ABSTRACT
Next comes 'demonstrative' or 'rational' knowledge, where a chain of intuitively perceived steps is necessary for us to perceive the relation between two ideas. The chain is constructed by means of 'intermediate ideas'. Locke gave the standard example of the Euclidean demonstration, employing diagrams, that the angles of a triangle are equal to two right angles." The 'intermediate ideas' are presumably the further lines and angles constructed in the diagram for the purpose of the proof. Their purpose is like that of a measuring rod: 'As a Man, by a Yard, finds two Houses to be of the same length, which could not be brought together ... by juxta-position."? Locke did not think of the need for such steps as variable from person to person, which would suggest that the need stems from the relation itself. That would be consistent with his view that propositions known intuitively could not conceivably require a proof for anyone who understood them; but once in the Essay, unfortunately, he indulged in the speculation that demonstrative knowledge for man might be intuitive for angels.24 For us, at any rate, prior to demonstration there is room for doubt, ignorance or belief about the conclusion, and so room for error. Even after demonstration, it seems, the sheer complexity of the proof can lead to 'a great abatement of that evident lustre and full assurance' enjoyed by intuition. In the course of demonstration, moreover, memory is commonly necessary.
Because in long proofs we may think we remember previously perceiving connections where there are none, 'this is more imperfect than intuitive Knowledge, and Men embrace often Falsehoods for Demonstrations'." Nevertheless, demonstrative knowledge is knowledge because it involves only what is known by intuition, even if remembered intuition.
