Quuat?

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Some Warning Signs…

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An Inductive Way of Thinking about the Modal Ontological Argument

P1. If philosophers of religion over the past 50+ years have successfully defended the coherence of the concept of a maximally great God, then probably a maximally great God is metaphysically possible.
P2. The metaphysical possibility of a maximally great God entails that a maximally great God exists.
P3. Philosophers of religion over the past 50+ years have successfully defended the coherence of the concept of a maximally great God.
C. Probably a maximally great God exists.

I think this argument also helps to distinguish between epistemic possibility (I think it is probable because of sustained intellectual scrutiny) and metaphysical possibility.

Also, I should note that by the coherence of the concept of a maximally great God, I mean more than mere consistency among the attributes, or even self-consistency of each attribute, but also the coherence of theism with other facts, necessary or contingent, e.g. evil or suffering.

 

A Dilemma for the The Problem of Evil

Just a little fun argument. Oh and by “God,” I mean an omnipotent, omniscience, and morally perfect being. I am just curious as to which premise the atheist might reject…

P1. If God’s existence implies that we should expect the amount of evil and suffering that we actually find in the world, then the problem of evil is a failure.

P2. If there are morally adequate reasons for God to allow the amount of evil and suffering that we actually find in the world, then the problem of evil is a failure.

P3. If God exists, there are morally adequate reasons for God to allow the amount of evil and suffering that we actually find in the world.

C. The problem of evil is a failure.

Defense of P1: The central claim of the problem of evil is that we should expect to see little to no evil, if there is a God, but if God’s existence actually implies precisely the amount of evil that we find in the world, our expectations are not defied when we look and see, and so evil would not really serve as a counter-example to the hypothesis that there is a God. In other words, if God’s existence implies exactly what we do find, what we find is not a reason to reject God’s existence.

Defense of P2: To say that there are morally adequate reasons for God to allow the amount of evil and suffering we actually find in the world is just to say that there is a successful theodicy that undercuts the problem of evil. If there is a successful undercutting theodicy, the problem of evil is a failure as an atheological argument.

Defense of P3: if it is true that God’s nature is all-powerful, all-knowing, and morally perfect, then God would have the knowledge, ability, and desire to eliminate any evil for which there is not some overriding moral reason for allowing, e.g. soul-building, free-will, moral meaning, etc. In other words, if should turn out that there is a God, God would be able to account for the evil we find in the world and adequately explain why He allowed for it.

From these three premises, it follows that the problem of evil, as an atheological argument, is a failure.

Here is the proof:

Let,

G ≝ God exists
F ≝ The problem of evil is a failure
M ≝ There are morally adequate reasons for God to allow the amoung of evil and suffering that we actually find in the world
W ≝ We should expect the amount of evil and suffering that we actually find in the world

1. (G ⊃ W) ⊃ F (P1)
2. M ⊃ F (P2)
3. G ⊃ M (P3)
4. ~[(G ⊃ W) ∨ ~(G ⊃ W)] (Assumption for Indirect Proof)
5. ~(G ⊃ W) ∧ ~~(G ⊃ W) (4 DeMorgan’s Theorem)
6. (G ⊃ W) ∨ ~(G ⊃ W) (4-5 Indirect Proof)
7. ~(G ⊃ W) (Assumption for Conditional Proof)
8. ~(~G ∨ W) (7 Material Implication)
9. ~~G ∧ ~W (8 DeMorgan’s Theorem)
10. ~~G (9 Simplification)
11. G (10 Double Negation)
12. ~(G ⊃ W) ⊃ G (7-11 Conditional Proof)
13. G ⊃ F (2,3 Hypothetical Syllogism)
14. ~(G ⊃ W) ⊃ F (12,13 Hypothetical Syllogism)
15. [(G ⊃ W) ⊃ F] ∧ [~(G ⊃ W) ⊃ F](1,14 Conjunction)
16. F ∨ F (6,15 Constructive Dilemma)
17. F (16 Tautology)

Do you even quus, bro?

