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Inlogic andmathematics, thelogical biconditional, also known asmaterial biconditional orequivalence orbidirectional implication orbiimplication orbientailment, is thelogical connective used to conjoin two statements${\displaystyle P}$ and${\displaystyle Q}$ to form the statement "${\displaystyle P}$if and only if${\displaystyle Q}$" (often abbreviated as "${\displaystyle P}$ iff${\displaystyle Q}$"^[1]), where${\displaystyle P}$ is known as theantecedent, and${\displaystyle Q}$ theconsequent.^[2][3]

Nowadays, notations to represent equivalence include${\displaystyle \leftrightarrow ,\iff ,\equiv }$.

${\displaystyle P\leftrightarrow Q}$ islogically equivalent to both${\displaystyle (P\to Q)\wedge (Q\to P)}$ and${\displaystyle (P\wedge Q)\vee (\mathrm{\neg }P\wedge \mathrm{\neg }Q)}$, and theXNOR (exclusive NOR)Boolean operator, which means "both or neither".

Semantically, the only case where a logical biconditional is different from amaterial conditional is the case where the hypothesis (antecedent) is false but the conclusion (consequent) is true. In this case, the result is true for the conditional, but false for the biconditional.^[2]

In the conceptual interpretation,P =Q means "AllP's areQ's and allQ's areP's". In other words, the setsP andQ coincide: they are identical. However, this does not mean thatP andQ need to have the same meaning (e.g.,P could be "equiangular trilateral" andQ could be "equilateral triangle"). When phrased as a sentence, the antecedent is thesubject and the consequent is thepredicate of auniversal affirmative proposition (e.g., in the phrase "all men are mortal", "men" is the subject and "mortal" is the predicate).

In the propositional interpretation,${\displaystyle P\leftrightarrow Q}$ means thatP impliesQ andQ impliesP; in other words, the propositions are logically equivalent, in the sense that both are either jointly true or jointly false. Again, this does not mean that they need to have the same meaning, asP could be "the triangle ABC has two equal sides" andQ could be "the triangle ABC has two equal angles". In general, the antecedent is thepremise, or thecause, and the consequent is theconsequence. When an implication is translated by ahypothetical (orconditional) judgment, the antecedent is called thehypothesis (or thecondition) and the consequent is called thethesis.

A common way of demonstrating a biconditional of the form${\displaystyle P\leftrightarrow Q}$ is to demonstrate that${\displaystyle P\to Q}$ and${\displaystyle Q\to P}$ separately (due to its equivalence to the conjunction of the two converseconditionals^[2]). Yet another way of demonstrating the same biconditional is by demonstrating that${\displaystyle P\to Q}$ and${\displaystyle \mathrm{\neg }P\to \mathrm{\neg }Q}$.

When both members of the biconditional are propositions, it can be separated into two conditionals, of which one is called atheorem and the other itsreciprocal.^{[citation needed]} Thus whenever a theorem and its reciprocal are true, we have a biconditional. A simple theorem gives rise to an implication, whose antecedent is thehypothesis and whose consequent is thethesis of the theorem.

It is often said that the hypothesis is thesufficient condition of the thesis, and that the thesis is thenecessary condition of the hypothesis. That is, it is sufficient that the hypothesis be true for the thesis to be true, while it is necessary that the thesis be true if the hypothesis were true. When a theorem and its reciprocal are true, its hypothesis is said to be thenecessary and sufficient condition of the thesis. That is, the hypothesis is both the cause and the consequence of the thesis at the same time.

Notations to represent equivalence used in history include:

• ${\displaystyle =}$ inGeorge Boole in 1847.^[4] Although Boole used${\displaystyle =}$ mainly on classes, he also considered the case that${\displaystyle x,y}$ are propositions in${\displaystyle x=y}$, and at the time${\displaystyle =}$ is equivalence.
• ${\displaystyle \equiv }$ inFrege in 1879;^[5]
• ${\displaystyle \sim }$ inBernays in 1918;^[6]
• ${\displaystyle \rightleftarrows }$ inHilbert in 1927 (while he used${\displaystyle \sim }$ as the main symbol in the article);^[7]
• ${\displaystyle \leftrightarrow }$ inHilbert andAckermann in 1928^[8] (they also introduced${\displaystyle \rightleftarrows ,\sim }$ while they use${\displaystyle \sim }$ as the main symbol in the whole book;${\displaystyle \leftrightarrow }$ is adopted by many followers such as Becker in 1933^[9]);
• ${\displaystyle E}$ (prefix) inŁukasiewicz in 1929^[10] and${\displaystyle Q}$ (prefix) inŁukasiewicz in 1951;^[11]
• ${\displaystyle \supset \subset }$ inHeyting in 1930;^[12]
• ${\displaystyle \iff }$ inBourbaki in 1954;^[13]
• ${\displaystyle \subset \supset }$ in Chazal in 1996;^[14]

and so on. Somebody else also use${\displaystyle \mathrm{EQ}}$ or${\displaystyle \mathrm{EQV}}$ occasionally.^{[citation needed][vague][clarification needed]}

Logical equality (also known as biconditional) is anoperation on twological values, typically the values of twopropositions, that produces a value oftrue if and only if both operands are false or both operands are true.^[2]

The following is a truth table for${\displaystyle A\leftrightarrow B}$:

${\displaystyle A}$	${\displaystyle B}$	${\displaystyle A\leftrightarrow B}$
F	F	T
F	T	F
T	F	F
T	T	T

