In other posts regarding cosmological arguments and the principle of sufficient reason , I have offered brief overview’s in respect to Gottfried Leibniz’s (1646-1716) cosmological argument. Before an explanation is offered of this argument, I first want to scheme the argument so that our thinking is clear and ready for reference:
- (1) Everything that exists, has an explanation of its existence.
- (2) If the universe has an explanation of its existence, that explanation is God.
- (3) The universe exists.
- (4) Therefore, the universe has an explanation of its existence. (from 1, 3)
- (5) Therefore, the explanation of the existence of the universe is God. (from 2, 4).
This schematization was taken from William Lane Craig (2010) in his presentation of Leibniz’s argument. At first glance, the observer must note that premise (1) can otherwise be understood as The Principle of Sufficient Reason (hereafter, PSR):
This premise is compatible with there being brute facts about the world. What it precludes is that there could exist things which just exist inexplicably. According to (1) there are two kinds of being: necessary beings, which exist of their own nature and so have no external cause of their existence, and contingent beings, whose existence is accounted for by causal factors outside themselves. 
However, looking at premise (2) of the argument, why should we believe that if the universe has an explanation of its existence (if even such an explanation is possible), that explanation is God? For instance, even Craig says that premise (2) “might seem at first blush to be a very bold assertion on the part of the theist” . However, premise (2) is the simple logical equivalence of the atheist responding to Leibniz by saying that on the theistic worldview the universe simply exists as a brute contingent thing. In other words, “[a]theists typically assert that, since there is no God, it is false that everything has an explanation of its existence, for the universe, in this case, just exists inexplicably” .
Thus, consider the two propositions:
- A. If atheism is true, then the universe has no explanation of its existence, atheists are also affirming the logically equivalent claim that
- A’. If the universe has an explanation of its existence, then atheism is not true, that is to say, that God exists. Hence, most atheists are implicitly committed to (2).
A further increase in the plausibility of (2) would be to understand that the universe, by definition, includes all of physical reality . Thus, the cause of the universe must be causally prior to the universe’s existence, therefore transcending space and time. Richard Swinburne at this points notes: “Thus for the theist, explanation stops at what, intuitively, is the most natural kind of stopping place for explanation – the choice of an agent” . This agent, is what most people have traditionally come to identify as “God.”
However, the argument doesn’t stop there. If there is to be an explanation for the existence of the universe, and that explanation is God, we must first understand what is meant by God and his causal relation to the universe. Since, if everything that exists has an explanation of its existence (from premise (1)), then what about God? Here some discernment should be made.
Leibniz distinguishes between necessary causes and external causes. Premise (1) can be restated as:
- (1′) Anything that exists has an explanation of its existence, either in the necessity of its own nature or in an external cause.
God, being a necessary cause of the universe, can be defined as:
The claim that there is a necessary being is the claim that there is a being whose nature entails existence, so that any possible world would involve the existence of such an entity. Such a being, we might say, is absolutely invulnerable to nonexistence. 
Another understanding of God as necessary can be found from Swinburne (2004): “To say that ‘God exists’ is necessary is, I believe, to say that the existence of God is a brute fact that is inexplicable – not in the sense that we do not know its explanation, but in the sense that it does not have one” . Thus, God need not have an explanation of his existence because God exists by the necessity of his own nature.
To conclude the argument, premise (3) is obviously true and therefore premises (4) and (5) follow.
-  You can see my post A Presentation of the Cosmological Arguments and The Relationship Between Existence and Explanation by clicking these links.
-  Alexander Pruss, “The Leibnizian Cosmological Argument,” in The Blackwell Companion to Natural Theology, ed. J.P. Moreland and William Lane Craig (Wiley-Blackwell: 2009) p. 24
-  Quoted from The Mystery of Existence, ed. John Leslie and Robert Lawrence Kuhn (Wiley-Blackwell: 2013) pp. 155-156
-  Ibid., p. 156
-  Ibid., pp. 156-157
-  Lee Smolin (2002), a theoretical physicist, comments in his book that the universe, “by definition, is a closed system.” Three Roads to Quantum Gravity (Basic Books: 2002) p. 17
-  Richard Swinburne in The Existence of God (2004), quoted from Leslie and Kuhn (2013), p. 152
-  Timothy O’Connor in Theism and Ultimate Explanation (2009), quoted from Leslie and Kuhn (2013) p. 153
-  Swinburne (2004), quoted from Leslie and Kuhn (2013) p. 149