Biconditional elimination

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.^[1] 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.

References

^ Cohen, S. Marc. "Chapter 8: The Logic of Conditionals" (PDF). University of Washington. Retrieved 8 October 2013.

[Cohen2007-1] Cohen, S. Marc. "Chapter 8: The Logic of Conditionals" (PDF). University of Washington. Retrieved 8 October 2013.

[1]

Biconditional elimination

Formal notation

See also

References