These tests always pop up in my feed.  Here is my response:

Vexing Links (8/12/17)

Check them out:

  1. My review of Jaworski’s Structure and the Metaphysics of Mind: How Hylomorphism Solves the Mind-Body Problem is on JBTS.
  2. Paul Draper has updated the “Atheism and Agnosticism” page on the SEP.
  3. Koji Tanaka has co-authored an excellent article on “Paraconsistent Logic” along with Graham Priest and Zach Weber on the SEP.
  4. Plantinga’s EAAN in a nutshell
  5. New article on “Religious Language” on the SEP
  6. Josh Rasmussen’s Worldview Design is one of my new favorite YouTube channels.
  7. New interviews of Eleonore Stump are on Closer to Truth: 1) Do Heaven and Hell Really Exist? 2) What is God’s Eternity? 3) What are Persons? 4) Do Persons have Souls?
  8. A classical theist explains the difference between Classical Theism and Theistic Personalism
  9. Pruss says that Naturalists should be Aristotelians (I agree).
  10. Read the first chapter of Justin Brierley’s Unbelievable? the Book.

A quick argument that I’ve been thinking about:

P1) There is a real distinction between metaphysical and nomological modalities.

P2) If there is a real distinction between metaphysical and nomological modalities, there is a metaphysically necessary, non-natural, indeterministic explanation that makes the distinction between metaphysical and nomological modalities intelligible.

P3) If there is a metaphysically necessary, non-natural, indeterministic explanation that makes the distinction between metaphysical and nomological modalities intelligible, then God exists.

C) God exists.

Some fun memes:




Wagering on Free Will

If you don’t think the evidence can decide the question on free will, you might run a wager style argument, as some studies have suggested that belief in free will encourages moral behavior (Vohs KD, et al. Psychol Sci. 2008). 

Ah, but you object that wager-style arguments cannot motivate belief because you think doxastic voluntarism is false. Well, give it a shot and try to believe on the basis of this wager. And if you succeed, you will have more than pragmatic reasons for holding your belief. As William James puts it, “[t]here are… cases where a fact cannot come at all unless a preliminary faith exists in its coming” (The Will to Believe, 1896).

Belief in free will may be just the sort of belief that verifies itself, if one is able to believe in it while admitting the evidence isn’t sufficient on its own to compel belief.  If one chooses to believe because one thinks it is not a possibility closed off by science, and assesses the merits of the belief from pragmatic concerns, then one has the sort of first-person experience of freedom that libertarians tout even in the face of Libet tests.

In other words, see if you can bring yourself to believe in free will by wagering on it, and thus experiencing direct evidence of doxastic voluntarism, i.e. direct control over your own beliefs.

A Slingshot from S4 to S5 establishing the Modal Ontological Argument?

…Or why the “strong” atheologian, i.e. the atheologian who holds that there is no omniscient, omnipotent, and omnibenevolent being, must say that ♢Θ semantically entails ☐Θ in S4.

Θ is the proposition that necessarily there is an omniscient, omnipotent, and omnibenevolent being.

That is:

Kx ≝ x is omniscient
Px ≝ x is omnipotent
Bx ≝ x is omnibenevolent
Θ ≝ ☐(∃x)[(Kx ∧ Px) ∧ Bx]

Consider the following:

1. It is false that ♢Θ semantically entails ☐Θ in S4.

If that is true, then:

2. There is a world in which the valuation of ♢Θ at that world in S4 is true, and the valuation of ☐Θ at that world in S4 is false.

But this is just to say…

3. ♢♢Θ

That is, there is a world in which it is true that ♢Θ.  Moreover, it is an axiom of S4 that ♢♢p → ♢p, and so:

4. ♢Θ

But given our definition for “Θ”, we can say:

5. ♢☐(∃x)[(Kx ∧ Px) ∧ Bx]

Since S5 is just an extension of S4, if something is possible in S4 it is also possible in S5.  Given that ♢☐p → ☐p is an axiom in S5:

6. ☐(∃x)[(Kx ∧ Px) ∧ Bx]

And since ☐p → p in S5 (axiom M/T), we can conclude:

7. (∃x)[(Kx ∧ Px) ∧ Bx]

Hence, the committed “strong” atheologian must say that ♢Θ semantically entails ☐Θ in S4.  Moreover, since S4 is strongly complete, the atheologian is committed to:

♢Θ ⊢S4 ☐Θ

I’d like to see that deduction.

