MOA Regained: A Critique of Symmetry Lost

 

Abstract

In a recent paper by Fritz, Lo and Schmid(Fritz et al 2025), published in Noûs demonstrated that in lower modal logics e.g. KT and S4, a case for atheism can be lodged. I will demonstrate that while their case is logically valid in these lower modal logics, I have potentially an undercutting defeater for their central claim that non-existence of God can be established using the Reverse Modal Ontological Argument (RMOA) in these lower logics. If successful I will show that even in lower modal logics, using the law of non-contradiction, the Modal Ontological Argument can be prioritised.

1.    Introduction

In Symmetry Lost: A Modal Ontological Argument For Atheism, the authors give a formal proof that on lower modal logics God (g) does not exist in the actual world. While this was logically valid in KT and S4 modal logics, their conclusion I will argue is metaphysically impossible. If successful, I will use the law of non-contradiction to show that if g is epistemically possible and no clear contradiction can be shown, this is sufficient to grant metaphysical possibility. If g is metaphysically possible ¬g cannot be simultaneously metaphysical without contradiction or showing g → ⊥ (the concept of a Maximally Great Being has a contradiction (⊥)). Without a demonstration that the concept of an MGB entails contradiction (i.e., g → ⊥), or that true symmetry obtains between g and ¬g, I argue that g regains logical and metaphysical priority to favour God’s existence in Modal Ontological Arguments.

In Section two I will summarise their ‘Symmetry Lost’ argument, showing how they ultimately come to their conclusion (the author strongly suggests reading their paper for full engagement). Section three will be my rebuttal, section four will be possible objections, section five will be concluding remarks, with a reference section and section six will feature appendices.

 

 

2.    Symmetry Lost 

 

2.1  Introduction

 

“Symmetry Lost: A Modal Ontological Argument for Atheism?” by Peter Fritz, Tien-Chun Lo, and Joseph C. Schmid presents a novel critique of the modal ontological argument (MOA) for God’s existence by highlighting a symmetry problem and proposing a reverse modal ontological argument (RMOA) that supports atheism. Below, I summarise the key points and questions raised in the paper and the implications the authors make.

 

2.2 Summary of the Argument

 

The MOA, rooted in Anselm’s ontological argument and formalized in modal logic (e.g., S5), is typically stated as:

 

P1: Necessarily, if God exists, then God necessarily exists (□(g → □g)).

P2: God possibly exists (♢g).

C: Therefore, God exists (g).

The symmetry problem arises because the motivation for P2 (e.g., conceivability implies possibility) equally supports the possibility of God’s non-existence (♢¬g).

This leads to the RMOA:

P1: □(g → □g).

P2*: God possibly does not exist (♢¬g).

C*: Therefore, God does not exist (¬g).

Since God cannot both exist and not exist, the validity of both arguments in S5 implies that at least one premise (P1, P2, or P2*) is false, undermining the MOA unless a symmetry breaker favours P2 over P2*. The authors explore what happens if no symmetry breaker is found and all three premises (P1, P2, P2*) are accepted, requiring a weaker modal logic than S5 where they are consistent.

2.3  Key Questions and Findings 

 

Question 1: Are there independent reasons for adopting a modal logic weak enough to allow P1, P2, and P2* to be consistent? The premises are inconsistent in S5 due to the principal Inc.: □(g → □g) → (♢g → □g), which implies the MOA’s validity in normal extensions of KT (modal logic with the T axiom: □p → p).

The authors identify independent metaphysical reasons for rejecting Inc., such as:

Higher-order metaphysics: Theories like Classicism (Bacon & Dorr, 2024) or essence-based reductions (Ditter, 2020, 2022) endorse S4 (which includes T and 4: □p → □□p) but reject B (p → □♢p), which implies rejecting Inc[1] in KT S4. 

Combinatorial principles: Roberts (2023) argues against the necessity of distinctness, which requires rejecting Inc. in KT, as Inc. derives this necessity.

