consequent, unlike the material conditional. strict implication A relation between propositions where the truth of the first (the antecedent) necessarily This is a glossary of logic. Logic is the study of the principles of valid reasoning and argumentation. material consequence The relationship between statements where the truth of one (the antecedent) results in the truth of another (the consequent), based on the content of the statements rather than their logical form. Contrast formal consequence. See also semantic consequence. Not to be confused with material implication.

states that the truth values of two statements are equal. It is paraphrased by the biconditional, a logical connective between statements. The biconditional In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") states that the truth values of two statements are equal. It is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of materia In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") states that the truth values of two statements are equal. It is paraphrased by the biconditional, a logical connective between statements. 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), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its converse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, P if and only if Q means that P is true whenever Q is true, and the only case in which P is true is if Q is also true, whereas in the case of P if Q, there could be other scenarios where P is true and Q is false. In writing, phrases commonly used as alternatives to P "if and only if" Q include: Q is necessary and sufficient for P, for P it is necessary and sufficient that Q, P is equivalent (or materially equivalent) to Q (compare with material implication), P precisely if Q, P precisely (or exactly) when Q, P exactly in case Q, and P just in case Q. Some authors regard "iff" as unsuitable in formal writing; others consider it a "borderline case" and tolerate its use. In logical formulae, logical symbols, such as ↔ {\displaystyle \leftrightarrow } and ⇔ {\displaystyle \Leftrightarrow } , are used instead of these phrases; see § Notation below.

by the use of a negating connective.) Afterthought connectives make it possible to construct all the important truth - functional relationships in a variety