Biconditional introduction

In propositional logic, biconditional introduction 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 into a logical proof. If ${\displaystyle (P\to Q)}$ is true, and ${\displaystyle (Q\to P)}$ then one may infer that ${\displaystyle (P\leftrightarrow Q)}$ 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.

Formal notation

The biconditional introduction rule may be written in sequent notation:

${\displaystyle (P\to Q),(Q\to P)\vdash (P\leftrightarrow Q)}$ 

where ${\displaystyle \vdash }$  is a metalogical symbol meaning that ${\displaystyle (P\leftrightarrow Q)}$  is a syntactic consequence when ${\displaystyle (P\to Q)}$  and ${\displaystyle (P\to Q)}$  are both in a proof; or in rule form:

${\displaystyle {\frac {(P\to Q),(Q\to P)}{\therefore (P\leftrightarrow Q)}}}$ 

where the rule is that wherever instances of "${\displaystyle (P\to Q)}$ " and "${\displaystyle (Q\to P)}$ " appear on lines of a proof, "${\displaystyle (P\leftrightarrow Q)}$ " can be placed on a subsequent line; or as the statement of a truth-functional tautology or theorem of propositional logic:

${\displaystyle ((P\to Q)\land (Q\to P))\to (P\leftrightarrow Q)}$ 

where ${\displaystyle P}$ , and ${\displaystyle Q}$  are propositions expressed in some formal system.

References