Proposition 2: Two substances having different attributes have nothing in common with one another. (In other words, if two substances differ in nature, then they have nothing in common).
Proposition 3: If things have nothing in common with one another, one of them cannot be the cause of the other.
Proposition 4: Two or more distinct things are distinguished from one another, either by a difference in the attributes [i.e., the natures or essences] of the substances or by a difference in their affections [i.e., their accidental properties].
Proposition 5: In nature, there cannot be two or more substances of the same nature or attribute.
Proposition 6: One substance cannot be produced by another substance.
Proposition 7: It pertains to the nature of a substance to exist.
Proposition 8: Every substance is necessarily infinite.
Proposition 9: The more reality or being each thing has, the more attributes belong to it.
Proposition 10: Each attribute of a substance must be conceived through itself.
Proposition 11: God, or a substance consisting of infinite attributes, each of which expresses eternal and infinite essence, necessarily exists. (The proof of this proposition consists simply in the classic "ontological proof for God's existence". Spinoza writes that "if you deny this, conceive, if you can, that God does not exist. Therefore, by axiom 7 [‘If a thing can be conceived as not existing, its essence does not involve existence’], his essence does not involve existence. But this, by proposition 7, is absurd. Therefore, God necessarily exists, q.e.d.")
Proposition 12: No attribute of a substance can be truly conceived from which it follows that the substance can be divided.
Proposition 13: A substance which is absolutely infinite is indivisible.
Proposition 14: Except God, no substance can be or be conceived.
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