Logical biconditional

File:Venn1001.svg

Venn diagram of <math>P \leftrightarrow Q</math>
(true part in red)

In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or bidirectional implication or biimplication or bientailment, is the logical connective used to conjoin two statements <math>P</math> and <math>Q</math> to form the statement "<math>P</math> if and only if <math>Q</math>" (often abbreviated as "<math>P</math> iff <math>Q</math>"<ref name=":2">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>), where <math>P</math> is known as the antecedent, and <math>Q</math> the consequent.<ref name=":1">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite book</ref>

Nowadays, notations to represent equivalence include <math>\leftrightarrow,\Leftrightarrow,\equiv</math>.

<math>P\leftrightarrow Q</math> is logically equivalent to both <math>(P \rightarrow Q) \land (Q \rightarrow P)</math> and <math>(P \land Q) \lor (\neg P \land \neg Q) </math>, and the XNOR (exclusive NOR) Boolean operator, which means "both or neither".

Semantically, the only case where a logical biconditional is different from a material 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.<ref name=":1" />

In the conceptual interpretation, Template:Math means "All Template:Mvar's are Template:Mvar's and all Template:Mvar's are Template:Mvar's". In other words, the sets Template:Mvar and Template:Mvar coincide: they are identical. However, this does not mean that Template:Mvar and Template:Mvar need to have the same meaning (e.g., Template:Mvar could be "equiangular trilateral" and Template:Mvar could be "equilateral triangle"). When phrased as a sentence, the antecedent is the subject and the consequent is the predicate of a universal affirmative proposition (e.g., in the phrase "all men are mortal", "men" is the subject and "mortal" is the predicate).

In the propositional interpretation, <math>P \leftrightarrow Q</math> means that Template:Mvar implies Template:Mvar and Template:Mvar implies Template:Mvar; 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, as Template:Mvar could be "the triangle ABC has two equal sides" and Template:Mvar could be "the triangle ABC has two equal angles". In general, 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 of the form <math>P \leftrightarrow Q</math> is to demonstrate that <math>P \rightarrow Q</math> and <math>Q \rightarrow P</math> separately (due to its equivalence to the conjunction of the two converse conditionals<ref name=":1" />). Yet another way of demonstrating the same biconditional is by demonstrating that <math>P \rightarrow Q</math> and <math>\neg P \rightarrow \neg Q</math>.

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.Template: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 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 that the thesis is the necessary 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 the necessary 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

Notations to represent equivalence used in history include:

• <math>=</math> in George Boole in 1847.<ref name="boole1847">Template:Cite book</ref> Although Boole used <math>=</math> mainly on classes, he also considered the case that <math>x,y</math> are propositions in <math>x=y</math>, and at the time <math>=</math> is equivalence.
• <math>\equiv</math> in Frege in 1879;<ref name="frege1879b">Template:Cite book</ref>
• <math>\sim</math> in Bernays in 1918;<ref name="bernays1918">Template:Cite book</ref>
• <math>\rightleftarrows</math> in Hilbert in 1927 (while he used <math>\sim</math> as the main symbol in the article);<ref name="hilbert1927">Template:Cite journal</ref>
• <math>\leftrightarrow</math> in Hilbert and Ackermann in 1928<ref name="hilbert-ackermann1928">Template:Cite book</ref> (they also introduced <math>\rightleftarrows,\sim</math> while they use <math>\sim</math> as the main symbol in the whole book; <math>\leftrightarrow</math> is adopted by many followers such as Becker in 1933<ref name="becker1933">Template:Cite book</ref>);
• <math>E</math> (prefix) in Łukasiewicz in 1929<ref name="lukasiewicz1929">Template:Cite book</ref> and <math>Q</math> (prefix) in Łukasiewicz in 1951;<ref name="lukasiewicz1951">Template:Cite book</ref>
• <math>\supset\subset</math> in Heyting in 1930;<ref name="heyting1929">Template:Cite journal</ref>
• <math>\Leftrightarrow</math> in Bourbaki in 1954;<ref name="bourbaki1954b">Template:Cite book</ref>
• <math>\subset\supset</math> in Chazal in 1996;<ref name="chazal1996">Template:Cite book</ref>

and so on. Somebody else also use <math>\operatorname{EQ}</math> or <math>\operatorname{EQV}</math> occasionally.Template:Citation neededTemplate:VagueTemplate:Clarify

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.<ref name=":1" />

Truth table

The following is a truth table for <math>A \leftrightarrow B</math>:

When more than two statements are involved, combining them with <math>\leftrightarrow</math> might be ambiguous. For example, the statement

<math>x_1 \leftrightarrow x_2 \leftrightarrow x_3 \leftrightarrow \cdots \leftrightarrow x_n</math>

may be interpreted as

<math>(((x_1 \leftrightarrow x_2) \leftrightarrow x_3) \leftrightarrow \cdots) \leftrightarrow x_n</math>,

or may be interpreted as saying that all Template:Math are jointly true or jointly false:

<math>(x_1 \land \cdots \land x_n) \lor (\neg x_1 \land \cdots \land \neg x_n)</math>

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:

File:Variadic logical XAND.svg

<math>~x_1 \leftrightarrow \cdots \leftrightarrow x_n</math>
meant as equivalent to
<math>\neg~(\neg x_1 \oplus \cdots \oplus \neg x_n)</math>

The central Venn diagram below,
and line (ABC  ) in this matrix
represent the same operation.

