Symmetry Regained or Asymmetry Reversed? A Reply to Fritz, Lo, and Schmid (philosophical journal submission)

Abstract
In modal system S5, Plantinga’s modal ontological argument for the existence of God and the corresponding reverse modal ontological argument for atheism are often regarded as dialectically symmetric. In this paper, I reply to Fritz, Lo, and Schmid, who suggest that, unlike Plantinga’s modal ontological argument, the reverse modal ontological argument becomes dialectically effective when moving from S5 to the weaker modal systems S4 and KT. I argue that the alleged dialectical asymmetry in favor of atheism in these systems depends on an incomplete characterization of the modal landscape. Once the full family of structurally parallel modal arguments is considered, and God’s modal nature is assumed to be relatively stable, the dialectical asymmetry in S4 and KT in favor of atheism dissolves. I further argue that, under a neo-Aristotelian, powers-based account of modality, the dialectical asymmetry may even reverse in favor of theism.

Keywords
philosophy of religion; modal logic; modal ontological argument; neo-Aristotelian modality; perfect being theology

1. Introduction
Plantinga’s (1974) modal ontological argument for the existence of God is often thought to admit a structurally parallel reverse modal ontological argument for atheism, thereby generating a dialectical symmetry between theism and atheism in S5 modal logic (Oppy, Rasmussen, & Schmid 2024). In this system, both arguments are logically valid and, since acceptance of either argument requires rejecting the modal possibility premise of its dialectical opponent in order to preserve consistency, neither side appears dialectically privileged.

Peter Fritz, Tien-Chun Lo, and Joseph C. Schmid (2026) show that, in the weaker modal systems S4 and KT, Plantinga’s original modal ontological argument for theism is invalid, whereas the corresponding reverse modal ontological argument for atheism remains valid. On the basis of this result together with further dialectical considerations, Fritz, Lo, and Schmid suggest that, unlike Plantinga’s modal ontological argument, the reverse modal ontological argument becomes dialectically effective in S4 and KT. In what follows, I argue that their discussion does not ultimately yield a dialectically successful argument for atheism within these weaker modal systems.

More specifically, I shall argue that Fritz, Lo, and Schmid do not consider the full family of structurally parallel modal arguments generated by their own framework. Once the complete modal landscape is taken into account, together with a plausible assumption concerning the relative modal stability of God’s metaphysical nature, the alleged dialectical asymmetry in S4 and KT in favor of atheism arguably dissolves, thereby restoring dialectical symmetry. I shall then argue that, under a neo-Aristotelian, powers-based interpretation of modal claims, the alleged dialectical asymmetry in favor of atheism may even reverse in favor of theism rather than merely disappear.

In this paper, I distinguish between formal symmetry and dialectical symmetry. By formal symmetry, I mean symmetry in the logical structure and validity relations of the arguments available to the opposing sides in the dispute. By dialectical symmetry, I mean symmetry in the dialectical force of the arguments and premises available to those sides. Formal symmetry does not necessarily entail dialectical symmetry, nor does dialectical symmetry necessarily entail formal symmetry.

2. Preliminaries
The modal systems S5, S4, and KT may be characterized semantically in terms of Kripke frames and accessibility relations between possible worlds. A Kripke frame consists of a set of possible worlds together with an accessibility relation between possible worlds specifying which worlds are considered possible relative to others. A model is obtained by supplementing such a frame with a valuation function that assigns truth-values to propositions at possible worlds.

The modal system KT is characterized by a reflexive accessibility relation: every possible world accesses itself. The system S4 strengthens KT by additionally requiring the accessibility relation to be transitive: if a possible world w accesses a possible world v, and v accesses a possible world u, then w also accesses u. Finally, S5 strengthens S4 by additionally requiring the accessibility relation to be symmetric: if a world w accesses a world v, then v likewise accesses w. An argument P ⊢ C in a given modal system is logically valid if and only if every model satisfying the conjunction of the premises P also satisfies the conclusion C. A model satisfies a proposition just in case it is true at every world in that model. Since S4 strengthens KT and S5 strengthens S4, every argument that is logically valid in KT is also valid in S4, and every argument valid in S4 is likewise valid in S5, while arguments valid in S5 or S4 need not be valid in the weaker systems.

