a success story of higher-order (automated) theorem...
TRANSCRIPT
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A Success Story ofHigher-Order (Automated) Theorem Proving
in Computational Metaphysics
Christoph Benzmüller1, Stanford (CSLI/Cordula Hall) & FU Berlinjww: B. Woltzenlogel Paleo (& L. Paulson , C. Brown, G. Sutcliffe and many others!)
Logic Colloquium, UC Berkeley, Feb 19, 2016
February 21, 2016
1Supported by DFG Heisenberg Fellowship BE 2501/9-1/2
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 1
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Vision of Leibniz (1646–1716): Calculemus!
If controversies were to arise, therewould be no more need of disputa-tion between two philosophers than be-tween two accountants. For it wouldsuffice to take their pencils in theirhands, to sit down to their slates, andto say to each other . . . : Let us calcu-late.
(Translation by Russell)
Quo facto, quando orientur controversiae, non magis dispu-tatione opus erit inter duos philosophos, quam inter duosComputistas. Sufficiet enim calamos in manus sumeresedereque ad abacos, et sibi mutuo . . . dicere: calculemus.(Leibniz, 1684)
Required:characteristica universalis and calculus ratiocinator
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 2
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Talk Outline
A: HOL as a Universal (Meta-)Logic via Semantic Embeddings
B: New Knowledge on the Ontological Argument from HOL ATPs
C: Reconstruction of the Inconsistency of Gödel’s Axioms
D: Recent Technical Improvements
(E: Other Related Work: Zalta’a Theory of Abstract Object)
(F: Other Related Work: Scott’s Free Logic)
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 3
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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.- . . .
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 4
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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.- . . .
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 4
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C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 5
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C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 5
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C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 5
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C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 5
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See more serious and funny news links athttps://github.com/FormalTheology/GoedelGod/blob/master/Press/LinksToNews.md
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 5
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Part A:HOL as a Universal (Meta-)Logic via Semantic Embeddings
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 6
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HOL as a Universal (Meta-)Logic via Semantic Embeddings
HOL
Logic LSyntax
Logic LSemantics
Examples for L we have already studied:Modal Logics, Conditional Logics, Intuitionistic Logics, Access Control Logics, NominalLogics, Multivalued Logics (SIXTEEN), Logics based on Neighborhood Semantics,(Mathematical) Fuzzy Logics, Paraconsistent Logics, . . .
Works also for (first-order & higher-order) quantifiers
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 7
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HOL as a Universal (Meta-)Logic via Semantic Embeddings
HOL
Logic LSyntax
Logic LSemantics
Examples for L we have already studied:Modal Logics, Conditional Logics, Intuitionistic Logics, Access Control Logics, NominalLogics, Multivalued Logics (SIXTEEN), Logics based on Neighborhood Semantics,(Mathematical) Fuzzy Logics, Paraconsistent Logics, . . .
Works also for (first-order & higher-order) quantifiers
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 7
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Embedding Approach — Idea
HOL (meta-logic) ϕ ::=
Your-logic (object-logic) ψ ::=
Embedding of in
=
=
=
=
Embedding of meta-logical notions on in
valid =
satisfiable =
... =
Pass this set of equations to a higher-order automated theorem prover
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 8
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Classical Higher-Order Logic (HOL)
Simple Types α ::= o | ι | µ | α1 � α2
HOL s, t ::= cα | xα | (λxαsβ)α�β | (sα�β tα)β |(¬o�o so)o | (so ∨o�o�o to)o | (∀(α�o)�o(λxαto))o
(note: binder notation ∀xαto as syntactic sugar for ∀(α�o)�o(λxαto))
HOL with Henkin semantics is (meanwhile) well understoodOrigin [Church,JSymbLog,1940]
Henkin semantics [Henkin,JSymb.Log,1950]
[Andrews, JSymbLog,1971,1972]
Extens./Intens. [BenzmüllerEtAl,JSymbLog,2004]
[Muskens,JSymbLog,2007]
Sound and complete provers do exists
interactive: Isabelle/HOL, PVS, HOL4, Hol Light, Coq/HOL, . . .
automated: TPS, LEO-II, Satallax, Nitpick, Isabelle/HOL, . . .
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 9
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Classical Higher-Order Logic (HOL)
Simple Types α ::= o | ι | µ | α1 � α2
HOL s, t ::= cα | xα | (λxαsβ)α�β | (sα�β tα)β |(¬o�o so)o | (so ∨o�o�o to)o | (∀(α�o)�o(λxαto))o
(note: binder notation ∀xαto as syntactic sugar for ∀(α�o)�o(λxαto))
HOL with Henkin semantics is (meanwhile) well understoodOrigin [Church,JSymbLog,1940]
Henkin semantics [Henkin,JSymb.Log,1950]
[Andrews, JSymbLog,1971,1972]
Extens./Intens. [BenzmüllerEtAl,JSymbLog,2004]
[Muskens,JSymbLog,2007]
Sound and complete provers do exists
interactive: Isabelle/HOL, PVS, HOL4, Hol Light, Coq/HOL, . . .
automated: TPS, LEO-II, Satallax, Nitpick, Isabelle/HOL, . . .
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 9
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Classical Higher-Order Logic (HOL)
Simple Types α ::= o | ι | µ | α1 � α2
HOL s, t ::= cα | xα | (λxαsβ)α�β | (sα�β tα)β |(¬o�o so)o | (so ∨o�o�o to)o | (∀(α�o)�o(λxαto))o
(note: binder notation ∀xαto as syntactic sugar for ∀(α�o)�o(λxαto))
HOL with Henkin semantics is (meanwhile) well understoodOrigin [Church,JSymbLog,1940]
Henkin semantics [Henkin,JSymb.Log,1950]
[Andrews, JSymbLog,1971,1972]
Extens./Intens. [BenzmüllerEtAl,JSymbLog,2004]
[Muskens,JSymbLog,2007]
Sound and complete provers do exists
interactive: Isabelle/HOL, PVS, HOL4, Hol Light, Coq/HOL, . . .
automated: TPS, LEO-II, Satallax, Nitpick, Isabelle/HOL, . . .
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 9
![Page 18: A Success Story of Higher-Order (Automated) Theorem ...logic.berkeley.edu/colloquium/BenzmuellerSlides.pdf · A Success Story of Higher-Order (Automated) Theorem Proving in Computational](https://reader033.vdocuments.us/reader033/viewer/2022060313/5f0b46c77e708231d42fb7b1/html5/thumbnails/18.jpg)
Classical Higher-Order Logic (HOL)
Simple Types α ::= o | ι | µ | α1 � α2
HOL s, t ::= cα | xα | (λxαsβ)α�β | (sα�β tα)β |(¬o�o so)o | (so ∨o�o�o to)o | (∀(α�o)�o(λxαto))o
(note: binder notation ∀xαto as syntactic sugar for ∀(α�o)�o(λxαto))
HOL with Henkin semantics is (meanwhile) well understoodOrigin [Church,JSymbLog,1940]
Henkin semantics [Henkin,JSymb.Log,1950]
[Andrews, JSymbLog,1971,1972]
Extens./Intens. [BenzmüllerEtAl,JSymbLog,2004]
[Muskens,JSymbLog,2007]
Sound and complete provers do exists
interactive: Isabelle/HOL, PVS, HOL4, Hol Light, Coq/HOL, . . .
automated: TPS, LEO-II, Satallax, Nitpick, Isabelle/HOL, . . .
