Formalization, Mechanization and Automation of...
Transcript of Formalization, Mechanization and Automation of...
Formalization, Mechanization and Automation ofGödel’s Proof of God’s Existence
Christoph Benzmüller and Bruno Woltzenlogel Paleo
November 1, 2013
A gift to Priest Edvaldo and his church in Piracicaba, Brazil
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 1
Germany- Telepolis & Heise- Spiegel Online- FAZ- Die Welt- Berliner Morgenpost- Hamburger Abendpost- . . .
Austria- Die Presse- Wiener Zeitung- ORF- . . .
Italy- Repubblica- Ilsussidario- . . .
India- DNA India- Delhi Daily News- India Today- . . .
US- ABC News- . . .
International- Spiegel International- Yahoo Finance- United Press Intl.- . . .
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 2
Introduction — Quick answers to your most pressing questions!
Are we in contact with Steve Jobs? No
Do you really need a MacBook to obtain the results? No
Is Apple sending us money? No(but maybe they should)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 3
Introduction — Quick answers to your most pressing questions!
Are we in contact with Steve Jobs? No
Do you really need a MacBook to obtain the results? No
Is Apple sending us money? No(but maybe they should)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 3
Introduction
Def: Ontological Argument/Proof
* deductive argument* for the existence of God* starting from premises, which are justified by pure reasoning, i.e.they do not depend on observation of the world.
Existence of God: different types of arguments/proofs
— a posteriori (use experience/observation in the world)—— teleological—— cosmological—— moral—— . . .
— a priori (based on pure reasoning, independent)—— ontological argument
—— definitional—— modal—— . . .
—— other a priori arguments
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 4
Introduction
Def: Ontological Argument/Proof
* deductive argument* for the existence of God* starting from premises, which are justified by pure reasoning, i.e.they do not depend on observation of the world.
Existence of God: different types of arguments/proofs
— a posteriori (use experience/observation in the world)—— teleological—— cosmological—— moral—— . . .
— a priori (based on pure reasoning, independent)—— ontological argument
—— definitional—— modal—— . . .
—— other a priori arguments
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 4
Introduction
Wohl eine jede Philosophie kreist um denontologischen Gottesbeweis
(Adorno, Th. W.: Negative Dialektik. Frankfurt a. M. 1966, p.378)
. . . . . .
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 5
Introduction
Rich history on ontological arguments (pros and cons)
. . . Anse
lmv.
C.
Gau
nilo
. . . Th. A
quin
as. . . . . . D
esca
rtes
Spin
oza
Leib
niz
. . . Hum
eKa
nt
. . . Heg
el
. . . Freg
e
. . . Har
tsho
rne
Mal
colm
Lew
isPl
antin
gaG
ödel
. . .
Anselm’s notion of God:“God is that, than which nothing greater can be conceived.”
Gödel’s notion of God:“A God-like being possesses all ‘positive’ properties.”
To show by logical reasoning:“(Necessarily) God exists.”
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 6
Introduction
Rich history on ontological arguments (pros and cons)
. . . Anse
lmv.
C.
Gau
nilo
. . . Th. A
quin
as. . . . . . D
esca
rtes
Spin
oza
Leib
niz
. . . Hum
eKa
nt
. . . Heg
el
. . . Freg
e
. . . Har
tsho
rne
Mal
colm
Lew
isPl
antin
gaG
ödel
. . .
Anselm’s notion of God:“God is that, than which nothing greater can be conceived.”
Gödel’s notion of God:“A God-like being possesses all ‘positive’ properties.”
To show by logical reasoning:“(Necessarily) God exists.”
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 6
Introduction
Different Interests in Ontological Arguments:
Philosophical: Boundaries of Metaphysics & EpistemologyWe talk about a metaphysical concept (God),but we want to draw a conclusion for the real world.
Necessary Existence (NE): metaphysical NE vs. logical NE vs. modal NE
Theistic: Successful argument should convince atheists
Ours: Can computers (theorem provers) be used . . .. . . to formalize the definitions, axioms and theorems?. . . to verify the arguments step-by-step?. . . to fully automate (sub-)arguments?
“Computer-assisted Theoretical Philosophy”
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 7
Introduction
Challenge: No provers for Higher-order Quantified Modal Logic (QML)
Our solution: Embedding in Higher-order Classical Logic (HOL)[BenzmüllerPaulson, Logica Universalis, 2013]
What we did (rough outline for remaining presentation!):
A: Pen and paper: detailed natural deduction proofB: Formalization: in classical higher-order logic (HOL)
Automation: theorem provers Leo-II and SatallaxConsistency: model finder Nitpick (Nitrox)
C: Step-by-step verification: proof assistant CoqD: Automation & verification: proof assistant Isabelle
Did we get any new results? Yes — let’s discuss this later!
