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The Dilemma Once More

P1. If it is possible that necessarily there is an omniscient, omnipotent, omnibenevolent being, necessarily there is an omniscient, omnipotent, omnibenevolent being. (From axiom 5 of S5)[1]

P2. Either the proposition “necessarily there is an omniscient, omnipotent, omnibenevolent being” entails the proposition “there is gratuitous evil and suffering” or it is not the case the proposition “necessarily there is an omniscient, omnipotent, omnibenevolent being” entails the proposition “there is gratuitous evil and suffering”. (From the Law of the Excluded Middle)[2]

P3. For all propositions p if there is some proposition q such that it is not the case that p entails q, then possibly p. (Contraposition of the Principle of Explosion)[3][4]

C1. If it is not the case the proposition “necessarily there is an omniscient, omnipotent, omnibenevolent being” entails the proposition “there is gratuitous evil and suffering”, it is possible that necessarily there is an omniscient, omnipotent, omnibenevolent being. [From P3][5]

C2. If it is not the case the proposition “necessarily there is an omniscient, omnipotent, omnibenevolent being” entails the proposition “there is gratuitous evil and suffering”, necessarily there is an omniscient, omnipotent, omnibenevolent being. [From P1 and C1, Hypothetical Syllogism][6]

P4. If the proposition “necessarily there is an omniscient, omnipotent, omnibenevolent being” entails the proposition “there is gratuitous evil and suffering”, gratuitous evil and suffering is not counter-evidence to the proposition “necessarily there is an omniscient, omnipotent, omnibenevolent being”.[7]

C3. Either necessarily there is an omniscient, omnipotent, omnibenevolent being, or gratuitous evil and suffering is not counter-evidence to the proposition “necessarily there is an omniscient, omnipotent, omnibenevolent being.” (From P2,C2,P4 Constructive Dilemma)[8][9]

[1] The axiom in S5 can be found here: https://en.m.wikipedia.org/wiki/S5_(modal_logic). So, given the axiom 5 of S5: ♢p → ☐♢p

Here is the proof for P1:

Let

Kx ≝ x is omniscient
Px ≝ x is omnipotent
Bx ≝ x is omnibenevolent

1 ~ ☐(∃x)[(Kx ∧ Px) ∧ Bx] (Assump. CP)
2 ~ ☐~~(∃x)[(Kx ∧ Px) ∧ Bx] (1 DN)
3 ♢~(∃x)[(Kx ∧ Px) ∧ Bx] (2 ME)
4 ☐♢~(∃x)[(Kx ∧ Px) ∧ Bx] (3 Axiom 5)
5 ☐~~♢~(∃x)[(Kx ∧ Px) ∧ Bx] (4 DN)
6 ☐~☐(∃x)[(Kx ∧ Px) ∧ Bx] (5 ME)
7 ~☐(∃x)[(Kx ∧ Px) ∧ Bx] → ☐~☐(∃x)[(Kx ∧ Px) ∧ Bx] (CP 1-6)
8 ~☐~☐(∃x)[(Kx ∧ Px) ∧ Bx] → ~~☐(∃x)[(Kx ∧ Px) ∧ Bx] (7 Contra)
9 ~☐~☐(∃x)[(Kx ∧ Px) ∧ Bx] → ☐(∃x)[(Kx ∧ Px) ∧ Bx] (8 DN)
10 ♢☐(∃x)[(Kx ∧ Px) ∧ Bx] → ☐(∃x)[(Kx ∧ Px) ∧ Bx] (9 ME)

[2] The Law of the Excluded Middle can be found here: https://en.m.wikipedia.org/wiki/Law_of_excluded_middle

[3] Contraposition can be found here: https://en.m.wikipedia.org/wiki/Contraposition

[4] The Principle of Explosion can be found here: https://en.m.wikipedia.org/wiki/Principle_of_explosion

Here is the proof that P3 is the contrapositive of the Principle of Explosion, which we will state as follows: (∀p)[~♢p → (∀q)(p ⊨ q)], for all propositions p, if p is impossible, then for all propositions q1, p entails q.

