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Evil is not evidence

Published online by Cambridge University Press:  24 March 2022

Mike Almeida*
Affiliation:
Department of Philosophy and Classics, University of Texas at San Antonio, San Antonio, Texas78249, USA
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Abstract

The article aims to show that, if S5 is the logic of metaphysical necessity, then no state of affairs in any possible world constitutes any non-trivial evidence for or against the existence of the traditional God. There might well be states of affairs in some worlds describing extraordinary goods and extraordinary evils, but it is false that these states of affairs constitute any (non-trivial) evidence for or against the existence of God. The epistemological and metaphysical consequences for philosophical theology of assuming that S4 or Kσρ is the logic of metaphysical necessity are equally untenable. S4 guarantees that God does not exist if there is the slightest evidence against the existence of God. And Kσρ guarantees that God might survive the loss or acquisition of any essential property at all.

Type
Original Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press

Introduction

Let ◻Fx be the conjunctive property of essential moral perfection, omnipotence, and omniscience. The traditional God exemplifies the conjunctive property ◻Fx. The standard epistemic view on intrinsic evil is that intrinsically evil states of affair constitute at least some evidence against ◻FG. Indeed, it is widely held that an intrinsic evil in some world constitutes at least some evidence against the truth of ◻FG in any world.

In the section ‘An evidential puzzle’, I offer an evidential puzzle. If S5 is the logic of metaphysical necessity, then a contradiction is derivable from the simple assumption that there are some states of affairs in some worlds that confirm ◻FG and some states of affairs in some worlds that confirm ~◻FG.Footnote 1 It is a consequence of the evidential puzzle that it is impossible that there is some evidence in favour of the traditional God and also some evidence against the traditional God.

In the section ‘The triviality solution’ I offer the triviality solution to the evidential puzzle. According to the triviality solution, independent evidence for or against the existence of God is impossible. If it is true that ◻FG, then every state of affairs in every possible world trivially entails ◻FG, and so every state of affairs in every possible world trivially confirms ◻FG. If it is true that ~◻FG, then every state of affairs in every possible world trivially entails ~◻FG and so every state of affairs in every possible world trivially confirms ~◻FG. There is no independent evidence for or against ◻FG in any possible world.

If it is true that ◻FG, then every intrinsic evil in every world trivially confirms ◻FG. If it is true that ~◻FG, then every intrinsic good in every world trivially confirms ~◻FG.

It is a consequence of the triviality solution that even epistemically possible states of affairs trivially confirm ◻FG or trivially confirm ~◻FG. Indeed every epistemically possible state of affairs trivially confirms ~◻FG or every epistemically possible state of affairs trivially confirms ◻FG. So, we could not discover any independent evidence for or against the existence of God.

In the section ‘On abandoning S5’ I discuss the epistemological and metaphysical consequences of assuming a weaker logic – S4 or Kσρ – is the logic of metaphysical necessity. I offer some concluding remarks in the final section.

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 ◻FG.Footnote 2 Since it is an S5 theorem that ◻◻FG v ◻~◻FG, 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 ◻◻FG v ◻~◻FG, 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(◻FG|S) meant to express the epistemic probability of ◻FG 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 → ◻FG) → ∃S◻︎(S → ~◻FG). 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 → ~◻FG). 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 → ◻FG) → ∀S◻︎(S → ~◻FG). So, again, we derive from (1) that ∀S◻︎(S → ~◻FG).

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 ◊~◻FG → ◻~◻FG 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 → ~◻FG) → ∃S◻︎(S → ◻FG). 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 → ◻FG). 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 → ~◻FG) → ∀S◻︎(S → ◻FG). So, (4) also entails ∀S◻︎(S → ◻FG).

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 ◊◻FG → ◻◻FG 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 ◻~◻FG ↔ ~◻◻FG. 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 ◻FG and some states of affairs, in some worlds, (even mildly) confirm ~◻FG. 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 ◻FG. 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 ◻FG or every state of affairs in every possible world constitutes conclusive evidence for ~◻FG. (7) rules out the possibility that some states of affairs provide some support for ◻FG and some states of affairs provide some support for ~◻FG. It cannot happen.