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 8
Part A:Informal Proof and Natural Deduction Proof
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 9
Gödel’s Manuscript: 1930’s, 1941, 1946-1955, 1970
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 10
Scott’s Version of Gödel’s Axioms, Definitions and Theorems
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 11
Proof Overview
T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 12
Proof Overview
C1: ^∃z.G(z)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 13
Proof Overview
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 14
Proof Overview
L2: ^∃z.G(z)→ �∃x.G(x)
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 15
Proof Overview
S5∀ξ.[^�ξ→ �ξ]
L2: ^∃z.G(z)→ �∃x.G(x)
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 16
Proof Overview
^∃z.G(z)→ ^�∃x.G(x)S5
∀ξ.[^�ξ→ �ξ]L2: ^∃z.G(z)→ �∃x.G(x)
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 17
Proof Overview
L1: ∃z.G(z)→ �∃x.G(x)^∃z.G(z)→ ^�∃x.G(x)
S5∀ξ.[^�ξ→ �ξ]
L2: ^∃z.G(z)→ �∃x.G(x)
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 18
Proof Overview
D1: G(x) ≡ ∀ϕ.[P(ϕ)→ ϕ(x)]
L1: ∃z.G(z)→ �∃x.G(x)^∃z.G(z)→ ^�∃x.G(x)
S5∀ξ.[^�ξ→ �ξ]
L2: ^∃z.G(z)→ �∃x.G(x)
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 19
Proof Overview
D1: G(x) ≡ ∀ϕ.[P(ϕ)→ ϕ(x)]
D3*: E(x) ≡ �∃y.G(y)
P(E)L1: ∃z.G(z)→ �∃x.G(x)^∃z.G(z)→ ^�∃x.G(x)
S5∀ξ.[^�ξ→ �ξ]
L2: ^∃z.G(z)→ �∃x.G(x)
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 20
Proof Overview
D1: G(x) ≡ ∀ϕ.[P(ϕ)→ ϕ(x)]
D3*: E(x) ≡ �∃y.G(y) (cheating!)
P(E)L1: ∃z.G(z)→ �∃x.G(x)^∃z.G(z)→ ^�∃x.G(x)
S5∀ξ.[^�ξ→ �ξ]
L2: ^∃z.G(z)→ �∃x.G(x)
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 20
Proof Overview
D1: G(x) ≡ ∀ϕ.[P(ϕ)→ ϕ(x)]
D3*: E(x) ≡ �∃y.G(y) D3: E(x) ≡ ∀ϕ.[ϕ ess x→ �∃y.ϕ(y)]
T2: ∀y.[G(y)→ G ess y] P(E)L1: ∃z.G(z)→ �∃x.G(x)^∃z.G(z)→ ^�∃x.G(x)
S5∀ξ.[^�ξ→ �ξ]
L2: ^∃z.G(z)→ �∃x.G(x)
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 21
Proof Overview
D1: G(x) ≡ ∀ϕ.[P(ϕ)→ ϕ(x)]
D3*: E(x) ≡ �∃y.G(y) D3: E(x) ≡ ∀ϕ.[ϕ ess x→ �∃y.ϕ(y)]
T2: ∀y.[G(y)→ G ess y]A5
P(E)L1: ∃z.G(z)→ �∃x.G(x)^∃z.G(z)→ ^�∃x.G(x)
S5∀ξ.[^�ξ→ �ξ]
L2: ^∃z.G(z)→ �∃x.G(x)
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 22
Proof Overview
D1: G(x) ≡ ∀ϕ.[P(ϕ)→ ϕ(x)]
D3*: E(x) ≡ �∃y.G(y) D3: E(x) ≡ ∀ϕ.[ϕ ess x→ �∃y.ϕ(y)]
T2: ∀y.[G(y)→ G ess y]A5
P(E)L1: ∃z.G(z)→ �∃x.G(x)^∃z.G(z)→ ^�∃x.G(x)
S5∀ξ.[^�ξ→ �ξ]
L2: ^∃z.G(z)→ �∃x.G(x)