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 9
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Embedding HOML in HOL
HOML ϕ, ψ ::= . . . | ¬ϕ | ϕ ∧ ψ | ϕ→ ψ | �ϕ | ^ϕ | ∀xγ ϕ | ∃xγ ϕ
I Kripke style semantics (possible world semantics)M, g, s |= ¬ϕ iff not M, g, s |= ϕM, g, s |= ϕ ∧ ψ iff M, g, s |= ϕ and M, g, s |= ψ. . .M, g, s |= �ϕ iff M, g, u |= ϕ for all u with r(s, u). . .M, g, s |= ∀xγ ϕ iff M, [d/x]g, s |= ϕ for all d ∈ Dγ
. . .
[BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
[Muskens, HandbookOfModalLogic, 2006]
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 10
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Embedding Approach — HOML in HOL (remember my talk at SRI in 2010!)
HOL s, t ::= cα | xα | (λxαsβ)α�β | (sα�β tα)β | ¬so | so ∨ to | ∀xα to
HOML ϕ, ψ ::= . . . | ¬ϕ | ϕ ∧ ψ | ϕ→ ψ | �ϕ | ^ϕ | ∀xγ ϕ | ∃xγ ϕ
HOML in HOL: HOML formulas ϕ are mapped to HOL predicates ϕµ�o
(explicit representation of labelled formulas)
¬ = λϕµ�oλwµ¬ϕw∧ = λϕµ�oλψµ�oλwµ(ϕw ∧ ψw)→ = λϕµ�oλψµ�oλwµ(¬ϕw ∨ ψw)
∀ = λhγ�(µ�o)λwµ∀dγ hdw∃ = λhγ�(µ�o)λwµ∃dγ hdw
� = λϕµ�oλwµ∀uµ (¬rwu ∨ ϕu)^ = λϕµ�oλwµ∃uµ (rwu ∧ ϕu)
valid = λϕµ�o∀wµϕw
Ax (polymorphic over γ)
The equations in Ax are given as axioms to the HOL provers!
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 11
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Embedding Approach — HOML in HOL (remember my talk at SRI in 2010!)
HOL s, t ::= cα | xα | (λxαsβ)α�β | (sα�β tα)β | ¬so | so ∨ to | ∀xα to
HOML ϕ, ψ ::= . . . | ¬ϕ | ϕ ∧ ψ | ϕ→ ψ | �ϕ | ^ϕ | ∀xγ ϕ | ∃xγ ϕ
HOML in HOL: HOML formulas ϕ are mapped to HOL predicates ϕµ�o
(explicit representation of labelled formulas)
¬ = λϕµ�oλwµ¬ϕw∧ = λϕµ�oλψµ�oλwµ(ϕw ∧ ψw)→ = λϕµ�oλψµ�oλwµ(¬ϕw ∨ ψw)
∀ = λhγ�(µ�o)λwµ∀dγ hdw∃ = λhγ�(µ�o)λwµ∃dγ hdw
� = λϕµ�oλwµ∀uµ (¬rwu ∨ ϕu)^ = λϕµ�oλwµ∃uµ (rwu ∧ ϕu)
valid = λϕµ�o∀wµϕw
Ax (polymorphic over γ)
The equations in Ax are given as axioms to the HOL provers!
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 11
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Embedding HOML in HOL
Example
HOML formula ^∃xG(x)HOML formula in HOL valid (^∃xG(x))µ�o
expansion (λϕ∀wµϕw)(^∃xG(x))µ�o
βη-normalisation ∀wµ((^∃xG(x))µ�o w)expansion ∀wµ(((λϕµ�oλwµ∃uµ (rwu ∧ ϕu))∃xG(x))µ�o w)βη-normalisation ∀wµ∃uµ(rwu ∧ (∃xG(x))µ�ou)syntactic sugar ∀wµ∃uµ(rwu ∧ (∃(λxG(x)))µ�ou)expansion ∀wµ∃uµ(rwu ∧ ((λhγ�(µ�o)λwµ∃dγ hdw)(λxG(x)))µ�ou)βη-normalisation ∀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 validϕµ�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 12
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Embedding HOML in HOL
Example
HOML formula ^∃xG(x)HOML formula in HOL valid (^∃xG(x))µ�o
expansion (λϕ∀wµϕw)(^∃xG(x))µ�o
βη-normalisation ∀wµ((^∃xG(x))µ�o w)expansion ∀wµ(((λϕµ�oλwµ∃uµ (rwu ∧ ϕu))∃xG(x))µ�o w)βη-normalisation ∀wµ∃uµ(rwu ∧ (∃xG(x))µ�ou)syntactic sugar ∀wµ∃uµ(rwu ∧ (∃(λxG(x)))µ�ou)expansion ∀wµ∃uµ(rwu ∧ ((λhγ�(µ�o)λwµ∃dγ hdw)(λxG(x)))µ�ou)βη-normalisation ∀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 validϕµ�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 12
![Page 24: A Success Story of Higher-Order (Automated) Theorem ...logic.berkeley.edu/colloquium/BenzmuellerSlides.pdf · A Success Story of Higher-Order (Automated) Theorem Proving in Computational](https://reader033.vdocuments.us/reader033/viewer/2022060313/5f0b46c77e708231d42fb7b1/html5/thumbnails/24.jpg)
Embedding HOML in HOL
Example
HOML formula ^∃xG(x)HOML formula in HOL valid (^∃xG(x))µ�o
expansion (λϕ∀wµϕw)(^∃xG(x))µ�o
βη-normalisation ∀wµ((^∃xG(x))µ�o w)expansion ∀wµ(((λϕµ�oλwµ∃uµ (rwu ∧ ϕu))∃xG(x))µ�o w)βη-normalisation ∀wµ∃uµ(rwu ∧ (∃xG(x))µ�ou)syntactic sugar ∀wµ∃uµ(rwu ∧ (∃(λxG(x)))µ�ou)expansion ∀wµ∃uµ(rwu ∧ ((λhγ�(µ�o)λwµ∃dγ hdw)(λxG(x)))µ�ou)βη-normalisation ∀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 validϕµ�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 12
![Page 25: A Success Story of Higher-Order (Automated) Theorem ...logic.berkeley.edu/colloquium/BenzmuellerSlides.pdf · A Success Story of Higher-Order (Automated) Theorem Proving in Computational](https://reader033.vdocuments.us/reader033/viewer/2022060313/5f0b46c77e708231d42fb7b1/html5/thumbnails/25.jpg)
Embedding HOML in HOL
Example
HOML formula ^∃xG(x)HOML formula in HOL valid (^∃xG(x))µ�o
expansion (λϕ∀wµϕw)(^∃xG(x))µ�o
βη-normalisation ∀wµ((^∃xG(x))µ�o w)expansion ∀wµ(((λϕµ�oλwµ∃uµ (rwu ∧ ϕu))∃xG(x))µ�o w)βη-normalisation ∀wµ∃uµ(rwu ∧ (∃xG(x))µ�ou)syntactic sugar ∀wµ∃uµ(rwu ∧ (∃(λxG(x)))µ�ou)expansion ∀wµ∃uµ(rwu ∧ ((λhγ�(µ�o)λwµ∃dγ hdw)(λxG(x)))µ�ou)βη-normalisation ∀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 validϕµ�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 12
![Page 26: A Success Story of Higher-Order (Automated) Theorem ...logic.berkeley.edu/colloquium/BenzmuellerSlides.pdf · A Success Story of Higher-Order (Automated) Theorem Proving in Computational](https://reader033.vdocuments.us/reader033/viewer/2022060313/5f0b46c77e708231d42fb7b1/html5/thumbnails/26.jpg)
Embedding HOML in HOL
Example
