The conditional operator of formal logic

Most of the operators of formal logic correspond in a fairly straightforward way to words or phrases in English. $\neg$ is “not” $\wedge$ is “and” and $\vee$ is “or”. The equivalence operator, $\leftrightarrow$, does not really correspond to a well-known English phrase, but it’s pretty easy to understand its meaning: for every pair of logical formulae $P$ and $Q$, $P \leftrightarrow Q$ is true if $P$ and $Q$ have the same truth value (i.e. they are both true or both false) and false if they have different truth values. In other words, $\leftrightarrow$ expresses equality of truth values.

The exception among these operators is the conditional operator, $\rightarrow$. This is usually thought of as corresponding to the English word “implies”, so for every pair of logical formulae $P$ and $Q$, $P \rightarrow Q$ means “$P$ implies $Q$”, which can also be expressed as “$Q$ if $P$” or “if $P$, then $Q$”. However, this does not tell the full story. It is clear, when you think of the meaning of $\rightarrow$ in this way, that if $P$ is true but $Q$ is false, $P \rightarrow Q$ is false, because if $P \rightarrow Q$ were true, that would mean “if $P$, then $Q$”, so, since $P$ is true, $Q$ would have to be true rather than false. It may also seem reasonable that if $P$ and $Q$ are both true, so is $P \rightarrow Q$, and in fact this is correct. But if you want the meaning of $P \rightarrow Q$ to correspond exactly to “if $P$, then $Q$”, then you’ll have a problem with this: what if $Q$ just happens to be true as well as $P$, and there is no causal link between the two formulae? In that case, the statement “if $P$, then $Q$” would be false, since it would be possible to conceive of a circumstance in which $P$ is true but $Q$ is false. For example, it is true that I am currently alive. And it’s true, at this particular moment, that Barack Obama is the President of the USA. Yet I wouldn’t say it’s true that if I am alive at this particular moment, then Barack Obama is the President of the USA. Indeed, this statement actually seems to be false, since if Mitt Romney had won the 2012 election rather than Obama, that would not necessarily have caused me to die1.

The problem is that the notion of causality is a key part of the meaning of the English word “if”—basically, “if $P$, then $Q$” means that $P$ being true causes $Q$ to be true. But the operators of formal logic are supposed to be truth-functional: if we know the truth value of $P$, and we know the truth value of $Q$, then we should be able to work out the truth value of $P \rightarrow Q$. If we defined $P \rightarrow Q$ to have exactly the same truth value as “if $P$, then $Q$”, then this would not be the case: if we knew that $P$ and $Q$ were both true, we would still need some further information about the causal link between the two formulae before we could conclude that $P \rightarrow Q$ was true.

Rather than thinking of the meaning of $P \rightarrow Q$ as “if $P$, then $Q$”, I suggest you think of it as “when $P$, then $Q$. The point of switching to the word “when” is that “when” has the same meaning as “if” except that it lacks the connotation of a causal link. It seems less ridiculous to say that, when I am alive at this particular moment, then Barack Obama is the President of the USA2. The scenario in which Mitt Romney won rather than Obama has no relevance to whether this statement is true.

Using this conception of the meaning of $P \rightarrow Q$ we can work out how to determine its truth value. As we already said, if $P$ is true, then $P \rightarrow Q$ has the truth vaue of $Q$ (it’s true if $Q$ is true, and false if $Q$ is false). What if $P$ is false? Well, then $P \rightarrow Q$ is actually always true in a vacuous sense. To see why this is the case, it might be best to think about what happens if $P$ is a disjunction of multiple logical formulae $P_1$, $P_2$, … and $P_n$ (so that the meaning of $P$ is “$P_1$, or $P_2$, … or $P_n$”). Then $P \rightarrow Q$ means “when $P_1$, $P_2$, … or $P_n$, $Q$”. In order to check whether $P \rightarrow Q$ is true, then, you have to check that all the statements $P_1 \rightarrow Q$ is true, $P_2 \rightarrow Q$, … and $P_n \rightarrow Q$ are true. But if $n$ is 0 (and the disjunction of an empty list of statements is false, since none of the statements in the list are true), there are no statements to check, so you’re already done: you can conclude straight away that $P \rightarrow Q$ is true.

Another way to think about it which may help you make sense of it is to think about quantified statements: those of the form “all Xs are Ys”, e.g. “all male worker bees are avid fans of Doctor Who”. For every object $x$, let $P(x)$ be the statement “$x$ is a male worker bee” and let $Q(x)$ be the statement “$x$ is an avid fan of Doctor Who”; then this statement can be expressed in formal logic as the conjunction of all the statements $P(x) \rightarrow Q(x)$ for each object $x$. Is this statement true? Well, are there any male worker bees that aren’t avid fans of Doctor Who? No, there aren’t, because there are no male worker bees, and therefore, all male worker bees really are avid fans of Doctor Who. That means that for every object $x$, $P(x) \rightarrow Q(x)$ is true. In particular, for every object $x$ which is not a male worker bee, so that $P(x)$ is true, $P(x) \rightarrow Q(x)$ is true, regardless of whether $x$ is actually an avid fan of Doctor Who or not.

Isn’t formal logic weird?

Footnotes

1. ^ Mitt Romney wouldn’t have been that bad a President.
2. ^ If you heard this statement in conversation, it would, however, acquire additional connotations due to Grice’s maxim of quantity. If the speaker merely wanted to say that Barack Obama was the President of the USA, they would just say that, and the fact that they didn’t say that suggests it is not in fact true, so perhaps if the speaker wasn’t alive at this particular moment, Barack Obama wouldn’t be the President of the USA. However, the statement is true in the same sense that it is true for a person with two eyes to say “I have one eye”.