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Gödel’s Proof of God’s Existence
Christoph Benzmüller and Bruno Woltzenlogel Paleo
Square of OppositionVatican, May 6, 2014
A gift to Priest Edvaldo in Piracicaba, Brazil
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 1
Contribution
First time mechanization and automation of(variants of) a modern ontological argument(variants of) higher-order modal logic
Work context/history:
Proposal: exploit classical higher-order logic (HOL) asuniversal meta-logic — cf. previous talks at UNILOG
for object-level reasoning (in embedded non-classical logics)for meta-level reasoning (about embedded non-classical logics)
Proof of concept: demonstrate practical relevance of theapproach by an interesting and relevant applicationExperiments: systematic study of Gödel’s argumentRelation to Square of Opposition: should be easy toanalyze variants of the Square within our approach
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 2
Introduction
Challenge: No provers for Higher-order Quantified Modal Logic (QML)
Our solution: Embedding in Higher-order Classical Logic (HOL)
What we did:
A: Pen and paper: detailed natural deduction proofB: Formalization: in classical higher-order logic (HOL)
Automation: theorem provers LEO-II(E) 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 Gödel’s Proof of God’s Existence 3
Introduction
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 Gödel’s Proof of God’s Existence 4
Introduction
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 Gödel’s Proof of God’s Existence 4
Introduction
Do you really need a MacBook to obtain the results? No
Did Apple send us some money? No
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 5
Introduction
Do you really need a MacBook to obtain the results? No
Did Apple send us some money? No
Christoph Benzmüller and Bruno Woltzenlogel Paleo 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 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 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.
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?
Towards: ‘Computer-assisted Theoretical Philosophy”
(cf. Leibniz dictum — Calculemus!)
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 7
Gödel’s Manuscript: 1930’s, 1941, 1946-1955, 1970
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 8
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 necessarilyimplying 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 individ. is the necessary exemplification of all itsessences: NE(x) ≡ ∀φ[φ ess. x ⊃ �∃yφ(y)]
Axiom A5 Necessary existence is a positive property: P(NE)
Thm. T3 Necessarily, God exists: �∃xG(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 9
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 necessarilyimplying 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 individ. is the necessary exemplification of all itsessences: NE(x) ≡ ∀φ[φ ess. x ⊃ �∃yφ(y)]
Axiom A5 Necessary existence is a positive property: P(NE)
Thm. T3 Necessarily, God exists: �∃xG(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 9
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 necessarilyimplying 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 individ. is the necessary exemplification of all itsessences: NE(x) ≡ ∀φ[φ ess. x ⊃ �∃yφ(y)]
Axiom A5 Necessary existence is a positive property: P(NE)
Thm. T3 Necessarily, God exists: �∃xG(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 9
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 necessarilyimplying 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 individ. is the necessary exemplification of all itsessences: NE(x) ≡ ∀φ[φ ess. x ⊃ �∃yφ(y)]
Axiom A5 Necessary existence is a positive property: P(NE)
Thm. T3 Necessarily, God exists: �∃xG(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 9
Remainder of this Talk
Embedding of QML in HOL and Proof Automation (myself)
Proof Overview (Bruno)Experiments and Results (Bruno)Conclusion and Outlook (Bruno)
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 10
Embedding of QML in HOL and Proof Automation
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 11
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 empirical findings:
Embedding of First-order Modal Logic in HOL works well[BenzmüllerOttenRaths, ECAI, 2012]
[BenzmüllerRaths, LPAR, 2013]
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 12
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 exist
interactive: Isabelle/HOL, HOL4, Hol Light, Coq/HOL, PVS, . . .
automated: TPS, LEO-II, Satallax, Nitpick, Isabelle/HOL, . . .
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 13
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 Gödel’s Proof of God’s Existence 14
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 Gödel’s Proof of God’s Existence 14
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 Gödel’s Proof of God’s Existence 14
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.
Soundness and Completeness: wrt. Henkin semantics
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 15
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.
Soundness and Completeness: wrt. Henkin semantics
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 15
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.
