It is sometimes objected to cosmological arguments for the existence of God that the conclusion drawn from them, even if all the premises are accepted, are too strong insofar as the conclusions involve the use of a definite article before the noun ‘God’ instead of an indefinite article. In other words, having argued one’s way to an intelligent, timeless, spaceless, immaterial, personal creator of the universe, the claim ought to be that there is at least one thing x such that x fits the above description; (∃x)(Ix & ((Tx & (Sx & IMx)) & (Px & Cx)). That does not tell you, however, that there is only one such thing. Perhaps there are many things fitting the above descriptions (assuming that more than one thing can cause the universe to exist). Moreover, even when we enrich this notion of ‘God’ by appending to our cosmological argument(s) other arguments, such as the Ontological argument, in order to build a more complete bridge from what is above described to the rich philosophical notion of ‘God’, some have maintained the same criticism, saying that all this only proves (at best) that there is a maximally great being (meaning that there is at least one), but it does not prove that ‘the‘ maximally great being exists. Just because there is some being whose essence involves existence, such that it is per se necessarium, does not entail that there is no more than one being whose essence involves existence such that it is per se necessarium.
Two responses can be given which come to my mind. The first from Bonaventure, and the second from Leibniz. First, from Bonaventure, I will quote from his Magnum Opus, his commentary on the sentences of the master Peter Lombard, at some length.
On the unity of the Divine Essence and the Plurality of the Persons.
Whether there is only one God.
ABOUT THE FIRST, that in God there is to be posited a unity of essence or nature, it seems (that it is so) by demonstrative reason [ratione ostensiva], having considered the threefold supposition, which is proper to be posited, both on account of His simplicity, and on account of a common conception of spirit, which is, that God is the one greater than whom (nothing) can be thought, and by reason of (His) status, which is not but in the highest and first (position).
(1.) The first supposition is, that God is the most simple. From this it argued, that since a thing [aliquid] can communicate with no thing diverse (from itself), because, if it does communicate, and it differs: therefore (it does so) not according to the same (respect); therefore there is a composition (of being). If it can communicate nothing, therefore (it can communicate) neither deity nor entity; therefore if there are two gods, since one is Being [ens], the other is not Being [non est ens], if one is God, the other is not God: therefore if there are two gods, there are not two gods.
2. Likewise, the second supposition is, that God is, because God is the most Omnipotent. From this it is argued: therefore He will be able to bring it about [poterit facere], that every power other than His own can (do) nothing: therefore if there are two gods diverse in nature, one of the two can bring it about, that the other can (do) nothing, and conversely. But the one whose power can be borne away, is not God: therefore if there are two gods, none is God.
3. Likewise, the third supposition is, that God is simply the Most High (Being) [summum]. From this it is argued: therefore all things are below Him: therefore all others (are) from Him and for Him. If therefore there are two gods, one is below the other, and conversely; one is from the other according to a diverse nature, and conversely; one is for the other, and conversely; but what is below the other in nature and from the other and for the other, is not God: ergo etc…
4. Likewise, this can be proved through deduction to the impossible. If there are two gods, either one is where the other (is), or (it is) not. If one (is) where the other (is): therefore one is in the other, since they are by the same manner of being [eodem modo essendi]: therefore one is material to the other: therefore the other is not God. If one is not where the other (is): therefore each are limited, therefore neither (is) infinite.
5. Likewise, if there are very many good gods, either one understands the other, or (it does) not. If not; therefore each is ignorant. If it does understand; therefore either (it does so) through a prior-sensing[praesentiam] or through appearance [species], or through its very self as through its exemplar. If through a prior-sensing: therefore one (is) in the other, therefore God is inserted into [illabitur] God and perfects God; if through appearance: therefore (there is) a composition [compositus]; if through an exemplar: therefore one is the exemplar of the other, therefore (it is) also (its) principle.
6. Likewise, if there are two diverse gods, of which each is the Most High Good; either one loves [diligit] the other, as it is to be loved, or not. If so, since each is the Most High Good, each is to be loved with the love of enjoying: therefore each enjoys the other; but what enjoys a good other than itself, is in need of it [indiget]: therefore one is in need of the other: therefore each is indigent, therefore neither (is) God.
That there are more gods is impossible, nay rather if it is rightly understood, what God is, it is not (even) intelligible.
It must be said, that it is impossible that there be more gods, and if the thing signified by this name God be rightly accepted, it is not only impossible, but also not intelligible: for God means simply the Most High (Being) both in reality [in re] and in the opinion of the one thinking. Because (He is such) in reality, for that reason all things (are) from Him and in Him and for Him, and in Him entirely is (their) stability [status]; for that reason it is impossible to understand, without violation of the intellect, that anything makes another equal to itself [sibi parificetur] from itself. Likewise, nothing greater than God can be thought nor even (anything) equal, because (He is) the Most High (Being) in the opinion (of rational creatures). For that reason it is impossible and not intelligible to posit more gods.
If one should think the previous demonstration(s) unsatisfying, either through a failure to understand it, or through some perceived flaw in it, then perhaps one can turn to Leibniz. Leibniz maintained something called the Principle of the Identity of Indiscernibles, which stipulated basically that if two things (substances) bear all the same predicates in common, then they are indiscernible, and thus identical (meaning here that they simply are the same thing, such that to speak of two different things becomes semantically bankrupt). There is no ground for cognizing them as two different things if by each we understand simply the very same thing, and thus by either we contemplate both. Notice that the Principle of the identify of indiscernibles is stronger than the indiscernibility of identicals (which is a different principle, a little more trivial, which says basically that if two things are identical then we cannot distinguish them). The Principle of the Identity of Indiscernibles says that where two things are indiscernible (i.e., bear all and only the same predicates), they are in fact identical (i.e., they are in fact the very same thing). Thus, if one wishes to say that one can establish that ‘a‘ maximally great being exists, but not that only one such being with all of the great-making predicates exists, then the Monotheist can simply charge the polytheist (or more likely the agnostic/atheistic interlocutor whose appeal to polytheism is merely rhetorical) with incoherence. If two things are exhaustively described with all and only the same predicates born by ‘another’, then it seems that we are speaking about one single thing and confusing it for two or more such things.
Finally, in the case where all the above arguments are thought to have failed, one could, I suppose, simply argue that Mono-theism is more parsimonious than any form of polytheism. That, however, would only demonstrate that we ought to prefer monotheism, and not that monotheism has been demonstrated by probative arguments.