These views provide plausible grounds for adopting a logic like S4 or weaker, where P1, P2, and P2* are consistent (e.g., as shown in a Kripke model in Figure 3b).

Question 2: If P1, P2, and P2 are true, what becomes of the MOA and RMOA? In a logic like KT or S4, where P1, P2, and P2* are consistent: The RMOA remains valid. Its logical form, □(g → □g) → (g → □g) (RMOA, is a substitution instance of T, derivable in KT. Thus, P1 and P2* entail ¬g.

The MOA is invalid unless the logic includes Inc. (e.g., in KTB or S5), which would make P1, P2, and P2* inconsistent.

This asymmetry arises because P1 (motivated by God’s perfection entailing necessary existence) has no parallel for P1* (□(¬g → □¬g)), breaking the symmetry without needing to reject P2 or P2*.

2.4 Implications for Atheism

 

The RMOA is more compelling than the MOA because: It requires weaker logical assumptions (valid in KT vs. KTB/S5 for MOA).

It avoids the symmetry problem, as P1’s motivation does not extend to P1*.

However, its strength as a case for atheism is limited by:

Controversial modal logic: Rejecting Inc./S5 is debated, with defenders of S5 (e.g., Pruss, 2011) offering counterarguments.

Premise plausibility: P2* (♢¬g) relies on conceivability, which may not entail possibility (Kripke, 1980). If God’s existence is non-contingent, modal arguments may be inconclusive.

The authors suggest the RMOA opens a new pathway to atheism but acknowledge its dependence on resolving these issues.

 

3.    Modal Ontological Argument Regained 

3.1 Introduction

In Section two, I took their conclusions and reasoning for granted to bring their arguments best features to the fore and to treat it with the utmost respect. However, in what follows, I will present what I take to be a decisive undercutting defeater to the core claim of Symmetry Lost, that while, g and ¬g can be modally symmetrical in KT/S4. Moreover, once g is understood as metaphysically possible, and what the proper understanding of what a MGB Would be like, ◇¬g is in fact metaphysically impossible, from the Law of non-contradiction. 

3.2 Definitions and Notation

g: An MGB exists, with properties of omnipotence, omniscience, and Omni-benevolence.

w0: Actual world.

w1: A possible world where g holds (for P2).

w2: A possible world where ¬g holds (for P2*).

◇p: p is possible (true in some accessible world).

□p: p is necessary (true in all accessible worlds).

R(w, w’): Accessibility relation (w accesses w’).

KT: Modal logic with reflexive accessibility (T: □p → p).

S4: KT + transitive accessibility (4: □p → □□p).

S5: S4 + symmetric and universal accessibility (◇□p → □p).

3.3 The Cost of Epistemic Possibility 

Epistemic possibility is a useful standard for certain limited fields. E.g. science fiction or film plot ideas. It can also be useful in philosophy to determine what could be possibly be true, in other words, epistemic possibility is what isn’t known to be false.

What we reason could be epistemically possible, if no clear contradiction is found, it could be deemed metaphysically possible, with metaphysical possibility being how reality could be.

For instance, it is epistemically possible that in a possible world, I am the king of England,  it is also metaphysically possible too, as it contains no clear contradiction. 

It may be epistemically possible that in one thousand years time, we could develop light speed travel, although it might be metaphysically impossible, due to laws of physics. 

However, it is neither epistemically nor metaphysically possible for a married bachelor or square circle to exist, as what it means to be a married man and what it is to be a bachelor are mutually exclusive. Similarly, what it means to have four sides (a square), can’t have 360 degrees (a circle). 

Additionally and simultaneously, we have the law of non-contradiction. So, something that we know to be metaphysically plausible e.g. Squares, circles, felines etc. exists. It would be rather radical if someone claimed that felines aren’t metaphysically possible, although undoubtedly, their existence is clearly contingent. But even if felines weren’t in the actual world, it would still be conceivable that felines could exist, just like it is conceivable unicorns could exist.

