Demonstrandum

Demonstrandum is a Latin term meaning “that which is to be demonstrated.” In logic, philosophy, and mathematics, it denotes the proposition or statement that a proof aims to establish as true. The corresponding element is the demonstrans, the premises and reasoning that support the conclusion. Together, demonstrandum and demonstrans structure a demonstration: from accepted axioms or premises, one derives the demonstrandum through valid inference.

The term is often used in historical or formal analyses of proofs and argument structure. In contemporary practice, the pair is usually described in English as “the proposition to be proven” and “the premises or argument” supporting it. Demonstrandum is typically labeled within a proof to indicate what justification is intended to establish.

Etymology: demonstrandum comes from the Latin gerundive form of demonstrare, meaning “to demonstrate” or “to show,” literally “that which must be demonstrated.”

Examples: In Euclid’s proof that there are infinitely many primes, the demonstrandum is the statement “There are infinitely many prime numbers,” and the demonstration consists of the steps that derive this conclusion from the axioms and known properties of numbers. While the Latin terminology is less common in modern texts, the concept of a conclusion to be proven and the supporting argument remains central to formal proofs and rigorous reasoning.