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 23
Proof Overview
D1: G(x) ≡ ∀ϕ.[P(ϕ)→ ϕ(x)]
D2: ϕ ess x ≡ ϕ(x) ∧ ∀ψ.(ψ(x)→ �∀x.(ϕ(x)→ ψ(x)))
D3*: E(x) ≡ �∃y.G(y) D3: E(x) ≡ ∀ϕ.[ϕ ess x→ �∃y.ϕ(y)]
A1b∀ϕ.[¬P(ϕ)→ P(¬ϕ)]
A4∀ϕ.[P(ϕ)→ � P(ϕ)]
T2: ∀y.[G(y)→ G ess y]A5
P(E)L1: ∃z.G(z)→ �∃x.G(x)^∃z.G(z)→ ^�∃x.G(x)
S5∀ξ.[^�ξ→ �ξ]
L2: ^∃z.G(z)→ �∃x.G(x)
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 24
Proof Overview
D1: G(x) ≡ ∀ϕ.[P(ϕ)→ ϕ(x)]
D2: ϕ ess x ≡ ϕ(x) ∧ ∀ψ.(ψ(x)→ �∀x.(ϕ(x)→ ψ(x)))
D3*: E(x) ≡ �∃y.G(y) D3: E(x) ≡ ∀ϕ.[ϕ ess x→ �∃y.ϕ(y)]
C1: ^∃z.G(z)
A1b∀ϕ.[¬P(ϕ)→ P(¬ϕ)]
A4∀ϕ.[P(ϕ)→ � P(ϕ)]
T2: ∀y.[G(y)→ G ess y]A5
P(E)L1: ∃z.G(z)→ �∃x.G(x)^∃z.G(z)→ ^�∃x.G(x)
S5∀ξ.[^�ξ→ �ξ]
L2: ^∃z.G(z)→ �∃x.G(x)
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 25
Proof Overview
D1: G(x) ≡ ∀ϕ.[P(ϕ)→ ϕ(x)]
D2: ϕ ess x ≡ ϕ(x) ∧ ∀ψ.(ψ(x)→ �∀x.(ϕ(x)→ ψ(x)))
D3*: E(x) ≡ �∃y.G(y) D3: E(x) ≡ ∀ϕ.[ϕ ess x→ �∃y.ϕ(y)]
P(G)C1: ^∃z.G(z)
A1b∀ϕ.[¬P(ϕ)→ P(¬ϕ)]
A4∀ϕ.[P(ϕ)→ � P(ϕ)]
T2: ∀y.[G(y)→ G ess y]A5
P(E)L1: ∃z.G(z)→ �∃x.G(x)^∃z.G(z)→ ^�∃x.G(x)
S5∀ξ.[^�ξ→ �ξ]
L2: ^∃z.G(z)→ �∃x.G(x)
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 26
Proof Overview
D1: G(x) ≡ ∀ϕ.[P(ϕ)→ ϕ(x)]
D2: ϕ ess x ≡ ϕ(x) ∧ ∀ψ.(ψ(x)→ �∀x.(ϕ(x)→ ψ(x)))
D3*: E(x) ≡ �∃y.G(y) D3: E(x) ≡ ∀ϕ.[ϕ ess x→ �∃y.ϕ(y)]
A3P(G)
C1: ^∃z.G(z)
A1b∀ϕ.[¬P(ϕ)→ P(¬ϕ)]
A4∀ϕ.[P(ϕ)→ � P(ϕ)]
T2: ∀y.[G(y)→ G ess y]A5
P(E)L1: ∃z.G(z)→ �∃x.G(x)^∃z.G(z)→ ^�∃x.G(x)
S5∀ξ.[^�ξ→ �ξ]
L2: ^∃z.G(z)→ �∃x.G(x)
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 27
Proof Overview
D1: G(x) ≡ ∀ϕ.[P(ϕ)→ ϕ(x)]
D2: ϕ ess x ≡ ϕ(x) ∧ ∀ψ.(ψ(x)→ �∀x.(ϕ(x)→ ψ(x)))
D3*: E(x) ≡ �∃y.G(y) D3: E(x) ≡ ∀ϕ.[ϕ ess x→ �∃y.ϕ(y)]
A3P(G) T1: ∀ϕ.[P(ϕ)→ ^∃x.ϕ(x)]
C1: ^∃z.G(z)
A1b∀ϕ.[¬P(ϕ)→ P(¬ϕ)]
A4∀ϕ.[P(ϕ)→ � P(ϕ)]
T2: ∀y.[G(y)→ G ess y]A5
P(E)L1: ∃z.G(z)→ �∃x.G(x)^∃z.G(z)→ ^�∃x.G(x)
S5∀ξ.[^�ξ→ �ξ]
L2: ^∃z.G(z)→ �∃x.G(x)
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 28
Proof Overview
D1: G(x) ≡ ∀ϕ.[P(ϕ)→ ϕ(x)]
D2: ϕ ess x ≡ ϕ(x) ∧ ∀ψ.(ψ(x)→ �∀x.(ϕ(x)→ ψ(x)))
D3*: E(x) ≡ �∃y.G(y) D3: E(x) ≡ ∀ϕ.[ϕ ess x→ �∃y.ϕ(y)]
A3P(G)
A2∀ϕ.∀ψ.[(P(ϕ) ∧ �∀x.[ϕ(x)→ ψ(x)])→ P(ψ)]
A1a∀ϕ.[P(¬ϕ)→ ¬P(ϕ)]
T1: ∀ϕ.[P(ϕ)→ ^∃x.ϕ(x)]C1: ^∃z.G(z)
A1b∀ϕ.[¬P(ϕ)→ P(¬ϕ)]
A4∀ϕ.[P(ϕ)→ � P(ϕ)]
T2: ∀y.[G(y)→ G ess y]A5
P(E)L1: ∃z.G(z)→ �∃x.G(x)^∃z.G(z)→ ^�∃x.G(x)
S5∀ξ.[^�ξ→ �ξ]
L2: ^∃z.G(z)→ �∃x.G(x)
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 29
Proof Overview
D1: G(x) ≡ ∀ϕ.[P(ϕ)→ ϕ(x)]
