Understanding the material conditional

by Issa Rice

August 22, 2021

Table of contents

Introduction and audience for this piece # ↩︎

In this essay we will explore the truth table of the material conditional (i.e., the “if …, then …” statement used in math). Students first learning formal logic or getting started with writing proofs are introduced to the idea of truth tables along with the usual logical connectives like \(\lnot\) (NOT), \(\land\) (AND), and \(\lor\) (OR). Everything goes well until the logical connective \(\implies\) (IMPLIES) for the material conditional is introduced. Why does a false statement imply everything? This is usually explained quite poorly, e.g., in terms of promises. For instance, “if the Moon is made of cheese, then \(1+1=3\)” is interpreted as the promise that \(1+1=3\) as long as the Moon is made of cheese; but since the Moon is not made of cheese, we can’t say that the promise was broken, and so we say that the promise was kept. In this way, a false antecedent (the part of the conditional appearing after “if …”) means the condition of the promise never arises, so one automatically keeps the promise.

As a beginner to math, I spent many years being confused about this. Just to give one aspect of my confusion, the explanation in terms of promises didn’t make sense because while it is true that one didn’t break the promise when the antecedent is false, it’s also not clear that one has actually kept the promise, or that one would have kept the promise. It also seemed suspicious that we were picking one specific way in which “if …, then …” was used in natural language and ignoring all the others. Why were we allowed to pick one specific meaning to use for “if …, then …”? But then again, I didn’t have the same problem with the logical OR connective: it always meant the inclusive OR in math, and I was fine to accept that. So why was I so confused about the material conditional? It seemed like I was not only confused about the truth table, but also, inside my own mind, confused about why I was confused; I had trouble articulating my own confusion.

Like many students of mathematics, I eventually decided to accept the usual truth table without understanding it because it just seemed to work. Many years later, I eventually hit upon the core of my confusion, and also a way to resolve this confusion. This is what I’d like to explain in this essay.

In keeping with the above, the audience for this essay is someone who has written some proofs before and has seen propositional logic and truth tables, but who still feels confused why the truth table for the material conditional is the way it is.

One small terminological note: in this essay I use “material conditional”, “conditional”, and “implication” interchangeably. In some contexts people distinguish the material conditional from other kinds of conditionals, and in such places these terms can’t be used interchangeably, but we will not be encountering any other conditionals in this essay, so it is safe to switch around if one phrase sounds more natural than the others. Similarly, we use “statement”, “sentence”, and “proposition” interchangeably.

Finally, I know how difficult it is to learn and retain math, so I have prepared some short questions for you to answer throughout this essay to check your understanding. If you sign up with your email address, you will get email reminders to review these questions in the future. Since the reminder emails will get more infrequent over time as your understanding becomes more durable (this is known as spaced repetition), this allows you to efficiently retain your understanding of this essay.

The material conditional # ↩︎

So why does the conditional statement \(P \implies Q\) have the truth table that it does? In order to answer this question, let’s think about what makes implication different from the other logical connectives like NOT, AND, and OR. After all, we have no problem with the other connectives so there must be some difference we can identify. And, in my opinion, it is this difference that makes people so uncomfortable about the truth table for the conditional. If you feel so inclined, I think this is a good exercise to attempt before reading on. It will give you practice peeking into your own mind and trying to pinpoint a subtle distinction your mind is making.

Now, for a logical connective like OR, there is no debate about whether it can be represented as a binary operation on the truth values of sentences. Sure, we might debate whether our “or” should be inclusive or exclusive, but in either case, it is clear that this concept can be represented as one of two columns in a truth table:

\(P\) \(Q\) \(P\) inclusive-or \(Q\) \(P\) exclusive-or \(Q\)
T T T F
T F T T
F T T T
F F F F

The same cannot be said for the conditional statement! It is true that “if \(P\), then \(Q\)” is a statement involving \(P\) and \(Q\), which we presume has some truth value. But is this statement really a function of the truth values of \(P\) and \(Q\)? Or could it depend on the contents of \(P\) and \(Q\)?

