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As a current atheist, who was a former theist, I feel that God is not a logically incoherent concept. However, many definitions of God, especially those that involve omnipotence, omniscience etc. are very difficult to do without logical contradictions. I am interested in definitions for these concepts that work as follows:

Define a binary relation as follows: Given two agents X and Y, X <= Y if and only if Y can do all the things X can do. An omnipotent agent is a greatest upper bound on the set of agents, under the above relation.

As it stands this definition itself still has paradoxes, but I wonder if someone has been able to take a definition like this, perhaps in a category theoretic context (instead of a mere poset) and use it to formalise concepts like omniscience, omnipotence etc. without contradictions?

  • I don't know but potentially interesting is that such a definition would avoid some of the paradoxes of omnipotence (e.g. the ability to create a stone one cannot move) – Quentin Ruyant Mar 1 '15 at 15:15
  • Like what paradoxes exactly? – Neil Meyer Mar 3 '15 at 8:01
  • possible duplicate of Russell's Paradox & Existence of God – Neil Meyer Mar 3 '15 at 12:29
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I can't refer to philosophers, but I have personally built a system like this to explore the consequences of such thinking. When I put it to some of my religious friends, they smiled, and politely said, "That's a neat model, but that isn't how my God is defined."

So yes, there is such a mathematical theoretical definition of God. However, one cannot assume that such a definition matches the definition used by others.

Consider, as a "proof by analogy," the definition of "finish" from Meriam Webster, of which I will only reproduce definition 2 and 3 of the transitive form of the verb:

transitive verb

2 a : to bring to completion or issue : hope to finish their new home before winter b : to provide with a finish; especially : to put a final coat or surface on finish a table with varnish

3 a : to defeat or ruin utterly and finally the scandal finished his career b : to bring about the death of

Needless to say, providing a mathematical definition for one of these forms of "to finish" does not help if you are conversing with someone who uses it in the other sense.

Even if you can show that behaviorally your model is identical to their, defining God within a religion is usually treated as an ontological issue, not a epistemology issue, so they generally will not accept any argument built from it through mere logical progression (which is usually the purpose of such a model).

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An order relation based definition of God is the basis for the ontological argument:

  • God is the greatest thing that we can conceive of.
  • Something that is real is greater than something imaginary.
  • The greatest thing we can conceive of is thus greater than all imaginary things we can conceive of, including an imaginary God,
  • Therefore God must be real.

A formal mathematical version of this argument was developed by Kurt Godel (and published posthumously) using modal logic. I don't know enough about modal logic to elaborate any further, but it sounds pretty close to what you are looking for.

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    Is this an argument taken seriously by any modern philosophers? I sort of doubt anyone with half a brain would agree with this argument. – 6005 Mar 3 '15 at 22:49
  • @C-S I share your sentiment, the argument makes no sense whatsoever to me. And yet it is mentioned in every serious discussion of philosophy of religion I have come across. Even Bertrand Russell mentioned that it was a sound argument (although obviously he wasn't convinced by it). – Alexander S King Mar 4 '15 at 0:21
  • The argument to me makes sense, the question is to what extent can it be true, or justified; and therein lies a lot of discussion. – Mozibur Ullah Mar 5 '15 at 19:57
  • @MoziburUllah I see 2 issues with it: First, we need an explicit measure of "greatness". In this argument the term greatness is subjective, one could argue that a greater God than the Abrahamic God can be conceived of, if you consider how flawed the latter's creations are. More importantly, using the same logic, all sorts of "greatest things that can be conceived of" can be willed (conceived) into existence simply because someone believed strongly enough in them. – Alexander S King Mar 6 '15 at 17:29
  • @king: sure, there's a lot of discussion around these points and more; for example, if perfection is solely identified with God, then his creation, in a sense, has to be less than perfect, because they are less than God. – Mozibur Ullah Mar 7 '15 at 14:15
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As the OP is not explicit in the paradoxes he mentions I will comment on what I perceive the paradoxes to be. If the OP can clarify in his question I will edit my post or delete it if necessary.

Can God create a stone so big that he cannot life it.

This at its core is an attack on one of his characteristics. His all enthralling power. His omnipotence. The word omnipotent comes from omni- meaning “all” and potent meaning “power.” but what about this statement what can we say about it.

This statement to me is not coherent. It is like the sentence a room full of emptiness. It is logically incoherent. This statement is to me tantamount to saying Can God create a polywog so big that even Janiray cannot life it? Your mouth is making sounds but the sounds have no objective meaning. Almost like baby jabber.

So why then would this pose a problem? God not being able to do the undefined or the absurd? Lets at least get a sentence with objective meaning and then ask whether or not he can do it and if he is omnipotent?

Omnipotence has generally (In the Christian tradition) not comprised of that which is logically incoherent and why should it? It is nonsense.

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You might be interested in the following paper of the logician Robert K. Meyer: http://www.jstor.org/discover/10.2307/2215186?sid=21106301893923&uid=2&uid=3737720&uid=4&uid=3739400. Sadly, it is behind a pay-wall but I've taken the liberty to provide the following abridged set of assumptions which are necessary for Meyer's proof to get off the ground.

(1) The set of all spatio-temporal events in the universe can coherently be formed into a set $-$ call it V. Specifically, V is a set whose existence is consistent with the axioms of Zermelo-Frankel Set-Theory. The idea here being that for every event, there exists a set whose members are either the event(s) themselves or sets of events, sets of sets of events, and so on.

(2) $\forall x, \exists y \in V$ such that $yRx$ where $R$ is a relation of ``x is causally anterior to y and x causes y to come into being".

(3) $\forall C \subseteq V$, $\exists z \in V$, $\forall x \in C$ zRx.

(4) From (1-3), we conclude that there exists a structure $$ such that $$ is a strict partial order with respect to $R$: $\forall x \in V$($\sim xRx$) $\& \ \forall x,y \in V (xRy \ \implies \ \sim yRx)$ \& $\forall x,y,z \in V ((xRy \ \& \ yRz) \ \implies \ (xRz))$.

(5) Every sequence $C \in V$ is bounded, by (3). Thus by Zorn's Lemma, there is a maximal element $G \in V$ such that $\forall x \in C \subseteq V (GRx)$ $-$ namely, God (`G') is causally anterior to every member of the domain of causation.

(6) By the rule of necessitation (in reasoning from modal logic), we can conclude that this first mover necessarily exists since we have given a proof of it.

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Since Will is a primary aspect of the notion of (a) God (since being obligated, or necessarily bound to obey said will pretty much defines a deistic religion), we can define (a) God in terms of His(/its) will.

Consider a logic, and apply a mode to it, like 'necessity', 'obligation', or 'is believed under Aristotelian philosophy'. That mode has a dual, the mode that is true of things that are not forbidden buy the original mode. For 'necessity', the dual is 'possibility', things are possible if is is not necessary for them to be absent. Things are permissible if it is not obligatory to avoid them. Things are considered under Aristotelian philosophy if there is not a belief in the philosophy that they are false...

I would propose that 'the will of (a) God (about a reality instantiating that logic)' is the(/a) maximal self-dual mode. We want God(s) to be consistent, but as powerful as possible. (If you have multiple gods, a 'God' is their perfect combination. Cultures that split God up in to multiple entities, generally still want the composite order of the universe to be consistent, complete and for the union of all gods to be sovereign over it.)

(Whether this mode is nontrivial or unique relative to any proposed system of thought, describing our reality or another, is a question to be established.)

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