Similarly, g’s existence, namely as a MGB possessing omnipotence, omniscience and Omni-benevolence is at least epistemically possible, as the claim ◇g (possibly God exists) is at least conceivable. Furthermore, the concept of a MGB is coherent and contains no clear contradiction, so could be metaphysically possible (Swinburne 2004,2016)(Craig 2008) (Plantinga 1974).

To further bolster the claim, we have the almost ubiquity of modern philosophers prior to Plantinga that either implicitly or explicitly use the properties of a MGB to critique g’s existence. If there were some impossibility to g, the philosophy of religion landscape would be very different. Atheistic and agnostic philosophers alike use a MGB like being to critique (Schellenberg 2017, 2017) (Sobel 2004) (Oppy 2009).

Now what about ♢¬g? 

If ◇g thought to be metaphysically possible, then at best ♢¬g can only be epistemically possible. Why? If we said that ♢¬g is equally metaphysically possible,  this would violate the law of non-contradiction. 

For instance, would it be coherent to say felines are metaphysically possible and simultaneously metaphysically not possible? I will argue it does not. Remember,  metaphysical possibility is about how reality could be. So, to say metaphysically it  possible for something not to exist, is to say no world could have such a thing. 

This isn’t to say that something that is metaphysically possible must exist,  as in the case of felines and unicorns, the former exists in the actual world, while the latter may only exist in some possible worlds. This means metaphysically possible things are at least contingently possible.

This seems to equally hold for g’s existence, so as it pertains to g, ♢g exists in at least one possible world, this gives at least prima facie evidence the concept of g is plausibly metaphysical.[2],[3]

Which by the Law of non-contradiction means ♢¬g can only be epistemically possible.

3.4 KT and S4

Even in these weaker modal logics, the law of non-contradiction must apply in any possible world. So, granting KT or S4 is correct, in some possible world where felines didn’t in fact exist, it would be absurd to assume that therefore felines are impossible. It must still be metaphysically possible that felines could have existed and do in fact exist in some possible world. 

Accessibility in these worlds, isn’t guaranteed. So, although in w0 (the actual world) felines, bears etc. Exist, in w2 (a possible world) felines, bears, g might not obtain in that world. 

More pertinently, in relation to g, if g obtains in w1, w1 may not have accessibility to w0, w2, w3...wn. So, g isn’t guaranteed necessary existence in these models.

Similarly, in these lower logics, if g is possible (◇g), then any conclusion can’t be ¬g, as there are at least one world where g obtains, at best the conclusion is ◇¬g and due to the accessibility limitations, in such worlds the authors claim should be ¬□g (not necessarily God exists) as they can’t know with any certainty which world we have (See Appendix 6.1 for formal logical proof and Appendix 6.2 for opposite conclusion g). What they have done is logically permissible in these logics, so not forbidden, only if they already stipulate w0 is a  ¬g world.

However in their attempt to prove ¬g in w0, additionally they inadvertently seem to allow, by the same reasoning, ¬g in w2...wn to obtain in almost every world they chose, by stipulating such. Similarly, the inverse holds for ◇g, proving g requiring the sceptic to show a contradiction in g to salvage their claims (g → ⊥)(see figures 1.1 and 1.2).

 

 

Figure 1.1 [4]

 

 

Figure 1.2 [5]

3.4.1 g’s existence amplified

I’ve demonstrated that in these lower logics, we can’t have certainty what accessibility relations a possible world has to another. But what would it mean to say g possibly exists (◇g), and what would g as a MGB be like?

G if such a being exists, must be maximally great, and typically is understood to have;

·        Omnipotence: The ability to do anything logically possible. 

·        Omniscience: Knowing all true facts including moral facts. 

·        Omni-benevolence: Being morally perfect. 

Focusing in on omnipotence, what would it be for a being to be omnipotent? 

