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Preprint; article to appear in Sophia.

A Simpliﬁed Variant of G¨odel’s Ontological Argument

Christoph Benzm¨uller

Abstract A simpliﬁed variant of G¨odel’s ontological argument is presented. The simpliﬁed argument is valid already in

basic modal logics K or KT, it does not suﬀer from modal collapse, and it avoids the rather complex predicates of essence

(Ess.) and necessary existence (NE) as used by G¨odel. The variant presented has been obtained as a side result of a

series of theory simpliﬁcation experiments conducted in interaction with a modern proof assistant system. The starting

point for these experiments was the computer encoding of G¨odel’s argument, and then automated reasoning techniques

were systematically applied to arrive at the simpliﬁed variant presented. The presented work thus exempliﬁes a fruitful

human-computer interaction in computational metaphysics. Whether the presented result increases or decreases the

attractiveness and persuasiveness of the ontological argument is a question I would like to pass on to philosophy and

theology.

Keywords Ontological argument ·Computational metaphysics ·Modal collapse

1 Introduction

G¨odel’s (1970) ontological argument has attracted signiﬁcant, albeit controversial, interest among philosophers, logi-

cians and theologians (Sobel, 2004). In this article I present a simpliﬁed variant of G¨odel’s argument that was developed

in interaction with the proof assistant system Isabelle/HOL (Nipkow et al., 2002), which is based on classical higher-

order logic (Benzm¨uller & Andrews, 2019). My personal interest in G¨odel’s argument has been primarily of logical

nature. In particular, this interest encompasses the challenge of automating and applying reasoning in quantiﬁed modal

logics using an universal meta-logical reasoning approach (Benzm¨uller, 2019) in which (quantiﬁed) non-classical logics

are semantically embedded in classical higher-order logic. The simpliﬁed ontological argument presented below is a

side result of this research, which began with a computer encoding of G¨odel’s argument so that it became amenable

to formal analysis and computer-assisted theory simpliﬁcation experiments; cf. Benzm¨uller (2020) for more technical

details on the most recent series of experiments. The simpliﬁed argument selected for presentation in this article has,

I believe, the potential to further stimulate the philosophical and theological debate on G¨odel’s argument, since the

simpliﬁcations achieved are indeed quite far-reaching:

–Only minimal assumptions about the modal logic used are required. The simpliﬁed variant presented is indeed

valid in the comparatively weak modal logics K or KT, which only use uncontroversial reasoning principles.1

C. Benzm¨uller

Otto-Friedrich-Universit¨at Bamberg, Kapuzinerstraße 16, 96047 Bamberg, Germany

Freie Universit¨at Berlin, Dep. of Mathematics and Computer Science, Arnimallee 7, 14195 Berlin, Germany

E-mail: christoph.benzmueller@uni-bamberg.de

1Some background on modal logic (see also Garson, 2018, and the references therein): The modal operators 2and 3are employed,

in the given context, to capture the alethic modalities “necessarily holds” and “possibly holds”, and often the modal logic S5 is used for

this. However, logic S5 comes with some rather strong reasoning principles, that could, and have been, be taken as basis for criticism

on G¨odel’s argument. Base modal logic K is comparably uncontroversial, since it only adds the following principles to classical logic: (i)

If sis a theorem of K, then so is 2s, and (ii) the distribution axiom 2(s→t)→(2s→2t) (if simplies tholds necessarily, then the

arXiv:2202.06264v1 [cs.LO] 13 Feb 2022

2 Christoph Benzm¨uller

–G¨odel’s argument introduces the comparably complex predicates of essence (Ess.) and necessary existence (NE),

where the latter is based on the former. These terms are avoided altogether in the simpliﬁed version presented

here.

