Logic 101 · Series

Necessary and Sufficient Conditions

This post is the first in a series of posts on deductive logic. If you aren’t familiar with what that means, please take a look at this post. For that matter, it wouldn’t hurt to go through all that I’ve covered so far in this series by going to the Series Index page.

Let’s dive right in.

Sufficient Conditions

One of the most useful and important distinctions in logical argumentation is the distinction between what counts as a necessary or sufficient condition for something else. Consider the following argument:

  1. If it is raining outside my house, then my front lawn is wet.
  2. It is raining outside my house.
  3. Therefore, my front lawn is wet.

Premise (1) of this argument is what we call a conditional statement. A conditional statement is worded like this: “If P, then Q.” (‘P’ and ‘Q’ can stand for any sentence that we want to insert.) The first part of the statement that says, “If P. . .” is called the antecedent. The second part of the statement that says, “. . . then Q” is called the consequent. Now we are ready to make our distinction.

Any conditional statement, like premise (1) above, states that if the antecedent is the case, then the consequent is also the case automatically. Think about it: if you know that it is raining outside of your house, isn’t it the case that you can also say that you know your front lawn is wet? Therefore, the fact that it is raining outside your house is a sufficient condition for your front lawn’s being wet. If you place the argument I gave above in logical form, it would look like this:

  1. If P, then Q.
  2. P.
  3. Therefore, Q.

This is a common logical argument form called modus ponens. It simply says that P is a sufficient condition for Q, P is the case, and therefore Q is also the case.

Necessary Conditions

Now consider an argument that is only slightly different from the one above:

  1. If it is raining outside my house, then my front lawn is wet.
  2. My front lawn is not wet.
  3. Therefore, it is not raining outside my house.

This argument makes the point that if your front lawn is not wet, it cannot be the case it is raining outside your house either. In logical form it looks like this:

  1. If P, then Q.
  2. Not-Q.
  3. Therefore, Not-P.

Therefore, the consequent of the conditional—the “. . . then Q” part—is a necessary condition for the antecedent. If you do not have Q, you do not have P either. This argument form is called modus tollens.

To sum up, if you have two things that are connected in an antecedent-consequent relationship by a conditional statement, you automatically have the consequent if you have the antecedent (sufficient condition), and you do not have the antecedent if you do not have the consequent (necessary condition). The antecedent is a sufficient condition for the consequent, and the consequent is a necessary condition for the antecedent.

Defining Terms with Necessary and Sufficient Conditions

In a previous post I said that one of the best ways to provide an analytical definition for a term is to provide the necessary and sufficient conditions for that term to apply in a particular case. Let’s apply that knowledge to another argument as an illustration:

  1. If Jesus Christ is God incarnate, then he is omniscient (i.e., all-knowing).
  2. Jesus Christ is God incarnate.
  3. Therefore, Jesus Christ is omniscient.

By looking at premise (1) we can learn two things. First, the fact that Jesus Christ is God incarnate is a sufficient condition for his also being omniscient (since “God” is here understood to mean a being that is omniscient). Second, in order for Jesus Christ to be God incarnate, he must also be omniscient; that is to say, if Jesus’ omniscience is a necessary condition for his being God incarnate. You will also note that premise (2) says that the sufficient condition for Christ being omniscient is the case, which means that this argument is of the modus ponens form.

Now how do we define what “God incarnate” means? As I’ve already said, one way is to list all of the necessary conditions for a human being to be considered God incarnate. If a person does not have these, he cannot be God incarnate. Here is a provisional list of these necessary conditions to illustrate:

  1. Possessing a physical body.
  2. Being omniscient.
  3. Being omnipotent (all-powerful).
  4. Being omnipresent (present everywhere).
  5. Being omni-benevolent (perfectly good).
  6. Being the creator of the universe.

This set of necessary conditions for being God incarnate can also be called a set of properties. Each one of properties (1)-(6) is, by itself, a necessary condition for being God incarnate. If any of them is missing, one is not God incarnate. But the whole set taken together is a sufficient condition (it is “jointly sufficient”) for being God incarnate. In other words, if all of the necessary conditions belong to a person, then the whole set of properties belongs to them. And if the whole set of properties belongs to them, then that is a sufficient condition for being God incarnate. Therefore, this set of properties is both necessary and sufficient for being God incarnate.

Biconditional Statements

The way that we usually say that a set of properties is necessary and sufficient for something else is by saying “X if and only if Y.” A statement that contains an “if and only if” statement is called a “bi-conditional” statement. That’s because there are actually two conditional statements that are joined together. They are these:

  1. If X, then Y.
  2. If Y, then X.

If we create a sentence that joins these two, we say “If X, then Y, AND If Y, then X.” That is a bi-conditional statement.

Following our previous example, we can say that “Jesus Christ is God incarnate if and only if he possesses the set of properties indicated by (1)-(6) above.” If Jesus has all of these properties, then he is God incarnate. But if he does not possess the whole set of properties (e.g., if he isn’t omnipotent) then he is not God incarnate. So, one way to give a definition of what it means to be God incarnate is to list a set of necessary conditions that, if taken altogether, are sufficient for being God incarnate. (As an exercise, try and give a set of necessary conditions for what the meaning of the term “God” is.)

Conclusion

I know that this material will be new to a lot of readers, but if you re-read this material and use your own examples you will quickly master the distinction between necessary and sufficient conditions. I have given examples for each concept individually, and an example that illustrates how something can be both a necessary and sufficient condition for something else.

When you master this distinction, you will be able to spot all sorts of fallacious arguments and definitions that individuals try to put forward in conversation. In the next post I will introduce logical fallacies. In the post after that, I will introduce you to two basic fallacies: denying the antecedent and affirming the consequent. By then you should have a solid grasp of what necessary and sufficient conditions are (if you don’t already).

Leave a Reply

Your email address will not be published. Required fields are marked *