An omnipotent being couldn’t do logically impossible things e.g. create a square circle, create a rock too heavy for them to lift, cause them self out of existence and admits no contingency, as this would be a limit in power, which is what many theistic philosophers articulate (Pruss/Rasmussen 2018) (Plantinga 1974) (Craig 2025). As the authors of Symmetry Lost articulates “God exists necessarily if at all.”, sharing a universal understanding that if g exists, then contingency isn’t compatible with omnipotence.

If g is metaphysically possible and therefore a way in which reality could be, and w1 is a possible world, and g is omnipotent, then in w1 g is necessary (□g). In w2, if w2 is possible and g in omnipotent, then it would contain a contradiction if g (g → ⊥) didn’t obtain in such a possible world, by the law of non-contradiction. Therefore, it follows even in KT and S4 if g is omnipotent, every possible world is pushed through by the law of non-contradiction. 

Even if in KT and S4 does not allow universal accessibility, the point of my claim is  ◇g and ◇¬g are formally consistent in KT/S4, but metaphysically incompatible once g is defined as a necessarily existing being (via □(g → □g)). Which means, as MOA advocates, like Plantinga and Craig have protested, a contradiction must be presented to show g in incoherent (g → ⊥).

3.5 S5

Given the limitations of KT and S4, as demonstrated in 3.3, and the indefinite conclusions that at best case for atheism on weaker modal logics is ¬□g  (God’s existence isn’t necessary) and simultaneously g. This is inadequate for philosophy of religion debates, that require more definite conclusion, like Symmetry Lost tried to prove, namely ¬g in w0. Therefore, it seems logical that S5 should be the standard modal logic in such debates, where necessity is required. Though this goes beyond this papers scope.

 

3.6 Closing Remarks

While the authors of Symmetry Lost construct a formally valid model in KT and S4, their conclusion, that ¬g holds in the actual world goes beyond what their own premises entail. What follows from ◇¬g and □(g → □g) is, at best, ¬□g (that God’s existence is not necessary), not ¬g simpliciter. This already weakens the force of the RMOA in KT/S4. But more significantly, I argue that once g is granted as metaphysically possible and coherently defined as a necessarily existing MGB, ◇¬g is no longer metaphysically possible without contradiction. Thus, the RMOA’s premises either under deliver (in weaker logics) or collapse (under metaphysical analysis), restoring asymmetry in favour of the MOA, requiring a clear contradiction for the RMOA to be preferred (g → ⊥) .

 

4 Objections 

4.1 Both ◇g and ◇¬g are epistemically possible only. Neither can be justified as metaphysically possible. 

Response:

If this is the case, the theist still has a non-contradictory definition of g, that requires serious attention by sceptics, which plausibly makes it metaphysically possible until it can be shown to be contradictory (g → ⊥). This still is reasons to prefer ◇g to ¬□g for the reasons given above.

But even granting favourable conditions for atheism in these lower modal logics, the best-case scenario is ¬□g, far from proving atheism and firmly leaves g to exist, which as 6.2 shows can equally prove g. This then forces the atheist to go back to S5 and show why g → ⊥).

4.2 Metaphysical Possibility isn’t Exclusive. 

Claim: You're implicitly treating g as if modal logic must respect its maximal greatness i.e., that g’s necessity (g → □g) propagates across all worlds, as in S5. But KT/S4 don’t support that structure. You can’t say g is metaphysically necessary in all worlds using logics that don’t support global necessity.

Response:

This isn’t the argument I was making. I agree KT and S4 doesn’t entail universal accessibility. The argument is once g is taken as a metaphysically serious claim, g as an omnipotent being can’t have any contingency so by necessity of what such a being would be like, must be necessary. 

In other words, the modal model fails to reflect the metaphysical commitments entailed by the content of what g would be as an omnipotent and necessary being would be like.

4.3 Need To Prove □g, Not Just Refute ¬g.

Claim: It’s not enough to say ¬g is a metaphysically inconsistent conclusion. Unless you can positively derive □g without assuming S5, your argument just blocks the RMOA but doesn’t establish MOA.