It is a consequence of (7) that some extremely evil states of affairs – earthquakes, hurricanes, genocides, etc. – constitute conclusive evidence for ◻FG or some extremely good states of affairs – vast improvements in well-being, the eradication of suffering, etc. – constitute conclusive evidence for ~◻FG. 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(~◻FG|S) < 1 then we know that ~◻(S → ~◻FG). But ~◻(S → ~◻FG) entails ◻(S → ◻FG), and ◻(S → ◻FG) entails P(◻FG|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(◻FG|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 ~◻FG certain, then every state of affairs makes ◻FG 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 → ◻FG) or ∀S◻︎(S → ~◻FG).

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, ◻◻FG, then trivially ∀S◻︎(S → ◻FG), and if the right disjunct is true ◻~◻FG then trivially ∀S◻︎(S → ~◻FG). But if ∀S◻︎(S → ◻FG), then every possible state of affairs – no matter how bad or good – trivially constitutes conclusive evidence for ◻FG. And if ∀S◻︎(S → ~◻FG) is true, then every possible state of affairs – no matter how bad or good – trivially constitutes conclusive evidence for ~◻FG. 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 ◻FG and states of affairs that (even mildly) confirm ~◻FG 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 ◻FG and that some states of affairs could non-trivially confirm ~◻FG. The triviality solution denies that extremely good states of affairs provide some non-trivial evidence for ◻FG and that extremely bad states of affairs provide some non-trivial evidence against ◻FG.

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

The mistake in assuming that some state of affairs might non-trivially confirm ◻FG 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 ◻FG.

There is a fascinating consequence of the triviality solution to the evidential puzzle. If it is true that ◻FG then every epistemically possible state of affairs provides conclusive evidence for ◻FG. For any state of affairs S that you might discover, S constitutes conclusive evidence for ◻FG. The discovery of intensely bad possible worlds, for instance, trivially entails ◻FG in exactly the same way that intensely bad possible worlds trivially entail ◻(Fa v ~Fa). Both ◻FG 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 ◻FG is true tells us nothing at all about the states of affairs we could discover. Knowing that ◻FG 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 ◻FG. Knowing that ◻FG 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 ◻FG or constitute independent evidence for ~◻FG. A state of affairs S constitutes evidence for ◻FG only if ◻FG is true. In this case S trivially entails ◻FG and so P(◻FG|S) = 1. And S constitutes trivial evidence for ~◻FG only if ~◻FG is true. In this case S trivially entails ~◻FG and so P(~◻FG|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 ◻FG or for ~◻FG, S5 entails that various agnostic positions are impossible. The proposition P(◻FG|S) = .5 & P(~◻FG|S) = .5 entails a contradiction, for instance, and so does every proposition of the form P(◻FG|S) = n, n(0 < n < 1). So agnosticism is impossible.

S4 is a logic of relative necessity, so the proposition ◻FG is true just if FG 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 ◻FG is true in some world is perfectly consistent with ◻FG being false in many others. But the highly unconventional thesis that God might exist in some worlds and not exist in others – ◊◻FG & ◊~◻FG – is not consistent with standard conceptions of the traditional God.

It is an epistemological consequence of S4 that P(~◻FG|S) > 0 → ~◻FG and also that ◻︎FG → ∀SP(◻︎FG|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 Kσρ, but the metaphysical consequences of Kσρ are even more extreme.Footnote 11 Kσρ 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 Kσρ 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 ◻FG. 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 ~◻FG. 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 ◻FG and the very best possible states of affairs might be conclusive evidence for ~◻FG. If every possible state of affairs trivially entails ◻FG then we could discover many extremely evil possible worlds all of which trivially entail ◻FG. If every possible state of affairs trivially entails ~◻FG we could discover many morally and naturally perfect worlds all of which trivially entail ~◻FG.

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 ◻FG, and S could be, for all we know, conclusive evidence in favour of ~◻FG. We cannot determine whether S is conclusive evidence for or against ◻FG until we know whether it is true that ◻FG or true that ~◻FG.

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 Kσρ. But the consequences of S4 and Kσρ 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 Kσρ guarantees that God might survive the loss or acquisition of any essential property at all.

Acknowledgements

Thanks to David Johnson, Josh Thurow, and Abraham Graber for discussion of earlier drafts of this article. Thanks also to two referees for Religious Studies.

