and that things equal to the same thing are equal to one another, that two contraries cannot coexist, and that two contradictories can neither both be false nor both be true. There is no universal proposition of this kind used as a premise in a demonstration whose conclusion cannot be known without that demonstration as an intermediary. Indeed, this frequently occurs.
Thus, if it is known that every unit is half of any two units, and that half of any two units is one unit, then it is known that this particular unit is half of these particular two units, without inferring this from the universal proposition. The same applies to all the other propositions. Likewise, with any part, one knows that this particular whole is greater than its part without recourse to the universal proposition. Similarly, whoever conceives of two particular contradictories as contradictories knows that they cannot coexist.
Everyone knows that this particular thing cannot both exist and not exist, just as he knows this of another particular thing. This does not require an inference from the proposition that nothing can both exist and not exist. The same applies to contraries: a person knows that this thing cannot be both black and white, or both moving and at rest, just as he knows the same of another thing. To know this, he does not need a universal proposition stating that nothing can be both black and white, or both moving and at rest.
The same applies to every other pair known to be contraries: if one knows that two particular things are contraries, one knows that they cannot coexist. Knowledge of the universal proposition yields knowledge of the major premise, which contains the major term. Yet this does not suffice without knowledge of the minor premise, which contains the minor term. Knowledge of the conclusion—that these two notions are contraries and therefore cannot coexist—is possible without knowledge of