Francis Beckwith mentions the Kalam Cosmological Argument in his latest The Catholic Thing article (7 January 2106):
1. Whatever begins to exist has a cause.
2. The universe began to exist.
3. Therefore, the universe has a cause of its existence.
Suppose that a Muslim and Christian come to believe that God exists on the basis of this Kalam argument and such ancillary philosophical arguments and considerations as are necessary to establish that the cause of the universe is uncreated, transcendent of the universe, unchanging, etc. The result is a conception of God achieved by reason without the aid of divine revelation. It is a conception common to the normative Muslim and the normative Christian. Crucial differences emerge when the core conception is fleshed out in competing ways by the competing (putative) revelations. But if we stick with the core philosophical conception, then all should agree that there is important overlap as between the Christian and Muslim God conceptions. The overlap is achieved by abstraction from the differences.
So far so good.
Beckwith then asks whether the Muslim and the Christian "believe in the same God" and he concludes that they do.
Permit me a quibble. 'Believe in' connotes 'trust in, have faith in, rely upon the utterances of,' and so on. I believe in my wife: I trust her, I am convinced of her fidelity. That goes well beyond believing that she exists. If I believe in a person, it follows that I believe that the person exists. But if I believe that a person exists, it does not follow that I believe in the person. Professor Beckwith is of course aware of this distinction.
At best, then, what the Christian and the Muslim are brought to by the Kalam argument and supplementary considerations is not belief in God, but belief that God exists. To be even more precise, the Kalam argument, at best, brings us to the belief that there exists a unique, transcendent, uncreated (etc.) cause of the beginning of the universe. In other words, both Christian and Muslim are brought to the belief and perhaps even the knowledge that a certain definite description is satisfied. The properties mentioned in this description are what constitute the shared philosophical understanding of 'God' by the Muslim and the Christian. At best, philosophy brings us to knowledge of God by description, not a knowledge by acquaintance. The common description is usefully thought of as a 'job description' inasmuch as God in brought in to do a certain explanatory job, that of explaining the beginning of the universe. As my teacher J. N. Findlay once said, "God has his uses."
But note that this common Christian-Muslim description leaves undetermined many properties an existent God must possess. (And it must be so given the finitude of our discursive, ectypal, intellects.) But in reality, outside the mind and outside language, God, like everything else, is completely determinate, or complete, for short. I am assuming the following existence entails completeness principle of general metaphysics (metaphysica generalis).
EX -->COMP: Necessarily, for any existent x, and for any non-intentional property P, either x instantiates P or x instantiates the complement of P.
What the principle states is that every real item, everything that exists, satisfies the property version of the Law of Excluded Middle. It rules out of reality incomplete objects. For example, God in reality is either triune or non-triune. He cannot be neither, any more than I can be neither a blogger nor not a blogger. The definite description(s) by means of which we have knowledge by description of God, however, are NECESSARILY (due to the finitude of our intellects) such that there are properties of God in reality that these descriptions do not mention. This is of course true of knowledge by description of everything. Everything is such that no description manageable by a finite mind makes mention of all of the thing's properties, intrinsic and relational.
Now suppose that Christianity is true and that God in reality is triune. Then the above common definite description is satisfied. The common Muslim-Christian conception is instantiated -- but it is instantiated by the Christian God which of course must exist to instantiate it.
The Christian and the Muslim both believe that God (understood as the unique uncreated creator of the universe) exists. That is: they believe that the common conception of God is instantiated, that the common definite description is satisfied. They furthermore believe that the common conception is uniquely instantiated and that the common description is uniquely satisfied. But they differ as to whether the instantiator/satisfier is the triune God or the non-triune God.
So we can answer our question as follows. The question, recall, is: Do Christians and Muslims believe in the same God?
Muslims and Christians believe in the same God, as Beckwith claims, in the following precise sense: they believe that the same God exists, which is to say: they believe that the common philosophical God concept is uniquely instantiated, instantiated by exactly one being. Call this the anemic sense of believing in the same God.
But this is consistent with saying that Muslim and Christian do not believe in the same God in the following precise sense: they don't believe that the wholly determinate being in reality that instantiates the common philosophical God concept is the triune God who sent his only begotten Son, etc. Call this the robust sense of believing in the same God.
Now we robustos will naturally go with the robust sense. So, to give a plain answer: Christians and Muslims do not believe in the same God. If Christianity is true, the Muslim God simply does not exist, and Muslims believe in an idol.
The mistake that some are making here is to suppose that the shared Muslim-Christian philosophical understanding enscapsulated in the common concept suffices to show that in reality one and the same God is believed in, and successfully referred to, and non-idolatrously worshipped by both Muslims and Christians. Not so!
The real (extramental, extralinguistic) existence of God cannot be identified with or reduced to the being instantiated of a concept that includes only some of the divine determinations (properties). 'Is instantiated' is a second-level predicate, but God exists in the first-level way. Equivalently, God is not identical to an instance of one of our concepts. God is transcendent of all our concepts. So if we know by revelation that God is a Trinity, then we know that the Muslim God, the non-triune God, does not exist.