DEMONSTRATION (demonstro, to point out; to cause to see).—(1) In old English writers this word was used to signify the pointing out of the connection between a conclusion and its premises, or between a phenomenon and its asserted cause; (2) it now denotes a necessary consequence, and is synonymous with proof from first principles. To draw from a necessary and universal truth consequences which necessarily follow, is demonstration. |
To connect a truth with a first principle, to show that it is this principle applied or realised in a particular case, is to demonstrate. The result is science, knowledge, certainty. Those general truths arrived at by induction in the sciences of observation are certain knowledge. But it is knowledge which is not definite or complete. It may admit of increase or modification by new discoveries, but the knowledge which demonstration gives is fixed and unalterable.
A demonstration may therefore be defined as a reasoning consisting of one or more arguments, by which some proposition brought into question is shown to be contained in some other proposition assumed, whose truth and certainty being evident and acknowledged, the proposition in question must also be admitted as certain.
Demonstration is direct or indirect. Direct demonstration is descending—when starting from a general truth we come to a particular conclusion, which we must affirm or deny; or ascending—when starting from the subject and its attributes, we arrive by degrees at a general principle, with which we connect the proposition in question. Both these are deductive, because they connect a particular truth with a general principle. Indirect demonstration is when we admit hypothetically a proposition contradictory of that which we wish to demonstrate, and show that this admission leads to absurdity, that is, to an impossibility or a contradiction. This is demonstratio per impossible, or reductio ad absurdum. It should only be employed when direct demonstration is unattainable.
The theory of demonstration is to be found in the Organon of Aristotle, «since whose time,» says Kant, «Logic, as to its foundation, has gained nothing.»
