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Automating Gdels Ontological Proof of Gods Existence with Higher-order Automated Theorem Provers Christoph Benzmller 1 and Bruno Woltzenlogel Paleo 20th of August 2014 ECAI 2014 1 Supported by DFG grant BE 2501/9-1 Christoph Benzmller

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Automating Gödel’s Ontological Proof of God’s Existence with Higher-order Automated Theorem Provers

Christoph Benzmüller1 and Bruno Woltzenlogel Paleo 20th of August 2014 ECAI 2014

1Supported by DFG grant BE 2501/9-1 Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 1

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Vision of Leibniz (1646–1716): Calculemus!

If controversies were to arise, there would be no more need of disputa- tion between two philosophers than between two accountants. For it would suffice to take their pencils in their hands, to sit down to their slates, and to say to each other . . . : Let us calculate. (Translation by Russell)

Quo facto, quando orientur controversiae, non magis dis- putatione opus erit inter duos philosophos, quam inter duos Computistas. Sufficiet enim calamos in manus sumere sedereque ad abacos, et sibi mutuo . . . dicere: calculemus. (Leibniz, 1684) Required: characteristica universalis and calculus ratiocinator

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 2

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Our Contribution: Towards a Computational Metaphysics

Ontological argument for the existence of God We focused on Gödel’s modern version in higher-order modal logic Automation with provers for higher-order classical logic (HOL) confirmation of known results detection of some novel results systematic variation of the logic settings exploited HOL as a universal metalogic (characteristica universalis)

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 3

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A Long History

pros and cons

. . . A n s e l m v . C . G a u n i l

  • . . .

T h . A q u i n a s . . . . . . D e s c a r t e s S p i n

  • z

a L e i b n i z . . . H u m e K a n t . . . H e g e l . . . F r e g e . . . H a r t s h

  • r

n e M a l c

  • l

m L e w i s P l a n t i n g a G ö d e l . . . Anselm’s notion of God (Proslogion, 1078): “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: “God exists.” ∃xG(x)

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 4

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A Long History

pros and cons

. . . A n s e l m v . C . G a u n i l

  • . . .

T h . A q u i n a s . . . . . . D e s c a r t e s S p i n

  • z

a L e i b n i z . . . H u m e K a n t . . . H e g e l . . . F r e g e . . . H a r t s h

  • r

n e M a l c

  • l

m L e w i s P l a n t i n g a G ö d e l . . . Anselm’s notion of God (Proslogion, 1078): “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: “God exists.” ∃xG(x)

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 4

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SLIDE 6

A Long History

pros and cons

. . . A n s e l m v . C . G a u n i l

  • . . .

T h . A q u i n a s . . . . . . D e s c a r t e s S p i n

  • z

a L e i b n i z . . . H u m e K a n t . . . H e g e l . . . F r e g e . . . H a r t s h

  • r

n e M a l c

  • l

m L e w i s P l a n t i n g a G ö d e l . . . Anselm’s notion of God (Proslogion, 1078): “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.” ∃xG(x)

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 4

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The Ontological Proof Today

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 5

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Gödel’s Manuscript: 1930’s, 1941, 1946-1955, 1970

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 6

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Scott’s Version of Gödel’s Axioms, Definitions and Theorems

Axiom A1 Either a property or its negation is positive, but not both: ∀φ[P(¬φ) ↔ ¬P(φ)] Axiom A2 A property necessarily implied by a positive property is positive: ∀φ∀ψ[(P(φ) ∧ ∀x[φ(x) → ψ(x)]) → P(ψ)]

  • Thm. T1 Positive properties are possibly exemplified:

∀φ[P(φ) → ∃xφ(x)]

  • Def. D1 A God-like being possesses all positive properties:

G(x) ↔ ∀φ[P(φ) → φ(x)] Axiom A3 The property of being God-like is positive: P(G)

  • Cor. C

Possibly, God exists: ∃xG(x) Axiom A4 Positive properties are necessarily positive: ∀φ[P(φ) → P(φ)]

  • Def. D2 An essence of an individual is a property possessed by it and necessarily

implying any of its properties: φ ess x ↔ φ(x) ∧ ∀ψ(ψ(x) → ∀y(φ(y) → ψ(y)))

