Inductive and Deductive Reasoning

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There are two approaches to furthering knowledge: reasoning from known ideas and synthesizing observations. In inductive reasoning you observe the world, and attempt to explain based on your observations. You start with no prior assumptions. Deductive reasoning consists of logical assertions from known facts.

Deductive Reasoning

There are a few forms of deductive logic. One of the most common deductive logical arguments is modus ponens, which states that:

p ⇒ q

p ∴ q

(If p, then q)

(p, therefore q)

An example of modus ponens:

If I stub my toe, then I will be in pain.

I stub my toe.

Therefore, I am in pain.

Another form of deductive logic is modus tollens, which states the following.

p ⇒ q

~q ∴ ~p

(If p, then q)

(not q, therefore not p)

Modus tollens is just as valid a form of logic as modus ponens. The following is an example which uses modus tollens.

If today is Thursday, then the cafeteria will be serving burritos.

The cafeteria is not serving burritos, therefore today is not Thursday.

Another form of deductive logic is known as the If-Then Transitive Property. Simply put, it means that there can be chains of logic where one thing implies another thing. The If-Then Transitive Property states:

p ⇒ q

(q ⇒ r) ∴ (p ⇒ r)

(If p, then q)

((If q, then r), therefore (if p, then r))

For example, consider the following chain of if-then statements.

If today is Thursday, then the cafeteria will be serving burritos.

If the cafeteria will be serving burritos, then I will be happy.

Therefore, if today is Thursday, then I will be happy.

Inductive Reasoning

Inductive reasoning is a logical argument which does not definitely prove a statement, but rather assumes it. Inductive reasoning is used often in life. Polling is an example of the use of inductive reasoning. If one were to poll one thousand people, and 300 of those people selected choice A, then one would infer that 30% of any population might also select choice A. This would be using inductive logic, because it does not definitively prove that 30% of any population would select choice A.

Because of this factor of uncertainty, inductive reasoning should be avoided when possible when attempting to prove geometric properties.