A (Simplified) Supreme Being Necessarily Exists — Says the Computer!

Christoph BenzmullerFreie Universitat Berlin & University of Luxembourg

c.benzmueller@fu-berlin.de

AbstractA simplified variant of Kurt Godel’s modal onto-logical argument is presented. Some of Godel’s,resp. Scott’s, premises are modified, others aredropped, and modal collapse is avoided. Theemended argument is shown valid already in quan-tified modal logic K.The presented simplifications have been compu-tationally explored utilising latest knowledge rep-resentation and reasoning technology based onhigher-order logic. The paper thus illustrates howmodern symbolic AI technology can contributenew knowledge to formal philosophy and theology.

1 IntroductionVariants of Kurt Godel’s [1970], resp. Dana Scott’s [1972],modal ontological argument have previously been analysed,studied and verified on the computer by Benzmuller andWoltzenlogel [2016] and Benzmuller and Fuenmayor [2020],and even some unknown flaws were revealed in these works.1

In this paper a simplified and improved variant of Godel’smodal ontological argument is presented. This simplifica-tion has been explored in collaboration with the proof as-sistant Isabelle/HOL [Nipkow et al., 2002b], while employ-ing Benzmuller’s [2019; 2013] shallow semantical embed-ding (SSE) approach as enabling technology. This technologysupports the reuse of theorem proving (ATP) systems for clas-sical higher-order logic (HOL) to represent and reason witha wide range of non-classical logics and theories, includinghigher-order modal logic (HOML) and Godel’s modal onto-logical argument, which are in the focus of this paper.

The new, simplified modal argument is as follows. Godeldefinition of a Godlike entity remains unchanged (P is anuninterpreted constant denoting positive properties):

A Godlike entity thus possesses all positive properties.

1E.g., the theorem prover LEO-II detected that Godel’s [1970]variant of the argument is inconsistent; this inconsistency had, un-knowingly, been fixed in the variant of Scott [1972]; cf. [Benzmullerand Woltzenlogel Paleo, 2016] for more details on this.

The three single axioms of the new theory are2

where the following defined terms are used (∀ is a possibilistquantifier and ∀E is an actualist quantifier for individuals):

Informally we have:A1’ Self-identity is a positive property, self-difference is not.A2’ A property entailed or necessarily entailed by a positive

property is positive.A3 The conjunction of any collection of positive properties

is positive. (Technical reading: ifZ is any set of positiveproperties, then the property X obtained by taking theconjunction of the properties in Z is positive.)

From these premises it follows, in a few argumentations stepsin base modal logic K, that a Godlike entity possibly and nec-essarily exists. Modal collapse, which expresses that there areno contingent truths and which thus eliminates the possibil-ity of alternative possible worlds, does not follow from theseaxioms. These observations should render the new theory in-teresting to formal philosophy and theology.

Compare the above with Godel’s premises of the modal on-tological argument (we give the consistent variant of Scott):

2An alternative to A1’ would be: The universal property (λx.>)is a positive property, and the empty property (λx.⊥) is not.

The third-order formalization of A3 given here has been proposedby Anderson and Gettings [1996], see also Fitting [2002]. AxiomA3, together with the definition of G, implies that being Godlike is apositive property. Since supporting this inference is the only role thisaxiom plays in the argument, P G could be (and has been; cf. Scott[1972]) taken as an alternative to our A3.

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Necessary existence (NE) and essence (E) are defined as (theother definitions are as above):

Informally: a property Y is the essence of an entity x if (i)x has property Y and (ii) Y necessarily entails every propertyof x. Moreover, an entity x has the property of necessaryexistence if the essence of x is necessarily instantiated.

We also give informal readings of Godel’s axioms: A1 saysthat one of a property or its complement is positive. A2 statesthat a property necessarily entailed by a positive property ispositive, and A3 is as before. A4 expresses that any positiveproperty is necessarily so. A5 postulates that necessary exis-tence is a positive property. Axiom B (symmetry of the acces-sibility relation associated with modal �-operator) is added toensure that we are in modal logic KB instead of just K.3

Using Godel’s premises as stated it can be proved that aGodlike entity possibly and necessarily exists, and this proofcan be verified with the computer. However, as is well known[Sobel, 1987], modal collapse is implied; see also Fitting[2002] and Sobel [2004] for further background information.

