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. 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

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.