HOML formula ^∃xG(x)HOML formula in HOL valid (^∃xG(x))µ�o
expansion (λϕ∀wµϕw)(^∃xG(x))µ�o
βη-normalisation ∀wµ((^∃xG(x))µ�o w)expansion ∀wµ(((λϕµ�oλwµ∃uµ (rwu ∧ ϕu))∃xG(x))µ�o w)βη-normalisation ∀wµ∃uµ(rwu ∧ (∃xG(x))µ�ou)syntactic sugar ∀wµ∃uµ(rwu ∧ (∃(λxG(x)))µ�ou)expansion ∀wµ∃uµ(rwu ∧ ((λhγ�(µ�o)λwµ∃dγ hdw)(λxG(x)))µ�ou)βη-normalisation ∀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 validϕµ�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 12
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Embedding HOML in HOL
Example
HOML formula ^∃xG(x)HOML formula in HOL valid (^∃xG(x))µ�o
expansion (λϕ∀wµϕw)(^∃xG(x))µ�o
βη-normalisation ∀wµ((^∃xG(x))µ�o w)expansion ∀wµ(((λϕµ�oλwµ∃uµ (rwu ∧ ϕu))∃xG(x))µ�o w)βη-normalisation ∀wµ∃uµ(rwu ∧ (∃xG(x))µ�ou)syntactic sugar ∀wµ∃uµ(rwu ∧ (∃(λxG(x)))µ�ou)expansion ∀wµ∃uµ(rwu ∧ ((λhγ�(µ�o)λwµ∃dγ hdw)(λxG(x)))µ�ou)βη-normalisation ∀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 validϕµ�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 12
![Page 28: A Success Story of Higher-Order (Automated) Theorem ...logic.berkeley.edu/colloquium/BenzmuellerSlides.pdf · A Success Story of Higher-Order (Automated) Theorem Proving in Computational](https://reader033.vdocuments.us/reader033/viewer/2022060313/5f0b46c77e708231d42fb7b1/html5/thumbnails/28.jpg)
Embedding HOML in HOL
Example
HOML formula ^∃xG(x)HOML formula in HOL valid (^∃xG(x))µ�o
expansion (λϕ∀wµϕw)(^∃xG(x))µ�o
βη-normalisation ∀wµ((^∃xG(x))µ�o w)expansion ∀wµ(((λϕµ�oλwµ∃uµ (rwu ∧ ϕu))∃xG(x))µ�o w)βη-normalisation ∀wµ∃uµ(rwu ∧ (∃xG(x))µ�ou)syntactic sugar ∀wµ∃uµ(rwu ∧ (∃(λxG(x)))µ�ou)expansion ∀wµ∃uµ(rwu ∧ ((λhγ�(µ�o)λwµ∃dγ hdw)(λxG(x)))µ�ou)βη-normalisation ∀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 validϕµ�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 12
![Page 29: A Success Story of Higher-Order (Automated) Theorem ...logic.berkeley.edu/colloquium/BenzmuellerSlides.pdf · A Success Story of Higher-Order (Automated) Theorem Proving in Computational](https://reader033.vdocuments.us/reader033/viewer/2022060313/5f0b46c77e708231d42fb7b1/html5/thumbnails/29.jpg)
Embedding HOML in HOL
Example
HOML formula ^∃xG(x)HOML formula in HOL valid (^∃xG(x))µ�o
expansion (λϕ∀wµϕw)(^∃xG(x))µ�o
βη-normalisation ∀wµ((^∃xG(x))µ�o w)expansion ∀wµ(((λϕµ�oλwµ∃uµ (rwu ∧ ϕu))∃xG(x))µ�o w)βη-normalisation ∀wµ∃uµ(rwu ∧ (∃xG(x))µ�ou)syntactic sugar ∀wµ∃uµ(rwu ∧ (∃(λxG(x)))µ�ou)expansion ∀wµ∃uµ(rwu ∧ ((λhγ�(µ�o)λwµ∃dγ hdw)(λxG(x)))µ�ou)βη-normalisation ∀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 validϕµ�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 12
![Page 30: A Success Story of Higher-Order (Automated) Theorem ...logic.berkeley.edu/colloquium/BenzmuellerSlides.pdf · A Success Story of Higher-Order (Automated) Theorem Proving in Computational](https://reader033.vdocuments.us/reader033/viewer/2022060313/5f0b46c77e708231d42fb7b1/html5/thumbnails/30.jpg)
Embedding HOML in HOL
Example
HOML formula ^∃xG(x)HOML formula in HOL valid (^∃xG(x))µ�o
expansion (λϕ∀wµϕw)(^∃xG(x))µ�o
βη-normalisation ∀wµ((^∃xG(x))µ�o w)expansion ∀wµ(((λϕµ�oλwµ∃uµ (rwu ∧ ϕu))∃xG(x))µ�o w)βη-normalisation ∀wµ∃uµ(rwu ∧ (∃xG(x))µ�ou)syntactic sugar ∀wµ∃uµ(rwu ∧ (∃(λxG(x)))µ�ou)expansion ∀wµ∃uµ(rwu ∧ ((λhγ�(µ�o)λwµ∃dγ hdw)(λxG(x)))µ�ou)βη-normalisation ∀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 validϕµ�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 12
![Page 31: A Success Story of Higher-Order (Automated) Theorem ...logic.berkeley.edu/colloquium/BenzmuellerSlides.pdf · A Success Story of Higher-Order (Automated) Theorem Proving in Computational](https://reader033.vdocuments.us/reader033/viewer/2022060313/5f0b46c77e708231d42fb7b1/html5/thumbnails/31.jpg)
Embedding HOML in HOL
Example
HOML formula ^∃xG(x)HOML formula in HOL valid (^∃xG(x))µ�o
expansion (λϕ∀wµϕw)(^∃xG(x))µ�o
βη-normalisation ∀wµ((^∃xG(x))µ�o w)expansion ∀wµ(((λϕµ�oλwµ∃uµ (rwu ∧ ϕu))∃xG(x))µ�o w)βη-normalisation ∀wµ∃uµ(rwu ∧ (∃xG(x))µ�ou)syntactic sugar ∀wµ∃uµ(rwu ∧ (∃(λxG(x)))µ�ou)expansion ∀wµ∃uµ(rwu ∧ ((λhγ�(µ�o)λwµ∃dγ hdw)(λxG(x)))µ�ou)βη-normalisation ∀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 validϕµ�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 12
![Page 32: A Success Story of Higher-Order (Automated) Theorem ...logic.berkeley.edu/colloquium/BenzmuellerSlides.pdf · A Success Story of Higher-Order (Automated) Theorem Proving in Computational](https://reader033.vdocuments.us/reader033/viewer/2022060313/5f0b46c77e708231d42fb7b1/html5/thumbnails/32.jpg)
Embedding HOML in HOL
Example
HOML formula ^∃xG(x)HOML formula in HOL valid (^∃xG(x))µ�o
expansion (λϕ∀wµϕw)(^∃xG(x))µ�o
βη-normalisation ∀wµ((^∃xG(x))µ�o w)expansion ∀wµ(((λϕµ�oλwµ∃uµ (rwu ∧ ϕu))∃xG(x))µ�o w)βη-normalisation ∀wµ∃uµ(rwu ∧ (∃xG(x))µ�ou)syntactic sugar ∀wµ∃uµ(rwu ∧ (∃(λxG(x)))µ�ou)expansion ∀wµ∃uµ(rwu ∧ ((λhγ�(µ�o)λwµ∃dγ hdw)(λxG(x)))µ�ou)βη-normalisation ∀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 validϕµ�o can be derived from Ax in HOL.
This can be done with interactive or automated HOL theorem provers.
