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Venn diagram of (true part in red) In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or biimplication or bientailment, is the logical connective used to conjoin two statements and to form the statement "if and only if" (often abbreviated as "iff " [1]), where is known as the antecedent, and the consequent.
The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), [2] and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of ...
material biconditional (material equivalence) if and only if, iff, xnor propositional logic, Boolean algebra: is true only if both A and B are false, or both A and B are true. Whether a symbol means a material biconditional or a logical equivalence, depends on the author’s style.
(As to equivalence, Howson calls it "truth-functional equivalence", while Cunningham calls it "logical equivalence".) Equivalence is symbolized with ⇔ and is a metalanguage symbol, while a biconditional is symbolized with ↔ and is a logical connective in the object language . Regardless, an equivalence or biconditional is true if, and only ...
In propositional logic, biconditional introduction [1] [2] [3] is a valid rule of inference. It allows for one to infer a biconditional from two conditional statements . The rule makes it possible to introduce a biconditional statement into a logical proof .
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 ↔ Q {\displaystyle P\leftrightarrow Q} is true, then one may infer that P → Q {\displaystyle P\to Q} is true, and also that Q → P {\displaystyle Q\to P} is true. [ 1 ]
An abbreviation for "if and only if," denoting a biconditional logical connective indicating mutual implication. ignoratio elenchi A logical fallacy where an argument misses the point or addresses an issue different from the one that was raised, also known as missing the point. image The set of all outputs of a function from a given set of inputs.
Randolph diagram that represents the logical statement (disjunction). A Randolph diagram ( R-diagram ) is a simple way to visualize logical expressions and combinations of sets. Randolph diagrams were created by mathematician John F. Randolph in 1965, during his tenure at the University of Arkansas .