1 (∀p)[~♢p → (∀q)(p ⊨ q)] (Principle of Explosion)
2 ~♢φ → (∀q)(φ ⊨ q) (1 UI)
3 ~(∀q)(φ ⊨ q) → ~~♢φ (2 Contra)
4 (∃q)~(φ ⊨ q) → ~~♢φ (3 QN)
5 (∃q)~(φ ⊨ q) → ♢φ (4 DN)
6 (∀p)(∃q)~(p ⊨ q) → ♢p] (5 UG)

[5] Here is the proof that C1 follows from P3:

Let

G ≝ ☐(∃x)[(Kx ∧ Px) ∧ Bx]
E ≝ ‘there is gratuitous evil and suffering’

1 (∀p)(∃q)~(p ⊨ q) → ♢p] (P3)
2 ~(G ⊨ E) (Assump. CP)
3 (∃q)~(G ⊨ q) → ♢G (1 UI)
4 (∃q)~(G ⊨ q) (2 EG)
5 ♢G (3,4 MP)
6 ~(G ⊨ E) → ♢G (205 CP)
7 ~(G ⊨ E) → ♢☐(∃x)[(Kx ∧ Px) ∧ Bx] (6 def. of ‘G’)

Thus Line 7 (C1) follows from Line 1 (P3), QED.

[6] Hypothetical Syllogism can be found here: https://en.m.wikipedia.org/wiki/Hypothetical_syllogism

[7] This premise is defended on given a Bayesian interpretation of counter-evidence:
(∀p)(∀q){[P(p|q)<P(p)] ⊃ Cqp} (read as: for all proposition p and q, if the probability of q given p is less than the probability of q unconditioned, then q is counter-evidence for p).

If we assume G ⊨ E, then by Logical Consequence P(E|G) = 1, but if E is counter-evidence to G, then it must be the case that P(G|E) < P(G). But both of these statements about probabilities cannot be true.

According to Bayes’ Theorem:

P(E|G) = [P(E)/P(G)] x P(G|E)

So given P(E|G) = 1

We can infer:

P(G)/P(G|E) = P(E)

But given 0 ≤ P(E) ≤ 1, it is not possible for P(G)/P(G|E) = P(E) and P(G|E) < P(G), as whenever the denominator is less than the numerator, the result is greater than 1.

Hence, we must reject the assumption that [P(E|G) = 1] ∧ [P(G|E) < P(G)]. This provides us with the following defense of P4:

1 ~{[P(E|G) = 1] ∧ [P(G|E) < P(G)]} (Result from the proof by contradiction above)
2 ~[P(E|G) = 1] ∨ ~[P(G|E) < P(G)] (1 DeM)
3 [P(E|G) = 1] → ~[P(G|E) < P(G)] (2 Impl)
4 [G ⊨ E] → [P(E|G) = 1] (by Logical Consequence)
5 [G ⊨ E] → ~[P(G|E) < P(G)] (3,4 HS)

And line 5 is just what is meant by P4.

[8] Constructive Dilemma can be found here: https://en.m.wikipedia.org/wiki/Constructive_dilemma

[9] The proof of the entire argument is as follows:

1 ♢☐(∃x)[(Kx ∧ Px) ∧ Bx] → ☐(∃x)[(Kx ∧ Px) ∧ Bx] (Premise)
2 (G ⊨ E) ∨ ~(G ⊨ E) (Premise)
3 (∀p)(∃q)~(p ⊨ q) → ♢p] (Premise)
4 [G ⊨ E] → ~[P(G|E) < P(G)] (Premise)
5 ~(G ⊨ E) (Assump CP)
6 (∃q)~(G ⊨ q) → ♢G (3 UI)
7 (∃q)~(G ⊨ q) (5 EG)
8 ♢G (6,7 MP)
9 ~(G ⊨ E) → ♢G (5-8 CP)
10 ~(G ⊨ E) → ♢☐(∃x)[(Kx ∧ Px) ∧ Bx] (9 definition of ‘G’)
11 ~(G ⊨ E) → ☐(∃x)[(Kx ∧ Px) ∧ Bx] (1,10 HS)
12 ☐(∃x)[(Kx ∧ Px) ∧ Bx] ∨ ~[P(G|E) < P(G)] (2,4,11 CD)

