Bill Vallicella has been arguing for the nuanced position that it's at least not obvious that Muslims and Christians worship the same God because it will depend on your theory of reference. I think he holds that taking the object of the worship as the same requires a Millian/Kripkean theory of reference; since I don't accept Millian/Kripkean theories of reference myself, and have often argued rather vehemently against them, I obviously don't think this is the case. My own theory of reference is, in Vallicellan terms, 'Fressellian', but I am entirely on the Same-God side of the argument. (But I may simply be misunderstanding his overall argument, which involves an approach I find rather foreign. In my view, a theory of reference is simply a model for facts about meaning, which are discovered by considering the actual use of language rather than applying theories, and therefore I don't regard facts about reference as so theory-laden as the Maverick apparently does.)
In any case, I think it needs to be noted that there are other assumptions that are in play here, as well. It is not an accident that such a significant portion of people on the Not-Same side are Evangelicals of one stripe or another, or that such a significant portion of people on the Same side are Catholics of one stripe or another. And we see this quite clearly in Bill's most recent post on the subject:
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.
There are a very great many reasons for a Catholic to disagree with this entirely, no matter what his or her theory of reference is. It is Catholic doctrine that the existence of God, the very same God Christians worship, can be proven by reason alone, in at least some broad sense of the word 'proven'. It is Catholic doctrine that this is not true of the doctrine of the Trinity; the doctrine of the Trinity is a mystery exceeding the capacity of natural reason to establish. It is foundational to the Catholic faith to believe that God is a Trinity; but it is inconsistent with the Catholic faith to hold that you cannot refer to God without knowing that He is a Trinity, because you can prove that God exists, the very same God Christians understand by faith to be Triune, without knowing that He is Triune. By the Catholic view, the sense of the term 'God' does not include the doctrine of the Trinity; we can know the sense of the term 'God' by reason alone, but we only get the doctrine of the Trinity by faith.
(There is, it should be noted, good reason for Muslims to disagree with the thesis, as well. The Qur'an pretty clearly commits the Muslim to saying that Muslims, Christians, and Jews all worship the same God; and the most reasonable interpretation of extended passages in the Qur'an is that the Christians worship the same God but that they commit the sin of shirk in doing so.)
If we look at the most influential and explicitly Catholic account of the meaning of the word 'God', that of Aquinas, it includes the following basic conclusions:
(1) We can use names that apply to God on the basis of reasoning from creatures.
(2) The names that genuinely apply to God are not necessarily synonymous.
(3) 'God' ('Deus') is applied to God on the basis of divine activity, in order to signify the divine nature.
(4) 'God' is not a proper name, precisely because it signifies the divine nature rather than the divine suppositum.
(5) Uses of 'God' improperly to talk about things that are not God have a respect or ordering to the proper use of 'God' to talk about God.
(6) Both Catholics and non-Catholics apply names to God, including 'God', on the basis of causation, excellence, or remotion.
None of this sits well with the notion that we can only apply 'God' to the God if we understand that God is Triune. To be sure, a Catholic could hold a different theory of divine names than Aquinas does; but any such theory is going to be constrained by some basic Catholic doctrines, one of which is that we can reasonably conclude that God, the one and only God, exists on the basis of reason alone, without any inkling of the doctrine of the Trinity.
Thus the Maverick's argument depends not only on issues of linguistic reference but on some fairly controvertible religious assumptions, as well. You can be Catholic while insisting on the Not-Same position; but the rational difficulty of doing so consistently is massively greater than it would be for many Protestants, who do not have the same emphasis on natural reason in religious matters that Catholics do. (It's a bit surprising that the assumption here is so Protestant, but he might be getting the assumption from Lydia McGrew's characterizations of the arguments.)
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As a minor side issue, one difficulty I have had with many of the Not-Same arguments, as well as Vallicella's Depends-on-Your-Theory arguments, is that it's so consistently put in terms of successful or unsuccessful reference, as if there were only these options. But in fact it's possible to have confused reference, ambiguous reference, obscure reference, wavering reference, and so forth. I don't know how this ultimately affects some of these arguments, but I would certainly argue that a theory of reference that has difficulty making sense of Cyrano de Bergerac is perhaps unlikely to be adequate for dealing with theology.