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 12
![Page 33: A Success Story of Higher-Order (Automated) Theorem ...logic.berkeley.edu/colloquium/BenzmuellerSlides.pdf · A Success Story of Higher-Order (Automated) Theorem Proving in Computational](https://reader033.vdocuments.us/reader033/viewer/2022060313/5f0b46c77e708231d42fb7b1/html5/thumbnails/33.jpg)
Advantages of the Embedding Approach
1. Pragmatics and convenience:I implementing new provers made simple (even for not yet automated logics)
2. Availability:I simply reuse and adapt our existing encodings (THF, Isabelle/HOL, Coq)
3. Flexibility:I rapid experimentation with logic variations and logic combinations
4. Relation to labelled deductive systems:I extra-logical labels vs. intra-logical labels (here)
5. Relation to standard translation:I extra-logical translation vs. extended intra-logical translation (here)
6. Meta-logical reasoning:I various examples already exist, e.g. verification of modal logic cube
7. Direct calculi and user intuition:I possible: tactics on top of embedding, hiding of embedding
8. Soundness and completeness:I already proven for many non-classical logics (wrt Henkin semantics)
9. Cut-elimination:I generic indirect result, since HOL enjoys cut-elimination (Henkin semantics)
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 13
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Advantage: 1. Pragmatics and convenienceimplementing new provers made simple (even for not yet automated logics)
A very “Lean” Prover for HOML K1 %----The base type $i (already built-in) stands here for worlds and2 %----mu for individuals; $o (also built-in) is the type of Booleans3 thf(mu_type,type,(mu:$tType)).4 %----Reserved constant r for accessibility relation5 thf(r,type,(r:$i>$i>$o)).6 %----Modal logic operators not, or, and, implies, box, diamond7 thf(mnot_type,type,(mnot:($i>$o)>$i>$o)).8 thf(mnot,definition,(mnot = (^[A:$i>$o,W:$i]:~(A@W)))).9 thf(mor_type,type,(mor:($i>$o)>($i>$o)>$i>$o)).
10 thf(mor,definition,(mor = (^[A:$i>$o,Psi:$i>$o,W:$i]:((A@W)|(Psi@W))))).11 thf(mand_type,type,(mand:($i>$o)>($i>$o)>$i>$o)).12 thf(mand,definition,(mand = (^[A:$i>$o,Psi:$i>$o,W:$i]:((A@W)&(Psi@W))))).13 thf(mimplies_type,type,(mimplies:($i>$o)>($i>$o)>$i>$o)).14 thf(mimplies,definition,(mimplies = (^[A:$i>$o,Psi:$i>$o,W:$i]:((A@W)&(Psi@W))))).15 thf(mbox_type,type,(mbox:($i>$o)>$i>$o)).16 thf(mbox,definition,(mbox = (^[A:$i>$o,W:$i]:![V:$i]:(~(r@W@V)|(A@V))))).17 thf(mdia_type,type,(mdia:($i>$o)>$i>$o)).18 thf(mdia,definition,(mdia = (^[A:$i>$o,W:$i]:?[V:$i]:((r@W@V)&(A@V))))).19 %----Quantifiers (constant domains) for individuals and propositions20 thf(mforall_ind_type,type,(mforall_ind:(mu>$i>$o)>$i>$o)).21 thf(mforall_ind,definition,(mforall_ind = (^[A:mu>$i>$o,W:$i]:![X:mu]:(A@X@W)))).22 thf(mforall_indset_type,type,(mforall_indset:((mu>$i>$o)>$i>$o)>$i>$o)).23 thf(mforall_indset,definition,(mforall_indset = (^[A:(mu>$i>$o)>$i>$o,W:$i]:![X:mu>$i>$o]:(A@X@W)))).24 thf(mexists_ind_type,type,(mexists_ind:(mu>$i>$o)>$i>$o)).25 thf(mexists_ind,definition,(mexists_ind = (^[A:mu>$i>$o,W:$i]:?[X:mu]:(A@X@W)))).26 thf(mexists_indset_type,type,(mexists_indset:((mu>$i>$o)>$i>$o)>$i>$o)).27 thf(mexists_indset,definition,(mexists_indset = (^[A:(mu>$i>$o)>$i>$o,W:$i]:?[X:mu>$i>$o]:(A@X@W)))).28 %----Definition of validity (grounding of lifted modal formulas)29 thf(v_type,type,(v:($i>$o)>$o)).30 thf(mvalid,definition,(v = (^[A:$i>$o]:![W:$i]:(A@W)))).
TPTP THF0 syntax: [SutcliffeBenzmüller, J.Formalized Reasoning, 2010]
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 14
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Advantage: 1. Pragmatics and convenienceimplementing new provers made simple (even for not yet automated logics)
Approach is competitive
I First-order modal logic: see experiments in[BenzmüllerOttenRaths, ECAI, 2012]
[BenzmüllerRaths, LPAR, 2013][Benzmüller, ARQNL, 2014]
I Higher-order modal logics:
There are no other systems yet!
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 14
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Advantage: 2. Availabilitysimply reuse and adapt our existing encodings (THF, Isabelle/HOL, Coq)
HOML in Isabelle/HOL
See formalisations at https://github.com/FormalTheology/GoedelGod/tree/master/FormalizationsC. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 15
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Advantage: 3. Flexibilityrapid experimentation with logic variations and logic combinations
Postulating modal axioms or semantical constraints
HOL
’Syntactical’ Sahlqvist axioms ’Semantical’ constraintsM: valid ∀ϕ(�rϕ→ ϕ) ↔ ∀x(rxx) (reflexivity)B: valid ∀ϕ(ϕ→ �r^rϕ) ↔ ∀x∀y(rxy→ ryx) (symmetry)D: valid ∀ϕ(�rϕ→ ^rϕ) ↔ ∀x∃y(rxy) (serial)4: valid ∀ϕ(�rϕ→ �r�rϕ) ↔ ∀x∀y∀z(rxy ∧ ryz→ rxz) (transitivity)5: valid ∀ϕ(^rϕ→ �r^rϕ) ↔ ∀x∀y∀z(rxy ∧ rxz→ ryz) (euclidean)
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 16
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Advantage: 3. Flexibilityrapid experimentation with logic variations and logic combinations
Possibilist vs. Actualist Quantification
∀ = λhγ�(µ�o)λwµ∀dγ hdw (constant domains)becomes∀va = λhγ�(µ�o)λwµ∀dγ (ExInW dw→ hdw) (varying domains)
where ExInW is an existence predicate(additional axioms: non-empty domains, denotation of constants & functions)
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 16
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Advantage: 4. Relation to labelled deductive systemsextra-logical labels vs. intra-logical labels (here)
^∃xG(x) worldlabel −→ ((^∃xG(x))µ�o worldlabelµ)
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 17
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Advantage: 5. Relation to standard translationextra-logical translation vs. extended intra-logical translation (here)
[BenzmüllerPaulson, LogicaUniversalis, 2013][BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
Intra-logical realisation of the standard translation
(�φ) a
−→ ((�φ)µ�o a)−→ (((λϕµ�oλwµ∀uµ (¬rwu ∨ ϕu)) φ)µ�o a)−→ (∀uµ (¬rau ∨ φµ�o u)
We have extended this also for first-order and higher-order quantifiers!
(∀x φ(x)) a
−→ ((∀x φ(x))µ�o a)−→ ((∀(λx φ(x)))µ�o a)−→ (((λhγ�(µ�o)λwµ∀dγ hdw)(λx φ(x)))µ�o a)−→ ∀d (φ(d)µ�o a)
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 18
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Advantage: 5. Relation to standard translationextra-logical translation vs. extended intra-logical translation (here)
[BenzmüllerPaulson, LogicaUniversalis, 2013][BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
Intra-logical realisation of the standard translation
(�φ) a
−→ ((�φ)µ�o a)−→ (((λϕµ�oλwµ∀uµ (¬rwu ∨ ϕu)) φ)µ�o a)−→ (∀uµ (¬rau ∨ φµ�o u)
We have extended this also for first-order and higher-order quantifiers!
