# Critical thinking problem with necessary and sufficient conditions

I'm reading a textbook about critical thinking but am having some difficulties with an exercise problem. Why are the following answers correct: a)true b)false c)true d)false e)true

The exercise problem:
Suppose we have a definition X=Y. Are the following statements correct about this definition? Why or why not?

a) If the definition is too wide, then X is not necessary for Y.
b) If the definition is too wide, then Y is not necessary for X.
c) If the definition is too narrow, then X is not sufficient for Y.
d) If the definition is too narrow, then Y is not sufficient for X.
e) If X is not necessary for Y, then the definition is too wide.

According to the book I'm using a definition is too wide if includes/applies to things that it should not, and a definition is too narrow if it does not include things it should.

• Is "=" supposed to be a biconditional? Or an actual equals sign? Assuming the former. a-d are about what is the case when the definition is inadequate, so when the biconditional is a wrong definition. Assuming we are trying to "define" X. Y is too wide when it can be the case while X isn't. According to this it's "X -> Y". This is fine for the statement a) because Y can be true while X doesn't have to be. b) is wrong with that. So it fits. e) is just a) turned around. – Marc H. Jul 26 '17 at 13:37

## 1 Answer

Since you are not sure why the answers are as such, let's first make sure you understand the definitions of the important terms. The definitions of the following ideas are standardized in any course on scientific reasoning or critical thinking:

1. Giving a definition means finding a necessary and sufficient condition. For example, the definition for "Brother" is "A male sibling."
2. A necessary condition for some state of affairs S is a condition that must be satisfied in order for S to obtain. For example, being male is necessary to be a brother.
3. A sufficient condition for some state of affairs S is a condition that, if satisfied, guarantees that S obtains. For example, being a dog is sufficient to be an animal.

I find most students are still confused by this verbal definition of the terms. So comes my trusty Venn Diagram. Visualized in this way, "Bulldogs are dogs" is a too-wide definition since it also includes poodles; "Dogs are bulldogs" is a too-narrow definition since it does not include poodles.

Now that we have the needed definitions in hand, let's examine your questions.

Suppose we have a definition X=Y. Are the following statements correct about this definition? Why or why not?

a) If the definition is too wide, then X is not necessary for Y.

To answer the question, first, find an example statement. "Bulldogs are dogs" is a too-wide definition. Here X is bulldog, and Y is dog. Being a bulldog is sufficient for being a dog, but not necessary since a poodle can be also a dog. Thus the given statement is true.

b) If the definition is too wide, then Y is not necessary for X.

Using the same example as a), "Bulldogs are dogs", being a dog is necessary to be a bulldog. Thus the given statement is false.

c) If the definition is too narrow, then X is not sufficient for Y.

"Dogs are bulldogs" is an example of a too-narrow definition. Here X is dogs and Y is bulldogs. Being a dog is not sufficient for being a bulldog. So the given statement is true.

d) If the definition is too narrow, then Y is not sufficient for X.

Using the same example as c), being a bulldog is sufficinet for being a dog. So the given statement is false.

e) If X is not necessary for Y, then the definition is too wide.

Being a bulldog is not necessary for being a dog, thus the definition, "bulldogs are dogs" is indeed a too-wide definition.