File:Variadic logical all or nothing.svg

<math>~x_1 \leftrightarrow \cdots \leftrightarrow x_n</math>
meant as shorthand for
<math>(~x_1 \land \cdots \land x_n~)</math>
<math>\lor~(\neg x_1 \land \cdots \land \neg x_n)</math>

The Venn diagram directly below,
and line (ABC  ) in this matrix
represent the same operation.

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

Venn diagrams

Red areas stand for true (as in File:Venn0001.svg for and).

File:Venn1001.svg The biconditional of two statements is the negation of the exclusive or: <math>~A \leftrightarrow B~~\Leftrightarrow~~\neg(A \oplus B)</math> File:Venn1001.svg <math>\Leftrightarrow \neg</math> File:Venn0110.svg

File:Venn 0110 1001.svg The biconditional and the exclusive or of three statements give the same result: <math>~A \leftrightarrow B \leftrightarrow C~~\Leftrightarrow</math> <math>~A \oplus B \oplus C</math> File:Venn 1001 1001.svg <math>\leftrightarrow</math> File:Venn 0000 1111.svg <math>~~\Leftrightarrow~~</math> File:Venn 0110 0110.svg <math>\oplus</math> File:Venn 0000 1111.svg <math>~~\Leftrightarrow~~</math> File:Venn 0110 1001.svg

File:Venn 1000 0001.svg But <math>~A \leftrightarrow B \leftrightarrow C</math> may also be used as an abbreviation for <math>(A \leftrightarrow B) \land (B \leftrightarrow C)</math> File:Venn 1001 1001.svg <math>\land</math> File:Venn 1100 0011.svg <math>~~\Leftrightarrow~~</math> File:Venn 1000 0001.svg

Properties

Commutativity: Yes

<math>A \leftrightarrow B</math>	<math>\Leftrightarrow</math>	<math>B \leftrightarrow A</math>
File:Venn1001.svg	<math>\Leftrightarrow</math>	File:Venn1001.svg

Associativity: Yes

<math>~A</math>	<math>~~~\leftrightarrow~~~</math>	<math>(B \leftrightarrow C)</math>	<math>\Leftrightarrow</math>			<math>(A \leftrightarrow B)</math>	<math>~~~\leftrightarrow~~~</math>	<math>~C</math>
File:Venn 0101 0101.svg	<math>~~~\leftrightarrow~~~</math>	File:Venn 1100 0011.svg	<math>\Leftrightarrow</math>	File:Venn 0110 1001.svg	<math>\Leftrightarrow</math>	File:Venn 1001 1001.svg	<math>~~~\leftrightarrow~~~</math>	File:Venn 0000 1111.svg

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

Idempotency: No

<math>~A~</math>	<math>~\leftrightarrow~</math>	<math>~A~</math>	<math>\Leftrightarrow</math>	<math>~1~</math>	<math>\nLeftrightarrow</math>	<math>~A~</math>
File:Venn01.svg	<math>~\leftrightarrow~</math>	File:Venn01.svg	<math>\Leftrightarrow</math>	File:Venn11.svg	<math>\nLeftrightarrow</math>	File:Venn01.svg

Monotonicity: No

<math>A \rightarrow B</math>	<math>\nRightarrow</math>			<math>(A \leftrightarrow C)</math>	<math>\rightarrow</math>	<math>(B \leftrightarrow C)</math>
File:Venn 1011 1011.svg	<math>\nRightarrow</math>	File:Venn 1101 1011.svg	<math>\Leftrightarrow</math>	File:Venn 1010 0101.svg	<math>\rightarrow</math>	File:Venn 1100 0011.svg

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

<math>A \land B</math>	<math>\Rightarrow</math>	<math>A \leftrightarrow B</math>
File:Venn0001.svg	<math>\Rightarrow</math>	File:Venn1001.svg

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

<math>A \leftrightarrow B</math>	<math>\nRightarrow</math>	<math>A \lor B</math>
File:Venn1001.svg	<math>\nRightarrow</math>	File:Venn0111.svg

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

Nonlinearity: 0 (the function is linear)

Rules of inference

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Like all connectives in first-order logic, the biconditional has rules of inference that govern its use in formal proofs.

Biconditional introduction

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Biconditional introduction allows one 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" 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

Biconditional elimination allows one to infer a conditional 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 breathing if and only if I'm alive, then it's true that if I'm breathing, then I'm alive; likewise, it's true that if I'm alive, then I'm breathing. Or more schematically:

 A ↔ B  
 ∴ A → B

 A ↔ B  
 ∴ B → A

Colloquial usage

One unambiguous way of stating a biconditional in plain English is to adopt the form "b if a and a if b"—if the standard form "a if and only if b" is not used. Slightly more formally, one could also say that "b implies a and a implies b", or "a is necessary and sufficient for b". The plain English "if'" may sometimes be used as a biconditional (especially in the context of a mathematical definition<ref>In fact, such is the style adopted by Wikipedia's manual of style in mathematics.</ref>). 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.

See also

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

External links

• Template:Commonscatinline

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