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In my MCCC philosophy of religion classes I begin with Anselm's Ontological Argument for God's existence. This proves to be a cool exercise in logic and a priori reasoning, and an introduction to the idea of God. For the most part theistic and atheistic philosophers share the same view of God, the former arguing for this God's existence, the latter arguing against it.
What do we mean when we say "God?" University of Notre Dame philosopher Peter van Inwagen's "The Idea of God" is a good place to begin. (In The Problem of Evil)
We are interested in the essential attributes of the idea of God. Note we're talking about the idea of God, and not assuming the reality of God. Every idea, every mental concept, has essential attributes. As does "God."
A person
Van Inwagen writes: "By a person, I mean a being who may be, in the most straightforward and literal sense, addressed - a being whom one may call 'thou'." (20)
Jews, Christians, and Muslims think of God this way. God "cannot possibly be thought of as impersonal, like Brahman or the Tao or the Absolute Idea or the Dialectic of History or, to descend to a more popular level, the Force." (Ib.)
Everyone who prays to God understands God as a person.
God is neither a 'he' or a 'she' since God is understood to not occupy physical space. More of this to come.
Omnipotent (or all-powerful or almighty)
An omnipotent being can do anything that is metaphysically possible. Note, philosophy of religion students, that I describe the attribute of omnipotence as the ability to do anything that is logically possible. Van Inwagen has difficulties with this, and uses metaphysical impossibility instead.
This still creates some difficulties, since it is generally agreed that God cannot lie, and lying is metaphysically possible. Van Inwagen devotes a good deal of attention to this.
'Omnipotence' concerns something like this: "The proposition that God cannot do X is consistent with the proposition that God is omnipotent of X is metaphysically impossible." (25) I take this to mean that God cannot, e.g., create 'square circles' or 'married bachelors' because such things are metaphysical nonsense. As is a stone so heavy that God cannot lift, or a burrito so hot that God cannot hold.
With these last two examples van Inwagen mentions Descartes' "strong omnipotence" view. "According to this conception, God is able to do anything, including (Descartes tells us) creating two mountains that touch at their bases and have no valley between them." (Ib.) Van Inwagen rightly rejects this because it is "incoherent because ability (the concept that is expressed by sentences of the form 'x is able to do y') is no more or less the power to choose among possible states of affairs, to determine which of various incompatible possible states of affairs are to be actual." (Ib.)
Omniscient (all-knowing)
"Here is the standard definition of omniscience: A being is omniscient if and only if that being knows the truth-value of every proposition." (26)
Van Inwagen prefers this definition of 'omniscience': "A being is omniscient if, for every proposition, that being believes either that proposition or its denial, and it is metaphysically impossible for that being to have false beliefs." (Ib.) [By 'proposition' Van Inwagen means what in logic is also called a 'statement'; viz., a sentence that is true or false, a statement that claims that a particular state of affairs obtains.]
Morally perfect (perfectly good)
"That is to say, God has no moral defect whatever. It follows that he is in no way a subject of possible oral criticism." (26)
Eternal
God has always existed and always will exist. And, God is outside time altogether.
If there is a God (which, of course, I believe true) then the Internet question "If God made the universe than who or what made God?" is a logically nonsensical question, since God did not begin to exist.
On God's relation to time, a very good book to read is: God and Time: Four Views.
Immutable
"That is, his attributes and other important properties do not and cannot change." (28)
"To speak the language of metaphysics, [God's] intrinsic or non-relational properties do not and cannot change with time... nothing in God's nature corresponds to the mutability that characterizes human existence and the existence of all things present to the senses." (Ib.)
Omnipresent
"To say that God is omnipresent is, obviously, to say that God is everywhere." (28)
Also obvious is that, "whatever omnipresence may come to, it is incompatible with God's having any sort of spatial or physical structure (and hence with his being either male or female)." (29)
Theists like myself affirm the actual existence of God, with the above essential attributes defining 'God'. Western philosophical atheists affirm that by the term 'God' the above-stated attributes apply, while of course working to deny the existence of God.