Reasoning: An Introduction to Logic, Sets, and Functions

Subsection 2.2.6 A Truth Table Definition of the Operator implies

The next operator is implies. Of all the operators that we are defining, this is the one that may seem the least natural to you. But let’s start with the definition that we are going to use. Then we’ll look at why we have chosen to define implies in this way and how you can get your head around it.

Here is the truth table for implies, also called material implication, (for which the symbol is \(\rightarrow \)):

When we have a statement of the form p → q, it’s common to call:

• p the antecedent, and

• q the consequent.

Back to the truth table. Here’s perhaps the easiest way to interpret it. Suppose I assert, “p → q”. For example, suppose I claim, “rain → (implies) wet sidewalks”. Under what circumstance(s) would you call me a liar? We are defining implies so that it returns False only in the case that I have lied. We are going to define it to return True in all other cases. So when have I lied? Surely if there is rain but not wet sidewalks, then I lied. Row 2 of the truth table describes that case. My claim is false. Are there any other cases in which you’d call me a liar? Surely an observation of rain and wet sidewalks (row 1) is consistent with my claim. But now suppose that there is no rain. Then you’ll probably say that my claim is irrelevant. Okay. But not a lie, right? So we’ll define (in rows 3 and 4) my claim to be true.

So we note that p → q is true in every situation except when (as shown in row 2) p is true and q is false.

Why is this definition so difficult for students (seeing it for the first time) to accept? Probably the biggest reason is that a very common interpretation of the English word “implies” is that there is some sort of causality. Thus, whenever we say, “p implies q”, many people read it as, “p causes q”. Think about the wet sidewalks example. Rain causes the sidewalks to be wet. When you think about it this way, the first two lines of our truth table make sense. The problem is the last two rows: it’s hard to know what truth value we ought to assign when p is false. The statement seems irrelevant. Causality is a very important phenomenon and we will want eventually to be able to reason about it. But, precisely because it is important and also complex, we need to separate it from many simpler notions.

Just as we are asking you to accept the meaning of or to be the inclusive version (even though the English word “or” often means the exclusive version), we ask you to remove all notion of causality from our usage of the word “implies”.

Meanwhile, keep in mind that assigning no truth value isn’t a choice. We must assign some truth value to the expression, “p implies q”, even when p is false and we may feel that the expression is irrelevant. If we don’t, then “p implies q” won’t be a logical statement (since a statement must have a truth value defined for all combinations of truth values of its constituents). Given that we’re in the business of defining operators that we can use to build statements, that won’t do. We must pick something. And the something we have picked is the “When have I lied?” criterion.

Another way to view our definition is this: Suppose that we want simply to claim that it’s not possible to have p without also having q (for whatever reason). There may be no obvious linkage at all between p and q except what we’ve observed about them. Our definition of implies captures this simple notion. Look again at the truth table we have given. It tells us that the only way for “p → q” to be false is that we do have p but do not have q.

Notice that, once we remove the notion of causality and look just at the simultaneous truth or falsity of two statements, it’s possible to make a logical claim about two completely unrelated statements (silly as that may at first seem).

For example, in the world in which we live, the claim, “The capital of Texas is Dallas implies that hydrogen atoms have two protons,” is true. Perhaps ridiculous. (Surely no one has ever said this before). But true. “The capital of Texas is Dallas,” is false. So is, “Hydrogen atoms have two protons”. So we use the last row of the truth table and determine that our claim is true. It could only be false if we lived in a world in which, “The capital of Texas is Dallas,” were true but, “Hydrogen atoms have two protons,” were still false. But we don’t.

Let us give one final example that may help you accept as reasonable the definition that we have given for implies: Consider the statement:

“If x is greater than \(1 \) , then x is greater than \(0 \) .”

Do you agree that this statement is true no matter what actual value x has? If you say “yes”, then ask yourself, “What about when x is -1?”. Both of the operand statements, “x is greater than 1” and “x is greater than 0,” are false. Yet, you still believe that “If x is greater than 1, then x is greater than 0” is true. You believe it probably because you don’t have an example of a value for x that makes it false. (Back to the “call me a liar” view of our definition.) In any case, we hope that you now have at least one situation in which you’re okay with the fact that we’ve defined “p \(\rightarrow\) q” to be true whenever p is false.

Big Idea

The logical operators mean what they mean because we have defined them to do so. We could have given other definitions. But we’ll see, as this course progresses, that these definitions give us the basis for a useful approach to discerning truth.

Finally, we’ll say simply that “p \(\rightarrow\) q” means what it means because we have defined it that way. We could possibly have avoided some of this discussion if we’d used the same truth table and claimed it as the definition for some logical operation quilpled. You’d then not have had any preconceived notion of what the definition ought to be. But trust us – we won’t lead you astray. Using this as the definition of implies will actually cause no problems. In fact, it will turn out to be very useful. And, as all students have done in the past, you will soon be accepting it.

English Aside

English gives us several ways to describe the logical relation implies. For example, we can say:

• p implies q. “Rain implies wet sidewalks.”

• if p then q. “If it’s raining then the sidewalks will be wet.”

• if p, q. “If it’s raining, the sidewalks will be wet.”

• p only if q. “It’s raining only if the sidewalks are wet.”

• q if p. “The sidewalks will be wet if it’s raining.”