(∀x φ(x)) a
−→ ((∀x φ(x))µ�o a)−→ ((∀(λx φ(x)))µ�o a)−→ (((λhγ�(µ�o)λwµ∀dγ hdw)(λx φ(x)))µ�o a)−→ ∀d (φ(d)µ�o a)
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 18
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Advantage: 5. Relation to standard translationextra-logical translation vs. extended intra-logical translation (here)
[BenzmüllerPaulson, LogicaUniversalis, 2013][BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
Intra-logical realisation of the standard translation
(�φ) a
−→ ((�φ)µ�o a)−→ (((λϕµ�oλwµ∀uµ (¬rwu ∨ ϕu)) φ)µ�o a)−→ (∀uµ (¬rau ∨ φµ�o u)
We have extended this also for first-order and higher-order quantifiers!
(∀x φ(x)) a
−→ ((∀x φ(x))µ�o a)−→ ((∀(λx φ(x)))µ�o a)−→ (((λhγ�(µ�o)λwµ∀dγ hdw)(λx φ(x)))µ�o a)−→ ∀d (φ(d)µ�o a)
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 18
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Advantage: 5. Relation to standard translationextra-logical translation vs. extended intra-logical translation (here)
[BenzmüllerPaulson, LogicaUniversalis, 2013][BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
Intra-logical realisation of the standard translation
(�φ) a
−→ ((�φ)µ�o a)−→ (((λϕµ�oλwµ∀uµ (¬rwu ∨ ϕu)) φ)µ�o a)−→ (∀uµ (¬rau ∨ φµ�o u)
We have extended this also for first-order and higher-order quantifiers!
(∀x φ(x)) a
−→ ((∀x φ(x))µ�o a)−→ ((∀(λx φ(x)))µ�o a)−→ (((λhγ�(µ�o)λwµ∀dγ hdw)(λx φ(x)))µ�o a)−→ ∀d (φ(d)µ�o a)
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 18
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Advantage: 5. Relation to standard translationextra-logical translation vs. extended intra-logical translation (here)
[BenzmüllerPaulson, LogicaUniversalis, 2013][BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
Intra-logical realisation of the standard translation
(�φ) a
−→ ((�φ)µ�o a)−→ (((λϕµ�oλwµ∀uµ (¬rwu ∨ ϕu)) φ)µ�o a)−→ (∀uµ (¬rau ∨ φµ�o u)
We have extended this also for first-order and higher-order quantifiers!
(∀x φ(x)) a
−→ ((∀x φ(x))µ�o a)−→ ((∀(λx φ(x)))µ�o a)−→ (((λhγ�(µ�o)λwµ∀dγ hdw)(λx φ(x)))µ�o a)−→ ∀d (φ(d)µ�o a)
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 18
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Advantage: 5. Relation to standard translationextra-logical translation vs. extended intra-logical translation (here)
[BenzmüllerPaulson, LogicaUniversalis, 2013][BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
Intra-logical realisation of the standard translation
(�φ) a
−→ ((�φ)µ�o a)−→ (((λϕµ�oλwµ∀uµ (¬rwu ∨ ϕu)) φ)µ�o a)−→ (∀uµ (¬rau ∨ φµ�o u)
We have extended this also for first-order and higher-order quantifiers!
(∀x φ(x)) a
−→ ((∀x φ(x))µ�o a)−→ ((∀(λx φ(x)))µ�o a)−→ (((λhγ�(µ�o)λwµ∀dγ hdw)(λx φ(x)))µ�o a)−→ ∀d (φ(d)µ�o a)
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Advantage: 6. Meta-logical reasoningvarious examples already exist, e.g. verification of modal logic cube
[Benzmüller, FestschriftWalther, 2010][BenzmüllerClausSultana, PxTP, 2015]
K
K4
K5
KB
K45 KB5
D
D4
D5
DB
D45
M
S4
BB
S5 ≡ M5 ≡ MB5 ≡ M4B5≡ M45 ≡ M4B ≡ D4B≡ D4B5 ≡ DB5
M: �P→ PB: P→ �^PD: �P→ ^P4: �P→ ��P5: ^P→ �^P
K
M
45
B
≡ K4B5 ≡ K4B
Verification of cube in less than 1 minute in Isabelle/HOL
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Advantage: 7. Direct calculi and user intuitionabstract level tactics (here in Coq) on top of embedding, hiding of embedding
[BenzmüllerWoltzenlogelPaleo, CSR’2015]
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Advantage: 8. Soundness and completenessalready proven for many non-classical logics (wrt Henkin semantics)
Soundness and Completeness|=L ϕ iff Ax |=HOL
Henkin valid ϕµ�o
Logic L:I Higher-order Modal Logics [BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
I First-order Multimodal Logics [BenzmüllerPaulson, LogicaUniversalis, 2013]
I Propositional Multimodal Logics [BenzmüllerPaulson, Log.J.IGPL, 2010]
I Quantified Conditional Logics [Benzmüller, IJCAI, 2013]
I Propositional Conditional Logics [BenzmüllerEtAl., AMAI, 2012]
I Intuitionistic Logics [BenzmüllerPaulson, Log.J.IGPL, 2010]
I Access Control Logics [Benzmüller, IFIP SEC, 2009]
I Logic Combinations [Benzmüller, AMAI, 2011]I . . . more is on the way . . . including:
I Description LogicsI Nominal LogicsI Multivalued Logics (SIXTEEN)I Logics based on Neighborhood SemanticsI (Mathematical) Fuzzy LogicsI Paraconsistent Logics
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 21
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Advantage: 9. Cut-eliminationgeneric indirect result, since HOL enjoys cut-elimination (Henkin semantics)
Soundness and Completeness|=L ϕ iff Ax |=HOL
Henkin valid ϕµ�o
Logic L:I Higher-order Modal Logics [BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
I First-order Multimodal Logics [BenzmüllerPaulson, LogicaUniversalis, 2013]
I Propositional Multimodal Logics [BenzmüllerPaulson, Log.J.IGPL, 2010]
I Quantified Conditional Logics [Benzmüller, IJCAI, 2013]
I Propositional Conditional Logics [BenzmüllerEtAl., AMAI, 2012]
I Intuitionistic Logics [BenzmüllerPaulson, Log.J.IGPL, 2010]
I Access Control Logics [Benzmüller, IFIP SEC, 2009]
I Logic Combinations [Benzmüller, AMAI, 2011]I . . . more is on the way . . . including:
I Description LogicsI Nominal LogicsI Multivalued Logics (SIXTEEN)I Logics based on Neighborhood SemanticsI (Mathematical) Fuzzy LogicsI Paraconsistent Logics
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 22
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Advantage: 9. Cut-eliminationgeneric indirect result, since HOL enjoys cut-elimination (Henkin semantics)
Soundness and Completeness and Cut-elimination|=L ϕ iff Ax |=HOL
Henkin valid ϕµ�o iff Ax `HOLcut-free valid ϕµ�o
Logic L:I Higher-order Modal Logics [BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
I First-order Multimodal Logics [BenzmüllerPaulson, LogicaUniversalis, 2013]
I Propositional Multimodal Logics [BenzmüllerPaulson, Log.J.IGPL, 2010]
I Quantified Conditional Logics [Benzmüller, IJCAI, 2013]
I Propositional Conditional Logics [BenzmüllerEtAl., AMAI, 2012]
I Intuitionistic Logics [BenzmüllerPaulson, Log.J.IGPL, 2010]
I Access Control Logics [Benzmüller, IFIP SEC, 2009]
I Logic Combinations [Benzmüller, AMAI, 2011]I . . . more is on the way . . . including:
I Description LogicsI Nominal LogicsI Multivalued Logics (SIXTEEN)I Logics based on Neighborhood SemanticsI (Mathematical) Fuzzy LogicsI Paraconsistent Logics
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 22
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Part B:New Knowledge on the Ontological Argument
from HOL ATPs
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Vision of Leibniz (1646–1716): Calculemus!
