An evidential puzzle

We let Fx be a conjunction of properties including omnipotence, omniscience, moral perfection and so on. On traditional theistic views God exemplifies Fx in absolutely every possible world or ◻F_G.Footnote ² Since it is an S5 theorem that ◻◻F_G v ◻~◻F_G, it is also true that God exemplifies Fx essentially in every possible world or God fails to exemplify Fx essentially in every possible world. There are, in general, no contingent essential properties in S5. It is impossible, for instance, that anything not essentially human should become essentially human and it is impossible that anything that is not essentially wine should become essentially wine. It is impossible, then, that the traditional God should acquire or lose any essential properties.Footnote ³

Since it is an S5 theorem that ◻◻F_G v ◻~◻F_G, an argument against the existence of the traditional God is fairly easy. There are many actual and possible states of affairs that do not entail the existence of the traditional God. Consider the state of affairs of Jones's suffering a prolonged illness, for instance, or the state of affairs of Smith's enduring a debilitating injury. These states of affairs seem to make it a bit less probable that there exists anything exemplifying essential omnipotence, omniscience, and moral perfection. In any event, they certainly do not seem to entail the existence of the traditional God.

We add quantifiers ∃S and ∀S to the language of the logic S5 meant to range over all possible states of affairs.Footnote ⁴ We also add sentences P(◻F_G|S) meant to express the epistemic probability of ◻F_G given the evidence in S.Footnote ⁵ (1) states that there are at least some possible states of affairs that do not entail the existence of the traditional God.

(1)$$\exists {\rm S}{\sim}\square( {\rm S} \to \square {\rm F}_{\rm G} )$$

There are also states of affairs that seem to be entirely irrelevant to the existence of a traditional God. The state of affairs of there being 228 billion trees in North America, for instance, and the state of affairs of there being roughly three thousand molecules in one millilitre of water provide no evidence at all for (or against) the existence of a traditional God. These states of affairs seem compatible with the existence of nothing exemplifying essential omnipotence, omniscience, and moral perfection. So there are many states of affairs satisfying the formula in (1).

But it follows immediately from (1) that the traditional God is impossible. It is an S5 theorem that ∃S~◻︎(S → ◻F_G) → ∃S◻︎(S → ~◻F_G). If some state of affairs S does not entail that the traditional God exists, then some S entails that the traditional God does not exist. So we derive from (1) that ∃S◻︎(S → ~◻F_G). Indeed, matters are worse, since if some possible state of affairs does not entail that the traditional God exists, then every possible state of affairs entails that the traditional God does not exist. It is also an S5 theorem that ∃S~◻︎(S → ◻F_G) → ∀S◻︎(S → ~◻F_G). So, again, we derive from (1) that ∀S◻︎(S → ~◻F_G).

So, from the fact that there are states of affairs that are evidentially irrelevant to the existence of God, we can prove in S5 that there are at least some worlds in which the traditional God does not exist.

(2)$$\lozenge {\sim}\square {\rm F}_{\rm G}$$

And from (2) it follows that the traditional God is impossible. Since it is true in S5 that ◊~◻F_G → ◻~◻F_G we derive (3) from (2).

(3)$$\square{\sim}\square {\rm F}_{\rm G}$$

So we can derive the conclusion that the traditional God is impossible from the fact that there are states of affairs that are evidentially irrelevant to the existence of the traditional God.

But notice that we can also derive the conclusion that the traditional God is necessary from the fact that there are some states of affairs that do not entail that the traditional God does not exist. Consider for instance the state of affairs of the actual world's including some great goods or the state of affairs of Jones's enjoying good health his entire life. These states of affairs do not entail that the traditional God does not exist. Indeed these states of affairs seem to make it at least a bit more probable that there exists something exemplifying essential omnipotence, omniscience, and moral perfection. Both of these states of affairs satisfy (4).

