On the one hand, those three religions (Christianity, Judaism, and Islam)trace themselves back, in an historical way, to the same people, places and events, and see themselves praying to the God of Abraham -- hence, the "same" God.

On the other hand, let's see.

You would understand if I said: A = A

You would also understand if I said: A does not = B.

You might understand if I added the caveat: A = B when certain conditions exist.

What conditions might exist to make A = B?

1. If the only "difference" between A and B is a semantic one.

In other words, if the essence of A and B are the same, but for one reason or another (culture, misunderstanding, perspective) that essence has taken on a different way of referring to it.

2. If A and B are a subset that is included within a larger domain. All robins are birds. Not all birds are robins. If you are talking about birds, then robin, or wren, or sparrow all fit the bill. But if you are talking about robins specifically, then wren or sparrow are outside the domain of the conversation.

So, it doesn't matter what you call God IF you are not making a difference of substance, and if by calling God a different name you are not by that name implying difference of character, nature, essence. Language does have meaning. For instance, when one is trying to be clear in their language one does not refer to a boy as "she". But…..

[b]But[/b]... if the question is, then, do all differing faiths believe in the same God but in a [i]different way[/i] ?", that is a different question altogether.

Now we're getting into a different "formula". This time the formula would be expressed…

A does not = non-A.

In other words, as long as the claims for the character of God are identical, then God, by any other name (or, presumably, by any other religion) is still God. But if the other religion's God makes distinct claims that are contradictory (non-A) to the premise of the God in question, then, to answer your question, no. Differing faiths do not believe in the same God.

Now, of course we all know that even within the single faith called "Christianity" Christians still hold contradictory views regarding the nature of God. This just means that the black-and-white of the above syllogisms are always tempered by other factors, factors like….

1. The knowledge that, try as one might, Christians [i]still[/i] believe that one cannot define God completely. They believe that he has revealed a sufficient amount of himself for His will and end, but Christians don't know but a fraction of Him.

2. The knowledge that, at least in the Christian faith, it is accepted that there are paradoxes in this world and beyond (God is one God – but he is three-in-one. Jesus was 100% God and 100% man. God is sovereign but man has free will). Christians live with these paradoxes (and denominationally describe them in nuanced ways) and so assume that there may be other ways that A might equal B (and MAYBE non-A).

These are a couple of reasons why Christians believe that they should remain humbly tolerant of others – and simultaneously, ever vigilant in their quest to more fully understand God.

Christians believe in an objective, transcendent God. He is who he is. He is not our invention – he transcends us (he would exist even if we never did).

That is my view on the subject, and my experience with Christianity as I know it and was taught it.