D2: ϕ ess x ≡ ϕ(x) ∧ ∀ψ.(ψ(x)→ �∀x.(ϕ(x)→ ψ(x)))
D3: E(x) ≡ ∀ϕ.[ϕ ess x→ �∃y.ϕ(y)]
A3P(G)
A2∀ϕ.∀ψ.[(P(ϕ) ∧ �∀x.[ϕ(x)→ ψ(x)])→ P(ψ)]
A1a∀ϕ.[P(¬ϕ)→ ¬P(ϕ)]
T1: ∀ϕ.[P(ϕ)→ ^∃x.ϕ(x)]
C1: ^∃z.G(z)
A1b∀ϕ.[¬P(ϕ)→ P(¬ϕ)]
A4∀ϕ.[P(ϕ)→ � P(ϕ)]
T2: ∀y.[G(y)→ G ess y]A5
P(E)
L1: ∃z.G(z)→ �∃x.G(x)
^∃z.G(z)→ ^�∃x.G(x)S5
∀ξ.[^�ξ→ �ξ]
L2: ^∃z.G(z)→ �∃x.G(x)
C1: ^∃z.G(z) L2: ^∃z.G(z)→ �∃x.G(x)
T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 30
Natural Deduction Calculus
A ∨ B
A....C
B....C
C∨E
A BA ∧ B
∧I
An
....B
A→ B→
nI
AA ∨ B
∨I1A ∧ B
A∧E1
BA→ B
→I
BA ∨ B
∨I2A ∧ B
B∧E2
A A→ BB
→E
A[α]∀x.A[x]
∀I∀x.A[x]
A[t]∀E
A[t]∃x.A[x]
∃I∃x.A[x]
A[β]∃E
¬A ≡ A→ ⊥¬¬A
A¬¬E
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 31
Natural Deduction CalculusRules for Modalities
α :
....A
�A�I
�A
t :
A....
�E
t :
....A
^A^I
^A
β :
A....
^E
^A ≡ ¬�¬A
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 32
Natural Deduction ProofsT1 and C1
A2∀ϕ.∀ψ.[(P(ϕ) ∧ �∀x.[ϕ(x)→ ψ(x)])→ P(ψ)]
∀E∀ψ.[(P(ρ) ∧ �∀x.[ρ(x)→ ψ(x)])→ P(ψ)]
∀E(P(ρ) ∧ �∀x.[ρ(x)→ ¬ρ(x)])→ P(¬ρ)
(P(ρ) ∧ �∀x.[¬ρ(x)])→ P(¬ρ)
A1a∀ϕ.[P(¬ϕ)→ ¬P(ϕ)]
∀EP(¬ρ)→ ¬P(ρ)
(P(ρ) ∧ �∀x.[¬ρ(x)])→ ¬P(ρ)
P(ρ)→ ^∃x.ρ(x)∀IT1: ∀ϕ.[P(ϕ)→ ^∃x.ϕ(x)]
A3P(G)
T1∀ϕ.[P(ϕ)→ ^∃x.ϕ(x)]
∀EP(G)→ ^∃x.G(x)→E
^∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 33
Natural Deduction ProofsT2 (Partial)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 34
Part B:
Formalization: in classical higher-order logic (HOL)Automation: theorem provers Leo-II and SatallaxConsistency: model finder Nitpick (Nitrox)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 35
Formalization in HOL
Challenge: No provers for Higher-order Quantified Modal Logic (QML)
Our solution: Embedding in Higher-order Classical Logic (HOL)Then use existing HOL theorem provers for reasoning in QML
[BenzmüllerPaulson, Logica Universalis, 2013]
Previous empiricial findings:
Embedding of First-order Modal Logic in HOL works well[BenzmüllerOttenRaths, ECAI, 2012]
[Benzmüller, LPAR, 2013]
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 36
Formalization in HOL
QML ϕ,ψ ::= . . . | ¬ϕ | ϕ ∧ ψ | ϕ→ ψ | �ϕ | ^ϕ | ∀xϕ | ∃xϕ | ∀Pϕ
Kripke style semantics (possible world semantics)
HOL s, t ::= C | x | λxs | s t | ¬s | s ∨ t | ∀x t
meanwhile very well understoodHenkin semantics vs. standard semanticsvarious theorem provers do exists
interactive: Isabelle/HOL, HOL4, Hol Light, Coq/HOL, PVS, . . .
automated: TPS, LEO-II, Satallax, Nitpick, Isabelle/HOL, . . .