The Dilemma Theodicy

  1. By definition, God is a maximally great being, i.e. an omnipotent, omniscience, morally perfect being in every possible world.
  2. Any argument against God’s existence that depends on a premise of the form “If God were to exist, then we would expect there to be x” (hereafter, the “counterfactual” premise) must have a justification, either by way of a trivial entailment, given the incoherence of the concept of God, and so the impossibility of the existence of God, or by way of the defense of a substantive counterfactual implication, given a thoroughgoing conceptual analysis of the concept of God, and the sorts of states of affairs implied by God’s existence.
  3. If the justification for the “counterfactual” premise is by way of a trivial entailment, given the incoherence of the concept of God, and so the impossibility of the existence of God, then the justification for the “counterfactual” premise begs the question of any argument against God’s existence that depends upon the “counterfactual” premise, which means the argument containing the “counterfactual” premise is informally fallacious.
  4. If the justification for the “counterfactual” premise is by way of a defense of a substantive counterfactual implication, given a thoroughgoing conceptual analysis of the concept of God, and the sorts of states of affairs implied by God’s existence, then the justification depends upon the metaphysical possibility of God, and the sorts of states of affairs that obtain in the nearest possible worlds where God exists, which also serves as a justification for the possibility premise of the modal ontological argument, by which the existence of God can be directly demonstrated from His metaphysical possibility, based upon an axiom of S5.
  5. But, a successful argument cannot be informally fallacious, nor can a successful argument depend on a justification that directly implies the contradictory of the its conclusion.
  6. So, no argument against God’s existence that depends on the “counterfactual” premise is successful.

Escaping the horns would require a substantive justification of the counterfactual premise that does not imply any real metaphysical possibility of God.  Would such a justification be compelling enough for a theist, or neutral party to accept the truth of the counterfactual premise? 

Vexing Links (6/13/2015)

Some recent links of note:

The Ontological Dilemma Against Gratuitous Evil

Here is a quick one, two, as it were… A good reason to think God exists and that the problem of evil is unsound1:
1. Either the concept of a maximally great being (a being that is omniscient, omnipotent, and morally perfect in every possible world) is self-consistent or not.
2. If the concept of a maximally great being is self-consistent, then there is at least one possible world where a maximally great being exists.
3. If there is at least one possible world where a maximally great being exists, then a maximally great being exists in every possible world.
4. If a maximally great being exists in every possible world, then a maximally great being exists.
5. If the concept of a maximally great being is not self-consistent, then the atheologian does not provide sufficient justification for arguing that the existence of a maximally great being entails the impossibility of gratuitous evil.
6. The atheologian provides sufficient justification for premise that the existence of a maximally great being entails the impossibility of gratuitous evil.
Therefore:
7. A maximally great being exists.
And…
8. If the atheologian provides sufficient justification for the premise that the existence of a maximally great being entails the impossibility of gratuitous evil, then the existence of a maximally great being entails the impossibility of gratuitous evil.
So…
8. Gratuitous evil is impossible.

I think the atheologian would have to object to (5) or (6). Giving up on (6) would mean that the atheologian abandons defending the problem of evil. I am more interested in denying (5). The denial of (5) means that the atheologian can provide sufficient justification for the premise of the problem of evil while the concept of a maximally great being is not self-consistent. This seems implausible to me, since most atheologians appeal to a conceptual analysis moral perfection, omnipotence, and omniscience in explaining what might be entailed by those properties. No atheologian whom I am aware of appeals to the non-self-consistence of a maximally great being in justifying those premises. So the practice of atheologians betrays the fact that they rely on a justification grounded in self-consistency rather than, say, the principle of explosion.