  • Thm. T2 Being God-like is an essence of any God-like being:

∀x[G(x) → G ess x]

  • Def. D3 Necessary existence of an individual is the necessary exemplification of all its

essences: E(x) ↔ ∀φ[φ ess x → ∃yφ(y)] Axiom A5 Necessary existence is a positive property: P(E)

  • Thm. T3 Necessarily, God exists:

∃xG(x)

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 7

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Scott’s Version of Gödel’s Axioms, Definitions and Theorems

Axiom A1 Either a property or its negation is positive, but not both: ∀φ[P(¬φ) ↔ ¬P(φ)] Axiom A2 A property necessarily implied by a positive property is positive: ∀φ∀ψ[(P(φ) ∧ ∀x[φ(x) → ψ(x)]) → P(ψ)]

  • Thm. T1 Positive properties are possibly exemplified:

∀φ[P(φ) → ∃xφ(x)]

  • Def. D1 A God-like being possesses all positive properties:

G(x) ↔ ∀φ[P(φ) → φ(x)] Axiom A3 The property of being God-like is positive: P(G)

  • Cor. C

Possibly, God exists: ∃xG(x) Axiom A4 Positive properties are necessarily positive: ∀φ[P(φ) → P(φ)]

  • Def. D2 An essence of an individual is a property possessed by it and necessarily

implying any of its properties: φ ess x ↔ φ(x) ∧ ∀ψ(ψ(x) → ∀y(φ(y) → ψ(y)))

  • Thm. T2 Being God-like is an essence of any God-like being:

∀x[G(x) → G ess x]

  • Def. D3 Necessary existence of an individual is the necessary exemplification of all its

essences: E(x) ↔ ∀φ[φ ess x → ∃yφ(y)] Axiom A5 Necessary existence is a positive property: P(E)

  • Thm. T3 Necessarily, God exists:

∃xG(x) Difference to Gödel (who omits this conjunct)

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 8

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Scott’s Version of Gödel’s Axioms, Definitions and Theorems

Axiom A1 Either a property or its negation is positive, but not both: ∀φ[P(¬φ) ↔ ¬P(φ)] Axiom A2 A property necessarily implied by a positive property is positive: ∀φ∀ψ[(P(φ) ∧ ∀x[φ(x) → ψ(x)]) → P(ψ)]

  • Thm. T1 Positive properties are possibly exemplified:

∀φ[P(φ) → ∃xφ(x)]

  • Def. D1 A God-like being possesses all positive properties:

G(x) ↔ ∀φ[P(φ) → φ(x)] Axiom A3 The property of being God-like is positive: P(G)

  • Cor. C

Possibly, God exists: ∃xG(x) Axiom A4 Positive properties are necessarily positive: ∀φ[P(φ) → P(φ)]

  • Def. D2 An essence of an individual is a property possessed by it and necessarily

implying any of its properties: φ ess x ↔ φ(x) ∧ ∀ψ(ψ(x) → ∀y(φ(y) → ψ(y)))

  • Thm. T2 Being God-like is an essence of any God-like being:

∀x[G(x) → G ess x]

  • Def. D3 Necessary existence of an individual is the necessary exemplification of all its

essences: E(x) ↔ ∀φ[φ ess x → ∃yφ(y)] Axiom A5 Necessary existence is a positive property: P(E)

  • Thm. T3 Necessarily, God exists:

∃xG(x) Modal operators are used

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 9

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Scott’s Version of Gödel’s Axioms, Definitions and Theorems

Axiom A1 Either a property or its negation is positive, but not both: ∀φ[P(¬φ) ↔ ¬P(φ)] Axiom A2 A property necessarily implied by a positive property is positive: ∀φ∀ψ[(P(φ) ∧ ∀x[φ(x) → ψ(x)]) → P(ψ)]

  • Thm. T1 Positive properties are possibly exemplified:

∀φ[P(φ) → ∃xφ(x)]

  • Def. D1 A God-like being possesses all positive properties:

G(x) ↔ ∀φ[P(φ) → φ(x)] Axiom A3 The property of being God-like is positive: P(G)

