  We needn’t suppose that God’s second-order properties are identical with his first-order properties to mount an argument from simplicity, however. Suppose that God is simple in the sense that, for any two of his first-order properties, P and Q, either P is identical with Q or P is logically equivalent to Q (that is, it is impossible for him to possess P without possessing Q, and vice versa). Let us also suppose that there are two gods. If both are God, then both possess the first-order properties essential to divinity. Call these D. If the two differ, each possesses at least one first-order property which the other lacks. Suppose, for example, that the first possesses a first-order property, H, and the second possesses its complement, non-H. Since each is God, each is simple. Hence, either H is identical with D and non-H is identical with D, or H is logically equivalent to D and non-H is logically equivalent to D. Therefore, either H is identical with non-H or H is logically equivalent to non-H. But this is incoherent and, even if it were not, the possession of H and non-H could not be used to distinguish the two, since either H and non-H are the same property or H and non-H are logically equivalent properties. It therefore seems that if God is simple, there can’t be two gods.