Footnotes

1. S5 is the most widely defended logic of metaphysical possibility. See, for instance, Kripke (Reference Kripke1972); Plantinga (Reference Plantinga1974); Forbes (Reference Forbes1986); Lewis (Reference Lewis1986); Linsky and Zalta (Reference Linsky and Zalta1994), (Reference Linsky and Zalta1996); Williamson (Reference Williamson2013); Pruss and Rasmussen (Reference Pruss and Rasmussen2018; Hale (Reference Hale2020).

S5 is also the most widely accepted logic among philosophers of religion. Its popularity is largely due to the fact that the logic is necessary to a variety of important theistic and atheistic arguments: for example, modal ontological arguments, modal arguments from evil, arguments from the perfections, various forms of cosmological arguments, etc. For a brief discussion of the consequences of adopting S4 or Kσρ as the logic of metaphysical possibility see section ‘On abandoning S5’ below.

2. On traditional theism, God exemplifies Fx essentially just if God exemplifies Fx in absolutely every possible world. In classical S5, everything exists in every possible world, so an object b is essentially Fx just if necessarily b is Fx. In non-classical S5, objects exemplify properties in worlds in which they do not exist. In either case, and certainly in the case of God, necessarily b is Fx just if b is essentially Fx.

3. This point concerns surviving the gain or loss of an essential property and not merely the gain or loss of such properties. In fact, it is impossible that God should do either in S5. If God exemplifies ◻︎Fx then God loses that essential property if there are worlds in which God is ~◻︎Fx. This can be true (depending on the logic involved) even if there are no possible worlds in which God is ~Fx.

4. I assume that states of affairs are abstract and necessarily existing objects. I use ‘S’ as a state of affairs variable in some contexts permitting quantification and a constant in other contexts. The usage should be clear from the context.

5. The epistemic probability in P(◻FG|S) is the probability of ◻FG given evidence S. It measures the extent to which S confirms ◻FG. Similarly, the probabilities of rival ‘big bang’ and steady state theories on the evidence is epistemic. They measure the extent to which the available physical evidence confirms each of the competing theories contexts. Mellor (Reference Mellor2005).

6. From (4) ∃S~◻︎(S → ~◻FG) and ∃S~◻︎(S → ~◻FG) → ∀S◻︎(S → ◻FG) we derive ∀S◻︎(S → ◻FG) or that every possible state of affairs entails that God exists. Let S be one of those states of affairs. From S and ∀S◻︎(S → ◻FG) it follows that ◻FG. And from ◻FG and S5 it follows that (5) ◊◻FG.

7. Theorem (7) ∀SP(◻FG|S) = 1 v ∀SP(~◻FG|S) = 1 follows from the S5 theorem ◻~◻FG v ◻◻FG. If the left disjunct is true, then (i) ∀S◻︎(S → ~◻FG) and if the right disjunct is true then (ii) ∀S◻︎(S → ◻FG. From (i) it follows that ∀SP(~◻FG|S) = 1 and from (ii) it follows that ∀SP(◻FG|S) = 1.

8. We assume that S is a possible state of affairs unless otherwise indicated and we assume that ∀S and ∃S range over all possible states of affairs. (8) is derived as follows. Since it is true that ◻︎(S → ~◻FG) → P(~◻FG|S) = 1, we know that the contrapositive is true P(~◻FG|S) < 1 → ~◻︎(S → ~◻FG). So, if P(~◻FG|S) < 1, then ~◻︎(S → ~◻FG). But since ~◻︎(S → ~◻FG) → ◻︎(S → ◻FG), we know that if P(~◻FG|S) < 1 then ◻︎(S → ◻FG). And since ◻︎(S → ◻FG) only if P(◻FG|S) = 1, it follows that (8) is true, viz. P(~◻FG|S) < 1 → P(◻FG|S) = 1.

9. In S5 the logical truth ◻(Fa v ~Fa) faces epistemological problems analogous to than ◻FG. If an agent assigns a purported logical truth L, P(L|S) = n, n(0 < n < 1), then ~◻(S → ~L) and since ~◻(S → ~L) → ◻(S → L), it follows from that assignment that P(L|S) = 1.

10. S4 does include the theorem ∀x(◻Fx → ◻◻Fx) so nothing could lose an essence or essential property that it does exemplify. God might acquire the essence of a beetle, but God could not thereby cease being divine. Still, it at least strains metaphysical credibility that God should acquire the essence of a beetle.

11. Models for Kσρ (or B) are symmetric and reflexive.

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