  • Cor. C

Possibly, God exists: ∃xG(x) Axiom A4 Positive properties are necessarily positive: ∀φ[P(φ) → P(φ)]

  • Def. D2 An essence of an individual is a property possessed by it and necessarily

implying any of its properties: φ ess x ↔ φ(x) ∧ ∀ψ(ψ(x) → ∀y(φ(y) → ψ(y)))

  • Thm. T2 Being God-like is an essence of any God-like being:

∀x[G(x) → G ess x]

  • Def. D3 Necessary existence of an individual is the necessary exemplification of all its

essences: E(x) ↔ ∀φ[φ ess x → ∃yφ(y)] Axiom A5 Necessary existence is a positive property: P(E)

  • Thm. T3 Necessarily, God exists:

∃xG(x) second-order quantifiers

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 10

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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: NE(x) ≡ ∀ϕ .[ϕ ess x → ∃y.ϕ(y)] A3 P(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)] C: ∃z.G(z) A1b ∀ϕ .[¬P(ϕ) → P(¬ϕ)] A4 ∀ϕ .[P(ϕ) → P(ϕ)] T2: ∀y .[G(y) → G ess y] A5 P(NE) L1: ∃z.G(z) → ∃x.G(x) ∃z.G(z) → ∃x.G(x) S5 ∀ξ .[ξ → ξ] L2: ∃z.G(z) → ∃x.G(x) C: ∃z.G(z) L2: ∃z.G(z) → ∃x.G(x) T3: ∃x.G(x)

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 11

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How to automate Higher-Order Modal Logic?

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 12

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Embedding HOML in HOL

Challenge: No provers for Higher-order Modal Logic (HOML) Our solution: Embedding in Higher-order Classical Logic (HOL) Then use existing HOL theorem provers for reasoning in HOML

[BenzmüllerPaulson, Logica Universalis, 2013]

Previous empirical findings: Embedding of First-order Modal Logic in HOL works well

[BenzmüllerOttenRaths, ECAI, 2012] [Benzmüller, LPAR, 2013]

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 13

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Embedding HOML in HOL

HOML ϕ, ψ ::= . . . | ¬ϕ | ϕ ∧ ψ | ϕ → ψ | ϕ | ϕ | ∀x ϕ | ∃x ϕ | ∀P ϕ Kripke style semantics (possible world semantics) HOL s, t ::= C | x | λxs | s t | ¬s | s ∨ t | ∀x t Church’s simple type theory

[Church, 1940], [Henkin, 1950]

various theorem provers exist interactive: Isabelle/HOL, HOL4, Hol Light, Coq/HOL, PVS, . . . automated: TPS, LEO-II, Satallax, Nitpick, Isabelle/HOL, . . .

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 14

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Embedding HOML in HOL

HOML ϕ, ψ ::= . . . | ¬ϕ | ϕ ∧ ψ | ϕ → ψ | ϕ | ϕ | ∀x ϕ | ∃x ϕ | ∀P ϕ HOL s, t ::= C | x | λxs | s t | ¬s | s ∨ t | ∀x t HOML in HOL: HOML formulas ϕ are mapped to HOL predicates ϕµo ¬ = λϕµoλwµ¬ϕw ∧ = λϕµoλψµoλwµ(ϕw ∧ ψw) → = λϕµoλψµoλwµ(¬ϕw ∨ ψw) ∀ = λhγ(µo)λwµ∀dγ hdw ∃ = λhγ(µo)λwµ∃dγ hdw

  • =

λϕµoλwµ∀uµ (¬rwu ∨ ϕu)

  • =

λϕµoλwµ∃uµ (rwu ∧ ϕu) valid = λϕµo∀wµ ϕw Ax The equations in Ax are given as axioms to the HOL provers!

