PROPOSITIONS. I. Substance is by nature prior to its modifications. >>>>>Proof--This is clear from Deff. iii. and v. II. Two substances, whose attributes are different, have nothing in common. >>>>>Proof--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. III. Things which have nothing in common cannot be one the cause of the other. >>>>>Proof--If they have nothing in common, it follows that one cannot be apprehended by means of the other (Ax. v.), and, therefore, one cannot be the cause of the other (Ax. iv.). Q.E.D. IV. Two or more distinct things are distinguished one from the other, either by the difference of the attributes of the substances, or by the difference of their modifications. >>>>>Proof--Everything which exists, exists either in itself or in something else (Ax. i.),-- that is (by Deff. iii. and v.), nothing is granted in addition to the understanding, except substance and its modifications. Nothing is, therefore, given besides the understanding, by which several things may be distinguished one from the other, except the substances, or, in other words (see Ax. iv.), their attributes and modifications. Q.E.D. V. There cannot exist in the universe two or more substances having the same nature or attribute. >>>>>Proof--If several distinct substances be granted, they must be distinguished one from the other, either by the difference of their attributes, or by the difference of their modifications (Prop. iv.). If only by the difference of their attributes, it will be granted that there cannot be more than one with an identical attribute. If by the difference of their modifications--as substance is naturally prior to its modifications (Prop. i.)--it follows that setting the modifications aside, and considering substance in itself, that is truly, (Deff. iii and vi.), there cannot be conceived one substance different from another--that is (by Prop. iv.), there cannot be granted several substances, but one substance only. Q.E.D. VI. One substance cannot be produced by another substance. >>>>>Proof--It is impossible that there should be in the universe two substances with an identical attribute, i.e. which have anything common to them both (Prop ii.), and, therefore (Prop. iii.), one cannot be the cause of the other, neither can one be produced by the other. Q.E.D.