Gödel s Proof of God s Existence

Transcription

1 Gödel s Proof of God s Existence Christoph Benzmüller and Bruno Woltzenlogel Paleo Square of Opposition Vatican, 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

2 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) as universal 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 the approach by an interesting and relevant application Experiments: systematic study of Gödel s argument Relation to Square of Opposition: should be easy to analyze variants of the Square within our approach Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 2

3 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 proof B: Formalization: in classical higher-order logic (HOL) Automation: theorem provers LEO-II(E) and Satallax Consistency: model finder Nitpick (Nitrox) C: Step-by-step verification: proof assistant Coq D: 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

4 Introduction Austria - Die Presse - Wiener Zeitung - ORF -... Italy - Repubblica - Ilsussidario -... India - DNA India - Delhi Daily News - India Today -... Germany - Telepolis & Heise - Spiegel Online - FAZ - Die Welt - Berliner Morgenpost - Hamburger Abendpost -... 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

5 Introduction Austria - Die Presse - Wiener Zeitung - ORF -... Italy - Repubblica - Ilsussidario -... India - DNA India - Delhi Daily News - India Today -... Germany - Telepolis & Heise - Spiegel Online - FAZ - Die Welt - Berliner Morgenpost - Hamburger Abendpost -... 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

6 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

7 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

8 Introduction Rich history on ontological arguments (pros and cons) Anselm v. C. Gaunilo Th. Aquinas Descartes Spinoza Leibniz... Hume Kant... Hegel... Frege... Hartshorne Malcolm Lewis Gödel Plantinga... 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

9 Introduction Rich history on ontological arguments (pros and cons) Anselm v. C. Gaunilo Th. Aquinas Descartes Spinoza Leibniz... Hume Kant... Hegel... Frege... Hartshorne Malcolm Lewis Gödel Plantinga... 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

10 Introduction Different Interests in Ontological Arguments: Philosophical: Boundaries of Metaphysics & Epistemology We 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

11 Gödel s Manuscript: 1930 s, 1941, , 1970 Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 8

12 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: Def. D1 A God-like being possesses all positive properties: Axiom A3 The property of being God-like is positive: φ[p(φ) xφ(x)] G(x) φ[p(φ) φ(x)] 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 individ. is the necessary exemplification of all its essences: NE(x) φ[φ ess. x yφ(y)] Axiom A5 Necessary existence is a positive property: Thm. T3 Necessarily, God exists: P(NE) xg(x) Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 9

13 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: Def. D1 A God-like being possesses all positive properties: Axiom A3 The property of being God-like is positive: φ[p(φ) xφ(x)] G(x) φ[p(φ) φ(x)] 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 individ. is the necessary exemplification of all its essences: NE(x) φ[φ ess. x yφ(y)] Axiom A5 Necessary existence is a positive property: Thm. T3 Necessarily, God exists: P(NE) xg(x) Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 9

14 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: Def. D1 A God-like being possesses all positive properties: Axiom A3 The property of being God-like is positive: φ[p(φ) xφ(x)] G(x) φ[p(φ) φ(x)] 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 individ. is the necessary exemplification of all its essences: NE(x) φ[φ ess. x yφ(y)] Axiom A5 Necessary existence is a positive property: Thm. T3 Necessarily, God exists: P(NE) xg(x) Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 9

15 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: Def. D1 A God-like being possesses all positive properties: Axiom A3 The property of being God-like is positive: φ[p(φ) xφ(x)] G(x) φ[p(φ) φ(x)] 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 individ. is the necessary exemplification of all its essences: NE(x) φ[φ ess. x yφ(y)] Axiom A5 Necessary existence is a positive property: Thm. T3 Necessarily, God exists: P(NE) xg(x) Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 9

16 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

17 Embedding of QML in HOL and Proof Automation Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 11

18 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

19 Formalization in HOL QML 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 meanwhile very well understood Henkin semantics vs. standard semantics various 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

20 Formalization in HOL QML 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 QML in HOL: QML formulas ϕ are mapped to HOL predicates ϕ ι o = λϕ ι o λs ι ϕs = λϕ ι o λψ ι o λs ι (ϕs ψs) = λϕ ι o λψ ι o λs ι ( ϕs ψs) = λϕ ι o λs ι u ι ( rsu ϕu) = λϕ ι o λs ι u ι (rsu ϕu) = λh µ (ι o) λs ι d µ hds = λh µ (ι o) λs ι d µ hds = λh (µ (ι o)) (ι o) λs ι d µ Hds Ax valid = λϕ ι o w ι ϕw 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

21 Formalization in HOL QML 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 QML in HOL: QML formulas ϕ are mapped to HOL predicates ϕ ι o = λϕ ι o λs ι ϕs = λϕ ι o λψ ι o λs ι (ϕs ψs) = λϕ ι o λψ ι o λs ι ( ϕs ψs) = λϕ ι o λs ι u ι ( rsu ϕu) = λϕ ι o λs ι u ι (rsu ϕu) = λh µ (ι o) λs ι d µ hds = λh µ (ι o) λs ι d µ hds = λh (µ (ι o)) (ι o) λs ι d µ Hds Ax valid = λϕ ι o w ι ϕw 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