(4)$$\exists {\rm S}{\sim}\square( {\rm S} \to {\sim}\square {\rm F}_{\rm G} )$$

But it follows immediately from (4) that the traditional God is necessary. It is an S5 theorem that ∃S~◻︎(S → ~◻F_G) → ∃S◻︎(S → ◻F_G). If some state of affairs does not entail that the traditional God does not exist, then some state of affairs entails that the traditional God does exist. So (4) entails ∃S◻︎(S → ◻F_G). But matters are again worse, since if some possible state of affairs does not entail the non-existence of the traditional God, then every possible state of affairs entails that the traditional God exists. It is also an S5 theorem that ∃S~◻︎(S → ~◻F_G) → ∀S◻︎(S → ◻F_G). So, (4) also entails ∀S◻︎(S → ◻F_G).

From the fact that there are states of affairs that satisfy (4), we can prove that there are at least some worlds in which the traditional God exists.Footnote ⁶

(5)$$\lozenge \square {\rm F}_{\rm G}$$

And from (5) it follows that the traditional God exists necessarily. Since it is true in S5 that ◊◻F_G → ◻◻F_G we derive (6) from (5).

(6)$$\square \square {\rm F}_{\rm G}$$

The argument for (6) requires only that there is some state of affairs in some world that does not entail the non-existence of the traditional God. But, indeed, there are many states of affairs in many worlds that increase the probability that there exists a traditional God. And the argument for (3) requires only that there is some state of affairs in some possible world that does not entail the existence of the traditional God. But there certainly are many states of affairs in many worlds that increase the probability that there does not exist a traditional God.

Of course, (6) and (3) cannot both be true, since they are inconsistent. It is an S5 theorem that ◻~◻F_G ↔ ~◻◻F_G. Since the arguments for (6) and (3) are both valid, some premises in these derivations must be false. It is a common assumption in arguments for and against the traditional God that we can adduce evidence S – say, intrinsically evil states of affairs – against the existence of God and we can adduce evidence Sʹ – say, intrinsically good states of affairs – for the existence of God and reach some conclusion about the existence of the traditional God based on our total evidence S and Sʹ. But this is false. The evidential puzzle shows that it is impossible that there should be evidence S against the existence of God and also evidence Sʹ for the existence of God. S and Sʹ together entail a contradiction. In the next section we consider the triviality solution to the evidential puzzle.

The triviality solution

In the previous section we showed that a contradiction could be generated in S5 from the fact that some states of affairs, in some worlds, (even mildly) confirm ◻F_G and some states of affairs, in some worlds, (even mildly) confirm ~◻F_G. We could also produce an argument showing that a contradiction is generated in S5 from the fact alone that some states of affairs are evidentially irrelevant to the truth of ◻F_G. The evidential puzzle is in part a result of theorem (7).Footnote ⁷

(7)$$\forall {\rm SP}( \square {\rm F}_{\rm G} \vert {\rm S}) = 1 \;{\rm v}\;\forall {\rm SP}( {\sim}\square {\rm F}_{\rm G} \vert {\rm S}) = 1$$

According to (7) every state of affairs in every possible world constitutes conclusive evidence for ◻F_G or every state of affairs in every possible world constitutes conclusive evidence for ~◻F_G. (7) rules out the possibility that some states of affairs provide some support for ◻F_G and some states of affairs provide some support for ~◻F_G. It cannot happen.

It is a consequence of (7) that some extremely evil states of affairs – earthquakes, hurricanes, genocides, etc. – constitute conclusive evidence for ◻F_G or some extremely good states of affairs – vast improvements in well-being, the eradication of suffering, etc. – constitute conclusive evidence for ~◻F_G. But how could devastating earthquakes and genocide constitute conclusive evidence in favour of God's existence? And how could vast improvements in well-being or the eradication of suffering constitute conclusive evidence against God's existence?

In S5, whether a states of affairs S constitutes evidence in favour of God's existence or against God's existence does not depend on the content of S. There are no states of affairs that constitute any independent evidence for or against the existence of God. Whether S constitutes evidence for or against the existence of God depends entirely and exclusively on whether or not God exists. Consider the state of affairs of Hume's having scratched his finger, S. S does not entail that God does not exist. But it is true in S5 that if a state of affairs S does not provide conclusive evidence against God's existence, then S provides conclusive evidence for God's existence.Footnote ⁸

(8)$${\rm P}( {\sim} \square {\rm F}_{\rm G} \vert {\rm S}) < 1 \to {\rm P}( \square {\rm F}_{\rm G} \vert {\rm S}) = 1$$