Soundness and Completeness: wrt. Henkin semantics
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 15
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.
Soundness and Completeness: wrt. Henkin semantics
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 15
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.
Soundness and Completeness: wrt. Henkin semantics
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 15
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.
Soundness and Completeness: wrt. Henkin semantics
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 15
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.
Soundness and Completeness: wrt. Henkin semantics
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 15
Automated Theorem Provers and Model Finders for HOL
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 16
Proof OverviewExperiments and Results
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 17
Gödel’s Manuscript: 1930’s, 1941, 1946-1955, 1970
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 18
Proof Overview
T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 19
Proof Overview
C1: ^∃z.G(z)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 20
Proof Overview
C1: ^∃z.G(z) L2: ^∃z.G(z) ⊃ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 21
Proof Overview
C1: ^∃z.G(z) L2: ^∃z.G(z) ⊃ �∃x.G(x)T3: �∃x.G(x)
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 22
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 Gödel’s Proof of God’s Existence 23
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 Gödel’s Proof of God’s Existence 24
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 Gödel’s Proof of God’s Existence 25
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 Gödel’s Proof of God’s Existence 26
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 Gödel’s Proof of God’s Existence 27
Proof Overview
D1: G(x) ≡ ∀ϕ.[P(ϕ) ⊃ ϕ(x)]
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 Gödel’s Proof of God’s Existence 28
Proof Overview
D1: G(x) ≡ ∀ϕ.[P(ϕ) ⊃ ϕ(x)]
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 Gödel’s Proof of God’s Existence 29
Proof Overview
D1: G(x) ≡ ∀ϕ.[P(ϕ) ⊃ ϕ(x)]
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 Gödel’s Proof of God’s Existence 30
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)]
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 Gödel’s Proof of God’s Existence 31
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)]
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 Gödel’s Proof of God’s Existence 32
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) 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 Gödel’s Proof of God’s Existence 33
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 Gödel’s Proof of God’s Existence 34
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 Gödel’s Proof of God’s Existence 35
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 Gödel’s Proof of God’s Existence 36
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 Gödel’s Proof of God’s Existence 37
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 Gödel’s Proof of God’s Existence 38
Natural Deduction ProofsT2 (Partial)
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 39
Implementations and Experiments
Formal encodings (in HOL) of:modal logic axiomsaxioms, definitions, and theorems in Scott’s proof script
Experiments using automated proversLEO-II, Satallax, AgsyHOL
Interactive proofs using proof assistantsIsabelle and Coq
Source files available at:
https://github.com/FormalTheology/GoedelGod/
Demos on request!
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 40
Implementations and Experiments
Formal encodings (in HOL) of:modal logic axiomsaxioms, definitions, and theorems in Scott’s proof script
Experiments using automated proversLEO-II, Satallax, AgsyHOL
Interactive proofs using proof assistantsIsabelle and Coq
Source files available at:
https://github.com/FormalTheology/GoedelGod/
Demos on request!
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 40
Results
Axioms and definitions are consistent.
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 41
Results
Axioms and definitions are consistent.Logic K is sufficient for proving T1, C and T2.Logic KB is sufficient for proving the final theorem T3.
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 41
Results
Axioms and definitions are consistent.Logic K is sufficient for proving T1, C and T2.Logic KB is sufficient for proving the final theorem T3.
Adresses criticisms: modal logic S5 is too strong
∀P.[^�P ⊃ �P]
If something is possibly necessary, then it is necessary.
S5 usually considered adequate(But KB is sufficient! — shown by HOL ATPs)
∀P.[P ⊃ �^P]
If something is the case, then it is necessarily possible.
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 41
Results
Axioms and definitions are consistent.Logic K is sufficient for proving T1, C and T2.Logic KB is sufficient for proving the final theorem T3.HOL-ATPs prove T1, C, and T2 from axioms quickly;succeed in proving T3 from axioms, C and T2;but fail in proving T3 from axioms alone.