3. Fritz, Lo, and Schmid's Modal Framework
The central framework employed by Fritz, Lo, and Schmid may be summarized as follows. Let g denote the proposition that God exists, and consider the following propositions:

P1: □(g→□g),
P1*: □(¬g→□¬g),
P2: ◇g,
P2*: ◇¬g.

P1 states that, necessarily, if God exists, then, necessarily, God exists. P1* states that, necessarily, if God does not exist, then, necessarily, God does not exist. P2 asserts that it is possible that God exists, whereas P2* asserts that it is possible that God does not exist. Fritz, Lo, and Schmid consider both Plantinga’s original modal ontological argument for the existence of God (MOA) and the reverse modal ontological argument for the non-existence of God (RMOA):

MOA: P1, P2 ⊢ g,
RMOA: P1, P2* ⊢ ¬g.

In modal logic S5, both MOA and RMOA are logically valid. Accordingly, theistic proponents of MOA must reject P2* in order to avoid contradiction, while atheistic proponents of RMOA must reject P2 in order to avoid contradiction. Hence, within S5, there is no non-question-begging argument either for the existence of God or for the non-existence of God. The resulting dialectical situation is therefore symmetric.

Fritz, Lo, and Schmid aim to break this dialectical symmetry by considering modal logics weaker than S5, namely S4 and KT. They show, notably, that in both S4 and KT, MOA is invalid, since it requires a symmetric accessibility relation, whereas RMOA remains valid, since the validity of RMOA depends only on reflexivity (2026, p. 12). Consequently, unlike the logical validity of Plantinga's original modal ontological argument, the logical validity of the reverse modal ontological argument survives in modal systems significantly weaker than S5. Building on this notable observation, Fritz, Lo, and Schmid (2026, pp. 11—13) suggest that the dialectical symmetry between MOA and RMOA characteristic of S5 collapses in S4 and KT, and that the resulting asymmetry in S4 and KT dialectically favors atheism over theism.

4. The Full Modal Landscape
However, the modal landscape explored by Fritz, Lo, and Schmid is incomplete, since they privilege one pairing of premises rather than considering all structurally parallel combinations. In order to uncover the full modal landscape, we must also consider the following arguments:

MOA*: P1*, P2 ⊢ g,
RMOA*: P1*, P2* ⊢ ¬g.

Once these arguments are taken into account, the following overall modal structure emerges. In S5, MOA, RMOA, MOA*, and RMOA* are all logically valid. This may be verified directly using the Kripke semantics outlined in Section 2. Accordingly, in S5 there are two valid arguments for the existence of God, namely MOA and MOA*, and two valid arguments for the non-existence of God, namely RMOA and RMOA*.

Each of these arguments is dialectically question-begging, since endorsement of P2 (in the case of MOA and MOA*) requires rejection of P2*, whereas endorsement of P2* (in the case of RMOA and RMOA*) requires rejection of P2, in order to avoid contradiction. Proponents of these arguments must therefore reject from the outset the modal possibility claim put forward by their dialectical opponents. Hence, the situation in S5 remains dialectically symmetric. The addition of MOA* and RMOA* therefore does not alter the dialectical situation in this modal system. Moreover, since P1 and P1* are logically equivalent in S5&emdash;as may again be verified using the Kripke semantics outlined in Section 2—MOA and MOA*, as well as RMOA and RMOA*, are likewise logically equivalent.

By contrast, in S4 and KT, MOA and RMOA* are invalid, whereas RMOA and MOA* remain valid. For example, the validity of MOA* may be shown as follows. Let w be a possible world in which it is necessarily the case that, if God does not exist, then God necessarily does not exist, and in which it is possible that God exists. Then there exists a possible world v accessible from w in which God exists. Assume, for reductio, that God does not exist in w. Since the accessibility relation in S4 and KT is reflexive, w is accessible from itself. It therefore follows from the first premise that, in w, God necessarily does not exist. Hence every world accessible from w, including v, must be a world in which God does not exist. But this contradicts the fact that God exists in v. Therefore, God exists in w. Since w was arbitrary, MOA* is valid in KT and therefore also in S4.

