A critique of Symmetry Lost: By Peter Fritz, Tien-Chun Lo, and Joseph C. Schmid

Formal Proof: The RMOA At Best Establishes ¬□g in w0

Objective:

My aim to prove that if g (an MGB exists) is metaphysically possible (◇g), the RMOA’s premises P1 (□(g → □g)), P2 (♢g), and P2* (♢¬g) in KT/S4 at best establish ¬□g (it is not necessary that God exists) in the actual world (w0), not ¬g (God does not exist in w0). The derivation of ¬g in w0 is formally valid but metaphysically overreaching, as ◇g implies g is possible in w0, making ¬g possibly false. The conclusion ¬g in w0 requires an unjustified assumption about w0’s metaphysical status, limiting the RMOA’s conclusion to ¬□g.

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).

Metaphysical Possibility: A proposition p is metaphysically possible if it is consistent with the laws of metaphysics, i.e., no contradiction in p’s properties.

Epistemic Possibility: A proposition p is epistemically possible if it is conceivable (not known to be false).

Law of Non-Contradiction: ¬(p ∧ ¬p) in any world.

RMOA Recap:

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

P2: ♢g (Possibly God exists).

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

C*: ¬g (God does not exist in w0).

Derivation in KT/S4:

From P1, □(g → □g) ⊢ g → □g in w0 (by T: □p → p).

From P2*, ♢¬g in w0 ⊢ ∃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, so g ⊢ □g in w0.

From (2), ¬□g in w0.

By modus tollens (g → □g, ¬□g ⊢ ¬g), ¬g in w0.

Authors’ Model (Figure 3b):

Worlds: w0 (¬g), w1 (g), w2 (¬g).

Accessibility: R(w0, w0), R(w0, w1), R(w0, w2), R(w1, w1), R(w2, w2).

P1 holds: In w1, g ⊢ □g (w1 accesses only itself); in w0 and w2, g is false, so g → □g is trivially true.

P2 holds: ♢g in w0 (R(w0, w1) ∧ g(w1)).

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

C*: ¬g in w0 follows.

Proof: The RMOA At Best Establishes ¬□g in w0

Step 1: Establish g’s Metaphysical Possibility

The MGB’s properties (omnipotence, omniscience, Omni benevolence) are coherent, with no implicit contradictions (Plantinga 1974, Swinburne 1993). For example, Plantinga’s free will defence counters the Logical Problem of Evil, showing consistency. Thus, ◇g is metaphysically plausible: ∃w1: g(w1).

P2 (♢g) reflects this: in the authors’ model, w1 has g, and R(w0, w1) holds.

Step 2: Implications of ◇g in S5

In S5 (universal accessibility):

If ◇g, ∃w1: g(w1).

By P1 (□(g → □g)), g(w1) ⊢ □g(w1) (g in all worlds w’ accessible from w1).

In S5, accessibility is universal, so □g(w1) ⊢ □g (g in all worlds), implying g in w0 (Modal Ontological Argument, MOA).

If ♢¬g (P2*), then □¬g (by axiom 5: ◇□¬g → □¬g), so ¬g in all worlds.

This contradicts ◇g, as g(w1) ∧ ¬g(w1) violates the law of non-contradiction (¬(p ∧ ¬p)). Thus, if ◇g is true, ♢¬g is false unless M → ⊥ (a contradiction in g’s properties) is shown.

Since g is coherent (no g → ⊥), P2* is false in S5, and ¬g cannot hold in w0. P1, P2, and P2* are inconsistent in S5.

Step 3: Implications of ◇g in KT/S4

In KT (reflexive) and S4 (reflexive and transitive), accessibility is non-universal:

If ◇g, ∃w1: g(w1). By P1, g(w1) ⊢ □g(w1) (g in all worlds accessible from w1). However, w1’s accessible worlds may not include w0, so ◇g does not imply g in w0.

