(_c_) Thirdly, essences considered apart from their actual existence, and compared with their own constitutive factors or with one another, reveal to the mind relations which the mind sees to be _necessary_, and which it formulates for itself in _necessary_ judgments,—judgments _in materia necessaria_. By virtue of the principle of identity an abstract essence is _necessarily_ what it is, what the mind conceives it to be, what the mind conceives as its definition. Man, as an object of thought, is _necessarily_ a rational animal, whether he actually exists or not. And if he is thought of as existing, he cannot at the same time be thought of as non-existing,—by the principle of contradiction. An existing man is necessarily an existing man,—by the principle of identity. These logical principles are rooted in the nature of reality, whether actual or possible, considered as an _object of thought_. There is thus a necessary relation between any complex object of thought and each of the constituent factors into which the mind can analyse it. And, similarly, there is a necessary negative relation—a relation of exclusion—between any object of thought and anything which the mind sees to be incompatible with that object as a whole, or with any of its constituent factors. Again, the mind sees necessary relations between abstract essences compared with one another. Five and seven are _necessarily_ twelve. Whatever begins to exist actually _must_ have a cause. Contingent being, if such exists, is _necessarily_ dependent for its existence on some other actually existing being. If potential being is actualized it _must_ be actualized by actual being. The three interior angles of a triangle are _necessarily_ equal to two right angles. And so on.