Logic: Conditionals – Foundations of Mathematics

So far our statements haven't been very interesting. In fact most mathematical statements of interest are things like

If a function is differentiable, then it is continuous.

These statements are known as conditional, because they depend on some criteria, or conditions, being satisfied.

Consider the following statement (first, convince yourself that it is a statement!):

If it rains, the laundry will be ruined.

What is this statement about? It's not really about whether or not it rained; it's not really about whether or not the laundry was ruined. Rather it's about the relationship between rain and laundry. That's what makes conditionals so interesting.

Definition. Given two propositions $P$ and $Q$ , the proposition $P\Rightarrow Q$ (read if $P$ then $Q$ ) is given by the truth table:

$P$	$Q$	$P\Rightarrow Q$
T	T	T	1
T	F	F	2
F	T	T	3
F	F	T	4

Exercise. Is $\Rightarrow$ commutative or associative?

A word of caution: this truth table causes consternation among most people when they first encounter it. In particular, lines 3 and 4 sometimes rub people the wrong way. Statements like

If $1=2$ , then $3=4$ .

don't exactly feel true. But we'll go over each line of the truth table to justify why these are the right choices. To make things easier, we'll label the two inputs to the conditional so we can refer to them. There are several labeling schemes:

$P\Rightarrow Q$
antecedent		consequent
hypothesis		thesis
assumption		conclusion

Now let's go through each line of the truth table for $\Rightarrow$ . There are two analogies that will help us understand this: the analogy of science, and the analogy of a contract.

The Science Analogy

Let's consider the conditional

If it rains, the laundry will be ruined.

as a scientific claim. As good laundrologists, we want to put this to the test. So we conduct an experiment. Line 1 of the truth table is the situation when our experimental variable (the rain) is present, and so is our measured variable (the ruin of the laundry). These particular experimental data support the claim. We'll record that as a T (though good experimental scientists know that a scientific claim is only ever provisionally established).

Line 2 is what would happen if the experimental variable were there, but the expected outcome (laundry being ruined) didn't happen. Then we would know for sure that the proposed theory is definitely false. We record a F.

What about lines 3 and 4? That's what scientists call the control condition; the experimental variable isn't present. So what do we record in our lab notebook? No matter what happens with the laundry, the control data don't tell us one way or the other about the claim. In particular, they leave open the possibility that the claim is true. So we record a provisional T in our lab book, just the same as the provisional T on line 1.

The Contract Analogy

Now let's consider a conditional like

If you do all the work in MA 225, you'll get an A.

Treat this as a contract or an agreement between us. Line 1 of the truth table is: you did the work, and you got the A. No problem: the agreement was carried out just fine. Line 2 of the truth table is: you did the work, but you didn't get the A. That's a clear violation of the agreement; in other words the agreement didn't hold. Hence the truth value of F.

What about lines 3 and 4? That's when you didn't do all of the work. I could still give you an A (if I'm feeling kind) or not give you an A, but either way -- you couldn't really say that I'd violated the agreement. The agreement is still in force the whole time. That's why we assign the truth value T to these lines.

In both analogies, a really solid way to think about this is what-if-not thinking: what would it mean for the conditional to be false? How could you know for sure that If it rains, the laundry will be ruined is false? Only if it it rains but the laundry isn't ruined. How could you know for sure that it's false that If you do all the work in MA 225, you'll get an A is false? Only if you do all the work in MA 225, but nevertheless you don't get the A. Let's record that as a proposition:

Proposition. $\sim (P\Rightarrow Q)$ is equivalent to $P\wedge (\sim Q)$

We also have the equivalence:

Proposition. $P\Rightarrow Q$ is equivalent to $(\sim P)\vee Q$

Exercise. Prove this proposition in two ways: first, from the proposition above it using one of De Morgan's Laws; and second, by comparing truth tables.

Notice that I used the words but and nevertheless to translate the logical symbol $\wedge$ . Normally $\wedge$ is translated and, but the word but can be very useful when you want to deny a conditional. When it comes to truth values, the following words all mean the same thing:

and, but, yet, nevertheless

When it comes to $\Rightarrow$ , there are even more possibilities for reasonable English translations; so many in fact that it's impossible to list them all. Here are a few that you will likely encounter. The following all mean the same thing, namely $\text{the King of France is bald } \Rightarrow \text{ roses are red}$ :

If the King of France is bald, then roses are red.
Roses are red if the King of France is bald.
The baldness of the King of France guarantees that roses are red.
The baldness of the King of France suffices for roses to be red.
For roses to be red, it is sufficient that the King of France be bald.
It is necessary for the King of France to be bald, that roses be red.
Roses' redness is necessary to the King of France's baldness.
Should the King of France be bald, roses will be red.
Supposing the King of France is bald, roses are red.
The King of France is bald only if roses are red.

I've departed from my normal typographical convention and put the keywords in italics, the antecedent in blue bold, and the consequent in green underline.

Some things to notice about these translations, in no particular order:

Sometimes the antecedent occurs first in the sentence, sometimes the consequent does.
Sometimes the antecedent occurs first in time, sometimes the consequent does.
The grammatical form of the antecedent and the consequent have to be adapted to read nicely.
The conditional keyword is sometimes split into more than one word.
If is different from only if.

Exercise. Create English translations of the conditional above, using the conditional keywords when and whenever.

How to keep it all straight? Use what-if-not-thinking. That is, ask yourself how you'd know the sentence was false. For example: what would it mean for Roses' redness is necessary to the King of France's baldness to be false? If the redness isn't truly necessary, that means there's a circumstance in which the King is bald but the roses aren't red.

biconditionals

There is also a kind of conditional called the biconditional, which we denote $P\Leftrightarrow Q$ :

$P$	$Q$	$P\Leftrightarrow Q$
T	T	T	1
T	F	F	2
F	T	F	3
F	F	T	4

Proposition. $P\Leftrightarrow Q$ is equivalent to $(P\Rightarrow Q)\wedge (Q\Rightarrow P)$

We therefore write and read $P\Leftrightarrow Q$ as $P$ if and only if $Q$ , or sometimes as $P$ iff $Q$ or $P$ exactly when $Q$ .

Exercise. Is $\Leftrightarrow$ commutative or associative?