Logical biconditional

220px|thumb|[[Venn diagram of <math>P \leftrightarrow Q</math><br />(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 or exclusive nor, 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>"), where <math>P</math> is known as the antecedent, and <math>Q</math> the consequent.

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. 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;
<math>\sim</math> in Bernays in 1918;
<math>\rightleftarrows</math> in Hilbert in 1927 (while he used <math>\sim</math> as the main symbol in the article);
<math>\leftrightarrow</math> in Hilbert and Ackermann in 1928 (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);
<math>E</math> (prefix) in Łukasiewicz in 1929 and <math>Q</math> (prefix) in Łukasiewicz in 1951;
<math>\supset\subset</math> in Heyting in 1930;
<math>\Leftrightarrow</math> in Bourbaki in 1954;
<math>\subset\supset</math> in Chazal in 1996;

and so on. Somebody else also use <math>\operatorname{EQ}</math> or <math>\operatorname{EQV}</math> occasionally.

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.). 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.

Logical biconditional

Definition

See also

References

External links