The reasoning to (8) is as described in the evidential puzzle. If P(~◻F_G|S) < 1 then we know that ~◻(S → ~◻F_G). But ~◻(S → ~◻F_G) entails ◻(S → ◻F_G), and ◻(S → ◻F_G) entails P(◻F_G|S) = 1. According to (8), if S does not provide conclusive evidence against God's existence, then S provides conclusive evidence for God's existence. So, Hume's having scratched his finger entails P(◻F_G|S) = 1. That is, Hume's having scratched his finger constitutes conclusive evidence for the existence of God. But there is a much stronger consequence that is also true. (9) states that if some state of affairs S does not make ~◻F_G certain, then every state of affairs makes ◻F_G certain.

(9)$${\rm P}( {\sim} \square {\rm F}_{\rm G} \vert {\rm S}) < 1 \to \forall {\rm SP}( \square {\rm F}_{\rm G} \vert {\rm S}) = 1$$

So, Hume's having scratched his finger entails that every possible state of affairs constitutes conclusive evidence for the existence of God. It follows that all of the extremely evil states of affairs noted above – earthquakes, hurricanes, genocides – all constitute conclusive evidence for the existence of God. But how could that be true?

The reason that Hume's having scratched his finger constitutes conclusive evidence for God's existence is because every possible state of affairs stands in a trivial evidential relation to the traditional God. And this is true since S5 guarantees that either ∀S◻︎(S → ◻F_G) or ∀S◻︎(S → ~◻F_G).

Every state of affairs in every possible world trivially entails that God exists or every state of affairs in every world trivially entails that God does not exist. (10) is a theorem.

(10)$$\square \square {\rm F}_{\rm G} \,{\rm v} \,\square {\sim} \square {\rm F}_{\rm G}$$

If the left disjunct is true, ◻◻F_G, then trivially ∀S◻︎(S → ◻F_G), and if the right disjunct is true ◻~◻F_G then trivially ∀S◻︎(S → ~◻F_G). But if ∀S◻︎(S → ◻F_G), then every possible state of affairs – no matter how bad or good – trivially constitutes conclusive evidence for ◻F_G. And if ∀S◻︎(S → ~◻F_G) is true, then every possible state of affairs – no matter how bad or good – trivially constitutes conclusive evidence for ~◻F_G. We can sum this up in (11) and (12).

(11)$$\forall {\rm S} \square ( {\rm S} \to \square {\rm F}_{\rm G}) \to \forall {\rm SP} ( \square {\rm F}_{\rm G} \vert {\rm S} ) = 1$$

(12)$$\forall {\rm S} \square ( {\rm S} \to {\sim}\square {\rm F}_{\rm G}) \to \forall {\rm SP} ( {\sim} \square {\rm F}_{\rm G} \vert {\rm S} ) = 1$$

It is (10)–(12) that result in (7) above. This explains how earthquakes, hurricanes, and genocides could constitute conclusive evidence for the existence of God. Earthquakes, hurricanes, and genocides constitute only trivial conclusive evidence for the existence of God. Indeed, it is a necessary truth that earthquakes, hurricanes, and genocides constitute trivial conclusive evidence for the existence of God or trivial conclusive evidence against the existence of God. These events are not independent evidence for or against the existence of God. Whether they constitute trivial evidence for or against the existence of God depends entirely on whether God exists.

The evidential puzzle states that if there are states of affairs that (even mildly) confirm ◻F_G and states of affairs that (even mildly) confirm ~◻F_G then we can derive a contradiction._. But the mistake in the evidential puzzle, according to the triviality solution, is the assumption that some states of affairs could non-trivially confirm ◻F_G and that some states of affairs could non-trivially confirm ~◻F_G. The triviality solution denies that extremely good states of affairs provide some non-trivial evidence for ◻F_G and that extremely bad states of affairs provide some non-trivial evidence against ◻F_G.

The states of affairs cited in the evidential puzzle that appear to provide non-trivial evidence for ◻F_G cannot in fact non-trivially confirm ◻F_G. And the states of affairs that appear to provide non-trivial evidence for ~◻F_G cannot in fact provide non-trivial confirmation for ~◻F_G. The assumption that some states of affairs provide independent and non-trivial evidence for or against ◻F_G produces the contradiction in the evidential puzzle.