Recent results of Benzmuller and Fuenmayor [2020] showthat different modal ultrafilter properties can be deduced fromGodel’s premises. These insights are key to the new argumentpresented in this paper: If Godel’s premises entail that posi-tive properties form a modal ultrafilter, then why not turningthings around, and start with an axiom U1 postulating thatpositive properties are an ultrafilter? Then use U1 instead ofA1 for proving that a Godlike entity necessarily exists, and onthe fly explore what further simplifications of the argumentare triggered. This research plan worked out and it led to thenew modal ontological argument presented above, where U1has been replaced by A1’ and where A2 has been strengthenedinto A2’ accordingly.

The proof assistant Isabelle/HOL and its integrated ATPsystems have supported our exploration work surprisinglywell, despite the undecidability and high complexity of theunderlying logic setting. As usual, we here only present themain steps of the exploration process, and various interestingeureka and frustration steps in between are dropped.

Paper structure: An SSE of HOML in HOL is introduced inSect. 2. The foundations outlined there ensure that the paperis sufficiently self-contained; readers familiar with the SSEapproach may simply skip it. Modal ultrafilter are defined inSect. 3. Section 4 recaps the Godel/Scott variant, and then anultrafilter-based modal ontological argument is presented inSect. 5. This new argument is further simplified in Sect. 6,leading to our new proposal based on axioms A1’, A2’ and A3as presented above. Related work is mentioned in Sect. 7.

Since we develop, explain and discuss our formal encod-ings directly in Isabelle/HOL [Nipkow et al., 2002a], somefamiliarity with this proof assistant and its underlying logicHOL [Andrews, 2002] is assumed. The entire sources of ourformalization are presented and explained.

Sections 2 and 3 have been adapted from [Benzmuller andFuenmayor, 2020] and [Kirchner et al., 2019].

3B’s counterpart ϕ � �♦ϕ is implied and could be used instead.

2 Modeling HOML in HOLRelated work focused on the development of various SSEs,cf. [Benzmuller, 2019; Kirchner et al., 2019] and the ref-erences therein. These contributions, among others, showthat the standard translation from propositional modal logicto first-order (FO) logic can be concisely modeled (i.e., em-bedded) within HOL theorem provers, so that the modal op-erator �, for example, can be explicitly defined by the λ-term λϕ.λw.∀v.(Rwv → ϕv), where R denotes the acces-sibility relation associated with �. Then one can constructFO formulas involving �ϕ and use them to represent andproof theorems. Thus, in an SSE, the target logic is inter-nally represented using higher-order (HO) constructs in a the-orem proving system such as Isabelle/HOL. Benzmuller andPaulson [2013] developed an SSE that captures quantifiedextensions of modal logic. For example, if ∀x.φx is short-hand in HOL for Π(λx.φx), then �∀xPx would be repre-sented as �Π′(λx.λw.Pxw), where Π′ stands for the λ-termλΦ.λw.Π(λx.Φxw), and the � gets resolved as above.

To see how these expressions can be resolved to producethe right representation, consider the following series of re-ductions:

�∀xPx≡ �Π′(λx.λw.Pxw)≡ �((λΦ.λw.Π(λx.Φxw))(λx.λw.Pxw))≡ �(λw.Π(λx.(λx.λw.Pxw)xw))≡ �(λw.Π(λx.Pxw))≡ (λϕ.λw.∀v.(Rwv → ϕv))(λw.Π(λx.Pxw))≡ (λϕ.λw.Π(λv.Rwv → ϕv))(λw.Π(λx.Pxw))≡ (λw.Π(λv.Rwv → (λw.Π(λx.Pxw))v))≡ (λw.Π(λv.Rwv → Π(λx.Pxv)))≡ (λw.∀v.Rwv → ∀x.Pxv)≡ (λw.∀vx.Rwv → Pxv)

Thus, we end up with a representation of �∀xPx in HOL. Ofcourse, types are assigned to each (sub-)term of the HOL lan-guage. We assign individual terms (such as variable x above)the type e, and terms denoting worlds (such as variable wabove) the type i. From such base choices, all other types inthe above presentation can actually be inferred.

