An infinite substance exists and exists alone

To preface this argument: Q.E.D. The first thing to be established is the necessity of causes. In any case wherein two contradictory states of affairs, x and not-x, are considered, the validity of one must preclude the validity of the other, as a result of the Law of Noncontradiction. As such, it cannot be said that both x and not-x are possible, since, if one is able to be actualized (the definition of possibility - something is never called possible when it cannot be actualized), and if that actualization conflicts with the actualization of another (in the sense that the actualization of the one makes it so that the other could never be simultaneously actualized), then it follows that, insofar as one is possible, caeteris paribus, the other is impossible. Imagine the absurdity which would result from the denial of this principle: both x and not-x could actualize, and thus a contradiction would be possible. If it is said that both do not have to actualize for both to be possible, only that both arepotentially able to actualize up until the point where one does, then the question has to be asked: what reason would there be for one of the two potentialities to actualize instead of the other? Consider x and not-x. Either these are equally possible, in which case there is no quality which makes one more conducive to actualization than the other, or there is, in fact, such a quality. In the first case, it is clear that the actualization of either would be contradictory to reason, for two futures which are identical in all qualities relevant to actualization must therefore be identically actualized - neither can be considered as being different than the other, so, if it holds that one has reason to be actualized, the other must as well, leading to a contradiction, and, likewise, if neither has reason to be actualized, then the claim that one will nevertheless actualize is the claim that two identicals can have different properties, which is a contradiction in terms. The only possibility left, then, is that there is some quality present only in one that makes that one actualize while the other does not, and, if this disparity between the two exists, then it could never be said that both could actualize, since actualization will only come to that option which has the quality which leads to it being actualized. Therefore, it is clear that only one of two conflicting "possibilities" is actually possible, and this possibility is necessarily actualized due to some quality it posesses. The quality that leads to this actualization can be thought of as its cause, and, as has been shown, every possibility which has actualized must have a quality in this vein, and therefore every possibility which has actualized must have a cause or reason for its existence (The Principle of Sufficient Reason). The natural extension of this law is that everything which does not exist must have a reason for its nonexistence, and this reason can be conceived as a restriction. Something which does not exist does not exist because it is absent from "logical space" (the field of all possibilities) as a result of the fact that anything which can existmust exist (as realized earlier), and this absense must be a result of the conflict between the thing in question being caused and things which do exist, which can be expressed as there being something which is the case that makes it so that the nonexistent thing cannot be the case, and this restrictor is the "cause of nonexistence" of the nonexistent thing. Q.E.D. Next, let us consider the nature of indistinguishable things. It is clear that, for one to maintain that multiple things are, in fact, distinct, those things must have unique qualities; things which are identical in every way cannot be said to be anything but one singular thing, for otherwise there would be distinctions without any basis for distinction, since there would be no qualities by which one could actually draw these distinctions from. Therefore, two separate things which are identical are inconceivable - they would have to be one and the same. This can be applied to substances. Two substances which share exactly the same attributes can only be said to be synonymous. The implications of this become clear if one considers the existence of a substance with infinite attributes. Such a substance would preclude the separate existence of any other substance, since it would, in itself, contain all conceivable attributes, so no substance outside of itself could have any attribute unique to it alone - all attributes would already be contained in the infinite substance, and, as such, multiple substances are impossible when one is infinite, since none could be distinguished from the infinite substance, as they would all, at the outset, be equal with the corresponding attributes in the infinite set. The definition of a substance is "that of which a conception can be formed independently of any other conception," meaning that the fact that all substances would be subsumed within the one infinite substances makes those supposed substances not substances at all, but rather things solely conceived within the infinite substance, and without the independence substancehood requires. An infinite substance, if it existed, must exist alone. Q.E.D. The only way, then, to support substance pluralism is to claim that an infinite substance does not exist. Quoting Proposition XI: "Of everything whatsoever a cause or reason must be assigned, either for its existence, or for its non-existence--e.g., if a triangle exist, a reason or cause must be granted for its existence; if, on the contrary, it does not exist, a cause must also be granted, which prevents it from existing, or annuls its existence. This reason or cause must either be contained in the nature of the thing in question, or be external to it. For instance, the reason for the non-existence of a square circle is indicated in its nature, namely, because it would involve a contradiction. On the other hand, the existence of substance follows also solely from its nature, inasmuch as its nature involves existence." It is clear that the definition of substance is not, in itself, self-contradictory, since, if this was so, no substance at all could exist, and this is absurd, since, as we must admit, "Cogito ergo sum" - something must exist in the first place. The only thing added to the broad definition of substance in this case is an infinitude of attributes, which is not contradictory in itself, for such an infinitude would preclude the existence of anything else, making it so that, for the cause of nonexistence to be the nature of the infinite substance, the infinite substance would not be nonexistent if not for itself (i.e. if not for its own existence negating itself), making it so that one would have to admit that it exists insofar as it negates itself, and, if it exists even to this miniscule degree, then, to this degree, nothing else exists besides it, so such an internal contradiction would, in effect, necessitate the nonexistence of reality as a whole. The nature of a thing can only preclude its existence if there is something outside of itself which it would contradict, since, if a wholly internal contradiction would be enough to stop a thing from existing, then there would be no restriction on its existence until it is already supposed to be existing. It is clear that, since the nature of an infinite substance makes it so that it must exist alone, any contradictions caused by its nature can only be caused by its contradicting itself and not anything outside of itself, and this state of contradiction would only exist if the substance itself existed. So much for a reason for nonexistence being drawn from within the substance. One may, then, try to find such a reason from without. As has already been made plain, however, there would be no "without" in relation to an infinite substance, so there would be nothing outside of itself to restrict its existence. Even without this fact, the nature of a substance is to be completely independent, meaning that two subtances, even if they coexisted, could not have causal influence on each other, since such an influence would put one in the position of being dependent on the other, rendering that thing decidedly not a substance. From these arguments, it follows, in fine, that there can be no reason for the nonexistence of an infinite substance, and, therefore, an infinite substance must exist, and, given the nature of an infinite substance, it must exist alone. Q.E.D.