Was thinking today about the modal ontological argument specifically as it relates to our multiple conceptions of God across civilizational history - wanted to formalize it.

Essentially the fundamental argument here is that, by necessity, either the S5 argument used in the modal ontological argument leads to a multiplicity of possible deities that, by the ontological argument itself, must exist.

Let me know if anyone finds any counterarguments/issues in the logic here - there are certainly axiomatic counterarguments, but I've workshopped a few of them and they seem weak?

Definition 1: Property P is positive iff it contributes to greatness.
Definition 2: X is God-like iff X has all positive properties.
Definition 3: X necessarily exists iff X exists in all possible worlds.

Axiom 1: If P is positive and P entails Q, then Q is positive.
Axiom 2: Either P is positive or ¬P is positive, not both.
Axiom 3: God-likeness is positive.
Axiom 4: Necessary existence is positive.

Definition 4: A divine conception is a set of attributes that could coherently belong to a greatest possible being.

Lemma 1 (Conception Multiplicity): Multiple coherent divine conceptions exist.
Proof:

1. Empirically observable across religious traditions (personal/impersonal, trinity/unity).
2. These conceptions differ substantively yet each maintains internal coherence.
3. Each can be articulated without logical contradiction.

Challenge 1: "But seemingly contradictory attributes (justice vs. mercy) must reconcile in a unique maximal being!"
Response:

1. Reconciliation attempts typically employ one of three strategies:
   a) Definitional gerrymandering (redefining justice until it's compatible with mercy).
   b) Mysterious transcendence ("They're reconciled beyond human understanding").
   c) Hierarchical subordination (one attribute trumps the other).
2. Each strategy fails:
   a) Creates vacuous concepts that lose normative force.
   b) Renders the conception unfalsifiable and meaningless.
   c) Admits multiple possible hierarchies → multiple conceptions.

Definition 5: A conception parameter φ is any attribute dimension admitting multiple coherent values.

Lemma 2 (Parameter Space): Divine conception space contains multiple independent parameters.
Proof:

1. Consider attributes: consciousness type, moral character, relation to time, aesthetics.
2. These vary independently (personal timeless being ≠ impersonal timeless being).
3. Each parameter admits multiple values without generating contradiction.

Axiom 5 (Modal Correspondence): If a divine conception C is coherent, then it's metaphysically possible.

Challenge 2: "Not all coherent conceptions are metaphysically possible!"
Response:

1. The modal ontological argument already accepts coherence → possibility for at least one conception.
2. Denying this for other conceptions requires a non-arbitrary criterion that doesn't beg the question.
3. No such criterion exists that doesn't presuppose the truth of a specific theological tradition.

Axiom 6 (Modal Plenitude): If a divine conception C is metaphysically possible, a God-like being instantiating C exists in some possible world.

Theorem 1 (Divine Multiplicity): Either the modal ontological argument fails OR multiple necessarily existing God-like beings exist.
Proof:

1. By Lemma 1 and Axiom 5, multiple divine conceptions are metaphysically possible.
2. By Axiom 6, each exists in some possible world.
3. By the ontological argument's S5 machinery, possibility entails necessity.
4. Therefore: Either the S5 inference is invalid OR divine multiplicity is true.

Definition 6: The True God Problem is determining which conception (if any) corresponds to the actual God.

Axiom 7 (Epistemic Neutrality): Absent privileged information, rational credence should be distributed across the space of coherent conceptions.

Theorem 2 (Epistemic Collapse): Rational credence in any specific divine conception approaches zero.
Proof:

1. With n parameters each admitting k values, we have kⁿ distinct conceptions.
2. Prior probability for any specific conception = 1/kⁿ.
3. Even with modest values (k = 3, n = 5), P(specific God) = 1/243 ≈ 0.004.
4. Realistic parameter space yields P → 0.

Challenge 3: "But revelation/miracles provide evidence favoring specific conceptions!"
Response:

1. Bayesian analysis: P(G|E) = P(E|G)P(G)/P(E).
2. Assume miracle claim with likelihood ratio = 1000:1.
3. With prior P(G) = 1/10⁶, posterior P(G|E) = 0.001.
4. Competing miracle claims from different traditions further dilute evidence.
5. Infinite parameter space → Miracles insufficient to overcome infinitesimal priors.

Theorem 3 (Sovereignty Paradox): Divine multiplicity precludes unique sovereignty.
Proof:

1. Sovereignty requires ultimate authority over all states of affairs.
2. Multiple necessarily existing Gods with different attributes entails conflicting wills.
3. Conflicting necessary wills create unsolvable metaphysical deadlock.
4. Therefore: Unique sovereignty is impossible under divine multiplicity.

Corollary 1: The modal ontological argument is self-defeating for monotheism.

Corollary 2: Even classical arguments (cosmological, teleological) succumb to multiplicity.
Proof:

1. Such arguments establish at most a first cause or designer.
2. They underdetermine specific attributes beyond what's necessary for causation/design.
3. Underdetermination generates the same epistemic problem as modal approaches.

Theorem 4 (Theological Agnosticism): If God exists, we cannot know which God.
Proof:

1. Assume God exists.
2. By Theorem 1, either multiple Gods exist OR modal logic fails.
3. If multiple Gods exist, by Theorem 2, identifying the correct one is impossible.
4. If modal logic fails, theistic arguments lose their demonstrative force.
5. Therefore: Theological certainty is impossible.

The monotheist is left with three equally devastating options:

1. Reject modal ontological arguments entirely.
2. Accept uncountable polytheism.
3. Retreat to pure apophaticism (negative theology).

..."How would you respond to the objection that "god-likeness" itself constrains the parameter space to a unique point?" 

That's just question begging. If you argue that god-likeness itself constrains the parameter space to a unique point, you are arguing that there are NO two possible coherent conceptions of God that are equally great. Consider: "the Christian god" and "the Christian god but his favorite color is produced by light at exactly 374.487nm" unless "favorite light color" is inherent to god-like-ness, those are two different entities. how can we demonstrate this? place each god in a situation in which it is forced to choose between light at 380nm and light at 374.487nm. The result (and thus the entire timeline) will now be quantitatively different, thus the two are separate entities.

This also introduces a 1:1 correspondence between possible deities and the real numbers, making the entire Bayesian section irrelevant:

Suppose maximal greatness uniquely determines a single being.

Then, for every contingent fact F (e.g., "How many Toms in Japan on a Tuesday?"), there exists a uniquely best value.

But many contingent facts F₁, F₂, ... are completely arbitrary in terms of maximal greatness.

If maximal greatness allows variation in any F, we can construct distinct maximally great beings that differ in their choices for F.

If we can do this for both continuous (real-valued preferences, e.g., photon wavelength) and discrete (integer-valued preferences, e.g., number of Toms) parameters, then we have an uncountable infinity of maximally great beings.

Therefore, either maximal greatness forces absurd metaphysical commitments to arbitrary facts or it permits infinite gods, making P(any one specific god) = 0.

Hence, modal ontological arguments fail.