Biconditional introduction

In propositional logic, biconditional introduction[1][2][3] is a valid rule of inference. It allows for one to infer a biconditional from two conditional statements. The rule makes it possible to introduce a biconditional statement into a logical proof. If is true, and if is true, then one may infer that 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 of biconditional elimination. The rule can be stated formally as:

Biconditional introduction
TypeRule of inference
FieldPropositional calculus
StatementIf is true, and if is true, then one may infer that is true.
Symbolic statement

where the rule is that wherever instances of "" and "" appear on lines of a proof, "" can validly be placed on a subsequent line.

Formal notation

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The biconditional introduction rule may be written in sequent notation:

 

where   is a metalogical symbol meaning that   is a syntactic consequence when   and   are both in a proof;

or as the statement of a truth-functional tautology or theorem of propositional logic:

 

where  , and   are propositions expressed in some formal system.

References

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  1. ^ Hurley
  2. ^ Moore and Parker
  3. ^ Copi and Cohen