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 "↔" 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 conditional, the result is true, yet in 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, that the sets ''a'' and ''b'' 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, ''a'' ⇔ ''b'' 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''.
When we have to demonstrate a biconditional we 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.

Contents

Definition

Truth table

Venn diagram

Properties

Colloquial usage

See also

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 iff B"

Properties

★ associativity: $(a\; equiv\; b)\; equiv\; c)\; equiv\; (a\; equiv\; (b\; equiv\; c)$

★ commutativity: $(a\; equiv\; b)\; equiv\; (b\; equiv\; a)$

★ reflexivity: $a\; equiv\; a$

★ '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

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.

See also

★ If and only if

★ Logical equivalence

★ Logical equality

★ XNOR gate

★ Biconditional elimination

★ Biconditional introduction