Deductive Reasoning

Mathematicians always write arguments, or proofs, using deductive reasoning.

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Definition: Deductive reasoning is the act of drawing conclusions from a set of premises.

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Example:

1. All cats are animals.
2. Fluffers is a cat.
3. Therefore, Fluffers is an animal.

In this example, statements 1 and 2 are premises. We believe all cats are animals, and that Fluffers is a cat. Statement 3 is deduced from statements 1 and 2.

There are two ways that this argument can fall apart. First, the premises may not be true. What if Fluffers is what I call my motorcycle? Second, the logic could be invalid.

In math, we usually do not argue the truth of the premises. In fact, we just assume some statements are true. Those statements are called axioms.

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Definition: An axiom is a statement that is assumed to be true.

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We are more concerned with making valid, deductive arguments. Given a set of axioms and a statement, we want to know whether the statement can be deduced.

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Definition: An statement is a sentence that is either true or false.

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Example:

Some sentences which are statements:

• I went to the store today.
• Green is the best color!
• An odd number times an odd number is an odd number.

Some sentences which are not statements:

• Wow!
• Where were you today?
• The door on the left.

Language

Mathematicians love precision. However, arguments are made with an imprecise tool: language. For mathematicians to communicate precisely what they mean, they have analyzed language and adopted some conventions.

We will often use a letter such as \(P\) or \(Q\) to represent a statement.

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The negation of the statement \(P\) is "not \(P\)".

For example, the negation of "I went to the store today" is "I did not go to the store today." Notice that in English the negation isn't "Not I went to the store today." We have to interpret the meaning of a statement to see if it is truly a negation.

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The conjunction of the statements \(P\) and \(Q\) is "\(P\) and \(Q\)".

For example, let \(P\) be the statement "Bears are strong." Let \(Q\) be the statement "Bears are not very smart." The following are all examples of conjunctions in English:

1. Bears are strong, and bears are not very smart.
2. Bears are strong and not very smart.
3. Bears are strong but not very smart.

Notice that the last conjunction uses "but". Both "and" and "but" are used to create conjunctions.

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The disjunction of the statements \(P\) and \(Q\) is "\(P\) or \(Q\)".

There are 2 uses of the word "or" in English.

1. Inclusive or: One, the other, or both.
2. Exclusive or: One or the other, but not both.

To be precise, mathematicians have agreed that "or" will always, always mean inclusive or. When they want to use an exclusive or, they need to use extra language to make it clear.

Example of inclusive or: "If you are a parent or teacher, you should encourage your kids to learn programming."
In this case, someone who is both a parent and teacher is being told that they should encourage their kids to learn programming. So, the or is inclusive.

Example of exclusive or: "We can get pizza or lasagna for dinner tonight."
This statement implies that you are supposed to get pizza, or lasagna, but not both. So, the or is exclusive.

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Translating from logic to English and back can get complicated. For example, "not \(P\) and not \(Q\)" is the same as "neither \(P\) nor \(Q\)." It will take some practice.

Why do we need both?

• We write mathematical arguments in English because we are not robots. It is much easier, and more pleasant, to read math in natural English instead of pure logic.
• We need to parse the logic in arguments made in English to see whether believe their conclusions. It is surprisingly easy to make an invalid argument sound logical in English.