Definitions

# Logical biconditional

In logic and mathematics, logical biconditional (sometimes also known as the material 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 "↔", an equality sign "=", an equivalence sign "≡", or EQV. It is logically equivalent to (p → q) ∧ (q → p), or the XNOR boolean operator. It is equivalent to (not p or q) and (not q or p). It is also logically equivalent to (not p and not q) or (p and q).

The hypothesis is sometimes also called "sufficient condition" while the conclusion may be called "necessary condition".

The only difference from material conditional is the case when the hypothesis is false but the conclusion is true. In that case, in the conditional, the result is true, yet in the biconditional the result is false.

In the conceptual interpretation, a = b means "All a 's are b 's and all b 's are a 's"; in other words, the sets a and b coincide: 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, ab means that a implies b and b implies a; 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.

A common way of demonstrating a biconditional is to use its equivalence to the conjunction of two converse conditionals, demonstrating these separately.

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.

## Definition

Logical equality (also known as biconditional) is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true.

### Truth table

The truth table for p EQ q (also written as p = q, p ↔ q, or p ≡ q) is as follows:

p q
T T T
T F F
F T F
F F T

### Venn diagram

The Venn Diagram of "A if and only if B" (red areas are true)

## Properties

• associativity: $\left(\left(a leftrightarrow b\right) leftrightarrow c\right) leftrightarrow \left(a leftrightarrow \left(b leftrightarrow c\right)\right)$
• commutativity: $\left(a leftrightarrow b\right) leftrightarrow \left(b leftrightarrow a\right)$
• reflexivity: $a leftrightarrow a$
• truth-preserving: The interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true' as a result of logical biconditional.
• linear

## Rules of Inference

Like all connectives in first-order logic, the biconditional has rules of inference that govern its use in formal proofs.

### Biconditional Introduction

Biconditional introduction allows you to infer that, if B follows from A, and A follows from B, then A if and only if B.

For example, from the statements "if I'm breathing, then I'm alive" and "if I'm alive, then I'm breathing", it can be inferred that "I'm breathing if and only if I'm alive".

` B → A   `
` A → B   `
` ∴  A ↔ B`

### Biconditional Elimination

Biconditional elimination allows one to infer a conditional from a biconditional: if (A B ) is true, then one may infer one direction of the biconditional, (A B ) and(B A ).

For example, if it's true that I'm breathing if and only if I'm alive, then it's true that if I'm breathing, I'm alive; likewise, it's true that if I'm alive, I'm breathing.

Formally:

` (A ↔ B )  `
` ∴ (A → B )`

also

` (A ↔ B )  `
` ∴ (B → A )`

## 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.