Response:

This treats my whole argument as a strict formal modal refutation; in 3.3, I show that by their own logic, their conclusion isn’t justified, namely, at best they show ¬□g, not ¬g, this is a modal critique. My second part of my argument is an analytical look at what the implications metaphysically would be the case, given what it would be like to be omnipotent. 

So, my undercutting defeater is a metaphysical implication, not a strict modal refutation. 

4.4 Assuming to Much of g’s Coherence.

Claim: You assume g (a necessarily existing, omnipotent being) is metaphysically coherent, but many philosophers (Oppy, Sobel, etc.) argue that maximal greatness is internally problematic (e.g., omnipotence paradoxes, incompatible properties). So your move from ◇g to ¬◇¬g is built on shaky ground.

Response:

I don’t assume g’s metaphysical possibility, I gave reasons why g is plausibly metaphysically possible. If my reasoning is faulty, my claims are falsifiable. Even Oppy, Sobel, etc. Will grant g’s properties to critique g’s existence, so the properties can’t be that problematic, or a clear contradiction be found.

4.5 Modal Scepticism.

Claim: Even if g is metaphysically possible, that doesn’t automatically settle the debate, it just tells us modal logic alone can’t do all the work.

 

Response:

This is part of my critique, even modal claims like ◇g exists has metaphysical implications. If g is metaphysically possibility, a being with such properties would have real implications to reality. 

Conceding ◇g is metaphysically possible has implications to ◇¬g, meaning ¬□g can only be epistemically possible, making ◇g a more significant claim. Furthermore,  if the content of g entails omnipotence and necessary existence,  ¬□g is an impossible state of affairs. 

 

5. Conclusions

5.1 Conclusion 

In Symmetry Lost, Fritz, Lo and Schmid, offers a formal valid model in KT and S4 that derives ¬g in the actual world, using the RMOA. While this is logically consistent in these lower modal logics, I show their conclusion overreaches and doesn’t demonstrate what they claim. 

Specifically:

If g (a Maximally Great Being) is epistemically and metaphysically possible, and no contradiction can be demonstrated in g’s concept, then ¬g cannot also be metaphysically possible without violating the law of non-contradiction.

The best that the RMOA establishes in KT/S4 is ¬□g (it is not necessary that God exists), not ¬g. Thus, the argument fails to reach its atheistic conclusion.

Moreover, once g is granted as possessing omniscience, omnipotence, and necessary existence, then any world in which g exists must entail g’s necessity meaning that even in KT/S4, these metaphysical implications override structural accessibility limitations.

Therefore, the modal symmetry claimed by the RMOA collapses, and the asymmetry favouring the MOA is restored.

Unless and until the concept of a Maximally Great Being is shown to be contradictory (i.e., g → ⊥), the RMOA cannot dislodge the force of the Modal Ontological Argument. At most, it demonstrates logical permissibility, not metaphysical plausibility.

 

6. Appendices

6.1 Formal Logical Proof: RMOA Limited to ¬□g

Objective: Show that in KT/S4, the RMOA’s premises (P1, P2, P2*) at best establish ¬□g in w0, not ¬g, if ◇g is true.

Proof:

P1: □(g → □g) ⊢ g → □g in w0 (T axiom: □p → p).

P2: ♢g ⊢ ∃w1: R(w0, w1) ∧ g(w1) (◇g: g is possible).

P2*: ♢¬g ⊢ ∃w2: R(w0, w2) ∧ ¬g(w2), so ¬□g in w0 (not all accessible worlds have g).

Assume g in w0 (for reductio).

From (1), g ⊢ □g in w0.

From (3), ¬□g in w0.

Modus tollens: g → □g, ¬□g ⊢ ¬g in w0.

Critique: If ◇g (2), then g is possible in w0 (♢g). Thus, ¬g in w0 is possibly false (¬□¬g), as w0 could be a g world.

Metaphysical Possibility: ◇g is true (no g → ⊥, Plantinga 1974). ¬g in w0 requires g → ⊥ to deny ◇g. Without g → ⊥, P2* is questionable.

