Biconditional elimination

Transformation rules
Propositional calculus
Rules of inference
Modus ponens / Modus tollens Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive / Hypothetical syllogism Constructive / Destructive dilemma Absorption Modus ponendo tollens
Rules of replacement
Associativity Commutativity Distributivity Double negation De Morgan's laws Transposition Material implication Exportation Tautology Negation introduction
Predicate logic
Universal generalization / instantiation Existential generalization / instantiation

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.

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Biconditional elimination

Formal notation

See also

References