P2* (♢¬g in w0) ⊢ ∃w2: R(w0, w2) ∧ ¬g(w2), so ¬□g in w0 (some accessible world lacks g).

The RMOA derives ¬g in w0 via modus tollens: g → □g (from P1), ¬□g (from P2*), ⊢ ¬g.

Key Issue: ◇g (w1: g, R(w0, w1)) implies g is possible relative to w0 (♢g). If w0 were a g world, ¬g would be false in w0. The derivation assumes w0 is a ¬g world, but ◇g means g is possible in w0, so ¬g in w0 is possibly false (♢g ⊢ ¬□¬g).

Thus, P2* guarantees ¬□g in w0 (some accessible world has ¬g), but ¬g in w0 is not necessitated, as g’s possibility allows w0 to be a g world.

Step 4: P2’s Limitation in KT/S4*

P2* (♢¬g) is motivated by epistemic possibility (conceivability of ¬g). In the authors’ model, it holds: R(w0, w2) ∧ ¬g(w2).

However, if ◇g is true, g is metaphysically possible (no g → ⊥). Deriving ¬g in w0 assumes ¬g is true in the actual world, but ◇g implies g is possible in w0 (♢g).

In KT/S4, ¬g in w0 is contingent on the model’s assignment (w0: ¬g). Since ◇g allows w0 to be a g world, ¬g in w0 is possibly false (¬□¬g). Thus, P2* establishes ¬□g (not all accessible worlds have g), but ¬g in w0 is not guaranteed.

If ◇g is metaphysically true, ¬g’s truth in w0 requires g to be impossible g → ⊥), as ◇g precludes ¬g’s metaphysical possibility without a contradiction. Since g → ⊥ is not shown, P2* may be false, undermining ¬g in w0.

Step 5: Flaw in the RMOA’s Conclusion

The RMOA’s derivation of ¬g in w0 is formally valid in KT/S4, as the authors’ model shows P1, P2, and P2* are consistent, and ¬g follows via modus tollens.

However, ◇g (P2) implies g is possible in w0 (♢g), so ¬g in w0 is not guaranteed, as w0 could be a g world. The derivation assumes w0 is a ¬g world, but this is model-specific, not metaphysically necessary.

P2*’s truth (♢¬g) guarantees ¬□g in w0, as ∃w2: ¬g(w2). The step to ¬g in w0 via P1 (g → □g) overreaches, as ◇g allows g in w0, making ¬g possibly false.

Thus, the RMOA’s strongest conclusion is ¬□g in w0 (God’s existence is not necessary), not ¬g (God does not exist in w0).

Step 6: KT/S4’s Inadequacy for Metaphysical Debates

KT/S4’s non-universal accessibility yields contingent conclusions (¬g in w0, g possible in w1), inadequate for theism-atheism debates, which concern metaphysical necessity (□g or □¬g).

In S5, ◇g ⊢ g, and ♢¬g ⊢ □¬g (requiring g → ⊥). The RMOA’s ¬g in w0 lacks S5’s decisiveness, as ◇g allows g in w0.

S5 is the standard for such debates (Plantinga 1974, Gödel 1970), as universal accessibility reflects metaphysical necessity. The RMOA’s use of KT/S4 sidesteps the need for g → ⊥ but fails to establish ¬g as metaphysically true.

Conclusion

If ◇g is metaphysically possible, the RMOA’s derivation of ¬g in w0 is formally valid in KT/S4 but metaphysically overreaching. P2* (♢¬g) establishes ¬□g in w0, but ◇g (P2) implies g is possible in w0, so ¬g is possibly false (¬□¬g). Thus, the RMOA at best concludes ¬□g (God’s existence is not necessary), not ¬g (God does not exist in w0). The claim ¬g in w0 assumes w0 is a ¬g world, but ◇g undermines this, as w0 could be a g world. The RMOA’s reliance on P2*’s epistemic possibility (conceivability) does not justify ¬g’s metaphysical truth, especially if ◇g is true (no g → ⊥).