Editing Logical biconditional (section)

{{Short description|If and only if relation}}
{{one source|date=June 2013}}
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[[File:Venn1001.svg|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''', 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">{{Cite web|url=http://mathworld.wolfram.com/Iff.html|title=Iff|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-25}}</ref>), where <math>P</math> is known as the ''[[antecedent (logic)|antecedent]]'', and <math>Q</math> the ''[[consequent]]''.<ref name=":1">{{Cite web|url=http://web.mnstate.edu/peil/geometry/Logic/4logic.htm|title=Conditionals and Biconditionals|last=Peil|first=Timothy|website=web.mnstate.edu|access-date=2019-11-25|archive-date=2020-10-24|archive-url=https://web.archive.org/web/20201024171606/http://web.mnstate.edu/peil/geometry/Logic/4logic.htm|url-status=dead}}</ref><ref>{{Cite book|title=Handbook of Logic|last=Brennan|first=Joseph G.|publisher=Harper & Row|year=1961|edition=2nd|pages=81}}</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 gate|XNOR]] (exclusive NOR) [[Logical connective|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, {{math|1=''P'' = ''Q''}} means "All {{mvar|P}}'s are {{mvar|Q}}'s and all {{mvar|Q}}'s are {{mvar|P}}'s". In other words, the sets {{mvar|P}} and {{mvar|Q}} coincide: they are identical. However, this does not mean that {{mvar|P}} and {{mvar|Q}} need to have the same meaning (e.g., {{mvar|P}} could be "equiangular trilateral" and {{mvar|Q}} 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 {{mvar|P}} implies {{mvar|Q}} and {{mvar|Q}} implies {{mvar|P}}; 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 {{mvar|P}} could be "the triangle ABC has two equal sides" and {{mvar|Q}} 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 [[Material conditional|conditional]]s<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''.{{Citation needed|date=August 2008}} 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.