When more than two statements are involved, combining them with${\displaystyle \leftrightarrow }$ might be ambiguous. For example, the statement

${\displaystyle x_1\leftrightarrow x_2\leftrightarrow x_3\leftrightarrow \cdots \leftrightarrow x_n}$

may be interpreted as

${\displaystyle (((x_1\leftrightarrow x_2)\leftrightarrow x_3)\leftrightarrow \cdots )\leftrightarrow x_n}$,

or may be interpreted as saying that allx_i arejointly true or jointly false:

${\displaystyle (x_1\wedge \cdots \wedge x_n)\vee (\mathrm{\neg }{x}_1\wedge \cdots \wedge \mathrm{\neg }{x}_n)}$

As it turns out, these two statements are only the same when zero or two arguments are involved. In fact, the following truth tables only show the same bit pattern in the line with no argument and in the lines with two arguments:

The left Venn diagram below, and the lines(AB    ) in these matrices represent the same operation.

Red areas stand for true (as in forand).

The biconditional of two statements is thenegation of theexclusive or: ${\displaystyle A\leftrightarrow B\iff \mathrm{\neg }(A\oplus B)}$ ${\displaystyle \iff \mathrm{\neg }}$

The biconditional and the exclusive or of three statements give the same result: ${\displaystyle A\leftrightarrow B\leftrightarrow C\iff }$ ${\displaystyle A\oplus B\oplus C}$ ${\displaystyle \leftrightarrow }$${\displaystyle \iff }$ ${\displaystyle \oplus }$${\displaystyle \iff }$

But${\displaystyle A\leftrightarrow B\leftrightarrow C}$ may also be used as an abbreviation for${\displaystyle (A\leftrightarrow B)\wedge (B\leftrightarrow C)}$ ${\displaystyle \wedge }$${\displaystyle \iff }$

Commutativity: Yes

${\displaystyle A\leftrightarrow B}$	${\displaystyle \iff }$	${\displaystyle B\leftrightarrow A}$
	${\displaystyle \iff }$

Associativity: Yes

${\displaystyle A}$	${\displaystyle \leftrightarrow }$	${\displaystyle (B\leftrightarrow C)}$	${\displaystyle \iff }$			${\displaystyle (A\leftrightarrow B)}$	${\displaystyle \leftrightarrow }$	${\displaystyle C}$
	${\displaystyle \leftrightarrow }$		${\displaystyle \iff }$		${\displaystyle \iff }$		${\displaystyle \leftrightarrow }$

Distributivity: Biconditional doesn't distribute over any binary function (not even itself), butlogical disjunction distributes over biconditional.

Idempotency: No

${\displaystyle A}$	${\displaystyle \leftrightarrow }$	${\displaystyle A}$	${\displaystyle \iff }$	${\displaystyle 1}$	${\displaystyle \nLeftrightarrow }$	${\displaystyle A}$
	${\displaystyle \leftrightarrow }$		${\displaystyle \iff }$		${\displaystyle \nLeftrightarrow }$

Monotonicity: No

${\displaystyle A\to B}$	${\displaystyle \nRightarrow }$			${\displaystyle (A\leftrightarrow C)}$	${\displaystyle \to }$	${\displaystyle (B\leftrightarrow C)}$
	${\displaystyle \nRightarrow }$		${\displaystyle \iff }$		${\displaystyle \to }$

Truth-preserving: Yes
When all inputs are true, the output is true.

${\displaystyle A\wedge B}$	${\displaystyle \Rightarrow }$	${\displaystyle A\leftrightarrow B}$
	${\displaystyle \Rightarrow }$

Falsehood-preserving: No
When all inputs are false, the output is not false.

${\displaystyle A\leftrightarrow B}$	${\displaystyle \nRightarrow }$	${\displaystyle A\vee B}$
	${\displaystyle \nRightarrow }$

Walsh spectrum: (2,0,0,2)

Nonlinearity: 0 (the function is linear)

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

Biconditional introduction allows one to infer that if B follows from A and A follows from B, then Aif 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" or equivalently, "I'm alive if and only if I'm breathing." Or more schematically:

 B → A    A → B    ∴ A ↔ B

 B → A    A → B    ∴ B ↔ A

Biconditional elimination allows one to infer aconditional from a biconditional: if A↔ B is true, then one may infer either A→ B, or B→ A.

For example, if it is true that I'm breathingif and only if I'm alive, then it's true thatif I'm breathing, then I'm alive; likewise, it's true thatif I'm alive, then I'm breathing. Or more schematically:

A ↔ B   ∴ A → B

A ↔ B   ∴ B → A

One unambiguous way of stating a biconditional in plain English is to adopt the form "b ifa anda ifb"—if the standard form "a if and only ifb" is not used. Slightly more formally, one could also say that "b impliesa anda impliesb", or "a is necessary and sufficient forb". The plain English "if'" may sometimes be used as a biconditional (especially in the context of a mathematical definition^[15]). In which case, one must take into consideration the surrounding context when interpreting these words.

For example, the statement "I'll buy you a new wallet if you need one" may be interpreted as a biconditional, since the speaker doesn't intend a valid outcome to be buying the wallet whether or not the wallet is needed (as in a conditional). However, "it is cloudy if it is raining" is generally not meant as a biconditional, since it can still be cloudy even if it is not raining.

• If and only if
• Logical equivalence
• Logical equality
• XNOR gate
• Biconditional elimination
• Biconditional introduction

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