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 37
Formalization in HOL
QML ϕ,ψ ::= . . . | ¬ϕ | ϕ ∧ ψ | ϕ→ ψ | �ϕ | ^ϕ | ∀xϕ | ∃xϕ | ∀Pϕ
HOL s, t ::= C | x | λxs | s t | ¬s | s ∨ t | ∀x t
QML in HOL: QML formulas ϕ are mapped to HOL predicates ϕι�o
¬ = λϕι�oλsι¬ϕs∧ = λϕι�oλψι�oλsι(ϕs ∧ ψs)→ = λϕι�oλψι�oλsι(¬ϕs ∨ ψs)� = λϕι�oλsι∀uι (¬rsu ∨ ϕu)^ = λϕι�oλsι∃uι (rsu ∧ ϕu)∀ = λhµ�(ι�o)λsι∀dµ hds∃ = λhµ�(ι�o)λsι∃dµ hds∀ = λH(µ�(ι�o))�(ι�o)λsι∀dµ Hds
valid = λϕι�o∀wιϕw
Ax
The equations in Ax are given as axioms to the HOL provers!(Remark: Note that we are here dealing with constant domain quantification)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 38
Formalization in HOL
QML ϕ,ψ ::= . . . | ¬ϕ | ϕ ∧ ψ | ϕ→ ψ | �ϕ | ^ϕ | ∀xϕ | ∃xϕ | ∀Pϕ
HOL s, t ::= C | x | λxs | s t | ¬s | s ∨ t | ∀x t
QML in HOL: QML formulas ϕ are mapped to HOL predicates ϕι�o
¬ = λϕι�oλsι¬ϕs∧ = λϕι�oλψι�oλsι(ϕs ∧ ψs)→ = λϕι�oλψι�oλsι(¬ϕs ∨ ψs)� = λϕι�oλsι∀uι (¬rsu ∨ ϕu)^ = λϕι�oλsι∃uι (rsu ∧ ϕu)∀ = λhµ�(ι�o)λsι∀dµ hds∃ = λhµ�(ι�o)λsι∃dµ hds∀ = λH(µ�(ι�o))�(ι�o)λsι∀dµ Hds
valid = λϕι�o∀wιϕw
Ax
The equations in Ax are given as axioms to the HOL provers!(Remark: Note that we are here dealing with constant domain quantification)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 38
Formalization in HOL
QML ϕ,ψ ::= . . . | ¬ϕ | ϕ ∧ ψ | ϕ→ ψ | �ϕ | ^ϕ | ∀xϕ | ∃xϕ | ∀Pϕ
HOL s, t ::= C | x | λxs | s t | ¬s | s ∨ t | ∀x t
QML in HOL: QML formulas ϕ are mapped to HOL predicates ϕι�o
¬ = λϕι�oλsι¬ϕs∧ = λϕι�oλψι�oλsι(ϕs ∧ ψs)→ = λϕι�oλψι�oλsι(¬ϕs ∨ ψs)� = λϕι�oλsι∀uι (¬rsu ∨ ϕu)^ = λϕι�oλsι∃uι (rsu ∧ ϕu)∀ = λhµ�(ι�o)λsι∀dµ hds∃ = λhµ�(ι�o)λsι∃dµ hds∀ = λH(µ�(ι�o))�(ι�o)λsι∀dµ Hds
valid = λϕι�o∀wιϕw
Ax
The equations in Ax are given as axioms to the HOL provers!(Remark: Note that we are here dealing with constant domain quantification)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 38
Formalization in HOL
Example
QML formula ^∃xG(x)QML formula in HOL valid (^∃xG(x))ι�oexpansion, βη-conversion ∀wι(^∃xG(x))ι�o wexpansion, βη-conversion ∀wι∃uι(rwu ∧ (∃xG(x))ι�ou)expansion, βη-conversion ∀wι∃uι(rwu ∧ ∃xGxu)
What are we doing?
In order to prove that ϕ is valid in QML,–> we instead prove that validϕι�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
Expansion: user or prover may flexibly choose expansion depth
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 39
Formalization in HOL
Example
QML formula ^∃xG(x)QML formula in HOL valid (^∃xG(x))ι�oexpansion, βη-conversion ∀wι(^∃xG(x))ι�o wexpansion, βη-conversion ∀wι∃uι(rwu ∧ (∃xG(x))ι�ou)expansion, βη-conversion ∀wι∃uι(rwu ∧ ∃xGxu)
What are we doing?
In order to prove that ϕ is valid in QML,–> we instead prove that validϕι�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
Expansion: user or prover may flexibly choose expansion depth
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 39
Formalization in HOL
Example
QML formula ^∃xG(x)QML formula in HOL valid (^∃xG(x))ι�oexpansion, βη-conversion ∀wι(^∃xG(x))ι�o wexpansion, βη-conversion ∀wι∃uι(rwu ∧ (∃xG(x))ι�ou)expansion, βη-conversion ∀wι∃uι(rwu ∧ ∃xGxu)
What are we doing?
In order to prove that ϕ is valid in QML,–> we instead prove that validϕι�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
Expansion: user or prover may flexibly choose expansion depth
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 39
Formalization in HOL
Example
QML formula ^∃xG(x)QML formula in HOL valid (^∃xG(x))ι�oexpansion, βη-conversion ∀wι(^∃xG(x))ι�o wexpansion, βη-conversion ∀wι∃uι(rwu ∧ (∃xG(x))ι�ou)expansion, βη-conversion ∀wι∃uι(rwu ∧ ∃xGxu)
What are we doing?
