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:

• In symbolic logic the conditional symbol is \(\to\).
• If \(P\) and \(Q\) are statements, then \(P\to Q\) (thought of as “If \(P\) then \(Q\)” or “\(P\) implies \(Q\)”) is the conditional statement with premise \(P\) and conclusion \(Q\).
• The truth table for the conditional is

• If \(x\) is \(3\), then \(3=3\) is true and \(3^2 = 9\) is true, so the conditional is true (top row).
• If \(x\) is \(2\), then \(2=3\) is false and \(2^2 = 9\) is also false, so the conditional is true (bottom row).
• If \(x\) is \(-3\), then \(-3=3\) is false and \((-3)^2 = 9\) is true, so the conditional is true (third row).
• In this example it is impossible for \(x=3\) to be true and \(x^2 = 9\) to be false.

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.

The truth table for the conditional is

”\(x+3 = 5\) if and only if \(x=2\)”

Converse, Inverse, and Contrapositive

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

• Converse: The converse of the above is \(Q\to P\) (reverse order).
• Inverse: The inverse of the above is \(\neg P \to \neg Q\) (reverse truth values).
• Contrapositive: The contrapositive of the above is \(\neg Q \to \neg P\) (reverse order and truth values).

For example consider the conditional statement:

• The converse of this statement is: If \(x\leq 1\), then \(x
• The inverse of this statement is: If \(x\geq 0\), then \(x>1\).
• The contrapositive of this statement is: If \(x>1\), then \(x\geq 0\).

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.