If controversies were to arise, therewould be no more need of disputa-tion between two philosophers than be-tween two accountants. For it wouldsuffice to take their pencils in theirhands, to sit down to their slates, andto say to each other . . . : Let us calcu-late.
(Translation by Russell)
Quo facto, quando orientur controversiae, non magis dispu-tatione opus erit inter duos philosophos, quam inter duosComputistas. Sufficiet enim calamos in manus sumeresedereque ad abacos, et sibi mutuo . . . dicere: calculemus.(Leibniz, 1684)
Required:characteristica universalis and calculus ratiocinator
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Our Contribution: Towards Computational Metaphysics
Ontological argument for the existence of God
I Focus on Gödel’s modern version in higher-order modal logic
I Experiments with HO provers and embedding approach
Different interests in ontological arguments
I Philosophical: Boundaries of metaphysics & epistemology
I Theistic: Successful argument could convince atheists?
I Ours: Computational metaphysics (Leibniz’ vision)
Related work: only for Anselm’s simpler argument
I first-order ATP PROVER9 [OppenheimerZalta, 2011]
I interactive proof assistant PVS [Rushby, 2013]
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 25
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Our Contribution: Towards Computational Metaphysics
Ontological argument for the existence of God
I Focus on Gödel’s modern version in higher-order modal logic
I Experiments with HO provers and embedding approach
Different interests in ontological arguments
I Philosophical: Boundaries of metaphysics & epistemology
I Theistic: Successful argument could convince atheists?
I Ours: Computational metaphysics (Leibniz’ vision)
Related work: only for Anselm’s simpler argument
I first-order ATP PROVER9 [OppenheimerZalta, 2011]
I interactive proof assistant PVS [Rushby, 2013]
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 25
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Our Contribution: Towards Computational Metaphysics
Ontological argument for the existence of God
I Focus on Gödel’s modern version in higher-order modal logic
I Experiments with HO provers and embedding approach
Different interests in ontological arguments
I Philosophical: Boundaries of metaphysics & epistemology
I Theistic: Successful argument could convince atheists?
I Ours: Computational metaphysics (Leibniz’ vision)
Related work: only for Anselm’s simpler argument
I first-order ATP PROVER9 [OppenheimerZalta, 2011]
I interactive proof assistant PVS [Rushby, 2013]
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 25
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A Long Historypros 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 (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, deductive reasoning:“God exists.”
∃xG(x)
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 26
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A Long Historypros 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 (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, deductive reasoning:“Necessarily, God exists.”
�∃xG(x)
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Gödel’s Manuscript: 1930’s, 1941, 1946-1955, 1970
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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 implyingany 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 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)
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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 implyingany 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 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)
Difference to Gödel (who omits this conjunct)
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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 implyingany 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 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)
Modal operators are used
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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 implyingany 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 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)
second-order quantifiers
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Gödel’s God in TPTP THF
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Gödel’s God in Isabelle/HOL
See verifiable Isabelle/HOL document (Archive of Formal Proofs) at:http://afp.sourceforge.net/entries/GoedelGod.shtml
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Gödel’s God in Coq
See verifiable Coq document at:https://github.com/FormalTheology/GoedelGod/tree/master/Formalizations/Coq
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Findings from our study[BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
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Main Findings [BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
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Main Findings [BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
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Main Findings [BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
Automating Scott’s proof script
T1: "Positive properties are possibly exemplified"proved by LEO-II and SatallaxI in logic: KI from assumptions:
I A1 and A2I A1(⊃) and A2
I notion of quantificationI possibilist quantifiers (constant dom.)I actualist quantifiers for individuals (varying dom.)
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Main Findings [BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
Automating Scott’s proof script
C: "Possibly, God exists”proved by LEO-II and SatallaxI in logic: KI from assumptions:
I T1, D1, A3I for domain conditions:
I possibilist quantifiers (constant dom.)I actualist quantifiers for individuals (varying dom.)
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Main Findings [BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
Automating Scott’s proof script
T2: "Being God-like is an ess. of any God-like being”proved by LEO-II and SatallaxI in logic: KI from assumptions:
I A1, D1, A4, D2I for domain conditions:
I possibilist quantifiers (constant dom.)I actualist quantifiers for individuals (varying dom.)
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Main Findings [BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
Automating Scott’s proof script
T3: "Necessarily, God exists”proved by LEO-II and SatallaxI in logic: KBI from assumptions:
I D1, C, T2, D3, A5I for domain conditions:
I possibilist quantifiers (constant dom.)I actualist quantifiers for individuals (varying dom.)
For logic K we got a countermodel by Nitpick
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Main Findings [BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
Automating Scott’s proof script
SummaryI proof verified and automatedI KB is sufficient (critisized logic S5 not needed!)I possibilist and actualist quantifiers (individuals)I exact dependencies determined experimentallyI ATPs have found alternative proofs
e.g. self-identity λx(x = x) is not needed
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Main Findings [BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
Consistency check: Gödel vs. Scott
I Scott’s assumptions are consistent;shown by Nitpick
I Gödel’s assumptions are inconsistent;shown by LEO-II (new philosophical result?)
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Main Findings [BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
Further Results
I Monotheism holdsI God is flawless
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Main Findings [BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
Modal Collapse (Sobel)
∀ϕ(ϕ ⊃ �ϕ)
I proved by LEO-II and SatallaxI for possibilist and actualist quantification (ind.)
Main critique on Gödel’s ontological proof:I there are no contingent truthsI everything is determined / no free will
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Main Findings [BenzmüllerWoltzenlogelPaleo, ECAI, 2014]
Observation
I good performance of ATPsI excellent match between
argumentation granularity inpapers and the reasoning strengthof the ATPs
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Avoiding the Modal Collapse: Recent Variants
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Avoiding the Modal Collapse: Some Emendations
Computer-supported Clarification of Controversy1st World Congress on Logic and Religion, 2015
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Results Obtained with Fully Automated ReasonersA controversy between Magari, Hájek and Anderson regarding the redundancy of some axioms
Leibniz (1646–1716)
characteristica universalis and calculus ratiocinatorIf controversies were to arise, there would be no more need of disputation between twophilosophers than between two accountants. For it would suffice to take their pencils intheir hands, to sit down to their slates, and to say to each other . . . : Let us calculate.
But: Intuitive proofs/models are needed to convince philosophers
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Results Obtained with Fully Automated ReasonersA controversy between Magari, Hájek and Anderson regarding the redundancy of some axioms
Leibniz (1646–1716)
characteristica universalis and calculus ratiocinatorIf controversies were to arise, there would be no more need of disputation between twophilosophers than between two accountants. For it would suffice to take their pencils intheir hands, to sit down to their slates, and to say to each other . . . : Let us calculate.