In order to prove that ϕ is valid in QML,–> we instead prove that validϕι�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
Expansion: user or prover may flexibly choose expansion depth
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 39
Formalization in HOL
Example
QML formula ^∃xG(x)QML formula in HOL valid (^∃xG(x))ι�oexpansion, βη-conversion ∀wι(^∃xG(x))ι�o wexpansion, βη-conversion ∀wι∃uι(rwu ∧ (∃xG(x))ι�ou)expansion, βη-conversion ∀wι∃uι(rwu ∧ ∃xGxu)
What are we doing?
In order to prove that ϕ is valid in QML,–> we instead prove that validϕι�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
Expansion: user or prover may flexibly choose expansion depth
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 39
Formalization in HOL
Example
QML formula ^∃xG(x)QML formula in HOL valid (^∃xG(x))ι�oexpansion, βη-conversion ∀wι(^∃xG(x))ι�o wexpansion, βη-conversion ∀wι∃uι(rwu ∧ (∃xG(x))ι�ou)expansion, βη-conversion ∀wι∃uι(rwu ∧ ∃xGxu)
What are we doing?
In order to prove that ϕ is valid in QML,–> we instead prove that validϕι�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
Expansion: user or prover may flexibly choose expansion depth
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 39
Formalization in HOL
Example
QML formula ^∃xG(x)QML formula in HOL valid (^∃xG(x))ι�oexpansion, βη-conversion ∀wι(^∃xG(x))ι�o wexpansion, βη-conversion ∀wι∃uι(rwu ∧ (∃xG(x))ι�ou)expansion, βη-conversion ∀wι∃uι(rwu ∧ ∃xGxu)
What are we doing?
In order to prove that ϕ is valid in QML,–> we instead prove that validϕι�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
Expansion: user or prover may flexibly choose expansion depth
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 39
Automated Theorem Provers and Model Finders for HOL
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 40
Proof Automation and Consistency Checking: Demo!
Provers are called remotely in Miami — no local installation needed!
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 41
Part C:Formalization and Verification in Coq
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 42
Coq ProofDemo
Goal: verification of the natural deduction proofStep-by-step formalizationAlmost no automation (intentionally!)
Interesting facts:Embedding is transparent to the userEmbedding gives labeled calculus for free
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 43
Coq ProofDemo
Goal: verification of the natural deduction proofStep-by-step formalizationAlmost no automation (intentionally!)
Interesting facts:Embedding is transparent to the userEmbedding gives labeled calculus for free
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 43
Coq ProofDemo
Goal: verification of the natural deduction proofStep-by-step formalizationAlmost no automation (intentionally!)
Interesting facts:Embedding is transparent to the userEmbedding gives labeled calculus for free
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 43
Coq ProofDemo
Goal: verification of the natural deduction proofStep-by-step formalizationAlmost no automation (intentionally!)
Interesting facts:Embedding is transparent to the userEmbedding gives labeled calculus for free
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 43
Part D:
automation & verification: proof assistant Isabelle
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 45
Automation & Verification in Proof Assistant Isabelle/HOL
Isabelle/HOL (Cambridge University/TU Munich)HOL instance of the generic Isabelle proof assistantUser interaction and proof automationAutomation is supported by Sledgehammer toolVerification of the proofs in Isabelle/HOL’s small proof kernel
What we did?Proof automation of Gödel’s proof script (Scott version)Sledgehammer makes calls to remote THF provers in MiamiThese calls the suggest respective calls to the Metis proverMetis proofs are verified in Isabelle/HOL’s proof kernel
See the handout (generated from the Isabelle source file).
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 47
Automation & Verification in Proof Assistant Isabelle/HOL
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 48
Part E:Criticisms
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 49
CriticismsS5
∀P.[^�P→ �P]
If something is possibly necessary, then it is necessary.
^�(A ∨ ¬A) �(A ∨ ¬A)
logical necessity ∼ validity logical possibility ∼ satisfiability
for all M,M |= F −→ �F exists M,M |= F −→ ^F
What about iterations?
^�^^F
weak intuitions⇒ dozens of modal logics
S5 is considered adequate
(But KB is sufficient!)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 50
CriticismsS5
∀P.[^�P→ �P]
If something is possibly necessary, then it is necessary.
^�(A ∨ ¬A) �(A ∨ ¬A)
logical necessity ∼ validity logical possibility ∼ satisfiability
for all M,M |= F −→ �F exists M,M |= F −→ ^F
What about iterations?
^�^^F
weak intuitions⇒ dozens of modal logics
S5 is considered adequate
(But KB is sufficient!)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 50
CriticismsS5
∀P.[^�P→ �P]
If something is possibly necessary, then it is necessary.
