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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.
The statement is true if and only if A is false. A slash placed through another operator is the same as ¬ {\displaystyle \neg } placed in front. The prime symbol is placed after the negated thing, e.g. p ′ {\displaystyle p'} [ 2 ]
Since the statement and the converse are both true, it is called a biconditional, and can be expressed as "A polygon is a quadrilateral if, and only if, it has four sides." (The phrase if and only if is sometimes abbreviated as iff.) That is, having four sides is both necessary to be a quadrilateral, and alone sufficient to deem it a quadrilateral.
(5) "∀x(Fx)" is true if, and only if, for all objects x, "Fx" is true. (6) "∃ x ( Fx )" is true if, and only if, there is an object x for which "Fx" is true. These explain how the truth conditions of complex sentences (built up from connectives and quantifiers ) can be reduced to the truth conditions of their constituents .
Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement which is not only true, but one which is true under all interpretations of its logical components (other than its logical constants).
The assertion that a statement is a "necessary and sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false. [4] [5] [6]
And we say that "Nixon is alive" is true if and only if the referent (or referent of) "Nixon" belongs to the set associated with "is alive", that is, if and only if Nixon is alive. In semantics, the truth condition of a sentence is almost universally considered distinct from its meaning. The meaning of a sentence is conveyed if the truth ...