The mistake in assuming that some state of affairs might non-trivially confirm ◻F_G is perfectly analogous to the mistake in assuming that some state of affairs might non-trivially confirm ◻(Fa v ~Fa).Footnote ⁹ (13) is also a theorem.

(13)$$ \square \square ( {\rm F}a \, {\rm v} \, {\sim} {\rm F}a ) \, {\rm v} \, \square {\sim} \square ( {\rm F}a \, {\rm v}\, {\sim }{\rm F}a )$$

So, every possible state of affairs S either trivially entails ◻(Fa v ~Fa) or trivially entails ~◻(Fa v ~Fa). But then, of course, every possible state of affairs trivially confirms ◻(Fa v ~Fa) or every possible state of affairs trivially confirms ~◻(Fa v ~Fa). (14) is also a theorem.

(14)$$ \forall {\rm SP} ( \square ( {\rm F}a \, {\rm v} \,{\sim} {\rm F}a ) \vert {\rm S} ) = 1 \, {\rm v} \, \forall {\rm SP} ({\sim}\square ( {\rm F}a \, {\rm v} \, {\sim} {\rm F}a ) \vert {\rm S} ) = 1$$

(14) ensures that there is no state of affairs that could provide non-trivial evidence for or against ◻(Fa v ~Fa), since non-trivial evidence for or against ◻(Fa v ~Fa) is impossible. All possible evidence for or against ◻(Fa v ~Fa) is both trivial and conclusive. There is, further, no way of knowing whether a particular state of affairs constitutes evidence for or against ◻(Fa v ~Fa) apart from knowing whether ◻(Fa v ~Fa) is true. If ◻(Fa v ~Fa) is true, then every possible state of affairs constitutes conclusive evidence for ◻(Fa v ~Fa) and if ◻(Fa v ~Fa) is false then every possible state of affairs constitutes evidence against ◻(Fa v ~Fa). Exactly the same conclusions hold for ◻F_G.

There is a fascinating consequence of the triviality solution to the evidential puzzle. If it is true that ◻F_G then every epistemically possible state of affairs provides conclusive evidence for ◻F_G. For any state of affairs S that you might discover, S constitutes conclusive evidence for ◻F_G. The discovery of intensely bad possible worlds, for instance, trivially entails ◻F_G in exactly the same way that intensely bad possible worlds trivially entail ◻(Fa v ~Fa). Both ◻F_G and ◻(Fa v ~Fa) are logically and probabilistically independent of any state of affairs we might discover.

So the discovery alone of intensely bad possible worlds would tell us nothing at all about the existence of the traditional God. Whether the discovery of intensely bad worlds constitutes evidence for or against the existence of God depends entirely on whether God exists. And whether it is true that God exists is logically and probabilistically independent of any possible state of affairs.

Similarly knowing that ◻F_G is true tells us nothing at all about the states of affairs we could discover. Knowing that ◻F_G is true does not, for instance, affect the discovery of extremely evil states of affairs in many possible worlds or the discovery that the actual world is among the worst possible worlds. It does not affect, even, the discovery of pointless evil in vast regions of metaphysical space. If we discover these states of affairs in metaphysical space, then they all trivially entail ◻F_G. Knowing that ◻F_G is true – like knowing that ◻(Fa v ~Fa) is true – does not reduce the kinds of worlds that are epistemically possible or reduce the states of affairs that are epistemically possible.

These consequences are unexpected. In S5 there are no states of affairs in any world that constitute independent evidence for ◻F_G or constitute independent evidence for ~◻F_G. A state of affairs S constitutes evidence for ◻F_G only if ◻F_G is true. In this case S trivially entails ◻F_G and so P(◻F_G|S) = 1. And S constitutes trivial evidence for ~◻F_G only if ~◻F_G is true. In this case S trivially entails ~◻F_G and so P(~◻F_G|S) = 1. The discovery that S is true, for any S whatsoever, does not itself constitute any evidence at all for or against the existence of the traditional God.