But: Intuitive proofs/models are needed to convince philosophers
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 49
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Part C:Reconstruction of the Inconsistency of Gödel’s Axioms
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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 implyingany 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 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)
��XX
Difference to Gödel (who omits this conjunct)
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Inconsistency (Gödel): Proof by LEO-II in KB
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Inconsistency (Gödel): Verification in Isabelle/HOL (KB)
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Inconsistency (Gödel): Verification in Isabelle/HOL (K)
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Inconsistency (Gödel): Informal Argument (in KB and K)
Def. D2∗ φ ess. x↔���XXXφ(x) ∧ ∀ψ(ψ(x)→ �∀y(φ(y)→ ψ(y)))
Lemma 1 The empty property is an essence of every entity. ∀x (∅ ess. x)
Theorem 1 Positive Properties are possibly exemplified. ∀φ[P(φ)→ ^∃xφ(x)]
Axiom A5 P(NE)I by T1, A5: ^∃x[NE(x)]
Def. D3 NE(x)↔ ∀φ[φ ess. x→ �∃yφ(y)]I ^∃x[∀φ[φ ess. x→ �∃y[φ(y)]]]I ^∃x[∅ ess. x→ �∃y[∅(y)]]I by L1 ^∃x[> → �∃y[∅(y)]]I by def. of ∅ ^∃x[> → �⊥]I ^∃x[�⊥]I ^�⊥
Inconsistency ⊥
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 55
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Inconsistency (Gödel): Informal Argument (in KB and K)
Def. D2∗ φ ess. x↔���XXXφ(x) ∧ ∀ψ(ψ(x)→ �∀y(φ(y)→ ψ(y)))
Lemma 1 The empty property is an essence of every entity. ∀x (∅ ess. x)
Theorem 1 Positive Properties are possibly exemplified. ∀φ[P(φ)→ ^∃xφ(x)]
Axiom A5 P(NE)I by T1, A5: ^∃x[NE(x)]
Def. D3 NE(x)↔ ∀φ[φ ess. x→ �∃yφ(y)]I ^∃x[∀φ[φ ess. x→ �∃y[φ(y)]]]I ^∃x[∅ ess. x→ �∃y[∅(y)]]I by L1 ^∃x[> → �∃y[∅(y)]]I by def. of ∅ ^∃x[> → �⊥]I ^∃x[�⊥]I ^�⊥
Inconsistency ⊥
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 55
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Inconsistency (Gödel): Informal Argument (in KB and K)
Def. D2∗ φ ess. x↔���XXXφ(x) ∧ ∀ψ(ψ(x)→ �∀y(φ(y)→ ψ(y)))
Lemma 1 The empty property is an essence of every entity. ∀x (∅ ess. x)
Theorem 1 Positive Properties are possibly exemplified. ∀φ[P(φ)→ ^∃xφ(x)]
Axiom A5 P(NE)I by T1, A5: ^∃x[NE(x)]
Def. D3 NE(x)↔ ∀φ[φ ess. x→ �∃yφ(y)]I ^∃x[∀φ[φ ess. x→ �∃y[φ(y)]]]I ^∃x[∅ ess. x→ �∃y[∅(y)]]I by L1 ^∃x[> → �∃y[∅(y)]]I by def. of ∅ ^∃x[> → �⊥]I ^∃x[�⊥]I ^�⊥
Inconsistency ⊥
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 55
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Inconsistency (Gödel): Informal Argument (in KB and K)
Def. D2∗ φ ess. x↔���XXXφ(x) ∧ ∀ψ(ψ(x)→ �∀y(φ(y)→ ψ(y)))
Lemma 1 The empty property is an essence of every entity. ∀x (∅ ess. x)
Theorem 1 Positive Properties are possibly exemplified. ∀φ[P(φ)→ ^∃xφ(x)]
Axiom A5 P(NE)
I by T1, A5: ^∃x[NE(x)]
Def. D3 NE(x)↔ ∀φ[φ ess. x→ �∃yφ(y)]I ^∃x[∀φ[φ ess. x→ �∃y[φ(y)]]]I ^∃x[∅ ess. x→ �∃y[∅(y)]]I by L1 ^∃x[> → �∃y[∅(y)]]I by def. of ∅ ^∃x[> → �⊥]I ^∃x[�⊥]I ^�⊥
Inconsistency ⊥
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 55
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Inconsistency (Gödel): Informal Argument (in KB and K)
Def. D2∗ φ ess. x↔���XXXφ(x) ∧ ∀ψ(ψ(x)→ �∀y(φ(y)→ ψ(y)))
Lemma 1 The empty property is an essence of every entity. ∀x (∅ ess. x)
Theorem 1 Positive Properties are possibly exemplified. ∀φ[P(φ)→ ^∃xφ(x)]
Axiom A5 P(NE)I by T1, A5: ^∃x[NE(x)]
Def. D3 NE(x)↔ ∀φ[φ ess. x→ �∃yφ(y)]I ^∃x[∀φ[φ ess. x→ �∃y[φ(y)]]]I ^∃x[∅ ess. x→ �∃y[∅(y)]]I by L1 ^∃x[> → �∃y[∅(y)]]I by def. of ∅ ^∃x[> → �⊥]I ^∃x[�⊥]I ^�⊥
Inconsistency ⊥
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 55
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Inconsistency (Gödel): Informal Argument (in KB and K)
Def. D2∗ φ ess. x↔���XXXφ(x) ∧ ∀ψ(ψ(x)→ �∀y(φ(y)→ ψ(y)))
Lemma 1 The empty property is an essence of every entity. ∀x (∅ ess. x)
Theorem 1 Positive Properties are possibly exemplified. ∀φ[P(φ)→ ^∃xφ(x)]
Axiom A5 P(NE)I by T1, A5: ^∃x[NE(x)]
Def. D3 NE(x)↔ ∀φ[φ ess. x→ �∃yφ(y)]
I ^∃x[∀φ[φ ess. x→ �∃y[φ(y)]]]I ^∃x[∅ ess. x→ �∃y[∅(y)]]I by L1 ^∃x[> → �∃y[∅(y)]]I by def. of ∅ ^∃x[> → �⊥]I ^∃x[�⊥]I ^�⊥
Inconsistency ⊥
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 55
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Inconsistency (Gödel): Informal Argument (in KB and K)
Def. D2∗ φ ess. x↔���XXXφ(x) ∧ ∀ψ(ψ(x)→ �∀y(φ(y)→ ψ(y)))
Lemma 1 The empty property is an essence of every entity. ∀x (∅ ess. x)
Theorem 1 Positive Properties are possibly exemplified. ∀φ[P(φ)→ ^∃xφ(x)]
Axiom A5 P(NE)I by T1, A5: ^∃x[NE(x)]
Def. D3 NE(x)↔ ∀φ[φ ess. x→ �∃yφ(y)]I ^∃x[∀φ[φ ess. x→ �∃y[φ(y)]]]
I ^∃x[∅ ess. x→ �∃y[∅(y)]]I by L1 ^∃x[> → �∃y[∅(y)]]I by def. of ∅ ^∃x[> → �⊥]I ^∃x[�⊥]I ^�⊥
Inconsistency ⊥
C. Benzmüller, 2016 —– A Success Story of Higher-Order (Automated) Theorem Proving in Computational Metaphysics 55
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Inconsistency (Gödel): Informal Argument (in KB and K)
Def. D2∗ φ ess. x↔���XXXφ(x) ∧ ∀ψ(ψ(x)→ �∀y(φ(y)→ ψ(y)))
Lemma 1 The empty property is an essence of every entity. ∀x (∅ ess. x)
Theorem 1 Positive Properties are possibly exemplified. ∀φ[P(φ)→ ^∃xφ(x)]
Axiom A5 P(NE)I by T1, A5: ^∃x[NE(x)]
Def. D3 NE(x)↔ ∀φ[φ ess. x→ �∃yφ(y)]I ^∃x[∀φ[φ ess. x→ �∃y[φ(y)]]]I ^∃x[∅ ess. x→ �∃y[∅(y)]]
