5.1 A Lesson on Conditionals

5.1    A Lesson on Conditionals

So far this semester you’ve been asked to become aware of much more of what goes on in everyday life –talking, writing, speaking—than you generally have ever been.  You may not even realize this (in fact, if you don’t, and are confused by this class, that might be part of why).

Let’s review.  I’ve asked you to note that sometimes we use words to talk about themselves, and that this is a bit different from the usual use of them.  Different enough that we’ve given it a name (“mention” rather than “use”), and we have agreed to a punctuational way to mark it (quotation marks).   And we have focused with deliberation on eight things people do with words: they report information, they express their opinions and beliefs, they give examples of general claims, they tell why things happen or are as they are, they clarify the meaning of words, objects and situations, they warn and advise, they state conditions for events or actions, and they make efforts to persuade themselves and others that things are true.   By the mere fact that you have largely committed to memory the terminology for these various elements of everyday language use, you are in a better position than you were before to be able to see with clarity (rather than guess) what is going on around you.  You are better able to sense and spot distinctions; you are a bit less oblivious than a lot of people are.  For example (illustration), you are less likely to think something is an argument when it’s an explanation, or to describe what someone is doing as “providing support” for a view if all she’s doing is giving an example of something.  (Though, please note in my comments on your TA’s that some of you are not being careful about keeping the language for these different things in their proper places.)

You’ve also learned that the effort to provide good reasons to believe that something is true (arguing) can be divided into two large styles, modes or families –inductive and deductive reasoning.  And you’ve become familiar with eleven patterns of arguing, and the language for evaluating argumentation (for talking about validity, soundness, strength, and cogency).

You may or may not have noticed that a common concern seems to be in play in all this distinction-making with regard to language and how people do things with words.  I’d propose that it’s a concern with truth: we care about whether or not we should believe things, such as things we hear others say. And when we realize we care about this, we realize we need some tools—and some training or practice in using them.  If you can’t name what’s going on, or even ask if something is going on because you lack the language to talk about it, you are at a disadvantage.

Consider the example from the other day:  If you’re at least 21, you can buy a drink, but you aren’t so you can’t.  This example is a bit disturbing because we want to say this is true.

(But we should be able to distinguish between saying that all three of the statements in the argument are true, and wanting to call the argument “valid” –remember that “true” and “false” only really apply to individual claims, but an argument is more than an individual claim.)

But I’ve told you it’s not a valid argument, because it’s just the same as If Washington was assassinated, he’s dead, but he wasn’t so he isn’t.  That is, they are both of the form:

If p then q.  But not p. Therefore not q.     And this form of reasoning, known as Denying the Antecedent, cannot be trusted.

You have to admit, most people are going to be surprised to hear that the argument about being able to buy a drink is invalid. And many are going to (want to) jump from hearing that to the utterly unwarranted conclusion that the conclusion is actually false, and that they can buy a drink.

The example teaches us how easy it is to be fooled, how easily manipulated we are. By the simple fact that people talk and we listen, we are manipulated, led to believe things, even things we should not believe.  Now, again, in this case, the thing we should not believe is not the conclusion of the invalid argument. No, it is that this extremely natural sounding pattern of reasoning and drawing a conclusion can be trusted. We have to learn to be on our guard, to recognize such things as that pattern, and when we do recognize it, we have to stop, point it out, and examine what is going on.  This is called identifying fallacies, and we’ll turn to it after we dig a little deeper into this example and into the logic of conditional statements.

In this case, and in many cases of this, what’s going on has to do with the fact that there are two kinds of conditions, and they each have their proper place in a conditional statement, but we are very often completely careless in how we express them.  The two kinds of conditions are called NECESSARY and SUFFICIENT.  The distinction is pretty easy, but it can confuse you in some cases.

Being a mammal is Necessary for being a cat, just as being a citizen is necessary for voting, and being at least 21 is necessary for buying a drink.  Those three examples all make good sense: one thing is necessary for another (is a necessary condition for another) if it has to be the case in order for the other thing to happen or be the case.

Again: nothing can be a cat unless it is a mammal; no one can vote unless s/he is a citizen; no one can buy a drink unless s/he is at least 21.  Something’s being necessary for something else can also be expressed with the words “ONLY IF.”  You’re a cat only if you’re a mammal, etc.

The other kind of condition, called Sufficient, is the kind that, if met, guarantees that something else is so.  For instance, Being a cat is sufficient for being a mammal.  That is, all we have to know about Socrates is that he’s a cat, and we can be guaranteed that he is a mammal (and not a fish or bird).  Being a wife is sufficient for being married.  That is, if I know that Jane is a wife, I know that Jane is married.   Being a Pope is sufficient for being a Catholic.  (If you tell me that Mark’s the Pope, that’s all I need to know to be certain that he’s a Catholic.)

Now look at the following conditionals and note where the sufficient conditions are located, and where the necessary conditions are located:

If Mark’s the Pope, then he’s Catholic.