An explicit encoding of HOML in Isabelle/HOL, follow-ing the above ideas, is presented in Fig. 1.4 In lines 4–5 thebase types i and e are declared. Note that HOL [Andrews,2002] comes with an inbuilt base type bool, the bivalent typeof Booleans. No cardinality constraints are associated withtypes i and e, except that they must be non-empty. To keepour presentation concise, useful type synonyms are intro-duced in lines 6–9. σ abbreviates the type i⇒bool, and termsof type σ can be seen to represent world-lifted propositions,i.e., truth-sets in Kripke’s modal relational semantics [Gar-son, 2018]. The explicit transition from modal propositions

4In Isabelle/HOL explicit type information can often be omitteddue the system’s internal type inference mechanism. This feature isexploited in our formalization to improve readability. However, forall new abbreviations and definitions, we usually explicitly declarethe types of the freshly introduced symbols. This supports a betterintuitive understanding, and it also reduces the number of polymor-phic terms in the formalization (heavy use of polymorphism maygenerally lead to decreased proof automation performance).

Figure 1: SSE of HOML in HOL.

to terms (truth-sets) of type σ is a key aspect in SSE tech-nique, and in the remainder of this article we use of phrasessuch “world-lifted” or “σ-type” terms to emphasize this con-version in the SSE approach. γ, which stands for e⇒σ (⇒ isthe function type constructor in HOL), is the type of world-lifted, intensional properties. µ and ν, which abbreviate σ⇒σand σ⇒σ⇒σ, are the types associated with unary and binarymodal logic connectives.

The modal logic connectives are defined in lines 11–24.In line 16, for example, the definition of the world-lifted∨-connective of type ν is given; explicit type informationis presented after the ::-token for ‘c5’, which is the ASCII-denominator for the (right-associative) infix-operator∨ as in-troduced in parenthesis shortly after. ϕσ ∨ψσ is then definedas abbreviation for the truth-set λw.(ϕσw) ∨ (ψσw), respec-tively. In the remainder we generally use bold-face symbolsfor world-lifted connectives (such as∨) in order to rigorouslydistinguish them from their ordinary counterparts (such as ∨)in meta-logic HOL.

Further modal logic connectives, such as⊥,>,¬,→, and↔ are introduced analogously. The operator ⇁, introduced

in lines 22, is inverting properties of types γ; this operationoccurs in some Godel’s axiom A1. = and 6= are defined inlines 23–24 as world-independent, syntactical equality.

The already discussed, world-lifted modal �-operator isintroduced in lines 19–20; accessibility relation R is nownamed r. The definition of ♦ in line 21 is analogous.

The world-lifted (polymorphic) possibilist quantifier ∀ asdiscussed before is introduced in line 27. In line 28, user-friendly binder-notation for ∀ is additionally defined. Insteadof distinguishing between ∀ and Π′ as in our illustrating ex-ample, these symbols are overloaded here. The introductionof the possibilist ∃-quantifier in lines 29–30 is analogous.

Further actualist quantifiers, ∀E and ∃E , are introducedin lines 33–37; their definition is guarded by an explicit,possibly empty, existsAt (infix @) predicate, which encodeswhether an individual object actually “exists” at a particularpossible world, or not. These additional actualist quantifiersare declared non-polymorphic, and they support quantifica-tion over individuals only. In the subsequent study of the on-tological argument we will indeed apply ∀ and ∃ for differenttypes in the type hierarchy of HOL, while we employ ∀E and∃E for individuals only.

Global validity of a world-lifted formula ψσ , denoted asbψc, is introduced in line 40 as an abbreviation for ∀wi.ψw.

Consistency of the introduced concepts is confirmed by themodel finder nitpick [Blanchette and Nipkow, 2010] in line43. Since only abbreviations and no axioms have been in-troduced so far, the consistency of the Isabelle/HOL theoryHOML as displayed in Fig. 1 is actually evident.

In line 44–47 its is studied whether instances of the Barcanand the converse Barcan formulas are implied. As expected,both principles are valid only for possibilist quantification,while they have counter models for actualist quantification.

Theorem 1. The SSE of HOML in HOL is faithful for basemodal logic K.

Proof. Follows [Benzmuller and Paulson, 2013].

Theory HOML thus successfully models base modal logicK in HOL. To arrive at logic KB the symmetry axiom B asshown earlier can be postulated.

