[Essay II xxv 4-8] What counts toward demonstrability...

[Essay II xxv 4-8] What counts toward demonstrability, on Locke's view, is the possibility of perceiving intermediate links between our ideas. Since the mixed modes of human action and the concepts of moral rules are both abstract ideas that serve as their own archetypes, it follows that the relations between them are fully demonstrable. [Essay IV iv 7-9] In this respect, at least, morality is on an equal footing with mathematics.

Where there is no Property, there is no Injustice, is a Proposition as certain as any Demonstration in Euclid: For the Idea of Property, being a right to any thing; and the Idea to which the name Injusticeis given, being the Invasion or Violation of that right; it is evident, that these Ideas being thus established, and these Names annexed to them, I can as certainly know this Proposition to be true, as that a Triangle has three Angles equal to two right ones.

Again, No Government allows absolute Liberty: The Idea of Government being the establishment of Society upon certain Rules or Laws, which require Conformity to them; and the Idea of absolute Liberty being for any one to do whatever he pleases; I am as capable of being certain of the Truth of this Proposition, as of any in Mathematicks.

[Essay IV iii 18] The apparent advantage of mathematical over moral reasoning, Locke speculated, rests only on the relative ease with which we can represent mathematical relationships in perspicuous diagrams and the relative absence of partisan concerns. Were we to approach moral reasoning with the same degree of objectivity we commonly bring to mathematical thinking, he argued, we would achieve the same quality of demonstrable certainty about substantive moral truths.

[Essay IV iii 19-20] Varieties of Moral Law In general, Locke held that the mental comparisons comprising our ideas of relations are significant for the practical conduct of life. Natural relations among human beings, like the various degrees of blood-kinship, for example, commonly carry the presumption of some special obligation toward other members of our families. Instituted relations based on social agreement ground the governance of human societies, as we'll see in detail in a few weeks.

But moral relations are most vital of all, so that the very descriptions of human action under our ideas of mixed modes commonly carry with them an implicit reference to the moral law under which they are commanded or proscribed.