If $P$ and $Q$ are not mutually exclusive, does that mean that if $P$ then $Q$ and vice versa?

Question

As far as I have understood, if statements $P$ and $Q$ are mutually exclusive then it is true that $\lnot \left(P\land Q\right).$ So then, if $P$ and $Q$ are not mutually exclusive, isn't it true that $\lnot \lnot \left(P\land Q\right)$ or, after simplification, that $P\land Q$. Given this, is the following sentence true: "Since $P$ and $Q$ are not mutually exclusive, if we make $P$ true then we obtain $Q$ as well and vice versa"?

Answer 1

The real issue here is that natural language is fuzzy, so that there are many things "$P$ and $Q$ are mutually exclusive" can mean in real life. These will correspond to mathematically different situations.

• Situation 1: $P$ and $Q$ are mere propositions, as in your question. Then "$P$ and $Q$ are mutually exclusive" indeed means $¬(P ∧ Q)$, so "$P$ and $Q$ are not mutually exclusive" does exactly mean that $P$ and $Q$ are both true.

  Given this, is the following sentence true: "Since P and Q are not mutually exclusive, if we make P true then we obtain Q as well and vice versa"?

  You could just say that $P$ and $Q$ are not mutually exclusive so they are both true. (You don't need to "make $P$ true" to obtain $Q$, you already have both.)

  However, this only applies if $P$ and $Q$ are closed propositions, with a defined truth value. In practice you would seldom use the expression "mutually exclusive" for this in real life. That's the reason why this equivalence might seem confusing.

• Situation 2: We have propositions $P(x)$ and $Q(x)$ with a free variable $x$. Then the meaning of "$P$ and $Q$ are mutually exclusive" is: $∀ x, ¬(P(x) ∧ Q(x))$, i.e., $P(x)$ and $Q(x)$ are never both true on the same $x$. The negation of this is: $∃ x, P(x) ∧ Q(x)$, i.e., there exists $x$ which makes both $P(x)$ and $Q(x)$ true. You cannot conclude more: for arbitrary $x$, either or both of $P(x)$ and $Q(x)$ might be false.

  This is already a more typical real-life use of "mutually exclusive". For example, being blue and being red are mutually exclusive properties of objects. Being blue and being made of porcelain are not mutually exclusive, which means you can find blue porcelain — it certainly doesn't mean that all porcelain is blue, or all blue objects are porcelain, or all objects are blue porcelain.

• Situation 3: In a probability context, $P$ and $Q$ could be events. Then "$P$ and $Q$ are mutually exclusive" would mean "the combination of $P$ and $Q$ at the same time has probability zero". In probability, the more common terminology for this is "$P$ and $Q$ are incompatible".

  This is also a possible meaning of "mutually exclusive" in natural language. For example, "candidate $A$ will win the election" and "candidate $B$ will win the election" are mutually exclusive. Each has a certain probability of happening, but the probability both will happen is zero.

A logician said to his kid: if you carry on acting up, you'll go without dessert! The kid stopped acting up. Yet he went without dessert. Indeed the logician had only specified what would happen if he kept acting up. (I read it here but it's probably folklore.)

Answer 2

You really need something like modal logic to analyze this. Two statements are mutually exclusive iff it is not possible for both to be true at the same time. So, not being mutually exclusive means that it is possible for both to be true. But that is of course quite different from saying that both statements are true, which is what $P \land Q$ says.

As a simple example: it is certainly possible for me to wear blue pants and a red shirt: these are not mutually exclusive. But that doesn’t mean that I am actually wearing blue pants and a red shirt right now. It also doesn't mean that if I wear blue pants, then I will wear a red shirt, or vice versa, whether meant as a weak material conditional, let alone as some kind of logical implication.

Answer 3

if statements $P$ and $Q$ are mutually exclusive then it is true that $\lnot \left(P\land Q\right).$

$$\forall x\,\lnot(P(x) \land Q(x))$$ means that $P(x)$ and $Q(x)$ have mutually exclusive truth sets, whereas $$\lnot \left(P\land Q\right)$$ means that $P$ and $Q$ are not both true.

(If $P(x) \land Q(x)$ is unsatisfiable—for example, when $Q(x)$ is defined as the $\lnot P(x)$—then we say that $P(x)$ and $Q(x)$ are inconsistent with each other.)

is the following sentence true: "Since $P$ and $Q$ are not mutually exclusive, if we make $P$ true then we obtain $Q$ as well and vice versa"?

If $P$ and $Q$ are not mutually exclusive, does that mean that if $P$ then $Q$ and vice versa?

If the truth sets of $P(x)$ and $Q(x)$ aren't mutually exclusive, then at least one object satisfies both $P(x)$ and $Q(x).$ In particular, this does imply that $P(x)$ and $Q(x)$ are equivalent for some object, that is, that $$\exists x\, \big(P(x)\leftrightarrow Q(x)\big).\tag1$$ However, the converse isn't true: sentence (1) doesn't imply that $P(x)$ and $Q(x)$ have mutually exclusive truth sets.

Answer 4

Your problem is that an implication is not a logical equivalence. As a now old dinosaur, I always say that the mathematical language is never ambiguous while the natural one (here English) can be.

Here what is true is exactly: $P$ and $Q$ mutually exclusive $\implies \lnot(P \land Q)$

And the negation gives: $P\land Q \implies$ $P$ and $Q$ are not mutually exclusive

But you cannot conclude on the other side on the implication...