3 Modal UltrafilterTheory ModalUltrafilter, see Fig. 2, imports theory HOML andadapts the topological notions of filter and ultrafilter to ourmodal logic setting. For an introduction to filter and ultrafiltersee the literature, e.g., [Burris and Sankappanavar, 1981].

Modal ultrafilter are introduced in lines 18–19 as world-lifted characteristic functions of type (γ⇒σ)⇒σ. A modalultrafilter is thus a world-dependent set of intensions of γ-type properties; in other words, a σ-subset of the σ-powersetof γ-type property extensions. An ultrafilter φ is defined as afilter satisfying an additional maximality condition: ∀ϕ.ϕ ∈φ ∨ (−1ϕ) ∈ φ, where ∈ is elementhood of γ-type objectsin σ-sets of γ-type objects (see line 4), and where −1 is therelative set complement operation on sets of entities (line 9).

A Filter φ, see lines 12–15, is required to

Figure 2: Definition of filter and ultrafilter (in modal context).

1. be large: U ∈ φ, where U denotes the full set of γ-typeobjects we start with,

2. exclude the empty set: ∅ 6∈ φ, where ∅ is the world-lifted empty set of γ-type objects,

3. be closed under supersets: ∀ϕψ.(ϕ ∈ φ ∧ ϕ ⊆ ψ)→ψ ∈ φ (world-lifted⊆-relation is defined in line 7), and

4. be closed under intersections: ∀ϕψ.(ϕ ∈ φ ∧ ψ ∈φ)→ (ϕu ψ) ⊆ φ (where u is defined in line 8).

Benzmuller and Fuenmayor [2020] have studied two dif-ferent notions of modal ultrafilter (called γ- and δ-ultrafilter)which are defined on intensions and extensions of properties,respectively. This distinction is not needed in this paper; whatwe call modal ultrafilter here corresponds to their γ-ultrafilter.

4 Godel’s Modal Ontological ArgumentThe full formalization of Scott’s variant of Godel’s argument,which relies on theories ModalUltrafilter and HOML, is pre-sented in Fig. 3. Line 3 starts out with the declaration ofthe uninterpreted constant symbol P , for positive properties,which is of type γ⇒σ. P is thus an intensional, world-depended concept.

The premises of Godel’s argument, as already discussedearlier, are stated in lines 5–24.5

An abstract level “proof net” for theorem T6, the neces-sary existence of a Godlike entity, is presented in lines 26–34. Following the literature the proof goes as follows: FromA1 and A2 infer T1: positive properties are possibly exempli-fied. From A3 and the defn. of G obtain T2: being Godlikeis a positive property (Scott actually directly postulated T2).Using T1 and T2 show T3: possibly a Godlike entity exists.Next, use A1, A4, the defns. of G and E to infer T4: beingGodlike is an essential property of any Godlike entity. Fromthis, A5, B and the defns. of G and NE have T5: the possible

5 Remark: whether we use actualist or possibilist quantifiers forindividuals in the definition of⊂ or T4 turned out irrelevant in thispaper, and we consistently use actualist quantifiers in the remainder.

Figure 3: Godel’s modal ontological argument; Scott’s variant.

existence of a Godlike entity implies its necessary existence.T5 and T3 then imply T6.

The six subproofs and their dependencies have been au-tomatically explored using state-of-the-art ATP system inte-grated with Isabelle/HOL via its sledgehammer tool; sledge-hammer then identified and returned the abstract level proofjustifications as displayed here, e.g. “using T1 T2 by simp”.The mentioned proof engines/tactics blast, metis, simp andsmt are trustworthy components of Isabelle/HOL’s, since theyinternally reconstruct and check each subproofs the proof as-sistants small and trusted proof kernel. Using the definitions

from Sect. 2, one can also reconstruct and formally verify allproofs with pen and paper.6

The presented theory is consistent, which is confirmed inline 37 by model finder nitpick; nitpick reports a model (notshown here) consisting of one world and one Godlike entity.

Validity of modal collapse (MC) is confirmed in lines 40–47; a proof net displaying the proofs main idea is shown.

Most relevant for this paper is that the ATP systems wereable to quickly prove that Godel’s notion of positive proper-ties P constitutes a modal ultrafilter, cf. lines 50–57. Thiswas key to the idea of taking the modal ultrafilter property ofP as an axiom U1; see the next section.

