.

I know the solution is that the conclusion is wrong, but I don't see why. I would say that the boss is either happy, or he is unhappy, but he can't be both. Because of this, it is impossible that the sales go up and the expenses go up, since this would entail a state of schizophrenia of the boss, and thus the conclusion is true.

Answer 1

Let $S$ be the proposition that sales will go up, $E$ the proposition that expenses will go up, $H$ the proposition that the boss will be happy.

Suppose we are given that $P$ is true, where $P = (S \wedge H) \vee (E \wedge \neg H)$.

With the stated definitions, it then is sound to say it is impossible that the boss will be both happy and unhappy; $\neg(H \wedge \neg H)$. But it is not sound to say that the boss's future happiness controls both sales and expenses simultaneously.

Suppose the boss will be happy ($H$ is true); then $(E \wedge \neg H)$ is false and so $(S \wedge H)$ must be true. It follows that $S$ is true.

Suppose the boss will not be happy ($H$ is false); then $(S \wedge H)$ is false and so $(E \wedge \neg H)$ must be true. It follows that $E$ is true.

But observe: knowing that $H$ is true tells us nothing about the truth or untruth of $E$. It tells us only about the truth of $S$. On the other hand, knowing that $H$ is false tells us nothing about the truth or untruth of $S$; it tells us only about the truth of $E$.

In other words, by knowing the truth value of $H$ you could make a conclusion about $S$ or a conclusion about $E$, but not both.

In fact, if both sales will go up and expenses will go up, that is, if $S \wedge E$, then $P$ is satisfied regardless of the boss's future happiness. Either $H$ is true, so $S \wedge H$ is true and $P$ likewise is true because its first clause is true; or $H$ is false, so $(E \wedge \neg H)$ is true and $P$ likewise is true because its second clause is true.

Answer 2

Surely you can have two different emotions about two different things at the same time? e.g. you could love dogs and hate cats simultaneously? This is the same kind of thing, because the event causes the emotion ABOUT THAT EVENT, but an overall emotion does NOT cause the event. You would be correct if the statement was 'either the boss will be happy so sales will go up, or the boss will be unhappy so expenses will go up. Therefore, sales and expenses will not both go up.'

Answer 3

You can verify this by a cursory search: the deduction is $$ \frac{(S\land H)\lor (E\land \neg H)}{\neg(S\land E)}, $$ where

• $S::=$ "Sales will go up."
• $H::=$ "The boss will be happy."
• $E::=$ "Expenses will go up."

This deduction is not sound. For example, $S\land E\land H$ is consistent with the antecedent but not with the conclusion.