The invalidity of RMOA* in S4 and KT may likewise be shown by a countermodel on a reflexive and transitive frame. Let w be a possible world in which God exists, and let v be a possible world in which God does not exist. Suppose that w accesses both itself and v, while v accesses only itself. The second premise of RMOA* is true in w, since v is accessible from w and God does not exist in v. Moreover, at w the antecedent of the conditional embedded in the first premise is false, since God exists in w, while at v the consequent of the conditional is true, since v accesses only itself and God does not exist in v. Hence, at every world accessible from w, if God does not exist, then God necessarily does not exist. The first premise of RMOA* is therefore true in w as well. Thus, both premises of RMOA* hold in w, while its conclusion is false in w. Accordingly, RMOA* is invalid in KT and, since the frame is also transitive, RMOA* is invalid in S4 as well.

The remaining invalidity and validity claims concerning MOA and RMOA in S4 and KT may likewise be verified directly using the Kripke semantics outlined in Section 2. Furthermore, by appeal to similar countermodels, it may be shown that, in S4 and KT, P1 neither entails nor is entailed by P1*. In addition, the conjunction of P1, P1*, P2, and P2* yields a contradiction in S4 and KT, since in both systems the conjunction of P1 and P2* entails ¬g, whereas the conjunction of P1* and P2 entails g. By contrast, endorsement of RMOA together with P2, as well as endorsement of MOA* together with P2*, remains jointly consistent in S4 and KT.

Thus, the theist may argue for theism by endorsing MOA* together with P2*&emdash;since rejecting P2* would render the argument question-begging&emdash;while rejecting P1 in order to avoid contradiction. Likewise, the atheist may argue for atheism by endorsing RMOA together with P2&emdash;since rejecting P2 would likewise render the argument question-begging&emdash;while rejecting P1* in order to avoid contradiction. Hence, once the full modal landscape is taken into account, the situation in S4 and KT is formally symmetric. For the theist, MOA* is valid and its premises are consistent with P2*, whereas MOA is invalid, while for the atheist, RMOA is valid and its premises are consistent with P2, whereas RMOA* is invalid. Thus, no formal asymmetry obtains in these weaker modal systems once the complete modal landscape is considered.

5. Modal Stability and Dialectical Symmetry
Because Fritz, Lo, and Schmid consider an incomplete modal landscape, their case for dialectical asymmetry in S4 and KT in favor of atheism builds on an incomplete dialectical landscape—namely one that privileges MOA and RMOA while neglecting the structurally parallel arguments MOA* and RMOA*.

As Fritz, Lo, and Schmid (2026, p. 11) argue, P1 enjoys an independent motivation grounded in perfect-being theology, whereas no parallel motivation appears to be available for P1*. This asymmetry in modal motivation plays a central role in their conclusion that S4 and KT dialectically favor atheism over theism. For the atheistic defender of RMOA would appear to face no pressure to accept P1*, and hence to reject P2 in order to preserve consistency. Therefore, whereas MOA is invalid in S4 and KT, it might seem that RMOA becomes a non-question-begging argument for atheism in those weaker modal systems.

The question, then, is what becomes of the dialectical situation in S4 and KT once the full modal landscape is taken into account. One might think that even if the valid theistic argument MOA* is considered, its defender in S4 and KT remains committed to P1 given the conception of God relevant to the modal ontological argument. After all, P1 is not merely a formal premise of MOA, but expresses a modal commitment traditionally associated with theism. This observation is not without force. It would seem that the theist must reject P2* in order to avoid contradiction, so that MOA* is not dialectically effective in S4 and KT. Since RMOA* is invalid in these weaker systems, it may thus seem that RMOA is the only logically valid non-question-begging argument in S4 and KT, thereby yielding a dialectical asymmetry in favor of atheism. I shall argue, however, that this appearance is misleading.

It is not unreasonable to expect a significant degree of stability in the modal characterization of God across accessible worlds. On a plausible conception of God in the present context—namely, as a being with a relatively stable metaphysical nature—modal profiles concerning God’s existence should not vary radically across accessible worlds. The question, then, is whether a stable conception of God’s modal nature can plausibly sustain endorsement of only one of P1 and P1* while rejecting the other.