One might object and say that the atheologian is agnostic towards whether a maximally-great being is self-consistent. Instead, they use the problem of evil to defeat the idea that a maximally excellent being exists (a maximally excellent being has omniscience, omnipotence, and moral perfection). They reason that if a maximally excellent being fails to exist in this world, then a maximally great being is impossible. A couple of responses might be ventured. 1) Given divine simplicity, God is identical to God’s attributes. This means that a maximally excellent being is essentially distinct from a maximally great being, if it is not the case that a maximally excellent being’s attributes are non-identical to necessary existence. So the non-existence of a maximally excellent being would not rule out the possible existence of a maximally great being. And 2) the self-consistency of a maximally great being should not be undermined by something external to it. Since gratuitous evil is said to be external to maximal greatness, it should not be a defeater for self consistency, and so no a defeater for the logical possibility of a maximally great being. A maximally great being, on the other hand, is a deafeater for gratuitous evil, if our atheologians have done their homework properly.

1The formulation of the argument in terms of consistency is inspired by the formulation of Plantinga’s modal ontological argument here:
K.E. Himma. “Anselm: Ontological Argument for God’s Existence” in The Internet Encyclopedia of Philosophy. http://www.iep.utm.edu/ont-arg/#H4

Why A Posteriori Arguments Against God’s Existence Fail, pt. 2

A few weeks ago I posted my argument against any and all a posteriori arguments. I thought I would revisit the argument again today and try to spell it out more clearly and concisely.

By a posteriori argument, I merely mean any argument that seeks to disprove God’s existence on the basis of some fact of the world, be it evil, evolution, “divine hiddenness”, the success of the natural sciences, or the failure of mind/body dualism. These arguments essentially boil down to something like:

1. If God exists, we would expect to observe X.

2. X is not the case.

3. Therefore, God does not exist.

The argument is in the form of a modus tollens deduction, which is a perfectly valid form of argumentation. My response is a kind of dilemma that puts the atheist in the awkward position of defending the soundness of the argument. I contend that any successful defense of the soundness of such arguments requires one to commit an informal fallacy of begging the question.

But first I think it is important to unpack my concept of God. I believe God is the greatest conceivable being, a definition common to classical theism. Here, Dr. William Lane Craig discusses this concept of God, and its implications. I would agree with what he says:

Dr. Craig points that since God is conceived of as a necessary being, if God possibly exists, then God actually exists. Now my argument is much more humble than this, as I do not even suppose that God’s existence is logically possible. What I offer is the following dilemma:

1. If the a posteriori argument is to be sound, then the truth of the conditional premise must be based on appeals to possibility or impossibility.

2. If the truth of the conditional premise is supported on the grounds of possibility, then there must be a possible world where the antecedent and consequent obtain, be it a fact of the actual world or counterfactually.

3. If God’s existence possibly obtains, then God actually exists.

4. If God actually exists, then either the first or second premise of the a posteriori argument must be false and the argument unsound.

5. If the truth of the conditional premise is supported on the ground of impossibility, then there is no possible world where the antecedent obtains and this counterpossible conditional is trivially true.

6. If the conditional premise is true because it is a counterpossible conditional, then the support for the first premise is nothing more than the assumption of the conclusion, which is to beg the question.

