Logical biconditional

In logical calculus of mathematics, logical biconditional is a logical operator connecting two statements to assert, p if and only if q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). The operator is denoted using a doubleheaded arrow "↔". It is logically equivalent to (p→q)<math>\land<math>(q→p).
The hypothesis is sometimes also called "necessary condition" while the conclusion may be called "sufficient condition".
It is defined using the following truth table:
p q  p ↔ q + T T  T T F  F F T  F F F  T
The only difference from logical conditional is the case when the hypothesis is false but the conclusion is true. In that case, in conditional, the result is true, yet in biconditional the result is false.
In the conceptual interpretation, <math>a = b<math> means "All <math>a<math>'s are <math>b<math>'s and all <math>b<math>'s are <math>a<math>'s"; in other words, that the sets <math>a<math> and <math>b<math> coincide, that they are identical. This does not mean that the concepts have the same meaning. Examples: "triangle" and "trilateral", "equiangular triangle" and "equilateral triangle". The antecedent is the subject and the consequent is the predicate of a universal affirmative proposition.
In the propositional interpretation, <math>a \Leftrightarrow b<math> means that <math>a<math> implies <math>b<math> and <math>b<math> implies <math>a<math>; in other words, that the propositions are equivalent, that is to say, either true or false at the same time. This does not mean that they have the same meaning. Example: "The triangle ABC has two equal sides", and "The triangle ABC has two equal angles". The antecedent is the premise or the cause and the consequent is the consequence. When an implication is translated by a hypothetical (or conditional) judgment the antecedent is called the hypothesis (or the condition) and the consequent is called the thesis.
When we shall have to demonstrate a biconditional we shall usually analyze it into two converse conditionals and demonstrate them separately. This analysis is sometimes made also when the equality is a datum (a premise).
When both members of the biconditional are propositions, it can be separated into two conditionals, of which one is called a theorem and the other its reciprocal. Thus whenever a theorem and its reciprocal are true we have a biconditional. A simple theorem gives rise to an implication whose antecedent is the hypothesis and whose consequent is the thesis of the theorem.
It is often said that the hypothesis is the sufficient condition of the thesis, and the thesis the necessary condition of the hypothesis; that is to say, it is sufficient that the hypothesis be true for the thesis to be true; while it is necessary that the thesis be true for the hypothesis to be true also. When a theorem and its reciprocal are true we say that its hypothesis is the necessary and sufficient condition of the thesis; that is to say, that it is at the same time both cause and consequence.
Colloquial usage
The only unambiguous way of stating a biconditional in plain English is of the form "b if a and a if b". Slightly more formal, one would say "b implies a and a implies b". The plain english "if'" may sometimes be used as a biconditional. One must weigh context heavily.
For example, "I'll buy you an ice cream if you pass the exam" is meant as a biconditional, since the speaker doesn't intend a valid outcome to be buying the ice cream whether or not you pass the exam (as in a conditional). However, "it is cloudy if it is raining" is not meant as a biconditional, since it can obviously be cloudy while not raining.