Let me stress this point by bringing in an unrelated example: Consider two functions, \(f\) and \(g\), both defined on pairs of numbers. We define \(f(x,y) = \operatorname{sgn}(xy)\), where \(\operatorname{sgn}\) is the sign function that returns \(1\) when a number is positive, \(0\) when it is the number \(0\), and \(-1\) when a number is negative. We define \(g(x,y) = xy\) to be the product of the two numbers. Thus for example \(f(-2,3) = -1\) and \(g(-2,3) = -6\). It can be shown that \(\operatorname{sgn}(xy) = \operatorname{sgn}(x)\operatorname{sgn}(y)\), so the value of \(f\) depends only on the sign of the inputs. This is analogous to how the OR connective depends only on the truth values of the input statements. On the other hand, the value of \(g\) depends on the actual values of the inputs, not just the sign. In an analogous way, a statement involving two other statements may depend on just the truth values of the input statements, or it may depend more specifically on the contents of the input statements—what those statements are actually saying.

A logical connective can only be represented in a truth table if it depends solely on the truth values of the inputs; this is because the rows of the truth table alternate between all the permutations of truth values that the input sentences can have, without reference to what those sentences are saying. So at the moment it is not clear whether the conditional statement can actually be represented in a truth table.

We can make the above point in a slightly different way: As humans, we tend to conflate similar-seeming things in order to simplify our thinking. But in math, it is important to pay attention to one’s mental representations of things and to not conflate things that are actually meaningfully distinct. This sort of thing happens in mathematics all the time. We might, for example, introduce a relation called \(\leq\) on some class of objects. But just because it looks like an inequality, we cannot assume that it is reflexive or transitive or has any of the properties we normally associate with an inequality relation! A mathematician has to think very carefully to avoid this kind of conflation.

Looking at the statement “if \(P\), then \(Q\)”, we currently have two different mental models of it:

  1. The mysterious new meaning textbooks are forcing on us: The logical connective \(\implies\), which takes two sentences and turns them into a third sentence with a specific truth value, as described by the truth table.
  2. A much more familiar meaning: As a rule of inference saying that once we have \(P\), we are allowed to have \(Q\).

Intuitively, we think of “if \(P\), then \(Q\)” as a rule telling us that whenever in an argument we’ve shown \(P\) to be true, then we also get to say that \(Q\) is true. A priori, this has very little to do with the statement \(P \implies Q\) as defined by the truth table! This is just another way of stating that we aren’t sure yet whether our intuitive notion of “if \(P\), then \(Q\)” can be represented in a truth table.

So, are these two meanings of “if \(P\), then \(Q\)” related? It turns out, there is a very satisfying connection: we can make the inference from \(P\) to \(Q\) exactly when the statement \(P \implies Q\) is true. In other words, our intuitive notion of “if \(P\), then \(Q\)” coincides perfectly with the truth-table definition of implication!

“Huh?” you might say. “How could our intuitive notion coincide with the truth table one? Didn’t we go on at length in the introduction about how unintuitive the implication is? How can it be both intuitive and unintuitive?” This is a good point, and we will come back to it soon! But first, let’s try to prove this result.

To prevent us from conflating the two meanings of “implies”, in our proof, let’s use \(\lnot P \lor Q\) instead of \(P \implies Q\). By inspecting the truth table, we see that the two are the same:

\(P\) \(Q\) \(\lnot P\) \(\lnot P \lor Q\) \(P \implies Q\)
T T F T T
T F F F F
F T T T T
F F T T T

To summarize, the result we are trying to show is the following: Our intuitive meaning of “if \(P\), then \(Q\)” (i.e., that from \(P\) we can derive \(Q\)) is true if and only if the material conditional \(P \implies Q\) (which is equivalent to \(\lnot P \lor Q\)) is true. If you feel excited at this point, I think it’s a good idea to try to prove this result yourself.

Let’s first suppose that our intuitive meaning of “if \(P\), then \(Q\)” is true. In other words, given \(P\), we can derive \(Q\). We want to show that \(\lnot P \lor Q\) is true. We have two cases, \(P\) or \(\lnot P\). Suppose first that \(P\) is true. Then by our assumption, we are allowed to derive \(Q\). Since we know that \(Q\) is true, we know that at least one of \(Q\) or \(\lnot P\) is true, so \(\lnot P \lor Q\) is true. Next suppose that \(\lnot P\) is true. This case is even simpler: since \(\lnot P\) is already true, we know that at least one of \(\lnot P\) or \(Q\) is true, so \(\lnot P \lor Q\) is true. In either case, we have shown that \(\lnot P \lor Q\) is true. This completes the first direction of the proof.

