A conditional statement is also an argument

The conditional statement is the logical “If..Then..” statement. The conditional is the basic statement used in logical arguments and is defined as follows:

\(P\) \(Q\) \(P \to Q\)
T T T
T F F
F T T
F F T

The above conditional statement is true for all possible values of \(x\). So even though the bottom two rows may seem strange, they are needed to make variable conditional statements in mathematics true.

Biconditional Statement

A biconditional statement (\(P \leftrightarrow Q\)) is a statement in which the implication of the conditional goes in both directions (think of it as a two way arrow instead of a one way arrow). In English, the biconditional is “if and only if”. In short the biconditional, if true, means that both statements are the same, either both true or both false, as seen in the truth table below.

\(P\) \(Q\) \(P \leftrightarrow Q\)
T T T
T F F
F T F
F F T
”\(x+3 = 5\) if and only if \(x=2\)”

Converse, Inverse, and Contrapositive

Given a conditional statement \(P\to Q\), the following are related conditional statements:

For example consider the conditional statement:

Logically the conditional is equivalent to the contrapositive, and the converse is equivalent to the inverse:

\[\begin P\to Q &\equiv \neg Q \to \neg P \\ Q\to P &\equiv \neg P \to \neg Q \end\]

Creating truth tables for all four statements will prove the equivalencies.