–Above all, a controversial side eﬀect of G¨odel’s argument, the so-called modal collapse, is avoided. Modal collapse

(MC), formally notated as ∀s(s→2s), expresses that “what holds that holds necessarily”, which can also be

interpreted as “there are no contingent truths” and that “everything is determined”. The observation that G¨odel’s

argument implies modal collapse has already been made by Sobel (1987), and Kovaˇc (2012) argues that modal

collapse may even have been intended by G¨odel. Indeed, the study of modal collapse has been the catalyst for much

recent research on the ontological argument. For example, variants of G¨odel’s argument that avoid modal collapse

have been presented by Anderson (1990, 1996) and Fitting (2002), among others, cf. also the formal veriﬁcation

and comparison of these works by Benzm¨uller and Fuenmayor (2020). In the following, however, it is shown that

modal collapse can in fact be avoided by much simpler means.

What I thus present in the remainder is a simple divine theory, derived from G¨odel’s argument, that does not

entail modal collapse.

Since G¨odel’s (1970) argument was shown to be inconsistent (Benzm¨uller & Woltzenlogel Paleo, 2016), the actual

starting point for the exploration of the simpliﬁed ontological argument has been Scott’s variant (1972), which is

consistent. The terminology and notation used in what follows therefore also remains close to Scott’s.

Only one single uninterpreted constant symbol Pis used in the argument. This symbol denotes “positive properties”,

and its meaning is restricted by the postulated axioms, as discussed below. Moreover, the following deﬁnitions (or

shorthand notations) were introduced by G¨odel, respectively Scott:

–An entity xis God-like if it possesses all positive properties.

G(x)≡∀φ(P(φ)→φ(x))

–A property φis an essence (Ess.) of an entity xif, and only if, (i) φholds for xand (ii) φnecessarily entails every

property ψof x(i.e., the property is necessarily minimal).

φEss. x ≡φ(x)∧ ∀ψ(ψ(x)→2∀y(φ(y)→ψ(y)))

Deviating from G¨odel, Scott added here the requirement that φmust hold for x. Scott found it natural to add

this clause, not knowing that it ﬁxed the inconsistency in G¨odel’s theory, which was discovered by an automated

theorem prover (Benzm¨uller & Woltzenlogel Paleo, 2016). G¨odel’s (1970) scriptum avoids this conjunct, although

it occurred in some of his earlier notes.

–A further shorthand notation, NE(x), termed necessary existence, was introduced by G¨odel. NE(x) expresses that

xnecessarily exists if it has an essential property.

NE(x)≡∀φ(φEss. x →2∃x φ(x))

The axioms of Scott’s (1972) theory, which constrain the meaning of constant symbol P, and thus also of deﬁnition

G, are now as follows:

AXIOM 1 Either a property or its negation is positive, but not both.2

∀φ(P(¬φ)↔ ¬P(φ))

AXIOM 2 A property is positive if it is necessarily entailed by a positive property.

∀φ∀ψ((P(φ)∧(2∀x(φ(x)→ψ(x)))) →P(ψ))

necessity of simplies the necessity of t). Modal logic KT additionally provides the T axiom: 2s→s(if sholds necessarily, then s),

respectively its dual s→3s(if s, then sis possible).

Model logics can be given a possible world semantics, so that 2scan be read as: for all possible worlds v, which are reachable from

a given current world w, we have that sholds in v. And its dual, 2s, thus means: there exists a possible world v, reachable from the

current world w, so that sholds in v.

2¬φis shorthand for λx ¬φ(x).

A Simpliﬁed Variant of G¨odel’s Ontological Argument 3

AXIOM 3 Being Godlike is a positive property.3

P(G)

AXIOM 4 Any positive property is necessarily positive (in Scott’s words: being a positive property is logical, hence,

necessary).

∀φ(P(φ)→2P(φ))

AXIOM 5 Necessary existence (NE) is a positive property.

P(NE)

From this theory the following theorems and corollaries follow; cf. Scott (1972) and Benzm¨uller and Woltzenlogel

Paleo (2014, 2016) for further details. Note that the proofs are valid already in (extensional) modal logic KB, which

extends base modal logic K with AXIOM B:∀φ(φ→23φ), or in words, if φthen φis necessarily possible.

THEOREM 1 Positive properties are possibly exempliﬁed.

∀φ(P(φ)→3∃x φ(x))

Follows from AXIOM 1 and AXIOM 2.

CORO Possibly there exists a God-like being.

3∃xG(x)

Follows from THEOREM 1 and AXIOM 3.

