# Conditionals

Conditionals are if-then statements. More precisely, conditionals are statements that say if one thing happens, another will follow. Conditionals are written as pq. The stands for "implies." Conditional are heavily used in computer programming. If you know any BASIC, you'll see what I mean.

The p stands for the antecedent. The antecedent, also called the hypothesis, the given or the problem, is the first part of the conditional. This is the part where something does happen.The q stands for the consequent. The consequent, also known as the conlcusion, the prove, or the answer, is the "then" part of the if-then statement. This is the part where something happens because of what happened or existed earlier. In an if-then statement, each part is a complete clause.

An instance of a conditional is a situation where it is true. A counterexample is a situation where the conditional is false. You only need one counterexample to prove a theorum false. However, if the antecedent is false, we label the whole thing as true because we don't know the difference. To paraphrase Piet Hein, if the universe didn't exist, it wouldn't be missed. Since the antecedent (the universe not existing) is false, we must assume the consequent to be true. In this case, it probably is.

A good definition should take the place of two conditionals. It can be separated into the two conditionals, which have separate names. They are the meaning half of a conditional and the sufficient condition half of a conditional. The meaning half tells what the term means and the term is in the antecedent. The sufficient condition half tells you when you can use the term, and the term is in the consequent. In other words, the two conditionals are converses of each other. See the section on bicionditionals.

### Converses

The converse of pq is qp. If the original conditional is false, its converse may or may not be true and vice versa. An example of where the conditional is false but its converse is true would be:

Your computer is on (antecedent). You are using it (consequent).
You just have to consider each one as a separate statement.

### Biconditionals

Biconditionals are conditionals and their converses when their converses are also true. A biconditional uses the symbol . It stands for if and only if. Many definitions and theorums are written like this.

### Inverses and Contrapositives

Inverses are the opposite of conditionals. Instead of being pq, inverses are ~p~q. ~ stands for "not", so an inverse is if not p, then not q. Contrapositives are the converse of inverses. In other words, they are ~q~p.

- Jaime III
"Never ruin the beauty of a thing by questioning its legality." - Theodore Roosevelt