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 15

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Embedding HOML in HOL

HOML ϕ, ψ ::= . . . | ¬ϕ | ϕ ∧ ψ | ϕ → ψ | ϕ | ϕ | ∀x ϕ | ∃x ϕ | ∀P ϕ HOL s, t ::= C | x | λxs | s t | ¬s | s ∨ t | ∀x t HOML in HOL: HOML formulas ϕ are mapped to HOL predicates ϕµo ¬ = λϕµoλwµ¬ϕw ∧ = λϕµoλψµoλwµ(ϕw ∧ ψw) → = λϕµoλψµoλwµ(¬ϕw ∨ ψw) ∀ = λhγ(µo)λwµ∀dγ hdw ∃ = λhγ(µo)λwµ∃dγ hdw

  • =

λϕµoλwµ∀uµ (¬rwu ∨ ϕu)

  • =

λϕµoλwµ∃uµ (rwu ∧ ϕu) valid = λϕµo∀wµ ϕw Ax The equations in Ax are given as axioms to the HOL provers!

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 15

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Embedding HOML in HOL

HOML ϕ, ψ ::= . . . | ¬ϕ | ϕ ∧ ψ | ϕ → ψ | ϕ | ϕ | ∀x ϕ | ∃x ϕ | ∀P ϕ HOL s, t ::= C | x | λxs | s t | ¬s | s ∨ t | ∀x t HOML in HOL: HOML formulas ϕ are mapped to HOL predicates ϕµo ¬ = λϕµoλwµ¬ϕw ∧ = λϕµoλψµoλwµ(ϕw ∧ ψw) → = λϕµoλψµoλwµ(¬ϕw ∨ ψw) ∀ = λhγ(µo)λwµ∀dγ hdw ∃ = λhγ(µo)λwµ∃dγ hdw

  • =

λϕµoλwµ∀uµ (¬rwu ∨ ϕu)

  • =

λϕµoλwµ∃uµ (rwu ∧ ϕu) valid = λϕµo∀wµ ϕw Ax The equations in Ax are given as axioms to the HOL provers!

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 15

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Embedding HOML in HOL

Example HOML formula ∃xG(x) HOML formula in HOL valid (∃xG(x))µo expansion, βη-conversion ∀wµ(∃xG(x))µo w expansion, βη-conversion ∀wµ∃uµ(rwu ∧ (∃xG(x))µou) expansion, βη-conversion ∀wµ∃uµ(rwu ∧ ∃xGxu) Expansion: user or prover may flexibly choose expansion depth What are we doing? In order to prove that ϕ is valid in HOML, –> we instead prove that valid ϕµo can be derived from Ax in HOL. This can be done with interactive or automated HOL theorem provers. For the experts: soundness and completeness wrt Henkin semantics

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 16

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Embedding HOML in HOL

Example HOML formula ∃xG(x) HOML formula in HOL valid (∃xG(x))µo expansion, βη-conversion ∀wµ(∃xG(x))µo w expansion, βη-conversion ∀wµ∃uµ(rwu ∧ (∃xG(x))µou) expansion, βη-conversion ∀wµ∃uµ(rwu ∧ ∃xGxu) Expansion: user or prover may flexibly choose expansion depth What are we doing? In order to prove that ϕ is valid in HOML, –> we instead prove that valid ϕµo can be derived from Ax in HOL. This can be done with interactive or automated HOL theorem provers. For the experts: soundness and completeness wrt Henkin semantics

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 16

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Embedding HOML in HOL

Example HOML formula ∃xG(x) HOML formula in HOL valid (∃xG(x))µo expansion, βη-conversion ∀wµ(∃xG(x))µo w expansion, βη-conversion ∀wµ∃uµ(rwu ∧ (∃xG(x))µou) expansion, βη-conversion ∀wµ∃uµ(rwu ∧ ∃xGxu) Expansion: user or prover may flexibly choose expansion depth What are we doing? In order to prove that ϕ is valid in HOML, –> we instead prove that valid ϕµo can be derived from Ax in HOL. This can be done with interactive or automated HOL theorem provers. For the experts: soundness and completeness wrt Henkin semantics

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 16

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Embedding HOML in HOL

Example HOML formula ∃xG(x) HOML formula in HOL valid (∃xG(x))µo expansion, βη-conversion ∀wµ(∃xG(x))µo w expansion, βη-conversion ∀wµ∃uµ(rwu ∧ (∃xG(x))µou) expansion, βη-conversion ∀wµ∃uµ(rwu ∧ ∃xGxu) Expansion: user or prover may flexibly choose expansion depth What are we doing? In order to prove that ϕ is valid in HOML, –> we instead prove that valid ϕµo can be derived from Ax in HOL. This can be done with interactive or automated HOL theorem provers. For the experts: soundness and completeness wrt Henkin semantics

