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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 ...
Venn diagram of (true part in red) In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or bidirectional implication 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.
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.
De Morgan's laws represented with Venn diagrams.In each case, the resultant set is the set of all points in any shade of blue. In propositional logic and Boolean algebra, De Morgan's laws, [1] [2] [3] also known as De Morgan's theorem, [4] are a pair of transformation rules that are both valid rules of inference.
Biconditional introduction / elimination; Conjunction introduction / elimination; Disjunction introduction / elimination; Disjunctive / hypothetical syllogism; Constructive / destructive dilemma; Absorption / modus tollens / modus ponendo tollens; Negation introduction; Rules of replacement
Conjunction Both A and B conjunct A and B are conjoined Disjunction Either A or B, or both disjunct A and B are disjoined Negation It is not the case that A negatum/negand A is negated Conditional If A, then B antecedent, consequent B is implied by A Biconditional A if, and only if, B equivalents A and B are equivalent
[35] [36] In English, these connectives are expressed by the words "and" (conjunction), "or" (disjunction), "not" , "if" (material conditional), and "if and only if" (biconditional). [1] [13] Examples of such compound sentences might include: Wikipedia is a free online encyclopedia that anyone can edit, and millions already have. (conjunction)
In logic, the term conditional disjunction can refer to: conditioned disjunction , a ternary logical connective introduced by Alonzo Church a rule in classical logic that the material conditional ¬ p → q is equivalent to the disjunction p ∨ q , so that these two formulae are interchangeable - see Negation