Biconditional introduction: Difference between revisions

Revision as of 20:35, 6 March 2012

Revision as of 21:54, 6 March 2012

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{{Transformation rules}}

In [[propositional calculus|propositional logic]], '''biconditional introduction''' is a [[validity|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 [[logical biconditional|biconditional]] into a [[formal proof|logical proof]]. If <math>(P \to Q)</math> is true, and <math>(Q \to P)</math> 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 ==

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:<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>(P \to Q)</math> are both in a proof;

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