Now, the defender of RMOA in S4 or KT must accept P1 while rejecting P1*, since acceptance of both P1 and P1*, together with P2 and P2*, results in inconsistency, whereas rejection of P2 would render RMOA dialectically ineffective. Without P1*, however, a world w in which God does not exist may access a world v in which God exists and which itself accesses only worlds in which God exists, so that God’s non-existence is impossible in v, thereby permitting radically divergent modal profiles concerning God’s existence across accessible worlds. Since this high degree of modal instability conflicts with the relevant conception of God, the defender of RMOA is thereby pressured toward accepting P1*. But once both P1 and P1* are accepted, the defender of RMOA must reject P2 in order to avoid contradiction, thereby undermining RMOA’s dialectical force as an argument for atheism.

Moreover, P1 is typically motivated by a significantly more robust conception of God than that of a being with a merely relatively stable metaphysical nature, namely as a necessarily existing being with a stable essential nature. Once such a conception of God is admitted, substantial modal divergence of God across accessible worlds becomes even more difficult to explain. Thus, although P1 does not entail P1* in S4 or KT, acceptance of P1 nevertheless generates still stronger pressure toward accepting P1*. This further indicates that RMOA remains dialectically ineffective in those weaker modal systems.

Parallel considerations apply to the theistic defender of MOA* in S4 and KT. Although acceptance of P1* need not by itself generate comparable pressure toward P1, the theist likewise has reason, given the stable conception of God’s nature assumed, to reject radically divergent modal profiles concerning God’s existence across accessible worlds. Without P1, however, a world w in which God exists may access a world v in which God does not exist and which itself accesses only worlds in which God does not exist, so that God’s existence is impossible in v, thereby permitting substantial modal instability concerning God’s existence across accessible worlds. Consequently, the defender of MOA* is similarly pressured toward accepting P1, in which case P2* must be abandoned in order to preserve consistency.

Hence, once a stable conception of God’s modal nature is admitted, endorsement of only one of P1 and P1* while rejecting the other becomes plausibly unsustainable on either side of the dialectic. The dialectical situation therefore collapses back into the same symmetry already present in S5, since the dispute ultimately returns to the question of which of P2 and P2* ought to be accepted. Thus, arguably, the alleged dialectical asymmetry in S4 and KT ultimately dissolves.

6. Neo-Aristotelian Modality and Asymmetry Reversal
But suppose, for the sake of argument, that radically divergent modal profiles concerning God's existence are in fact acceptable, in which case asymmetric endorsement of P1 and P1* is prima facie dialectically permissible. We thereby return to the situation of formal symmetry reached at the end of Section 4. In S4 and KT, the theist argues for theism by endorsing MOA*, together with P2*, while rejecting P1, whereas the atheist argues for atheism by endorsing RMOA, together with P2, while rejecting P1*. Rather than assuming modal stability concerning God’s nature, as in the previous section, let us instead consider the dialectical consequences of adopting a neo-Aristotelian, dispositional or powers-based account of modality of the sort developed by Vetter (2015) and referred to by Fritz, Lo, and Schmid (2026, p. 10).

On such an account, something is possible in a possible world just in case that world possesses the resources to bring it about—or to bring about something that can bring it about, and so on. Within a dispositional or powers-based modal framework, there emerges an independent reason to accept P1*. For if God does not exist in some possible world, then, plausibly, nothing in that world possesses the resources to bring about God’s existence—or to bring about something that can bring about God’s existence, and so on. It follows that God’s non-existence in that world is necessary, and hence that P1* holds. This would once again render RMOA in S4 and KT question-begging, since accepting P1* requires the defender of RMOA to reject P2 in order to preserve consistency.

Importantly, Fritz, Lo, and Schmid (2026, p. 11) themselves briefly acknowledge that a dispositional or powers-based account of modality may provide independent motivation for P1*. They note that, on such a view, if God does not exist, God's existence may be possible only if something possesses the relevant iterated potentiality for bringing about God’s existence, and perhaps only God could possess such a potentiality. In that case, necessarily, if God does not exist, nothing possesses the required potentiality to bring about God’s existence, so that it follows on a potentiality theory of modality that God’s non-existence is necessary, thereby yielding P1*.

What about P1? As Schmid has pointed out to me in personal correspondence, a neo-Aristotelian modal framework may also be used to independently motivate P1. The reasoning is structurally parallel. If God exists in some possible world, then nothing in that world possesses the resources to iteratively bring about God’s non-existence. It follows on a powers-based account of modality that God exists necessarily, and thus that P1 holds. Accordingly, if a neo-Aristotelian, powers-based account of modality generates comparable pressure toward both P1 and P1*, the formal symmetry present in S4 and KT is transformed into dialectical symmetry.