5 Ultrafilter Modal Ontological ArgumentTaking U1 as an axiom for Godel’s theory in fact leads to asignificant simplification of the modal ontological argument;this is shown in lines 16–28 in Fig. 4: not only Godel’s axiomA1 can be dropped, but also axioms A4 and A5, together withdefns. E and NE . Even logic KB can be given up, since K isnow sufficient for verifying the proof argument.

The proof is similar to before: Use U1 and A2 to infer T1(positive properties are possibly exemplified). From A3 anddefn. of G have T2 (being Godlike is a positive property). T1and T2 imply T3 (a Godlike entity possibly exists). From U1,A2, T2 and the defn. of G have T5 (possible existence of aGodlike entity implies its necessary existence). Use T5 andT3 to conclude T6 (necessary existence of a Godlike entity).

Consistency of the theory is confirmed in line 31; again amodel with one world and one Godlike entity is reported.

Most interestingly, modal collapse MC now has a simplecounter model as nitpick informs us. This counter modelconsists of a single entity e1 and two worlds i1 and i2 withr = {〈i1, i1〉, 〈i2, i1〉, 〈i2, i2〉} Trivially, formula Φ is suchthat Φ holds in i2 but not in i1, invalidates MC at world i2.e1 is the Godlike entity in both worlds, i.e., G is the prop-erty that holds for e1 in i1 and i2, which we may denote asλe.λw.e=e1∧ (w=i1∨w=i2). Using tuple notation we maywrite G = {〈e1, ii〉, 〈e1, i2〉}

Remember that P , which is of type γ⇒σ, is an inten-sional, world-depended concept. In our counter model forMC the extension of P for world i1 has the above G andλe.λw.e=e1 ∧ w=i1 as its elements, while in world i2 wehave G and λe.λw.e=e1 ∧ w=i2. Using tuple notation wenote P as

{〈{〈e1, i1〉, 〈e1, i2〉}, ii〉, 〈{〈e1, i1〉}, ii〉,〈{〈e1, i1〉, 〈e1, i2〉}, i2〉, 〈{〈e1, i2〉}, i2〉}

In order to verify that P is a modal ultrafilter we have to ver-ify that the respective modal ultrafilter conditions are satis-fied in both worlds. U ∈ P in i1 and also in i2, since both〈{〈e1, i1〉, 〈e1, i2〉}, i1〉 and 〈{〈e1, i1〉, 〈e1, i2〉}, i2〉 are in P;∅ 6∈ P in i1 and also in i2, since both 〈{}, i1〉 and 〈{}, i2〉are not in P . Is also easy to verify that P is closed undersupersets and intersection in both worlds.

6Reconstruction of proofs from such proof nets within di-rect proof calculi for quantified modal logics, cf. Kanckos andWoltzenlogel-Paleo [2017] or Fitting [2002], is ongoing work.

Figure 4: Ultrafilter variant of Godel’s ontological argument.

Note that in this counter model for MC, also Godel’s axiomA4 is invalidated. Consider X = λe.λw.e=e1 ∧ w=i2, i.e.,X is true for e1 in i2, but false for e1 in i1. We have P X ini2, but we do not have �P X in i1, since P X does not holdin i1, which is reachable from i2.

nitpick is e.g. capable of computing and displaying all par-tial modal ultrafilters in this counter model: out of 512 can-didates, nitpick identifies 32 structures of form 〈F, i〉, fori ∈ {1, 2}, in which F satisfies the filter conditions in thespecified world i. An example for such an 〈F, i〉 is

〈{〈{〈e1, i1〉}, ii〉, 〈{}, ii〉,〈{〈e1, i1〉, 〈e1, i2〉}, i2〉, 〈{〈e1, i2〉}, i2〉}, i2〉

F is not a proper modal ultrafilter, since 〈F, i1〉 does not hold.