22 Formalization in HOL QML 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 QML in HOL: QML formulas ϕ are mapped to HOL predicates ϕ ι o = λϕ ι o λs ι ϕs = λϕ ι o λψ ι o λs ι (ϕs ψs) = λϕ ι o λψ ι o λs ι ( ϕs ψs) = λϕ ι o λs ι u ι ( rsu ϕu) = λϕ ι o λs ι u ι (rsu ϕu) = λh µ (ι o) λs ι d µ hds = λh µ (ι o) λs ι d µ hds = λh (µ (ι o)) (ι o) λs ι d µ Hds Ax valid = λϕ ι o w ι ϕw 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

23 Formalization in HOL Example: QML formula xg(x) QML formula in HOL valid ( xg(x)) ι o expansion, βη-conversion w ι ( xg(x)) ι o w expansion, βη-conversion w ι u ι (rwu ( xg(x)) ι o u) expansion, βη-conversion w ι u ι (rwu xgxu) What are we doing? In order to prove that ϕ is valid in QML, > we instead prove that valid ϕ ι 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

24 Formalization in HOL Example: QML formula xg(x) QML formula in HOL valid ( xg(x)) ι o expansion, βη-conversion w ι ( xg(x)) ι o w expansion, βη-conversion w ι u ι (rwu ( xg(x)) ι o u) expansion, βη-conversion w ι u ι (rwu xgxu) What are we doing? In order to prove that ϕ is valid in QML, > we instead prove that valid ϕ ι 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

25 Formalization in HOL Example: QML formula xg(x) QML formula in HOL valid ( xg(x)) ι o expansion, βη-conversion w ι ( xg(x)) ι o w expansion, βη-conversion w ι u ι (rwu ( xg(x)) ι o u) expansion, βη-conversion w ι u ι (rwu xgxu) What are we doing? In order to prove that ϕ is valid in QML, > we instead prove that valid ϕ ι 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

26 Formalization in HOL Example: QML formula xg(x) QML formula in HOL valid ( xg(x)) ι o expansion, βη-conversion w ι ( xg(x)) ι o w expansion, βη-conversion w ι u ι (rwu ( xg(x)) ι o u) expansion, βη-conversion w ι u ι (rwu xgxu) What are we doing? In order to prove that ϕ is valid in QML, > we instead prove that valid ϕ ι 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

27 Formalization in HOL Example: QML formula xg(x) QML formula in HOL valid ( xg(x)) ι o expansion, βη-conversion w ι ( xg(x)) ι o w expansion, βη-conversion w ι u ι (rwu ( xg(x)) ι o u) expansion, βη-conversion w ι u ι (rwu xgxu) What are we doing? In order to prove that ϕ is valid in QML, > we instead prove that valid ϕ ι 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

28 Formalization in HOL Example: QML formula xg(x) QML formula in HOL valid ( xg(x)) ι o expansion, βη-conversion w ι ( xg(x)) ι o w expansion, βη-conversion w ι u ι (rwu ( xg(x)) ι o u) expansion, βη-conversion w ι u ι (rwu xgxu) What are we doing? In order to prove that ϕ is valid in QML, > we instead prove that valid ϕ ι 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

29 Formalization in HOL Example: QML formula xg(x) QML formula in HOL valid ( xg(x)) ι o expansion, βη-conversion w ι ( xg(x)) ι o w expansion, βη-conversion w ι u ι (rwu ( xg(x)) ι o u) expansion, βη-conversion w ι u ι (rwu xgxu) What are we doing? In order to prove that ϕ is valid in QML, > we instead prove that valid ϕ ι 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

30 Automated Theorem Provers and Model Finders for HOL Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 16

31 Proof Overview Experiments and Results Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 17

32 Gödel s Manuscript: 1930 s, 1941, , 1970 Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 18

33 Proof Overview T3: x.g(x) Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 19

34 Proof Overview C1: z.g(z) T3: x.g(x) Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 20

35 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

36 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

37 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

38 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

39 Proof Overview S5 z.g(z) x.g(x) 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

40 Proof Overview L1: z.g(z) x.g(x) S5 z.g(z) x.g(x) 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

41 Proof Overview D1: G(x) ϕ.[p(ϕ) ϕ(x)] L1: z.g(z) x.g(x) S5 z.g(z) x.g(x) 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

42 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) S5 z.g(z) x.g(x) 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

43 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) S5 z.g(z) x.g(x) 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

44 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) S5 z.g(z) x.g(x) 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

45 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)] T2: y.[g(y) G ess. y] A4 ϕ.[p(ϕ) P(ϕ)] A5 P(E) L1: z.g(z) x.g(x) S5 z.g(z) x.g(x) 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