If Jane’s a wife, she’s married.

If you can buy a drink, then you’re at least 21.

If you’re a cat, you’re a mammal.

If you can vote, you’re a citizen.

Note that the sufficient condition is in the antecedent each time, and the necessary condition is in the consequent each time.

I said a minute ago, that a necessary condition can be expressed as “only if.”  Restate these conditions using “only if,” and notice what happens:

Mark’s the Pope only if he’s Catholic.

Jane’s a wife only if she’s married.

You can by a drink only if you’re at least 21.

You’re a cat only if you’re a mammal.

You can vote only if you’re a citizen.

What happened is we dropped “if” off the beginning and inserted “only if” where “then” might have been before, namely, in the role of introducing the consequent.

Three things should be clear:

a) a sufficient condition is quite different from a necessary condition,

b) a sufficient condition goes in the antecedent, a necessary condition goes in the consequent, and

c) “if” introduces a sufficient condition, but “only if” or “then” introduces a necessary condition.

And something follows from all this:  “if” does not mean the same thing as “only if.”   And if we use “if” as though it meant the same thing as “only if,” we deceive ourselves and others (without even knowing it).  Let’s go back to the example that started all this trouble:

If you’re at least 21, you can buy a drink, but you aren’t so you can’t.

Does anything now appear suspicious about that?   The conditional premise expresses a necessary condition as though it were a sufficient condition. And that means that that premise is false after all (in that other conditions also have to met, such as having some money).  The truth is: If you can buy a drink, you’re at least 21. (You can buy a drink only if you’re at least 21; you can’t buy a drink unless you’re at least 21; being at least 21 is a necessary condition for buying a drink.)

If we rebuild the argument expressing the necessary condition where it should be expressed, in the consequent, the invalidity goes away:

If you can buy a drink, you’re at least 21, but you’re not, so you can’t.

The form here is  If p, then q.  Not q. Therefore not p.

Instead of an antecedent being denied as before, it is the consequent that is denied.  This form of argument (which you could call Denying the Consequent, but we’ll call it by its Latin name “modus tollens”) is thoroughly reliable, it is valid. You will never build a counterexample to it.

Denying the Antecedent is an example of what’s called a Formal Fallacy, and it’s easy to see why. The problem is due strictly to the form of the argument. It is the form of an argument that makes it valid if it is valid, and it is the form that makes it invalid if it is invalid.  The content has nothing to do with whether the argument is valid or invalid –the content only helps us see the invalidity through an example: a counter-example is an illustration of the claim that an argument form is invalid. You illustrate this by providing a case of the form in which the premises are all true and the conclusion is false, which violates the definition of “valid.”

Having had a taste of the issue of validity and form, we’re now going to leave it for a short while in favor of a more general study of Fallacies, which is an even more explicit interest in or concern with wanting to not be deceived (and wanting to not deceive ourselves).

 

For some practice with this material, rewrite each of the following as conditional statements, deciding which is the necessary condition and which is sufficient. Answers are provide below.

1. No one can get in unless they have a ticket.

2. Only people with enormous amounts of money or support can get elected.

3. You’ll do well only if you do the homework exercises.

4. You have to stop whenever there’s a red light.

5. You can’t smoke here.

6. Being a guitarist means being a musician.

7. Every good boy does fine.

8. Corporations are people.

9. You can’t get through college these days without winding up with debt.

10. You’re either in the top 1% or in the other 99%.

11.  You can’t be Commander-in-Chief without being President.

 

Answers.

1. No one can get in unless they have a ticket.

If you can get in, you have a ticket.

 

2. Only people with enormous amounts of money or support can get elected.

If you can get elected, you have enormous amounts of money or support.

 

3. You’ll do well only if you do the homework exercises.

If you do well, you do the homework exercises.   If you don’t do the homework exercises, you won’t do well.

 

4. You have to stop whenever there’s a red light.

If there’s a red light, you stop.

 

5. You can’t smoke here.

If you’re here, you are not smoking.

 

6. Being a guitarist means being a musician.

If you’re a guitarist, you’re a musician.

 

7. Every good boy does fine.

If you’re a good boy, then you do fine.

 

8. Corporations are people.

If you’re a corporation, you’re a person.

 

9. You can’t get through college these days without winding up with debt.

If you get through college these day, then you have some debt.

 

10. You’re either in the top 1% or in the other 99%.

If you’re in the 1%, you’re not in the 99%.  AND

If you’re in the 99%, you’re not in the 1%.

You’re in the 1% if and only if you’re not in the 99%.

 

11.  You can’t be Commander-in-Chief without being President.

If you’re Commander-in-Chief, you’re President, AND if you’re President, you’re Commander-in –Chief.

You are Commander in Chief if and only if you are President.

 

 

 

 

One Response to 5.1 A Lesson on Conditionals

  1. Blake says:

    I think the answer for question 11 is wrong; Even if you can’t A without B, you might still be able to B without A.

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