THEOREM 2 Being God-like is an essence of any God-like being.

∀xG(x)→G Ess. x

Follows from AXIOM 1 and AXIOM 4 using the deﬁnitions of Ess.and G.

THEOREM 3 Necessarily, there exists a God-like being.

2∃xG(x)

Follows from AXIOM 5,CORO,THEOREM2,AXIOM B using the deﬁnitions of Gand NE.

THEOREM 4 There exists a God-like being.

∃xG(x)

Follows from THEOREM 3 together with CORO and AXIOM B.

All claims have been veriﬁed with the higher-order proof assistant system Isabelle/HOL (Nipkow et al., 2002) and

the sources of these veriﬁcation experiments are presented in Fig. 2 in the Appendix. This veriﬁcation work utilised

the universal meta-logical reasoning approach (Benzm¨uller, 2019) in order to obtain a ready to use “implementation”

of higher-order modal logic in Isabelle/HOL’s classical higher-order logic.

In these experiments only possibilist quantiﬁers were initially applied and later the results were conﬁrmed for a

modiﬁed logical setting in which ﬁrst-order actualist quantiﬁers for individuals were used, and otherwise possibilist

quantiﬁers. It is also relevant to note that, in agreement with G¨odel and Scott, in this article only extensions of (positive)

properties paper are considered, in contrast to Fitting (2002), who studied the use of intensions of properties in the

context of the ontological argument.

3Alternatively, we may postulate A3’: The conjunction of any collection of positive properties is positive. Formally, ∀Z.(Pos Z→

∀X(XdZ→PX)), where Pos Zstands for ∀X(ZX→PX) and XdZis shorthand for 2∀u.(X u ↔(∀Y. ZY→Y u)).

4 Christoph Benzm¨uller

2 Simpliﬁed Variant

Scott’s (1972) theory from above has interesting further corollaries, besides modal collapse MC and monotheism

(cf. Benzm¨uller and Woltzenlogel Paleo, 2014, 2016),4and such corollaries can be explored using automated theorem

proving technology. In particular, the following two statements are implied.

CORO 1 Self-diﬀerence is not a positive property.

¬P(λx (x6=x))

Since the setting in this article is extensional, we alternatively get that the empty property, λx ⊥, is not a positive

property.

¬P(λx ⊥)

Both statements follow from AXIOM 1 and AXIOM 2. This is easy to see, because if λx (x6=x) (respectively, λx ⊥)

was positive, then, by AXIOM 2, also its complement λx (x=x) (respectively, λx >) to be so, which contradicts

AXIOM 1. Thus, only λx (x=x) and λx >can be and indeed are positive, but not their complements.

CORO 2 A property is positive if it is entailed by a positive property.

∀φ∀ψ((P(φ)∧(∀x(φ(x)→ψ(x)))) →P(ψ))

This follows from AXIOM 1 and THEOREM 4 using the deﬁnition of G. Alternatively, the statement can be proved

using AXIOM 1,AXIOM B and modal collapse MC.

The above observations are core motivation for our simpliﬁed variant of G¨odel’s argument as presented next; see

Benzm¨uller (2020) for further experiments and explanations on the exploration on this and further simpliﬁed variants.

Axioms of the Simpliﬁed Ontological Argument

CORO 1 Self-diﬀerence is not a positive property.

¬P(λx (x6=x))

(Alternative: The empty property λx ⊥is not a positive property.)

CORO 2 A property entailed by a positive property is positive.

∀φ∀ψ((P(φ)∧(∀x(φ(x)→ψ(x)))) →P(ψ))

AXIOM 3 Being Godlike is a positive property.

P(G)

As before, an entity xis deﬁned to be God-like if it possesses all positive properties:

G(x)≡∀φ(P(φ)→φ(x))

From the above axioms of the simpliﬁed theory the following successive argumentation steps can be derived in base

modal logic K:

LEMMA 1 The existence of a non-exempliﬁed positive property implies that self-diﬀerence (or, alternatively, the empty

property) is a positive property.

