Biconditional elimination

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Transformation rules
Propositional calculus
Rules of inference Modus ponens Modus tollens Biconditional introduction Biconditional elimination Conjunction introduction Simplification Disjunction introduction Disjunction elimination Disjunctive syllogism Hypothetical syllogism Constructive dilemma Destructive dilemma Absorption Rules of replacement Associativity Commutativity Distributivity Double negation De Morgan's laws Transposition Material implication Material equivalence Exportation Tautology
Predicate logic
Universal generalization Universal instantiation Existential generalization Existential instantiation
v t e

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". University of Washington. Retrieved 8 October 2013.

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Biconditional elimination

Formal notation

See also

References