Definition:Conditional/Necessary Condition

Definition

Let $p \implies q$ be a conditional statement.

Then $q$ is a necessary condition for $p$.

That is, if $p \implies q$, then it is necessary that $q$ be true for $p$ to be true.

This is because unless $q$ is true, $p$ cannot be true.

1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}.9$: The Use of Implication
1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $4$ The Biconditional
1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): Chapter $1$: Sets, Functions, and Relations: $\S 2$: Some Remarks on the Use of the Connectives and, or, implies
1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.5$: Theorems and Proofs
1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): necessary condition
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): necessary condition