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Spinoza's proof for Proposition 2 of part I of the Ethics, "Two substances whose attributes are different have nothing in common", is confusing. This is the proof:

1, prop 2, demo - Also evident from Def. iii. For each must exist in itself, and be conceived through itself; in other words, the conception of one does not imply the conception of the other.

The proof looks like Spinoza simply proves the following proposition 2*: "Two substances have nothing in common", as the proof only refers to the definition of substance and does not refer to Def. iv, which is the definition of attribute. So it must be evident for Spinoza, that proposition 2 and proposition 2* are equivalent. This equivalence seems to involve that it goes without saying that two substances have different attributes. Nonetheless, Spinoza delivers a proof for this conclusion in proposition 5: "There cannot exist in the universe two or more substances having the same nature or attribute." I conclude that either proposition 2 and 2* are not equivalent or the proof for proposition 5 should better be located before proposition 2, because 2 seems to effectively make use of 5. Furthermore, it seems strange to put forth some proposition as evident in one place (i.e. that substances can only have different attributes in 2) and delivering a proof for the same proposition in another place (i.e. in 5), indicating by this that the proposition is not that evident at all. How can one explain these difficulties of proposition 2?

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We have to take into account the substance-attribute-mode basic ontology and the way they interact with intellect.

Substance (Def.III) is "that which is in itself, and is conceived through itself: in other words, that of which a conception can be formed independently of any other conception."

The intellect can conceive the substance (e.g. mind) without any reference to other ideas (e.g. matter).

Compare with Descartes:

Descartes states in the Principles of Philosophy that attributes are the essence of a thing, so the essence of mind is thought or thinking, and the essence of body is to be extended (Principles, I, §53).

An attribute (Def.IV) is "that which the intellect perceives as constituting the essence of substance" (thinking is the attribute of the substance mind): thus, substance is identified by essence (this is pretty traditional) that coincides with attribute.

Essences of different substances (if any) must be different (obvious, in the traditional ontology); but if essence=attribute, there are no common attribute to different substances.

But the interpretative issues are many... See Spinoza’s Theory of Attributes.

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  • So the argument goes like this: Consider, hypothetically, two different substances. As substance is identical with its attribute, two substances need to have different attributes (because otherwise both substances/attributes would be identical). But as attributes are the essence of substance and two things with a different essence have nothing in common, two substances of different attributes have nothing in common. – Moritz Wolff Feb 3 at 14:33
  • @MoritzWolff - IMO, yes. – Mauro ALLEGRANZA Feb 3 at 14:41
  • Would you also agree that Spinoza sees 2 and 2* as equivalent? And that he assumes that this equivalence is obvious for his contemporaries? This would explain why he simply contents himself with the very simple proof of 2*. – Moritz Wolff Feb 3 at 14:44
  • Maybe :-) See Substance: Descartes, Spinoza and Leibniz as well as Spinoza’s Theory of Attributes: "Attributes sit at the very heart of Spinoza’s metaphysics." – Mauro ALLEGRANZA Feb 3 at 14:56
  • The problem is that 2* is not obvious at all - at least for disciples of Descartes. For Descartes it goes without saying that two substances can have the same attribute, so for him they can and usually do have something in common. On the other hand, a Cartesian could have accepted 2. So either Spinoza could not assume that 2 and 2* would be seen as equivalent in the eyes of his readers or Spinoza already in Proposition 2 takes for granted the strong assumption that no two substances can have the same attribute. Both interpretive options seem to me unpromising. – Moritz Wolff Feb 3 at 15:04

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