Geometry/Inductive and Deductive Reasoning

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.

Basic Terms edit

Before one can start to understand logic, and thereby begin to prove geometric theorems, one must first know a few vocabulary words and symbols.

Conditional: a conditional is something which states that one statement implies another. A conditional contains two parts: the condition and the conclusion, where the former implies the latter. A conditional is always in the form "If statement 1, then statement 2." In most mathematical notation, a conditional is often written in the form p ⇒ q, which is read as "If p, then q" where p and q are statements.

Converse: the converse of a logical statement is when the conclusion becomes the condition and vice versa; i.e., p ⇒ q becomes q ⇒ p. For example, the converse of the statement "If someone is a woman, then they are a human" would be "If someone is a human, then they are a woman." The converse of a conditional does not necessarily have the same truth value as the original, though it sometimes does, as will become apparent later.

AND: And is a logical operator which is true only when both statements are true. For example, the statement "Diamond is the hardest substance known to man AND a diamond is a metal" is false. While the former statement is true, the latter is not. However, the statement "Diamond is the hardest substance known to man AND diamonds are made of carbon" would be true, because both parts are true.

OR: If two statements are joined together by "or," then the truth of the "or" statement is dependent upon whether one or both of the statements from which it is composed is true. For example, the statement "Tuesday is the day after Monday OR Thursday is the day after Saturday" would have a truth value of "true," because even though the latter statement is false, the former is true.

NOT: If a statement is preceded by "NOT," then it is evaluating the opposite truth value of that statement. The symbol for "NOT" is For example, if the statement p is "Elvis is dead," then ¬p would be "Elvis is not dead." The concept of "NOT" can cause some confusion when it relates to statements which contain the word "all." For example, if r is "¬". "All men have hair," then ¬r would be "All men do not have hair" or "No men have hair." Do not confuse this with "Not all men have hair" or "Some men have hair." The "NOT" should apply to the verb in the statement: in this case, "have." ¬p can also be written as NOT p or ~p. NOT p may also be referred to as the "negation of p."

Inverse: The inverse of a conditional says that the negation of the condition implies the negation of the conclusion. For example, the inverse of p ⇒ q is ¬p ⇒ ¬q. Like a converse, an inverse does not necessarily have the same truth value as the original conditional.

Biconditional: A biconditional is conditional where the condition and the conclusion imply one another. A biconditional starts with the words "if and only if." For example, "If and only if p, then q" means both that p implies q and that q implies p.

Premise: A premise is a statement whose truth value is known initially. For example, if one were to say "If today is Thursday, then the cafeteria will serve burritos," and one knew that what day it was, then the premise would be "Today is Thursday" or "Today is not Thursday."

⇒: The symbol which denotes a conditional. p ⇒ q is read as "if p, then q."

Iff: Iff is a shortened form of "if and only if." It is read as "if and only if."

⇔: The symbol which denotes a biconditonal. p ⇔ q is read as "If and only if p, then q."

∴: The symbol for "therefore." p ∴ q means that one knows that p is true (p is true is the premise), and has logically concluded that q must also be true.

∧: The symbol for "and."

∨: The symbol for "or."

Deductive Reasoning edit

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 edit

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.

Truth Tables edit

Truth tables are a way that one can display all the possibilities that a logical system may have when given certain premises. The following is a truth table with two premises (p and q), which shows the truth value of some basic logical statements. (NOTE: T = true; F = false)