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 16

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Embedding HOML in HOL

Example HOML formula ∃xG(x) HOML formula in HOL valid (∃xG(x))µo expansion, βη-conversion ∀wµ(∃xG(x))µo w expansion, βη-conversion ∀wµ∃uµ(rwu ∧ (∃xG(x))µou) expansion, βη-conversion ∀wµ∃uµ(rwu ∧ ∃xGxu) Expansion: user or prover may flexibly choose expansion depth What are we doing? In order to prove that ϕ is valid in HOML, –> we instead prove that valid ϕµo can be derived from Ax in HOL. This can be done with interactive or automated HOL theorem provers. For the experts: soundness and completeness wrt Henkin semantics

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 16

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Embedding HOML in HOL

Example HOML formula ∃xG(x) HOML formula in HOL valid (∃xG(x))µo expansion, βη-conversion ∀wµ(∃xG(x))µo w expansion, βη-conversion ∀wµ∃uµ(rwu ∧ (∃xG(x))µou) expansion, βη-conversion ∀wµ∃uµ(rwu ∧ ∃xGxu) Expansion: user or prover may flexibly choose expansion depth What are we doing? In order to prove that ϕ is valid in HOML, –> we instead prove that valid ϕµo can be derived from Ax in HOL. This can be done with interactive or automated HOL theorem provers. For the experts: soundness and completeness wrt Henkin semantics

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 16

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Embedding HOML in HOL

Example HOML formula ∃xG(x) HOML formula in HOL valid (∃xG(x))µo expansion, βη-conversion ∀wµ(∃xG(x))µo w expansion, βη-conversion ∀wµ∃uµ(rwu ∧ (∃xG(x))µou) expansion, βη-conversion ∀wµ∃uµ(rwu ∧ ∃xGxu) Expansion: user or prover may flexibly choose expansion depth What are we doing? In order to prove that ϕ is valid in HOML, –> we instead prove that valid ϕµo can be derived from Ax in HOL. This can be done with interactive or automated HOL theorem provers. For the experts: soundness and completeness wrt Henkin semantics

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 16

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Embedding HOML in HOL

Example HOML formula ∃xG(x) HOML formula in HOL valid (∃xG(x))µo expansion, βη-conversion ∀wµ(∃xG(x))µo w expansion, βη-conversion ∀wµ∃uµ(rwu ∧ (∃xG(x))µou) expansion, βη-conversion ∀wµ∃uµ(rwu ∧ ∃xGxu) Expansion: user or prover may flexibly choose expansion depth What are we doing? In order to prove that ϕ is valid in HOML, –> we instead prove that valid ϕµo can be derived from Ax in HOL. This can be done with interactive or automated HOL theorem provers. For the experts: soundness and completeness wrt Henkin semantics

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 16

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Embedding HOML in HOL

Example HOML formula ∃xG(x) HOML formula in HOL valid (∃xG(x))µo expansion, βη-conversion ∀wµ(∃xG(x))µo w expansion, βη-conversion ∀wµ∃uµ(rwu ∧ (∃xG(x))µou) expansion, βη-conversion ∀wµ∃uµ(rwu ∧ ∃xGxu) Expansion: user or prover may flexibly choose expansion depth What are we doing? In order to prove that ϕ is valid in HOML, –> we instead prove that valid ϕµo can be derived from Ax in HOL. This can be done with interactive or automated HOL theorem provers. For the experts: soundness and completeness wrt Henkin semantics

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 16

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Automated Theorem Provers and Model Finders for HOL

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 17

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Proof Automation and Consistency Checking with HOL-P

Provers are called remotely in Miami — no local installation needed! Download our experiments from https://github.com/ FormalTheology/GoedelGod/tree/master/Formalizations/THF

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 18

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Interaction and Automation in Proof Assistant Isabelle/HOL