Fritz, Lo, and Schmid (2026, pp. 11–12) nevertheless set dispositional or powers-based modal considerations aside as lying beyond the scope of their article. Yet, once a powers-based framework is incorporated into the dialectical landscape, the formal symmetry obtained at the end of Section 4 may not merely be dialectically preserved. Rather, adopting a neo-Aristotelian modal framework may reverse Fritz, Lo, and Schmid’s alleged dialectical asymmetry in S4 and KT, thereby favoring theism instead of atheism. For, as I shall argue, within such a framework P1* is more plausibly motivated than P1.

Compare P1* to P1 on a neo-Aristotelian account of modality. It appears more plausible that, if God does not exist, nothing possesses the resources to iteratively bring about God’s existence than that, if God exists, nothing possesses the resources to iteratively bring about God’s non-existence. For within a powers-based framework, if God exists, it appears plausible that no non-divine entity could iteratively bring it about that God does not exist, whereas it is less evident that no power to iteratively bring about God's non-existence exists once God itself exists as the most powerful entity.

By contrast, in the absence of God, nothing appears sufficiently powerful to bring about God’s existence. Indeed, not even God could bring about God's existence, since God would already have to exist in order to cause God to exist. Moreover, God is traditionally conceived as a first cause, and a first cause, by definition, cannot itself be caused. If God, conceived as first cause, cannot itself be caused, then in the absence of God nothing could bring about God’s existence, thereby providing further support for P1*. Furthermore, the conception of God as first cause is deeply embedded within traditional theism, whereas the claim that God exists necessarily has been disputed even within theistic tradition, where contingent conceptions of God have been defended (Swinburne 2004). These considerations suggest that P1* may be firmly motivated on a powers-based modal framework, whereas no comparable plausible motivation for P1 appears available. Given these considerations, P1* appears prima facie more plausible than P1 within such a framework.

Consequently, once asymmetric endorsement of P1 and P1* is permitted within S4 and KT, the modal ontological argument for the existence of God in S4 and KT—proceeding from P1*, P2, and P2* while rejecting P1 in order to avoid contradiction—appears prima facie dialectically better motivated than the reverse modal ontological argument for the non-existence of God in S4 and KT, which proceeds from P1, P2*, and P2 while rejecting P1* in order to avoid contradiction. Thus, contrary to what Fritz, Lo, and Schmid suggest, the weaker modal systems S4 and KT need not favor atheism rather than theism. On the contrary, within a neo-Aristotelian, powers-based modal framework, Fritz, Lo, and Schmid's alleged dialectical asymmetry in S4 and KT in favor of atheism may plausibly be reversed in favor of theism rather than merely neutralized.

7. Conclusion
Fritz, Lo, and Schmid have shown that the reverse modal ontological argument for atheism, unlike Plantinga’s original modal ontological argument for theism, remains logically valid in the weaker modal systems S4 and KT. However, I have argued that the alleged dialectical asymmetry in S4 and KT in favor of atheism that Fritz, Lo, and Schmid develop from this result is considerably less tenable than they suggest. Once the wider modal landscape is considered, together with the dialectical pressure exerted by a stable conception of God’s modal nature on both sides of the dialectic toward accepting both P1 and P1*, the alleged dialectical asymmetry in favor of atheism in S4 and KT appears to dissolve. Moreover, as I have argued, taking into account the independent motivation for P1* within a neo-Aristotelian, powers-based account of modality, and permitting asymmetric endorsement of P1 and P1*, the alleged dialectical asymmetry in S4 and KT favoring atheism may even plausibly reverse in favor of theism rather than merely disappear.

References
Fritz, P., Lo, T.-C., & Schmid, J. C. (2026). Symmetry lost: A modal ontological argument for atheism? Noûs, 60(2), 313–327.
Oppy, G., Rasmussen, J., & Schmid, J. (2024). Ontological arguments. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.
Plantinga, A. (1974). The Nature of Necessity. Oxford: Clarendon Press.
Swinburne, R. (2004). The Existence of God. (2nd ed.). Oxford: Oxford University Press.
Vetter, B. (2015). Potentiality: From Dispositions to Modality. Oxford: Oxford University Press.