7. Therefore, either the a posteriori argument is unsound or question-begging.

Now one might challenge my argument by saying that the atheist need not assume that God is possible or impossible at the outset. However, if the argument is to be any good, the atheist must provide reasons that compel her interlocutor to think there is epistemic warrant for accepting the truth of the premises. After all, the conditional premise might be false. If the atheist draws out implications from the concept of God, then we must ask how it could be that the concept of God supplies those implications. The atheist is limited here. She cannot admit that the implications are drawn from a coherent conceptual analysis of God, since this would suggest that in at least one possible world where God exists and such implications follow. That would mean that God is logically possible, and so actual. Thus, the only alternative is that the implication is drawn out of an incoherent or impossible concept. But then anything could be said to be implied by the impossible, so the implication is merely a trivial one. The atheist would have to admit that “If God exists, gratuitous evil would not exist” is just as true as “If God exists, light bulbs wouldn’t exist”. If that is all the atheist is saying, then the argument is no more interesting than its theistic cousin “Either God exists, or 2+2 is 5, and since 2+2 is not 5, God exists.” More to the point, if the conditional premise is trivially true because it is counterpossible, then the atheist’s argument is simply fallacious. Clearly the atheist intends to disprove God on a posteriori concerns, but she ends up appealing to the impossibility of God, a priori, as epistemic warrant for the truth of the conditional premise. Since no argument is given for the impossibility of God, and it is just smuggled into the first premise, this is textbook question-begging.

Let’s suppose that the atheist admits this much, but then claims that she will back up her a priori assumption that God is impossible by way of an argument. But then she is arguing about God’s existence purely on a priori concerns, and that is not the original argument offered at all. So, I think this adequately demonstrates that the atheist is limited to presenting a priori arguments in favor of that position and that a posteriori arguments are useless in disproving God’s existence.

Why A Posteriori Arguments Against God’s Existence Fail

[The argument is further developed here]

Many argue that God does not exist on a posteriori, or empirical, grounds.  Consider, for instance, the problem of evil:

  1.  If God were to exist, there would not be any gratuitous evil in the world.
  2. There is gratuitous evil in the world.
  3. Therefore, God does not exist.

The argument is certainly a valid modus tollens deduction.  Much ink has been spilled by theists to defend against this argument.  But when the argument is presented to a theist, he should ask the proponent to put in the leg-work. Ask why theists ought to think the premises are true.  In particular, ask whythe first premise is true.

There are only so many conditions under which Premise 1 would be true.  Either the antecedent is true and the consequent is true, or the antecedent is false.  But it would be odd to defend the truth of the first premise by arguing that the antecedent is false, since that would beg the question of the argument.  I take the first premise to be a counterfactual statement suggesting that in possible worlds were God exists, gratuitous evil does not exist.  But unless the concept of God is logically coherent, how would we know that in a possible world where God exists, it necessarily would be the case that gratuitous evil would not? It seems to me that one could only be justified in making that supposition if, at the very least, one presumes that God is logically possible and that his existence is relevant to a certain set of states of affairs. Otherwise, one would have to admit that just about any state of affairs is implied by the supposition that God exists, including contradictory states of affairs, e.g. gratuitous evil would exist in great quantities.

It seems to me that if one wants to defend the soundness of an argument like the argument from evil in a non-question begging way, then it must be supposed that God, a necessary being, is logically possible. If it is argued that the first premise is true because God is logically impossible, then any defense for the truth of Premise 1 would beg the question for the conclusion, i.e. the defense would amount to arguing that the antecedent “God exists” is necessarily false, which is clearly the conclusion that the argument is trying to reach.

If the non-theist wants to argue in a non-question begging manner, she must explicate precisely what relevant aspects of the concept of God entails the state of affairs she suggests.  But in the process, she must admit that there is at least a possible world where God exists along with those states of affairs. And if God is logically possible, then God necessarily exists.   But if God, a necessary being, is possible, then God exists in all possible worlds including the actual world.  This means that the first premise is false since all possible states of affairs can obtain given the existence of a necessary being.  God’s existence would have to cohere with any possible state of affairs, including those that exist in this world.  Therefore, no state of affairs could count against God’s existence, including gratuitous evil, if gratuitous evil is logically possible.

So the non-theist, in defending an a posteriori argument like the problem of evil, must either fallaciously beg the question, or if she argues that the counterfactual premises is substantive, then it is clear that the premise is false.  Either way, the argument will not run.

Note that this dilemma would work against any a posteriori argument where it is assumed that God’s existences necessarily entails some state of affairs.

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