Does the cosmological argument "prove that anything that exists has a cause of its existence", or is that just a premise of the argument?
There is no such thing as the cosmological argument. Rather, there is a class of arguments that share similar themes and (sometimes) logical structure, but that rely on different premises that are all referred to as cosmological arguments. For instance, a cosmological argument defended by Aquinas depends on the impossibility of an essentially ordered infinite regress. On the other hand, the Leibnizian cosmological argument depends mostly on the Principle of Sufficient Reason.
Given the wording of your question, you are most likely referring to the kalam cosmological argument, which has a premise that is similar to, but importantly different from, what you stated. The argument is usually formulated as follows:
- Whatever begins to exist has a cause.
- The universe began to exist.
- Therefore, the universe has a cause.
So, instead of "anything that exists has a cause of its existence," the premise is closer in meaning to "anything that begins to exist has a cause." The kalam cosmological argument does not even attempt to prove that "everything that begins to exist has a cause," but assumes this as a premise. This premise is often defended as self-evident or as following from a commonly accepted proposition (such as the Causal Principle), but these defenses are not part of the kalam cosmological argument per se.
If I understand your question correctly, then the answer is no. The cosmological argument takes as its premise that anything that exists must have a cause either in fieri (like me causing these words to appear by typing them) or in esse (like the words showing up on my computer screen is caused in part by the electricity supplied to it and will go away when the electricity does). The conclusion is that there must, therefore at least have been a God in order for the cosmos to exist. The fact that all things have a cause was derived by induction from our experience that such a law seems to be in place.
In fact arguments against the cosmological argument come in part directly from the fact that this is simply presumed to be true (such as Russell's counter argument which cites the Problem of Induction in that we cannot deduce that all things have a cause from the fact that all things we have experienced have a cause).
OP: Does the cosmological argument "prove that anything that exists has a cause of its existence", or is that just a premise of the argument?
There is no need to prove that anything that exists has a cause : the existence of a thing is the rationalisation of how it appears. Rationalisation is the process of making sense of causes, so for example, the coffee cup before me exists because I acquired it (after someone made it) to drink coffee and it is right here because I am drinking coffee now.
In line with the trait of modern thinking that moves in the realm of Reason, Kant also thinks within and in terms of the dimension of Reason. As the faculty of fundamental principles, reason generally is the faculty of representing something as something. "I place something as something in front of myself" is the stricter formulation of the ego cogito of Descartes, of the "I think." Therefore the dimension of a critique, of theoretical, practical, and technical Reason is the I-ness of the I: the subjectivity of the subject. It is in relation to the I as subject that beings, placed before the I in representation, have the character of an Object for a subject. (Heidegger, The Principle of Reason, p. 77)
The principle of sufficient reason relates directly to existence as the discrimination of objects beyond raw sensation and undifferentiated matter. The latter are beyond grasp.
The term “Principle of Sufficient Reason [principe de raison suffisante/principium reddendae rationis]” was coined by Leibniz, though Spinoza is thought by many scholars to have preceded Leibniz in appreciating the importance of the Principle and placing it at the center of his philosophical system. ...[in] the 1663 geometrical exposition of Descartes’ Principles of Philosophy (which may or may not reflect Spinoza’s own views at the time). The eleventh axiom of Part I of the book states:
Nothing exists of which it cannot be asked, what is the cause (or reason) [causa (sive ratio)], why it exists.
– quoted from § 1. & 2. SEP - Principle of Sufficient Reason