[Update]: One objection that I have encountered is that the move from 5 to 6 seems to switch frameworks from S4 to S5, and so the argument is invalid. The argument does not presume S4 as the framework, but rather attempts to exploit an intuition about what is semantically entailed about ♢Θ in S4. In other words, if you grant that such entailment doesn’t hold in S4, I think it follows that you are committed to ♢♢Θ in S4 and S5, which of course is just to say that you are committed to ♢Θ in S5. So from the framework of S5, and its related axioms, you would have to be committed to Θ.

In an attempt to more clearly show how I am not applying axioms of S5 in S4, here is a more formal representation of the argument. Add to our key, the following:

T ≝ true
F ≝ false
V(ω)M(P) = … the valuation at ω in model M of proposition p equals…

1. (∀p)(∀q)~[p ⊨S4 q] → (∃ω){[V(ω)S4(p) = T] ∧ [V(ω)S4(q) = F] (premise)
2. (∃ω){[V(ω)S4(p) = T] ∧ [V(ω)S4(q) = F] → ⊨S4♢p} (premise)
3. (∀p){⊨S4♢p → (∃ω){[V(ω)S5(p) = T]} (premise)
4. (∀p)(∃ω){[V(ω)S5(p) = T] → ⊨S5♢p} (premise)
5. (∀p)[⊨S5♢♢☐p → ⊢S5☐p] (premise)
6. ~[♢Θ ⊨S4 ☐Θ] (premise)
7. (∀q)~[♢Θ ⊨S4 q] → (∃ω){[V(ω)S4(♢Θ) = T] ∧ [V(ω)S4(q) = F] (1 UI)
8. ~[♢Θ ⊨S4 ☐Θ] → (∃ω){[V(ω)S4(♢Θ) = T] ∧ [V(ω)S4(☐Θ) = F] (7 UI)
9. (∃ω){[V(ω)S4(♢Θ) = T] ∧ [V(ω)S4(☐Θ) = F] (6,8 MP)
10. [V(w)S4(♢Θ) = T] ∧ [V(w)S4(☐Θ) = F (9 EI)
11. [V(w)S4(♢Θ) = T] (10 Simp)
12. (∃ω)S4(♢Θ) = T] (11 EG)
13. (∃ω){[V(ω)S4(♢Θ) = T] → ⊨S4♢♢Θ (2 UI)
14. ⊨S4♢♢Θ (12,13 MP)
15. ⊨S4♢♢Θ → (∃ω){[V(ω)S5(♢Θ) = T] (3 UI)
16.(∃ω){[V(ω)S5(♢♢Θ) = T] → ⊨S5♢♢Θ (4 UI)
17. ⊨S4♢♢Θ → ⊨S5♢♢Θ (15,16 HS)
18. ⊨S5♢♢Θ (14,17 MP)
19. ⊨S5♢♢☐(∃x)[(Kx ∧ Px) ∧ Bx] (18 Def “Θ”)
20. ⊨S5♢♢☐(∃x)[(Kx ∧ Px) ∧ Bx] → ⊢S5☐(∃x)[(Kx ∧ Px) ∧ Bx] (5 UI)
21. ⊢S5☐(∃x)[(Kx ∧ Px) ∧ Bx](19,20 MP)

Self-Referential Unsound Modus Ponens 


[Image Source Credit: TeX]

An argument is sound if and only if it is valid and the premises are true. If those conditions are met, the conclusion must be true.

Consider the following argument:

P1. If God does not exists, this argument is unsound.
P2. God does not exist.
C. Therefore, this argument is unsound.

The argument is valid (Modus Ponens), so it is sound if the premises are true. But, if both premises are true, the conclusion is would have to be true, and the argument would both be sound and unsound. So consistency demands that we deny the soundness of the argument. At lease one of the premises must be false.  
Consider whether P1 is false. It is a material conditional, and so it is false when the antecedent is true (it is true that God does not exist) and when the consequent is false (it is false that this argument is unsound).[1] So P1 is false only if the argument is sound, which means that the falsity of P1 leads to a contradiction, since the soundness of the argument entails P1 is true. So, P1 cannot be false.  

P2 is the only premise that can be false. So given that the argument must be unsound, we must conclude that it is false that God does not exist.

So this unsound modus ponens proves the contradictory of the minor premise, whatever it might be!

I am probably not the first to note this, but it is new to me.

[1]The truth-table for the Material Conditional is as follows:

    p  q | p → q
1. T  T        T
2. T  F        F*
3. F  T        T
4. F  F        T
*The material conditional is only false on line 2.

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)

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