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 41
Results
Axioms and definitions are consistent.Logic K is sufficient for proving T1, C and T2.Logic KB is sufficient for proving the final theorem T3.HOL-ATPs prove T1, C, and T2 from axioms quickly;succeed in proving T3 from axioms, C and T2;but fail in proving T3 from axioms alone.Gödel’s original axioms and definitions, omitting conjunctφ(x) in the definition of essence, seem inconsistent.
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 41
Results
Axioms and definitions are consistent.Logic K is sufficient for proving T1, C and T2.Logic KB is sufficient for proving the final theorem T3.HOL-ATPs prove T1, C, and T2 from axioms quickly;succeed in proving T3 from axioms, C and T2;but fail in proving T3 from axioms alone.Gödel’s original axioms and definitions, omitting conjunctφ(x) in the definition of essence, seem inconsistent.∃x.G(x) can be proved without first proving �∃x.G(x).
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 41
Results
Axioms and definitions are consistent.Logic K is sufficient for proving T1, C and T2.Logic KB is sufficient for proving the final theorem T3.HOL-ATPs prove T1, C, and T2 from axioms quickly;succeed in proving T3 from axioms, C and T2;but fail in proving T3 from axioms alone.Gödel’s original axioms and definitions, omitting conjunctφ(x) in the definition of essence, seem inconsistent.∃x.G(x) can be proved without first proving �∃x.G(x).Equality is not necessary to prove T1.
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 41
Results
Axioms and definitions are consistent.Logic K is sufficient for proving T1, C and T2.Logic KB is sufficient for proving the final theorem T3.HOL-ATPs prove T1, C, and T2 from axioms quickly;succeed in proving T3 from axioms, C and T2;but fail in proving T3 from axioms alone.Gödel’s original axioms and definitions, omitting conjunctφ(x) in the definition of essence, seem inconsistent.∃x.G(x) can be proved without first proving �∃x.G(x).Equality is not necessary to prove T1.A2 may be used only once to prove T1.
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 41
Results
Gödel’s axioms imply the modal collapse: ∀φ.(φ ⊃ �φ)
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 42
Results
Gödel’s axioms imply the modal collapse: ∀φ.(φ ⊃ �φ)
Fundamental criticism against Gödel’s argument.
Everything that is the case is so necessarily.
Follows from T2, T3 and D2 (as shown by HOL ATPs).
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 Gödel’s Proof of God’s Existence 42
Results
Gödel’s axioms imply the modal collapse: ∀φ.(φ ⊃ �φ)Fundamental criticism against Gödel’s argument.
Everything that is the case is so necessarily.
Follows from T2, T3 and D2 (as shown by HOL ATPs).
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 Gödel’s Proof of God’s Existence 42
Results
God is flawless: ∀x.G(x) ⊃ (∀ϕ.¬P(ϕ) ⊃ ¬ϕ(x)).Monotheism: ∀x.∀y.G(x) ∧ G(y) ⊃ x = y.
All results hold for both- constant domain semantics- varying domain semantics
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 43
Results
God is flawless: ∀x.G(x) ⊃ (∀ϕ.¬P(ϕ) ⊃ ¬ϕ(x)).Monotheism: ∀x.∀y.G(x) ∧ G(y) ⊃ x = y.
All results hold for both- constant domain semantics- varying domain semantics
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 43
Results
God is flawless: ∀x.G(x) ⊃ (∀ϕ.¬P(ϕ) ⊃ ¬ϕ(x)).Monotheism: ∀x.∀y.G(x) ∧ G(y) ⊃ x = y.
All results hold for both- constant domain semantics- varying domain semantics
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 43
Conclusions
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 44
Conclusion
Achievements:
Infra-structure for automated higher-order modal reasoningVerification of Gödel’s ontological argument with HOL provers
experiments with different parameters
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 criticisms and proposed improvements
Own contributions — supported by theorem provers
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 45
Conclusion
Achievements:
Infra-structure for automated higher-order modal reasoningVerification of Gödel’s ontological argument with HOL provers
experiments with different parameters
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 criticisms and proposed improvements
Own contributions — supported by theorem provers
Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel’s Proof of God’s Existence 45