Conditionals in English

‘If … then … ’ sentences in English (and related sentences in other languages) are called conditionals. As we’ve noted already, arrow sentences are classed as conditionals.

There do seem to be parallels between arrow and at least some uses of ‘If … then … ’. Take a look at the following arguments.

• If it’s raining, then we’re getting wet. It’s raining. Therefore, we’re getting wet.
• If it’s raining, then we’re getting wet. It’s not the case that we’re getting wet. Therefore, it’s not the case that it’s raining.

These both seem to be valid, and formally valid: it seems like any argument with one of these shapes would be valid.

The case for the validity of the first argument seems strong and obvious. We want to say that the first premise says that where the antecedent is true, the consequent is true too. The second premise is just the antecedent. It seems that if they’re both true, then the consequent must be true too.

We can press for the validity of the second argument in the following way. We want to say that the first premise says that where the antecedent (‘It’s raining’) is true, the consequent (‘We’re getting wet’) is true too. Now try supposing that the antecedent is true: that it is raining. On that supposition, the consequent would have to be true too (we’d be getting wet). But our second premise states that it’s not the case that we’re getting wet. So our supposition (that it’s raining) isn’t compatible with the truth of our two premises; so where they are true it must be false.

In both cases it seems like any argument with the same shape would be valid for the kinds of reasons we’ve laid out.

Now, note the parallels between arrow arguments and ‘If … then …’ arguments. Look a the first form:

• (P (rightarrow) Q), P; therefore, Q *If it’s raining, then we’re getting wet. It’s raining. Therefore, we’re getting wet.

And now look at the second form:

• (P (rightarrow) Q), ~Q; therefore, ~P
• If it’s raining, then we’re getting wet. It’s not the case that we’re getting wet. Therefore, it’s not the case that it’s raining.

The parallels are striking. And this gives some support to the idea that arrow means the same as ‘If … then … ’. But this idea can be questioned, as we’ll see.