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Subsection 1.3.3 Definitions

Before we can make claims, in English or in logic, we need to define basic sets of terms.

In our arguments, definitions will have the same status as premises. We assert, up front, that they are true. We generally craft our definitions so that they capture properties that we want to reason about. Of course, if two people write different definitions for the same term, they are likely to be able to prove different things about the objects to which the definitions apply.

Mathematicians are particularly careful when they define the terms that they’ll use.

Activity 1.3.5.

Consider two possible definitions for “prime number”:

  1. A prime number is a positive integer that has, as positive divisors, only itself and 1.

  2. A prime number is an integer greater than 1 that has, as positive divisors, only itself and 1.

Now consider the sentence:

  1. The integer 1 is a prime number.

Clearly 1 is true if we use definition 1 . It is false if we use definition 2.

Note: 2 is the standard definition (and the one that we will use).

Activity 1.3.6.

Let’s consider the all-important question, “Is a tomato a fruit or a vegetable”? The answer is that it depends on the definitions that we start with.

  1. Biologists’ definition: A fruit is a plant part that develops in the ovary of the plant’s flower and contains the plant’s seeds.

  2. Cooks’ definition: A fruit is a fleshy plant part, with seeds, used in sweet (rather than savory) dishes. If used in savory dishes, such plants are called “vegetables”.

  3. The U.S. Supreme Court’s definition (in Nix v. Hedden, 1893): A tomato is a vegetable.

Now consider the sentence:

  1. A tomato is a fruit.

We see that 1 is true if we use definition 1. It is false if we use definition 2 or 3.

The key for us is that logic can’t resolve definitional issues. We must choose definitions and write them down. Then we can reason with them.

Problems 1.3.7.

Consider the following proposed definitions:

  1. A food is healthy if it has no chemical additives.

  2. A food is healthy if it contains fewer than 10 grams of carbohydrates per serving.

  3. A food is healthy if it gets fewer than half its calories from fat.

  4. A food is healthy if it is a fruit or vegetable.

Now consider the claim: A baked potato is healthy.

(a)

Using definition [1]:

  1. This claim is true.

  2. This claim is false.

  3. This claim could be either true or false depending on one or more other definitions.

Answer.

i

(b)

Using definition [2]:

  1. This claim is true.

  2. This claim is false.

  3. This claim could be either true or false depending on one or more other definitions.

Answer.

ii is correct.

(c)

Using definition [3]:

  1. This claim is true.

  2. This claim is false.

  3. This claim could be either true or false depending on one or more other definitions.

Answer.

i is correct.

(d)

(Part 4) Using definition [4]:

  1. This claim is true.

  2. This claim is false.

  3. This claim could be either true or false depending on one or more other definitions.

Answer.

C is Correct

Solution.

Explanation: To decide on Part 4, we need to refer to another definition: What is a vegetable? Botanists classify potatoes as vegetables. But nutritionists generally don’t; they classify potatoes as starches.

Problems 1.3.8.

Consider the following proposed definitions:

  1. A natural number is a whole number greater than or equal to 0.

  2. A natural number is a whole number greater than or equal to 1.

Note: We will use definition [1]. But we should point out that both of these definitions are in fairly common use.

Now consider the claim: The ratio of any two natural numbers is a rational number

(a)

Using definition [1]:

  1. This claim is true.

  2. This claim is false.

Answer.

ii is correct.

Solution.
The claim is false if we use definition 1 because, in that case, it is possible for the denominator to be 0. In that case, the ratio is undefined.

(b)

Using definition [2]:

  1. This claim is true.

  2. This claim is false.

Answer.

i

Solution.
Now the claim is true. It follows from the definition (another one) of the rational numbers as being those numbers that can be expressed as the ratio of two integers.