Conclusion: P2* guarantees ¬□g in w0 (3), but ¬g in w0 is contingent. The RMOA’s strongest conclusion is ¬□g, not ¬g.

6.2 Kripke Model Counterexample

Consider a KT/S4 model where w0 is a g world:

Worlds: w0 (g), w2 (¬g).

Accessibility: R(w0, w0), R(w0, w2), R(w2, w2).

P1 holds: In w0, g ⊢ □g (w0 accesses w0: g); in w2, g is false, so g → □g is true.

P2 holds: ♢g in w0 (R(w0, w0) ∧ g(w0)).

P2* holds: ♢¬g in w0 (R(w0, w2) ∧ ¬g(w2)).

¬□g in w0 (from P2*), but g in w0, contradicting ¬g. Thus, ¬g is not guaranteed.

 

References and Notes

Fritz, Peter ; Lo, Tien-Chun & Schmid, Joseph C. (forthcoming). Symmetry Lost: A Modal Ontological Argument for Atheism? Noûs.

Swinburne, Richard, The Existence of God, 2004, Oxford University Press, second edition, pp 4-8

Swinburne, Richard,  The Coherence of Theism,  2016, Oxford University Press, second edition. 

Craig, William Lane, Reasonable Faith: Christian Faith and Apologetics, 2008, Crossway Books, third edition. 

Plantinga, Alvin, The Nature of Necessity, 1974, Oxford University, pp 221.

Schellenberg, John. L, Divine Hiddenness: Part One, 2017, DOI: 10.1111/phc3.12355

Schellenberg , John. L, Divine Hiddenness: Part Two, 2017, DOI: 10.1111/phc3.12413

Sobel, J. H. Logic and theism: Arguments for and against beliefs in God. 2004, Cambridge University Press. 

Oppy, Graham,  Arguing About Gods, 2009, Cambridge University Press. 

Pruss, Alexander R., and Rasmussen, Joshua,  Necessary Existence, 2018, Oxford University Press, p58

Plantinga, Alvin, The Nature of Necessity, 1974, Oxford University, p132.

Craig, William Lane, Systematic Philosophical Theology On God: Attributes of God Volume IIa, 2025, Wiley Blackwell. 

 

[1] The authors Fritz et al. Restrict their paper to what they call “normal modal logics”, The Inc. principle,  short for “Inclusion Axiom”, is a formal expression of what happens in S5 logic, where modal operators collapse into necessity. □(g → □g) → (◇g → □g). This leap from possibility to necessity hinges on the logic accepting Inc. which S5 does, but KT and S4 reject. When they reject S5 as ‘not normal’, they mean they imply that S5 is philosophically extravagant or unjustified for certain metaphysical debates.

[2] For a more in depth analysis read Alvin Plantinga’s ‘The Nature of Necessity’ p9-13.

[3] Ibid p150. “Now the Ontological Principle does have a certain attractiveness and plausibility. But (as presently stated, anyway) it exploits our tendency to overlook the difference between (5') and (5"). Its plausibility, I suggest, has to do with predicative rather than impredicative singular propositions; with propositions like (5') rather than ones like (5"). It is plausible to say that any world in which a predicative singular proposition is true, is one in which the subject of that proposition has being or existence. Call this The Restricted Ontological Principle. Not only is this plausible; I think it is true.” In this case 5’ is a positive claim like ◇g, while 5” is ♢¬g, for Plantinga, existence should be preferentially prioritises over non-existence. Ultimately concluding The Restricted Ontological Principle in the case where 5’ and 5” are both possible, existence should be treated more plausible (ibid p151). 

[4] The point isn’t to show in KT and S4 that ¬g holds in all worlds, rather due to Fritz et al. conclusion we cannot know what accessibility if any can be shown in their model, if ¬g is in w0, equally it seems it could obtain in w1...wn.

[5] Similarly, the counter is true.