On abandoning S5

The epistemological consequences of S5 for traditional theism and traditional atheism are extremely unconventional. In addition to the fact that no state of affairs in any world constitutes any non-trivial evidence for either ◻F_G or for ~◻F_G, S5 entails that various agnostic positions are impossible. The proposition P(◻F_G|S) = .5 & P(~◻F_G|S) = .5 entails a contradiction, for instance, and so does every proposition of the form P(◻F_G|S) = n, n(0 < n < 1). So agnosticism is impossible.

S4 is a logic of relative necessity, so the proposition ◻F_G is true just if F_G is true in every relatively possible world. So, the fact that the traditional God necessarily exists does not entail that the traditional God exists in absolutely every possible world. Assuming S4, the fact that ◻F_G is true in some world is perfectly consistent with ◻F_G being false in many others. But the highly unconventional thesis that God might exist in some worlds and not exist in others – ◊◻F_G & ◊~◻F_G – is not consistent with standard conceptions of the traditional God.

It is an epistemological consequence of S4 that P(~◻F_G|S) > 0 → ~◻F_G and also that ◻︎F_G → ∀SP(◻︎F_G|S) = 1. The latter states that the traditional God exists only if it is certain given any possible state of affairs that the traditional God exists. The former states that if some state of affairs constitutes some evidence against the traditional God, then the traditional God does not exist. Neither of these epistemic propositions is any more plausible than the epistemic consequences of S5.

There are also metaphysical consequences of S4 that are untenable. Since it is not a theorem of S4 that ∀x(◊◻Fx → ◻Fx), it is perfectly possible that an object should acquire an essence that it does not now exemplify. Plantinga might have acquired the essence of an alligator in addition to his human essence. But could God acquire the essence of a beetle? If S4 is the logic of metaphysical possibility, then anything can acquire another essence – so God might indeed become essentially a beetle.Footnote ¹⁰

It is of course possible to weaken S4 to K_σρ, but the metaphysical consequences of K_σρ are even more extreme.Footnote ¹¹ K_σρ validates neither ∀x(◊◻Fx → ◻Fx) nor ∀x(◻Fx → ◻◻Fx), so the traditional God could survive the loss of every essential property – essential omnipotence, omniscience, and moral perfection. So the consequences of K_σρ for traditional theism and traditional atheism are less plausible than the consequences of S5.

Concluding remarks

It does seem that there are states of affairs in the pluriverse that provide at least some independent evidence in favour of ◻F_G. Extremely valuable states of affairs and extremely valuable worlds are standardly taken to provide some independent evidence that God exists. There are also states of affairs in the pluriverse that seem to provide at least some independent evidence in favour of ~◻F_G. Extremely bad states of affairs are standardly taken to provide at least some independent evidence that God does not exist.

But the logic of S5 ensures, on the contrary, that there is evidence for and against the existence of God only if a contradiction is true. So, we are left in the unexpected position of denying that there are various states of affairs in the pluriverse providing various degrees of evidence for and against the traditional God.

It follows from the triviality solution to the evidential puzzle that no states of affairs provide any independent evidence for or against the existence of God. The very worst possible states of affairs might be conclusive evidence for ◻F_G and the very best possible states of affairs might be conclusive evidence for ~◻F_G. If every possible state of affairs trivially entails ◻F_G then we could discover many extremely evil possible worlds all of which trivially entail ◻F_G. If every possible state of affairs trivially entails ~◻F_G we could discover many morally and naturally perfect worlds all of which trivially entail ~◻F_G.

In general, for any state of affairs S – no matter how good or bad S happens to be – S could be, for all we know, conclusive evidence in favour of ◻F_G, and S could be, for all we know, conclusive evidence in favour of ~◻F_G. We cannot determine whether S is conclusive evidence for or against ◻F_G until we know whether it is true that ◻F_G or true that ~◻F_G.

S5 is the most widely defended logic of metaphysical possibility, but the metaphysical and epistemological consequences of S5 might warrant abandoning S5 for S4 or K_σρ. But the consequences of S4 and K_σρ for philosophical theology are equally unconventional. S4 guarantees that God does not exist if there is the slightest evidence against the existence of God. And K_σρ guarantees that God might survive the loss or acquisition of any essential property at all.