Monthly Archives: May 2017

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] ∧ Bx] (5 UI)
21. ⊢S5☐(∃x)[(Kx ∧ Px) ∧ Bx] ∧ 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) (Astump. 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? 

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