^�(A ∨ ¬A) �(A ∨ ¬A)
logical necessity ∼ validity logical possibility ∼ satisfiability
for all M,M |= F −→ �F exists M,M |= F −→ ^F
What about iterations?
^�^^F
weak intuitions⇒ dozens of modal logics
S5 is considered adequate
(But KB is sufficient!)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 50
CriticismsS5
∀P.[^�P→ �P]
If something is possibly necessary, then it is necessary.
^�(A ∨ ¬A) �(A ∨ ¬A)
logical necessity ∼ validity logical possibility ∼ satisfiability
for all M,M |= F −→ �F exists M,M |= F −→ ^F
What about iterations?
^�^^F
weak intuitions⇒ dozens of modal logics
S5 is considered adequate
(But KB is sufficient!)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 50
CriticismsS5
∀P.[^�P→ �P]
If something is possibly necessary, then it is necessary.
^�(A ∨ ¬A) �(A ∨ ¬A)
logical necessity ∼ validity logical possibility ∼ satisfiability
for all M,M |= F −→ �F exists M,M |= F −→ ^F
What about iterations?
^�^^F
weak intuitions⇒ dozens of modal logics
S5 is considered adequate
(But KB is sufficient!)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 50
CriticismsS5
∀P.[^�P→ �P]
If something is possibly necessary, then it is necessary.
^�(A ∨ ¬A) �(A ∨ ¬A)
logical necessity ∼ validity logical possibility ∼ satisfiability
for all M,M |= F −→ �F exists M,M |= F −→ ^F
What about iterations?
^�^^F
weak intuitions⇒ dozens of modal logics
S5 is considered adequate
(But KB is sufficient!)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 50
CriticismsS5
∀P.[^�P→ �P]
If something is possibly necessary, then it is necessary.
^�(A ∨ ¬A) �(A ∨ ¬A)
logical necessity ∼ validity logical possibility ∼ satisfiability
for all M,M |= F −→ �F exists M,M |= F −→ ^F
What about iterations?
^�^^F
weak intuitions⇒ dozens of modal logics
S5 is considered adequate
(But KB is sufficient!)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 50
CriticismsS5
∀P.[^�P→ �P]
If something is possibly necessary, then it is necessary.
^c�c(A ∨ ¬A) �c(A ∨ ¬A)
logical necessity ∼ validity logical possibility ∼ satisfiability
for all M,M |= F −→ �F exists M,M |= F −→ ^F
What about iterations?
^�^^F
weak intuitions⇒ dozens of modal logics
S5 is considered adequate
(But KB is sufficient!)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 50
CriticismsS5
∀P.[^�P→ �P]
If something is possibly necessary, then it is necessary.
^c�c(A ∨ ¬A) �c(A ∨ ¬A)
logical necessity ∼ validity logical possibility ∼ satisfiability
for all M,M |= F −→ �F exists M,M |= F −→ ^F
What about iterations?
^�^^F
weak intuitions⇒ dozens of modal logics
S5 is considered adequate
(But KB is sufficient!)
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 50
CriticismsModal Collapse
∀P.[P→ �P]
Everything that is the case is so necessarily.
Follows from T2, T3 and D2.
There are no contingent “truths”.Everything is determined.
There is no free will.
Many proposed solutions: Anderson, Fitting, Hájek, . . .
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 51
CriticismsModal Collapse
∀P.[P→ �P]
Everything that is the case is so necessarily.
Follows from T2, T3 and D2.
There are no contingent “truths”.Everything is determined.
There is no free will.
Many proposed solutions: Anderson, Fitting, Hájek, . . .
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 51
CriticismsModal Collapse
∀P.[P→ �P]
Everything that is the case is so necessarily.
Follows from T2, T3 and D2.
There are no contingent “truths”.Everything is determined.
There is no free will.
Many proposed solutions: Anderson, Fitting, Hájek, . . .
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 51
CriticismsModal Collapse
∀P.[P→ �P]
Everything that is the case is so necessarily.
Follows from T2, T3 and D2.
There are no contingent “truths”.Everything is determined.
There is no free will.
Many proposed solutions: Anderson, Fitting, Hájek, . . .
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 51
CriticismsModal Collapse
∀P.[P→ �P]
Everything that is the case is so necessarily.
Follows from T2, T3 and D2.
There are no contingent “truths”.Everything is determined.
There is no free will.
Many proposed solutions: Anderson, Fitting, Hájek, . . .
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 51
CriticismsModal Collapse
∀P.[P→ �P]
Everything that is the case is so necessarily.
Follows from T2, T3 and D2.
There are no contingent “truths”.Everything is determined.
There is no free will.
Many proposed solutions: Anderson, Fitting, Hájek, . . .
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 51
CriticismsModal Collapse
∀P.[P→ �P]
Everything that is the case is so necessarily.
Follows from T2, T3 and D2.
There are no contingent “truths”.Everything is determined.
There is no free will.
Many proposed solutions: Anderson, Fitting, Hájek, . . .