6 Simplified Modal Ontological ArgumentWhat modal ultrafilters properties of P are actually needed tosupport T6? Which ones can be dropped? Experiments withthe computer confirm that, in modal logic K, the ultrafilterconditions 1-3 from Sect. 3 must be upheld for P , while 4can be dropped. However, it is possible to merge condition3 (closed under supersets) for P with Godel’s A2 into A2’ asshown in line 18 of Fig. 5. Moreover, instead of requiring theuniversal set/property U = λx.> to be a positive property,we postulate that self-identity λx.x=x, which is extension-ally equal to U, is in P . Analogously, we replace ∅ = λx.⊥

Figure 5: Simplified variant of Godel’s ontological argument.

in ultrafilter condition 2 for P by self-difference λx.x 6=x.Self-identity and self-difference have used frequently in thehistory of the ontological argument, which is part of the mo-tivation for this switch.

Now, from the definition of G (line 13) and the axioms A1’,A2’ and A3 (lines 17–19) theorem T6 immediately follows: inline 22 several theorem provers integrated with sledgeham-mer report a proof in about one second when running the ex-periments on a standard notebook. Moreover, a more detailed“proof net” is presented in lines 23–28; the proof argument isanalogous to what has been discussed before.

Consistency is confirmed by nitpick in line 31, and acounter model (similar to the one discussed in the previoussection) is reported to MC in line 34.

In lines 37–43 further questions are answered experimen-tally: neither A1, nor A4 or A5, of the premises we droppedfrom Godel’s theory are implied anymore. Also Monotheismis not implied (line 44); by postulating A1 it can be enforced.Since some of these axioms, e.g. the strong A1, have been dis-cussed controversially in the history of Godel’s argument and

since also MC is independent, it is justified to claim that wehave arrived at a philosophically and theologically potentiallyrelevant simplification of the modal ontological argument.

7 Related WorkFitting [2002] has suggested to carefully distinguish betweenintensions and extensions of positive properties in the con-text of Godel’s modal ontological argument, and, in orderto do so within a single framework, he introduces a suffi-ciently expressive HOML enhanced with means for the ex-plicit representation of intensional terms and their extensions;see also the intensional operations formalized by Fuenmayorand Benzmuller [2017; 2020] in the course of their analysisand verification of the variants of Fitting and Anderson [1990;1996] contributions.

The application of computational methods to philosophicalproblems was initially limited to first-order theorem provers.Fitelson and Zalta [2007] used PROVER9 to find a proof ofthe theorems about situation and world theory in [Zalta, 1993]and they found an error in a theorem about Plato’s Forms thatwas left as an exercise in [Pelletier and Zalta, 2000]. Oppen-heimer and Zalta [2011] discovered, using Prover9, that 1 ofthe 3 premises used in their reconstruction of Anselm’s on-tological argument (in [Oppenheimer and Zalta, 1991]) wassufficient to derive the conclusion. The first-order conversiontechniques that were developed and applied in these worksare outlined in some detail in [Alama et al., 2015].

More recent related work makes use of higher-order proofassistants. Besides some already given references to the workof Benzmuller and colleagues, this includes John Rushby’s[2018] study on the Anselm’s ontological argument in thePVS proof assistant and Blumson’s [2017] related study inIsabelle/HOL.

8 ConclusionGodel’s modal ontological argument stands in a prominenttradition of western philosophy. The ontological argumenthas its roots in the Proslogion (1078) of Anselm of Canter-bury and it has been picked up in the Fifth Meditation (1637)of Descartes and in the works of Leibniz, which in turn in-spired and informed the work of Godel.

In this paper we have linked Godel’s theory to a suitablyadapted mathematical theory (modal ultrafilter), and subse-quently we have developed a significantly simplified modalontological argument that avoids some axioms and conse-quences in the new theory, including modal collapse, thathave led to criticism in the past.

While data scientists apply subsymbolic AI techniques toobtain approximating and rather opaque models in their appli-cation domains of interest, we have in this paper applied mod-ern symbolic AI techniques to arrive at sharp and explainablemodels of the metaphysical concepts we have studied. In par-ticular, we have illustrated how state of the art theorem prov-ing systems, in combination with latest knowledge represen-tation and reasoning technology, can be fruitfully utilized toexplore and contribute new knowledge to theoretical philoso-phy and theology.

AcknowledgementsFor comments and/or support of this work and prior pa-pers I am grateful to, among others, D. Fuenmayor, B.Woltzenlogel-Paleo, D. Scott, E. Zalta, R. Rojas, A. Steen,D. Kirchner, L. van der Torre, X. Parent, E. Weydert.

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