Next we show that if \(\lnot P \lor Q\) is true, then given \(P\) we can derive \(Q\). So let’s suppose we are given \(P\). Our goal now is to show that \(Q\) is true. Can \(Q\) be false? Suppose for the sake of contradiction that \(Q\) is false. Then we have both \(P\) and \(\lnot Q\). This contradicts \(\lnot P \lor Q\), which states that at least one of \(\lnot P\) or \(Q\) is true. This contradiction shows that our assumption that \(Q\) is false was in error, so \(Q\) must be true. This completes the other direction of the proof, and hence we have the result.

The above result justifies calling \(\implies\) the “implies” connective, and treating the sentence \(P \implies Q\) the same as we would the claim that “if \(P\), then \(Q\)”. In a way, it is amazing that “if \(P\), then \(Q\)” can be captured by a logical connective! The statement “if \(P\), then \(Q\)” at first seems much more fuzzy, something we use in an argument but that can’t be formalized so simply—something that we would expect depends on the meanings of the sentences and not just their truth values.

Let’s return to the example from the start of this essay, “if the Moon is made of cheese, then \(1+1=3\)”. Since the antecedent is false, this is an example of a vacuous truth. Can we now make sense of this statement given the result we showed above? One approach is to run through part of the proof again, but using the concrete statements that we have instead of the letters \(P\) and \(Q\). We know that the Moon is not made of cheese, so “the Moon is not made of cheese or \(1+1=3\)” is true. But now suppose in addition that the Moon were made of cheese. Could we now say that \(1+1\ne 3\)? If we did, this would contradict what we said above, that either the Moon is not made of cheese or \(1+1=3\). So we have no choice but to conclude that \(1+1=3\). But now notice that we started with the assumption that the Moon is made of cheese, and derived \(1+1=3\), so we have shown “if the Moon is made of cheese, then \(1+1=3\)”!

Why propositional logic? # ↩︎

Now that we’ve hopefully demystified the material conditional, let’s zoom out a bit and look at propositional logic more generally. What have we accomplished? Why should we care?

Something mathematicians love to do is to discover and talk about recursive things like the Fibonacci sequence. If you reflect for a moment, you will see that propositional logic is a striking example of recursion. Mathematicians have been reasoning about math and writing proofs for thousands of years, leading to many fascinating results about numbers, geometry, and equations. But one way to view propositional logic is that it is pointing this machinery at itself: we are analyzing mathematical reasoning using mathematical reasoning. This is like using a microscope not to study cells, but to study microscopes themselves!

Why might we want to do this? Well, mathematicians are a weird and curious people, and delight in getting all self-referential and “meta” to see what happens. It’s also a fairly natural thought: as mathematicians, we have been reasoning in a particular way and have gotten used to it. It has become our “hammer” and we see “nails” everywhere. From this perspective, our own mathematical reasoning is itself just another mathematical object we could be analyzing using our existing “hammer”. But there are other reasons why studying mathematical reasoning using mathematical reasoning is useful. In math, we often want to find the negation of a complicated statement, or find the contrapositive of an implication, or perform some other operation. If we have the tools of propositional logic at hand, we no longer need to rely on our verbal ability to perform these manipulations. Instead, we can be on “autopilot”, simply following the formal rules of propositional logic. Once we invest in mathematically analyzing something, in a sense we get to mindlessly use that thing without having to pay too much attention to it every time.

So how does this all connect to the material conditional? Well, by showing that the conditional statement is amenable to being summarized by a truth table, we have brought it from the realm of fuzzy human thought down to the realm of mindless computational stuff. By doing so, we can quickly negate or otherwise manipulate complicated expressions involving “if …, then …” statements. This is a powerful ability to have when writing proofs!

Conclusion # ↩︎

Earlier we brought up the point that the implication seems to be both intuitive and unintuitive, and that this seems contradictory. But a little reflection shows that this sort of thing happens all the time in math. In fact, one way of thinking of math is to take things that seem awfully unintuitive or confusing, and then stare at them in just the right way so as to make them totally obvious.

It’s a fascinating fact that the conditional statement (and mathematical reasoning more generally) has such a concise formal description. We can easily imagine alternative worlds in which this was not the case, where mathematical reasoning itself is very complicated and messy even though the objects we study are simple formal systems.

Do you have feedback for this essay? Feel free to suggest an edit or post in the discussion!

Acknowledgments: Thanks to Satira and Vipul Naik for comments on a draft version of this essay.