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 51
CriticismsNo Neutral Properties
∀φ[P(¬φ)↔ ¬P(φ)]
Either a property is positive or its negation is (but never both)
Are the following properties positive or negative?
λx.G(x) λx.E(x) λx.x = x λx.>
λx.blue(x) λx.white(x) λx.human(x)
Solution:“. . . positive in the moral aesthetic sense (independently of the accidental
structure of the world). Only then the ax. true. . . . ”- Gödel, 1970
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 52
CriticismsNo Neutral Properties
∀φ[P(¬φ)↔ ¬P(φ)]
Either a property is positive or its negation is (but never both)
Are the following properties positive or negative?
λx.G(x) λx.E(x) λx.x = x λx.>
λx.blue(x) λx.white(x) λx.human(x)
Solution:“. . . positive in the moral aesthetic sense (independently of the accidental
structure of the world). Only then the ax. true. . . . ”- Gödel, 1970
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 52
CriticismsNo Neutral Properties
∀φ[P(¬φ)↔ ¬P(φ)]
Either a property is positive or its negation is (but never both)
Are the following properties positive or negative?
λx.G(x) λx.E(x) λx.x = x λx.>
λx.blue(x) λx.white(x) λx.human(x)
Solution:“. . . positive in the moral aesthetic sense (independently of the accidental
structure of the world). Only then the ax. true. . . . ”- Gödel, 1970
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 52
CriticismsNo Neutral Properties
∀φ[P(¬φ)↔ ¬P(φ)]
Either a property is positive or its negation is (but never both)
Are the following properties positive or negative?
λx.G(x) λx.E(x) λx.x = x λx.>
λx.blue(x) λx.white(x) λx.human(x)
Solution:“. . . positive in the moral aesthetic sense (independently of the accidental
structure of the world). Only then the ax. true. . . . ”- Gödel, 1970
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 52
CriticismsNo Neutral Properties
∀φ[P(¬φ)↔ ¬P(φ)]
Either a property is positive or its negation is (but never both)
Are the following properties positive or negative?
λx.G(x) λx.E(x) λx.x = x λx.>
λx.blue(x) λx.white(x) λx.human(x)
Solution:“. . . positive in the moral aesthetic sense (independently of the accidental
structure of the world). Only then the ax. true. . . . ”- Gödel, 1970
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 52
CriticismsNo Neutral Properties
∀φ[P(¬φ)↔ ¬P(φ)]
Either a property is positive or its negation is (but never both)
Are the following properties positive or negative?
λx.G(x) λx.E(x) λx.x = x λx.>
λx.blue(x) λx.white(x) λx.human(x)
Solution:“. . . positive in the moral aesthetic sense (independently of the accidental
structure of the world). Only then the ax. true. . . . ”- Gödel, 1970
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 52
Part F:Conclusions
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 53
Summary of Results
The (new) insights we gained from experiments include:
Logic K sufficient for T1, C and T2Logic S5 not needed for T3Logic KB sufficient for T3 (not well known)We found a simpler new proof of CGödel’s axioms (without conjunct φ(x) in D2) are inconsistentScott’s axioms are consistentFor T1, only half of A1 (A1a) is neededFor T2, the other half (A1b) is needed
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 54
Summary of Results
Our novel contributions to the theorem proving community include
Powerful infrastructure for reasoning with QMLA new natural deduction calculus for higher-order modal logicDifficult new benchmarks problems for HOL proversHuge media attention
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 55
Conclusion
What have we achieved
Verification of Gödel’s ontological argument with HOL proversexact parameters known: constant domain quantification, Henkin Semanticsexperiments with different parameters could be performed
Gained some novel results and insightsMajor step towards Computer-assisted Theoretical Philosophy
see also Ed Zalta’s Computational Metaphysics project at Stanford Universitysee also John Rushby’s recent verification of Anselm’s proof in PVSremember Leibniz’ dictum — Calculemus!
Interesting bridge between CS, Philosophy and Theology
Ongoing and future work
Formalize and verify literature on ontological arguments. . . in particular the criticism and improvements to Gödel
Own contributions — supported by theorem provers
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 56
Conclusion
What have we achieved
Verification of Gödel’s ontological argument with HOL proversexact parameters known: constant domain quantification, Henkin Semanticsexperiments with different parameters could be performed
Gained some novel results and insightsMajor step towards Computer-assisted Theoretical Philosophy
see also Ed Zalta’s Computational Metaphysics project at Stanford Universitysee also John Rushby’s recent verification of Anselm’s proof in PVSremember Leibniz’ dictum — Calculemus!
Interesting bridge between CS, Philosophy and Theology
Ongoing and future work
Formalize and verify literature on ontological arguments. . . in particular the criticism and improvements to Gödel
Own contributions — supported by theorem provers
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 56
Some Comments and Reactions
. . . find more on the internet . . .
Christoph Benzmüller and Bruno Woltzenlogel Paleo On Gödel’s Proof of God’s Existence 57