Biconditional elimination
| Type | Rule of inference |
|---|---|
| Field | Propositional calculus |
| Statement | If is true, then one may infer that is true, and also that is true. |
| Symbolic statement |
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 is true, then one may infer that is true, and also that 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:
and
where the rule is that wherever an instance of "" appears on a line of a proof, either "" or "" can be placed on a subsequent line.
Formal notation
The biconditional elimination rule may be written in sequent notation:
and
where is a metalogical symbol meaning that , in the first case, and in the other are syntactic consequences of in some logical system;
or as the statement of a truth-functional tautology or theorem of propositional logic:
where , and are propositions expressed in some formal system.
See also
References
- ^ Cohen, S. Marc. "Chapter 8: The Logic of Conditionals" (PDF). University of Washington. Archived (PDF) from the original on 2022-10-09. Retrieved 8 October 2013.