Biconditional elimination

Biconditional elimination

Type	Rule of inference
Field	Propositional calculus
Statement	If $P\leftrightarrow Q$ is true, then one may infer that $P\to Q$ is true, and also that $Q\to P$ is true.
Symbolic statement	${\frac {P\leftrightarrow Q}{\therefore P\to Q}}$ ${\frac {P\leftrightarrow Q}{\therefore Q\to P}}$

Biconditional elimination is the name of two valid rules of inference of propositional logic. It allows for one to infer a conditional from a biconditional. If $P\leftrightarrow Q$ is true, then one may infer that $P\to Q$ is true, and also that $Q\to P$ is true. For example, if it's true that I'm breathing if and only if I'm alive, then it's true that if I'm breathing, I'm alive; likewise, it's true that if I'm alive, I'm breathing. The rules can be stated formally as:

${\frac {P\leftrightarrow Q}{\therefore P\to Q}}$

and

${\frac {P\leftrightarrow Q}{\therefore Q\to P}}$

where the rule is that wherever an instance of "$P\leftrightarrow Q$" appears on a line of a proof, either "$P\to Q$" or "$Q\to P$" can be placed on a subsequent line.

Formal notation

The biconditional elimination rule may be written in sequent notation:

$(P\leftrightarrow Q)\vdash (P\to Q)$

and

$(P\leftrightarrow Q)\vdash (Q\to P)$

where $\vdash$ is a metalogical symbol meaning that $P\to Q$, in the first case, and $Q\to P$ in the other are syntactic consequences of $P\leftrightarrow Q$ in some logical system;

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

$(P\leftrightarrow Q)\to (P\to Q)$

$(P\leftrightarrow Q)\to (Q\to P)$

where $P$, and $Q$ are propositions expressed in some formal system.

See also

• Logical biconditional