Editing Biconditional introduction

{{Short description|Inference in propositional logic}}
{{Infobox mathematical statement
| name = Biconditional introduction
| type = [[Rule of inference]]
| field = [[Propositional calculus]]
| statement = If <math>P \to Q</math> is true, and if <math>Q \to P</math> is true, then one may infer that <math>P \leftrightarrow Q</math> is true.
| symbolic statement = <math>\frac{P \to Q, Q \to P}{\therefore P \leftrightarrow Q}</math>
}}
{{Transformation rules}}

In [[propositional calculus|propositional logic]], '''biconditional introduction'''<ref>Hurley</ref><ref>Moore and Parker</ref><ref>Copi and Cohen</ref> is a [[Validity (logic)|valid]] [[rule of inference]]. It allows for one to [[inference|infer]] a [[Logical biconditional|biconditional]] from two [[Material conditional|conditional statements]]. The rule makes it possible to introduce a biconditional statement into a [[formal proof|logical proof]]. If <math>P \to Q</math> is true, and if <math>Q \to P</math> is true, then one may infer that <math>P \leftrightarrow Q</math> is true. 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". Biconditional introduction is the [[Converse (logic)|converse]] of [[biconditional elimination]]. The rule can be stated formally as:

:<math>\frac{P \to Q, Q \to P}{\therefore P \leftrightarrow Q}</math>

where the rule is that wherever instances of "<math>P \to Q</math>" and "<math>Q \to P</math>" appear on lines of a proof, "<math>P \leftrightarrow Q</math>" can validly be placed on a subsequent line.

== Formal notation ==
The ''biconditional introduction'' rule may be written in [[sequent]] notation:
:<math>(P \to Q), (Q \to P) \vdash (P \leftrightarrow Q)</math>

where <math>\vdash</math> is a [[metalogic]]al symbol meaning that <math>P \leftrightarrow Q</math> is a [[logical consequence|syntactic consequence]] when <math>P \to Q</math> and <math>Q \to P</math> are both in a proof;

or as the statement of a truth-functional [[Tautology (logic)|tautology]] or [[theorem]] of propositional logic:

:<math>((P \to Q) \land (Q \to P)) \to (P \leftrightarrow Q)</math>

where <math>P</math>, and <math>Q</math> are propositions expressed in some [[formal system]].

==References==
{{Reflist}}

{{DEFAULTSORT:Biconditional Introduction}}
[[Category:Rules of inference]]
[[Category:Theorems in propositional logic]]