It depends.
Suppose the true God is the triune God. Then two possibilities. One is that Muslims worship the true God, but not as triune, indeed as non-triune; they worship the true God all right, the same one the Christians worship; it is just that the Muslims have one or more false beliefs about the true God. The other possibility is that Muslims do not worship the true God; they worship a nonexistent God, an idol. We are assuming the truth of monotheism: there is a God, but only one.
Now worship entails reference in the following sense: Necessarily, if I worship the true God, then I successfully refer to the true God. (The converse does not hold). So either (A) the (normative) Muslim successfully refers to the true God under one or more false descriptions, or else (B) he does not successfully refer to the true God at all.
Now which is it, (A) or (B)?
The answer depends on your theory of reference.
Consider this 'Kripkean' scenario. God presents himself to Abraham in person. All of Abraham's experiences on this marvellous occasion are veridical. Abraham 'baptizes God' with the name Yahweh or YHWH. The same name (though in different transliterations and translations) is passed on to people who use it with the intention of preserving the direct reference the name got when Abraham first baptized God with it. The name passes down eventually to Christians and Muslims. Of course the conceptions of God are different for Abraham, St. Paul, and Muhammad. To mention one striking difference: for Paul God became man in Jesus of Nazareth; not so for Muhammad, for whom such a thing is impossible.
If you accept a broadly Millian-Kripkean theory of reference, then it is reasonable to hold that (A) is true. For if the reference of 'God' is determined by an initial baptism or tagging and a causal chain of name transmission, then the reference of 'God' will remain the same even under rather wild variation in the concept of God. The Christian concept includes triunity; the Muslim conception excludes it. That is a radical difference in the conceptions. And yet this radical difference is consistent with sameness of referent. This is because the reference is not routed though the conception: it is not determined by the conception. The reference is determined by the initial tagging and the subsequent name transmission.
Now consider a 'Fressellian' scenario. The meaning of a proper name is not exhausted by its reference. Names are more than Millian tags. It is not just that proper names have senses: they have reference-determining senses. On a descriptivist or 'Fressellian' semantics, a thoughtful tokening by a person P of a proper name N successfully refers to an individual x just in case there exists an x such that x uniquely satisfies the definite descriptions associated with N by P and the members of his linguistic community.
So when a Christian assertively utters a token of 'God is almighty,' his use of 'God' successfully refers to God only if there is something that satisfies the sense the Christian qua Christian associates with 'God.' Now that sense must include being triune. The same goes for the Muslim except that the sense that must be satisfied for the Muslim reference to be successful must include being non-triune.
It should now be clear that, despite the considerable overlap in the Christian and Muslim conceptions of God, they cannot be referring to the same being on the 'Fressellian' theory of reference. For on this theory, sense determines reference, and no one thing can satisfy two senses one of which includes while the other excludes being triune. So we have to conclude, given the assumption of monotheism, that the Christian and Muslim do not refer to one and the same God. Given that the true God is triune, the Christian succeeds in referring to the true God while the Muslim fails. The Muslim does not succeed in referring to anything.
So I continue to maintain that whether Christians and Muslims worship the same God depends on one's theory of reference. This is why the question has no easy answer.
Those who simple-mindedly insist that Christians and Muslims worship numerically the same God are uncritically presupposing a dubious Millian-Kripkean theory of reference.
Exercise for the reader: explain what is wrong with Juan Cole's article below.
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