Leibniz cosmological argument
The Leibniz cosmological argument is a variant of the cosmological argument proposed by Gottfried Leibniz. It is lesser known than the Kalam version. It is based on the principle that things that exist must have an explanation and that explanation is ultimately God.
The argument is as follows:
- Everything which exists has a sufficient explanation of its existence (the principle of sufficient reason)
- If the universe had an explanation of its existence, then the explanation must be God
- The universe exists
- The universe has an explanation of its existence
- Therefore, God exists
- "A sufficient explanation is an explanation that provides a sufficient condition for the fact it explains. In other words, a sufficient explanation is sufficient to rule out other possibilities given that the condition holds. "
What is an explanation?
- Main Article: Explanation
An explanation is a set of statements constructed to describe a set of facts which clarifies the causes, context, and consequences of those facts in terms of known phenomena. God is an unknown entity so does not explain anything at all.
Contingent brute facts
- Main Article: Contingent brute facts
Most scientists and philosophers generally presume that things have sufficient explanation. The assumption has practical merits: abandoning it would potentially stop investigations into phenomena that might eventually be explained. However, this presumption that an explanation must exist does not necessarily reflect reality. It may be that some facts do not have explanations at all and it is difficult to eliminate this possibility by evidence.
A brute fact does not rely on more fundamental facts: it just is. "We normally cannot give a full account why the [brute] fact should be what it is, but must accept it without explanation."  If contingent brute facts (also known as "brute contingencies") exist, this would be a strong argument against PSR because contingent facts are not fully explained by existing "of their own nature". If brute facts exist and are contingent, PSR is false. A physical laws and initial conditions of the universe may be a contingent (or not) brute fact of our universe.
The existence of brute facts cannot be demonstrated with certainty without incurring an argument from ignorance. However, the possibility of contingent brute facts seems plausible.
William Lane Craig argued that brute facts would be compatible with the PSR but "inexplicably facts" are distinct and would be incompatible with PSR. He claims "brute facts" relate to necessary beings that "exist of their own nature".  He uses a variant of the first premise, in a similar fashion to his formulation of Kalam, to avoid objections:
- "Anything that exists has an explanation of its existence, either in the necessity of its own nature or in an external cause"
This addresses non-contingent brute facts, which necessarily exist. Arguably, a fact about something that "exist of their own nature" has a (begging the question) explanation in its nature and is therefore not a brute fact at all. This is a long winded way of denying the existence of contingent brute facts by implicit assumption. However, it does not really address the objection.
Principle of sufficient reason rules out contingent facts
The possibility of the universe being other than it is seems to be ruled out by PSR: 
- Everything which exists has a sufficient explanation of its existence (PSR)
- Different possible worlds may or may not exist
- One actual world exists and its existence may or may not be contingent
- The actual world has a reason (S) for its existence
- S must be true for the actual world. S is false for any other possible non-existent world.
- The explanation of S rules out the possibility of other possible worlds
- Therefore the actual world is necessary (and not contingent).
- Everything which exists has a sufficient explanation of its existence (PSR)
- Consider all contingent facts A, B, C, etc.
- BCCF (Big Contingent Conjunctive Fact) is the conjugation of all contingent facts.
- BCCF has explanation which is contingent or necessary
- If the explanation of BCCF is contingent, it would be included in BCCF
- If the explanation of BCCF is non-contingent and sufficient, than BCCF would also be necessary
- There are no contingent facts
This hard determinism is sometimes considered an unacceptable or absurd conclusion. If so, PSR is therefore considered to be false.
Did God have freedom to create the universe "as is", a different universe or nothing at all? If there are no contingent facts, God was not free to choose to create a different universe or to not create the universe.
- Main Article: Infinite regress does not occur
One may ask "what is the explanation for God?" and suggest this argument is prone to infinite regress. The assumption in the premise "If the universe has an explanation of its existence, that explanation is God." is arbitrarily assuming there is no infinite regress. The second premise contradicts the first premise that everything has an explanation. Apologists usually respond saying God's existence is explained by his necessarily existent nature: 
- "The explanation of God's existence lies in the necessity of His own nature."
A related problem is the Münchhausen trilemma which observes that if explanations require further explanations, this leads to infinite regress, axiomatic assumptions or circular arguments.
Assuming God is the explanation
- Main Article: Which God?
The argument has a premise that if there is an explanation, it is God. This assumes metaphysical beings can be explanations and there is no other possible non-divine necessary causes. The argument should be specified by recourse to an unspecified first cause. Calling this unspecified first cause "God" is not properly established.
No particular religion or specific God is suggested by the argument.
A poor explanation
- Main Article: Ultimate 747 gambit
The argument is grounded on providing explanations for entities. However, God is arguably an invalid explanation since it is more mysterious than the mystery it seeks to explain!
Modified principle of sufficient reason
Much of the criticism has focused on the first premise of the principle of sufficient reason (PSR). In response, alternative arguments have been proposed with weaker variants of PSR in an attempt to avoid these objections.
- "PSR3: There is a sufficient reason why some concrete objects exist rather than none at all."
Based on this premise, argument is briefly: some objects exist and their existence (rather than their non existence) has a sufficient explanation, God is that explanation. While this PSR might support the argument, it is difficult to establish if it is true. It also allows for some contingent facts to exist, although it still implies that the existence of some of the universe was non-contingent.
- "PSR3’: Possibly, there is a sufficient reason why some contingent concrete objects exist as rather than none at all."
This states that W-PSR is true in some possible world but not necessarily in the actual world. From this statement of WW-PSR, they derive the original W-PSR. As before, it is difficult to establish if this premise is true.
Crafting argument to avoid objections
The premises are specifically crafted to avoid previous objections while retaining the conclusion. The argument might be accused of begging the question. Richard M. Gale and Alexander R. Pruss reject this saying: 
- "Many atheists would be willing to grant W-PSR before we gave our argument, but once they see what follows from it in conjunction with some other seemingly innocent premises, they will no longer grant it to us and will charge it with begging the question. This move looks dogmatic, unless they can muster some grounds for doubting W-PSR. It appears as if they are dogmatically committed to rejecting any deductive theistic argument by rejecting some one of its premises."
This shifts the burden of proof or disproof on to critics. Since the PSR premise is disputed, it is up to the claimant to demonstrate its validity. Any version of PSR is difficult to establish by evidence since we would need direct experience of the cause of the universe and the origin of physical laws.
- ↑ 1.0 1.1 1.2 Tom Senor, An Argument Against the Principle of Sufficient Reason, retrieved 22 Apr 2014 
- ↑ The Blackwell Dictionary of Western Philosophy
- ↑ W. L. Craig, Reasonable Faith, 3rd edn. (Wheaton: 2006) see pp. 106-111
- ↑ William Lane Craig, Leibniz’s Cosmological Argument and the PSR 
- ↑ William Lane Craig, On Guard: Defending Your Faith with Reason and Precision
- ↑ 6.0 6.1 De Weese and Rasmussen, chapter Hume and the Kalam Cosmological Argument, In Defense of Natural Theology: A Post-Humean Assessment, 2005
- ↑ 7.0 7.1 Alexander Pruss, May 20, 2011 The Weak Weak Principle of Sufficient Reason 
- ↑ 8.0 8.1 Richard M. Gale and Alexander R. Pruss, A New Cosmological Argument, Religious Studies 35 (1999) 461–476