In contemporary usage, a logically valid proof from axioms or other accepted premises. It is extremely important that in the period between Descartes and J. S. Mill a demonstration is not this, but a chain of ‘intuitive’ comparisons of ideas, whereby a principle or maxim can be established by reason alone. It is in this sense that Locke believed that theological and moral principles are capable of demonstration, and Hume denies that they are, and also denies that scientific enquiry proceeds by demonstrating its results. See also matter of fact, proof, relations of ideas.