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Spinoza's style in the Ethics is very different from anything else you will read this quarter. It is very hard to get very much from the reading unless you think through some the proofs with some care. If you just try to read this text from beginning to end, you will get very little from it. So to help you think through the proofs, you will be required to reproduce a few of them according to the following rules.

Think of the proof of a proposition as the argument for the proposition. Every proposition needs an argument, but the definitions and axioms do not need arguments. (Why?) We will think of these arguments as having levels. The first level of an argument is the level expressed in the proof for the proposition, and if the proof appeals only to definitions and axioms, then there is no deeper level of the argument. If however the proof for the proposition appeals to another proposition, then there will be another level, (since all propositions need to be proven) and we'll call that next level--level 2. If the proof at level 2 appeals to any propositions, then there will be a level 3 to the proof. You will never be expected to provide a proof for a proposition to a level deeper than level 3.

The best way to represent the different levels of a proof is by indenting the different premises and intermediate conclusions to signal which level each one pertains to.

Here is an example of the proof of Proposition V. The black circle is level one, the white circle is level two.

Prop. V: There cannot exist in the universe two or more substances having the same nature or attribute.

• If there were two or more distinct substances, they would have to be distinguished from one another by difference of attributes or difference of modifications. Prop. IV
  • Everything which exists, exists either in itself or in something else. Axiom I
  • There is nothing outside of the intellect but substances and their modifications. Defs. III and V
  • An attribute is that which the intellect perceives of substance as constituting its essence. Def. IV
  • So, there is nothing outside the intellect by which a number of things can be distinguished one from another but substance, or, in other words, their attributes and their modifications

• If only by the difference of their attributes, it will be granted that there cannot be more than one with an identical attribute.
• So there cannot be two or more distinct substances having the same attribute that are distinguished from one another by different attributes.
• Substance is prior by nature to its modifications. Prop. I
  • Since a modification is that which is in another thing through which also it is conceived. Def. V, and a substance does not need the conception of another thing Def. III, then substance is prior by nature to its modifications.
• Since substance is prior by nature to its modifications, two substances cannot be distinguished by a difference in modifications.

So there cannot be two or more substances possessing the same nature or attribute.