See verifiable Isabelle/HOL journal article at: http://afp.sourceforge.net/entries/GoedelGod.shtml

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 19

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Interaction in Proof Assistant Coq

See verifiable Coq document at: https://github.com/ FormalTheology/GoedelGod/tree/master/Formalizations/Coq

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 20

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Main Findings

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 21

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Main Findings

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 22

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Main Findings

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 23

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Main Findings

Automating Scott’s proof script T1: "Positive properties are possibly exemplified" proved by LEO-II and Satallax in logic: K from axioms:

A1 and A2

for domain conditions:

constant domains

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 24

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Main Findings

Automating Scott’s proof script T1: "Positive properties are possibly exemplified" proved by LEO-II and Satallax in logic: K from axioms:

A1 and A2 A1(⊃) and A2

for domain conditions:

constant domains

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 24

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Main Findings

Automating Scott’s proof script T1: "Positive properties are possibly exemplified" proved by LEO-II and Satallax in logic: K from axioms:

A1 and A2 A1(⊃) and A2

for domain conditions:

constant domains varying domains (individuals)

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 24

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Main Findings

Automating Scott’s proof script C: "Possibly, God exists” proved by LEO-II and Satallax in logic: K from assumptions:

T1, D1, A3 A1, A2, D1, A3

for domain conditions:

constant domains varying domains (individuals)

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 25

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Main Findings

Automating Scott’s proof script T2: "Being God-like is an ess. of any God-like being” proved by LEO-II and Satallax in logic: K from assumptions:

A1, D1, A4, D2 A1, A2, D1, A3, A4, D2

for domain conditions:

constant domains varying domains (individuals)

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 26

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Main Findings

Automating Scott’s proof script T3: "Necessarily, God exists” proved by LEO-II and Satallax in logic: KB from assumptions:

D1, C, T2, D3, A5

for domain conditions:

constant domains varying domains (individuals)

For logic K we got a countermodel by Nitpick

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 27

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Main Findings

Automating Scott’s proof script Summary proof verified and automated KB is sufficient (critisized logic S5 not needed!) proof works for constant and varying domains exact dependencies determined experimentally ATPs have found alternative proofs (shorter)

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 28

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Main Findings

Consistency check: Gödel vs. Scott Scott’s assumptions are consistent; shown by Nitpick Gödel’s assumptions are inconsistent; shown by LEO-II (new philosophical result!)

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 29

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Main Findings

Further Results Monotheism holds God is flawless

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 30

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Main Findings

Modal Collapse ∀ϕ(ϕ ⊃ ϕ) proved by LEO-II and Satallax for constant and varying domains Main critique on Gödel’s ontological proof: there are no contingent truths everything is determined / no free will why using modal logic in the first place?

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 31

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Avoiding the Modal Collapse: Very recent work (not yet published)

Variants of Gödel’s proof that avoid the modal collapse [Frode Bjørdal, Understanding Gödel’s Ontological Argument, 1998] (verified and automated) [Anthony Anderson, Some emendations of Gödel’s ontological proof, 1990] (verified and automated) [Melvin Fitting, Types, Tableaux and Gödel’s God, 2002] (ongoing) Future work [André Fuhrmann, 2005] [Petr Hajek, 1996, 2001, 2002, 2008, 2011] [Szatkowski, 2011] . . .

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 32

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Conclusion

Achievements significant contribution towards a Computational Metaphysics HOL very fruitfully exploited as a universal metalogic systematic study of a prominent philosophical argument even some novel results were found by HOL-ATPs infrastructure can be adapted for other logics and logic combinations Relevance (wrt foundations and applications) Theoretical Philosophy, Artificial Intelligence, Computer Science, Maths Little related work: only for Anselm’s simpler argument first-order ATP PROVER9

[OppenheimerZalta, 2011]

interactive proof assistant PVS

[Rushby, 2013]

Future work continuation of systematic study of the ontological argument further studies in Computational Metaphysics

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 33

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SLIDE 48

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.
  • . . .

See links at https://github.com/FormalTheology/GoedelGod/tree/master/Press

Christoph Benzmüller and Bruno Woltzenlogel Paleo Automating Gödel’s Ontological Proof of God’s Existence 34