I by L1 ^∃x[> → �∃y[∅(y)]]I by def. of ∅ ^∃x[> → �⊥]I ^∃x[�⊥]I ^�⊥
Inconsistency ⊥
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Inconsistency (Gödel): Informal Argument (in KB and K)
Def. D2∗ φ ess. x↔���XXXφ(x) ∧ ∀ψ(ψ(x)→ �∀y(φ(y)→ ψ(y)))
Lemma 1 The empty property is an essence of every entity. ∀x (∅ ess. x)
Theorem 1 Positive Properties are possibly exemplified. ∀φ[P(φ)→ ^∃xφ(x)]
Axiom A5 P(NE)I by T1, A5: ^∃x[NE(x)]
Def. D3 NE(x)↔ ∀φ[φ ess. x→ �∃yφ(y)]I ^∃x[∀φ[φ ess. x→ �∃y[φ(y)]]]I ^∃x[∅ ess. x→ �∃y[∅(y)]]I by L1 ^∃x[> → �∃y[∅(y)]]
I by def. of ∅ ^∃x[> → �⊥]I ^∃x[�⊥]I ^�⊥
Inconsistency ⊥
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Inconsistency (Gödel): Informal Argument (in KB and K)
Def. D2∗ φ ess. x↔���XXXφ(x) ∧ ∀ψ(ψ(x)→ �∀y(φ(y)→ ψ(y)))
Lemma 1 The empty property is an essence of every entity. ∀x (∅ ess. x)
Theorem 1 Positive Properties are possibly exemplified. ∀φ[P(φ)→ ^∃xφ(x)]
Axiom A5 P(NE)I by T1, A5: ^∃x[NE(x)]
Def. D3 NE(x)↔ ∀φ[φ ess. x→ �∃yφ(y)]I ^∃x[∀φ[φ ess. x→ �∃y[φ(y)]]]I ^∃x[∅ ess. x→ �∃y[∅(y)]]I by L1 ^∃x[> → �∃y[∅(y)]]I by def. of ∅ ^∃x[> → �⊥]
I ^∃x[�⊥]I ^�⊥
Inconsistency ⊥
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Inconsistency (Gödel): Informal Argument (in KB and K)
Def. D2∗ φ ess. x↔���XXXφ(x) ∧ ∀ψ(ψ(x)→ �∀y(φ(y)→ ψ(y)))
Lemma 1 The empty property is an essence of every entity. ∀x (∅ ess. x)
Theorem 1 Positive Properties are possibly exemplified. ∀φ[P(φ)→ ^∃xφ(x)]
Axiom A5 P(NE)I by T1, A5: ^∃x[NE(x)]
Def. D3 NE(x)↔ ∀φ[φ ess. x→ �∃yφ(y)]I ^∃x[∀φ[φ ess. x→ �∃y[φ(y)]]]I ^∃x[∅ ess. x→ �∃y[∅(y)]]I by L1 ^∃x[> → �∃y[∅(y)]]I by def. of ∅ ^∃x[> → �⊥]I ^∃x[�⊥]
I ^�⊥
Inconsistency ⊥
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Inconsistency (Gödel): Informal Argument (in KB and K)
Def. D2∗ φ ess. x↔���XXXφ(x) ∧ ∀ψ(ψ(x)→ �∀y(φ(y)→ ψ(y)))
Lemma 1 The empty property is an essence of every entity. ∀x (∅ ess. x)
Theorem 1 Positive Properties are possibly exemplified. ∀φ[P(φ)→ ^∃xφ(x)]
Axiom A5 P(NE)I by T1, A5: ^∃x[NE(x)]
Def. D3 NE(x)↔ ∀φ[φ ess. x→ �∃yφ(y)]I ^∃x[∀φ[φ ess. x→ �∃y[φ(y)]]]I ^∃x[∅ ess. x→ �∃y[∅(y)]]I by L1 ^∃x[> → �∃y[∅(y)]]I by def. of ∅ ^∃x[> → �⊥]I ^∃x[�⊥]I ^�⊥
Inconsistency ⊥
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Inconsistency (Gödel): Informal Argument (in KB and K)
Def. D2∗ φ ess. x↔���XXXφ(x) ∧ ∀ψ(ψ(x)→ �∀y(φ(y)→ ψ(y)))
Lemma 1 The empty property is an essence of every entity. ∀x (∅ ess. x)
Theorem 1 Positive Properties are possibly exemplified. ∀φ[P(φ)→ ^∃xφ(x)]
Axiom A5 P(NE)I by T1, A5: ^∃x[NE(x)]
Def. D3 NE(x)↔ ∀φ[φ ess. x→ �∃yφ(y)]I ^∃x[∀φ[φ ess. x→ �∃y[φ(y)]]]I ^∃x[∅ ess. x→ �∃y[∅(y)]]I by L1 ^∃x[> → �∃y[∅(y)]]I by def. of ∅ ^∃x[> → �⊥]I ^∃x[�⊥]I ^�⊥
Inconsistency ⊥
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Gödel’s Manuscript: Identifying the Inconsistent Axioms
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Gödel’s Manuscript: Identifying the Inconsistent Axioms
Inconsistency Scott
∀φ[P(¬φ)→ ¬P(φ)] A1(⊃)∀φ∀ψ[(P(φ) ∧ �∀x[φ(x)→ ψ(x)])→ P(ψ)] A2φ ess. x↔ ∀ψ(ψ(x)→ �∀y(φ(y)→ ψ(y))) D2∗
NE(x)↔ ∀φ[φ ess. x→ �∃yφ(y)] D3P(NE) A5
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Part D:Recent Technical Improvements
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Usability: More Intuitive Syntax for Embedded Logics in Isabelle
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Improved Embedding of Modal Logic S5: S5U
Modal Logic S5I Reflexivity: ∀x.(r x x)I Symmetry: ∀x.∀y.(r x y)→ (r y x)I Transitivity: ∀x.∀y.∀z.(r x y) ∧ (r y z)→ (r x z)
Modal Logic S5U: with universal accessibilityI Universality: ∀x.∀y.(r x y)
S5�ϕ ≡ λw.∀v.���
�XXXXr(w, v)→ϕ(v) and ^ϕ ≡ λw.∃v.����XXXXr(w, v)∧ϕ(v)
S5U�ϕ ≡ λw.∀v.ϕ(v) and ^ϕ ≡ λw.∃v.ϕ(v)
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LEO-II proves T3 (in 2,5s) directly from the Axioms in S5U!
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Inconsistency in S5U
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Conclusion
Overall Achievements
I significant contribution towards a Computational MetaphysicsI novel results contributed by HOL-ATPsI infrastructure can be adapted for other logics and logic combinationsI basic technology works well; however, improvements still needed
Relevance (wrt foundations and applications)
I Philosophy, AI, Computer Science, Computational Linguistics, Maths
Related work: only for Anselm’s simpler argument
I first-order ATP PROVER9 [OppenheimerZalta, 2011]
I interactive proof assistant PVS [Rushby, 2013]
Ongoing/Future work
I Landscape of verified/falsified ontological argumentsI You may consider to contribute:https://github.com/FormalTheology/GoedelGod.git
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Discussion
I LEO-II detected relevant new knowledge:Inconsistency in Gödel’s original ontological argumentKey step: ’non-analytic’ instantiation of a second-order variable!
I LEO-II’s proof object actually contains the proof ideaI first: failed to identify the relevant puzzle piecesI only later (discussion with Brown): reconstructed abstract-level proofI Once a beautiful structure has been revealed it can’t be missed anymoreI Unmated low-level formal proofs, in contrast, are lacking persuasive
powerCut-introduction instead of cut-elimination!
We need (better) tools and means to bridge between machine-oriented andhuman-intuitive proofs and (counter-)models
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