Rick Love and John Piper have reinvigorated the debate over whether Muslims worship the same God as Christians. See Justin Taylor's summary of the reasons for the Piper position. I'm of course on record taking the opposite view (see here), but in contributing to the comments on Justin's post I ended up putting my reasoning in a different enough way that I wanted to post it here as well. What follows is a slightly modified version of my comment on Justin's post.
First, let me present an issue in the philosophy of language. There's some difference of opinion about how words acquire their reference, i.e. how it is that a word comes to refer to the thing that it does. The dominant view in philosophy of language today is that a word comes to refer to what it refers to because of an initial "baptism" that declares what it refers to, along with various processes that happen along its continued usage. But there's a causal chain back to the original "baptism".
The name "George W. Bush" refers to the guy who happens to be the current president because his parents gave him that name and continued to use it to refer to him without changing it, and he continued to use the name without changing it. Its reference is because of that causal chain back to when his parents declared it to be his name.
Now suppose someone comes along and enters into the causal chain, calling him George W. Bush and engaging in the normal process of using the name. But this person starts claiming that the guy called George W. Bush is a clone of the original and has only existed for a few years. That amounts to denying an essential property of George W. Bush, i.e. his origin. Someone can't be him without having that origin. Nonetheless, the person with the cloning theory successfully refers to the real George W. Bush, despite having a view that denies one of his essential properties. So it can't be that denying an essential property of a being means you're not referring to that being. Some claim that because one of God's essential properties, according to Christianity, is his existence in three persons, then someone who denies that element of God's nature must be talking about a different (and non-existent) being. Not so. That's not how language works.
Muslims use certain words to refer to the being they worship (to remain neutral at this point). The linguistic practice that involves those words referring to the being they worship traces back to the time of Muhammad, who wrote a series of Surahs that ended up becoming the Qur'an. In these writings, Muhammad claimed to have received them from an angel, and they spoke of the being worshiped by the Christians and Jews. The word 'Allah' was initially a description for a divine being in Arabic, not a name, although perhaps it now functions in a namelike way, much like 'God' in English. 'Allah' thus referred explicitly to the God that so far had been worshiped by Jews and Christians. Muhammad went on to say a whole bunch of things about God that Christians would deny, including some things that amount to denying some essential properties of God. Islam is a false religion that is worthless in terms of knowing God, according to Christian teaching, and the worship of this being under Islam does not count as genuine worship.
Nevertheless, it seems completely ludicrous to me to claim that this being that is falsely and ungenuinely worshiped by Muslims is not God. Muhammad intended to refer to the God long worshiped by Jews and Christians that Muhammad when he said all those false things about God. The being he misrepresented and twisted all sorts of things about is the God of the Bible. I don't know how the historical facts can get around that.
There is an issue of how a Christian should make this point. Perhaps Love didn't go far enough in distancing himself from how people might hear it. But that doesn't mean what he says is false.
I want to say that not only is Love's claim indeed true, but it's true in the same way the Bible is true when it speaks this way. See II Kings 17, which makes it clear that syncretists can fear God and other gods at the same time. It's true that they're thus not really fearing God by not serving him fully. The passage even goes on to say that. But it's still God that they're not really fearing. The name 'YHWH' is even used. It's him that they're not fearing properly when they fear both him and other gods. So why is it a problem if Muslims also worship God in one sense but don't really worship God in another sense? It's God they're not worshiping properly. The term still refers to him. It's just not genuine worship of him (which is what Love perhaps could have emphasized to avoid misleading people).
One further thought. Those who say that denying the Trinity means one's terms for God don't refer to God have to say the same about Jews. Do Christians want to say that we don't worship the God of the Jews? It's very clear that the New Testament authors thought they were worshiping the God of the Jews, and they didn't think the Jews suddenly were worshiping some fictional being once Jesus appeared and Jews started denying essential properties of God that hadn't previously been revealed. The way the NT speaks of this is that they were worshiping God but just not doing so properly. It was still God that they weren't worshiping properly.
Now I don't want to extend this to all monotheists. Being a mere monotheist may not be sufficient to be worshiping God (albeit wrongly). If the origin of a particular monotheistic religion connects its worship of one being up with God in some linguistically-appropriate way, then they might worship God even if they do so wrongly. But if the religion begins by picking out something that clearly isn't God, calling it God, and then worshiping it, then I wouldn't say they worship God, even wrongly. They worship that other thing. So I don't agree that any old monotheistic religion worships the true God but wrongly. It has to have the appropriate historical-linguistic connection. But Islam so clearly does that it amazes me that anyone could deny it. The only reason people are giving for denying it seems like such a non sequitur, as my Bush example above shows, that I have to think people are confusing improper worship with worship of the wrong thing. You can improperly treat anything, getting even its essential properties wrong, while nevertheless still referring to it when one speaks of it.