46 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)] C1: z.g(z) T2: y.[g(y) G ess. y] A4 ϕ.[p(ϕ) P(ϕ)] A5 P(E) L1: z.g(z) x.g(x) S5 z.g(z) x.g(x) 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

47 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)] A3 P(G) C1: z.g(z) T1: ϕ.[p(ϕ) x.ϕ(x)] A1b ϕ.[ P(ϕ) P( ϕ)] T2: y.[g(y) G ess. y] A4 ϕ.[p(ϕ) P(ϕ)] A5 P(E) L1: z.g(z) x.g(x) S5 z.g(z) x.g(x) 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

48 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)] A3 P(G) A2 ϕ. ψ.[(p(ϕ) x.[ϕ(x) ψ(x)]) P(ψ)] C1: z.g(z) T1: ϕ.[p(ϕ) x.ϕ(x)] A1a ϕ.[p( ϕ) P(ϕ)] A1b ϕ.[ P(ϕ) P( ϕ)] T2: y.[g(y) G ess. y] A4 ϕ.[p(ϕ) P(ϕ)] A5 P(E) L1: z.g(z) x.g(x) S5 z.g(z) x.g(x) 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

49 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)] A3 P(G) A2 ϕ. ψ.[(p(ϕ) x.[ϕ(x) ψ(x)]) P(ψ)] C1: z.g(z) T1: ϕ.[p(ϕ) x.ϕ(x)] A1a ϕ.[p( ϕ) P(ϕ)] A1b ϕ.[ P(ϕ) P( ϕ)] T2: y.[g(y) G ess. y] A4 ϕ.[p(ϕ) P(ϕ)] L1: z.g(z) x.g(x) z.g(z) x.g(x) A5 P(E) L2: z.g(z) x.g(x) C1: z.g(z) L2: z.g(z) x.g(x) T3: x.g(x) S5 ξ.[ ξ ξ] Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 35

50 Natural Deduction Calculus Ạ... A B C C Ḅ... C E A B A B I A n. B A B n I A A B I 1 A B A E 1 B A B I B A B I 2 A B B E2 A A B B 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

51 Natural Deduction Calculus Rules for Modalities. α : A A I t : A Ạ... E. t : A A I β : A Ạ... E A A Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 37

52 Natural Deduction Proofs T1 and C1 A2 ϕ. ψ.[(p(ϕ) x.[ϕ(x) ψ(x)]) P(ψ)] ψ.[(p(ρ) x.[ρ(x) ψ(x)]) P(ψ)] (P(ρ) x.[ρ(x) ρ(x)]) P( ρ) (P(ρ) x.[ ρ(x)]) P( ρ) E E (P(ρ) x.[ ρ(x)]) P(ρ) P(ρ) x.ρ(x) T1: ϕ.[p(ϕ) x.ϕ(x)] A1a ϕ.[p( ϕ) P(ϕ)] I P( ρ) P(ρ) E A3 P(G) T1 ϕ.[p(ϕ) x.ϕ(x)] P(G) x.g(x) x.g(x) E E Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 38

53 Natural Deduction Proofs T2 (Partial) Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 39

54 Implementations and Experiments Formal encodings (in HOL) of: modal logic axioms axioms, definitions, and theorems in Scott s proof script Experiments using automated provers LEO-II, Satallax, AgsyHOL Interactive proofs using proof assistants Isabelle and Coq Source files available at: Demos on request! Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 40

55 Implementations and Experiments Formal encodings (in HOL) of: modal logic axioms axioms, definitions, and theorems in Scott s proof script Experiments using automated provers LEO-II, Satallax, AgsyHOL Interactive proofs using proof assistants Isabelle and Coq Source files available at: Demos on request! Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 40

56 Results Axioms and definitions are consistent. Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 41

57 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

58 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

59 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

60 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 conjunct φ(x) in the definition of essence, seem inconsistent. Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 41

61 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 conjunct φ(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

62 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 conjunct φ(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

63 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 conjunct φ(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

64 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

65 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

66 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

67 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

68 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

69 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

70 Conclusions Christoph Benzmüller and Bruno Woltzenlogel Paleo Gödel s Proof of God s Existence 44

71 Conclusion Achievements: Infra-structure for automated higher-order modal reasoning Verification of Gödel s ontological argument with HOL provers experiments with different parameters Novel results and insights Major step towards Computer-assisted Theoretical Philosophy see also Ed Zalta s Computational Metaphysics project at Stanford University see also John Rushby s recent verification of Anselm s proof in PVS remember 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

72 Conclusion Achievements: Infra-structure for automated higher-order modal reasoning Verification of Gödel s ontological argument with HOL provers experiments with different parameters Novel results and insights Major step towards Computer-assisted Theoretical Philosophy see also Ed Zalta s Computational Metaphysics project at Stanford University see also John Rushby s recent verification of Anselm s proof in PVS remember 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