Definitions and biconditional statements

 Introduction

A biconditional statement is a statement that contains the phrase if and only if. Writing a biconditional statement is equivalent to writing a conditional statement and its converse.

A conditional statement has two parts, a hypothesis and a conclusion . When the statement is written in if - then form, the if part contains the hypothesis and the then part contains the conclusion. A statement is called biconditional when it expresses the idea that the presence of some property is a necessary and sufficient condition for the presence of some other property. Such a statement is usually phrased in the terms "P, if and only if Q." The phrase "if and only if is often abbreviated as iff.

Thus: He will be president if and only if he wins the presidential election is a biconditional statement. It is unusual that people use statements with such precision in ordinary language, but they are vital in philosophy. The label biconditional arises from the fact that the statement p iff q is really an abbreviated way of saying two conditional statements: if p then q and if q then p. Thus, the above biconditional really means:

Explanation through example

If he wins the election, he will be president, and if he is president, he has won the election.

We can explain through the following example.

Three lines are coplanar if and only if they lie in the same plane.

Conditional statement: If three lines are coplanar, then they lie in the same plane.

Converse: If three lines lie in the same plane, then they are coplanar.

Properties

A biconditional statement can be either true or false. To be true, both the conditional statement and its converse must be true. This means that a true biconditional statement is true both forward and backward. All definitions can be written as true biconditional statements.

If p and q are statements, the conditional of q by p is if p then q denoted p ! q. We call p the hypothesis of the conditional and q the conclusion.

A conditional statement is false only when the hypothesis is true and the conclusion is false. Though it is clear that a conditional statement is false only when the hypothesis is true and the conclusion is false, it is not clear why when the hypothesis is false, the conditional statement is always true.

If p and q are statements, then the biconditional of p and q is p if only if q and is denoted p $ q. It is true if both p and q have the same truth values and is false if they have opposite truth values.

The decomposition rule for biconditional statements is dictated by the truth functional assignments needed to make biconditional statements true. The statement form p * q is true if and only if p and q have the same truth value; in other words, it is true when either p and q are both true or p and q are both false.Therefore,p⁰q is logically equivalent to the disjunction [(p · q) v (~p · ~q)].