(∃φ(P(φ)∧ ¬∃x φ(x))) →P(λx (x6=x))

This follows from CORO 2, since such a φwould entail λx (x6=x).

4Monotheism results are of course dependent on the assumed notion of identity. This aspect should be further explored in future

work.

A Simpliﬁed Variant of G¨odel’s Ontological Argument 5

LEMMA 2 A non-exempliﬁed positive property does not exist.

¬∃φ(P(φ)∧ ¬∃x φ(x))

Follows from CORO 1 and the contrapositive of LEMMA 1.

LEMMA 3 Positive properties are exempliﬁed.

∀φ(P(φ)→ ∃x φ(x))

This is just a reformulation of LEMMA 2.

THEOREM 3’ There exists a God-like being.

∃xG(x)

Follows from AXIOM 3 and LEMMA 3.

THEOREM 3 Necessarily, there exists a God-like being.

2∃xG(x)

From THEOREM 3’ by necessitation.

The model ﬁnder nitpick Blanchette and Nipkow, 2010 available in Isabelle/HOL can be employed to verify the

consistency of this simple divine theory. The smallest satisfying model returned by the model ﬁnder consists of one

possible world with one God-like entity, and with self-diﬀerence, resp. the empty property, not being a positive property.

However, the model ﬁnder also tell us that it is impossible to prove CORO:3∃xG(x), expressing that the existence of

a God-like being is possible. The simplest countermodel consists of a single possible world from which no other world

is reachable, so that CORO, i.e. 3∃xG(x), obviously cannot hold for this world, regardless of the truth of THEOREM

3’:∃xG(x) in it. However, the simple transition from the basic modal logic K to the logic KT eliminates this defect.

To reach logic KT, AXIOM T:∀s(2s→s) is postulated, that is, a property holds if it necessarily holds. This postulate

appears uncontroversial. AXIOM T is equivalent to AXIOM T’:∀s(s→3s), which expresses that a property that holds

also possibly holds. Within modal logic KT we can thus obviously prove CORO from THEOREM 3’ with the help of

AXIOM T’.

As an alternative to the above derivation of THEOREM 3, we can also proceed in logic KT analogously to the

argument given in the introduction.

THEOREM 1 Positive properties are possibly exempliﬁed.

∀φ(P(φ)→3∃x φ(x))

Follows from CORO 1,CORO 2 and AXIOM T’.

CORO Possibly there exists a God-like being.

3∃xG(x)

Follows from THEOREM 1 and AXIOM 3.

THEOREM 2 The possible existence of a God-like being implies its necessary existence.

3∃xG(x)→2∃xG(x)

Follows from AXIOM 3,CORO 1 and CORO 2.

THEOREM 3 Necessarily, there exists a God-like being.

2∃xG(x)

Follows from CORO and THEOREM2.

THEOREM 3’ There exists a God-like being.

∃xG(x)

Follows from THEOREM 3 with AXIOM T.

6 Christoph Benzm¨uller

Interestingly, the above simpliﬁed divine theory avoids modal collapse. This is conﬁrmed by the model ﬁnder nitpick,

which reports a countermodel consisting of two possible worlds with one God-like entity.5

The above statements were all formally veriﬁed with Isabelle/HOL. As with Scott’s variant, only possibilist quan-

tiﬁers were used initially, and later the results were conﬁrmed also for a modiﬁed logical setting in which ﬁrst-order

actualist quantiﬁers for individuals were used, and possibilist quantiﬁers otherwise. The Isabelle/HOL sources of the

conducted veriﬁcation studies are presented in Figs. 1-4 in the Appendix.

In the related exploratory studies (Benzm¨uller, 2020), a suitably adapted notion of a modal ultraﬁlter was addi-

tionally used to support the comparative analysis of diﬀerent variants of G¨odel’s ontological argument, including those

proposed by Anderson and Gettings (1996) and Fitting (2002), which avoid modal collapse. These experiments are a

good demonstration of the maturity that modern theorem proving systems have reached. These systems are ready to

fruitfully support the exploration of metaphysical theories.

The development of G¨odel’s ontological argument has recently been addressed by Kanckos and Lethen (2019).

They discovered previously unknown variants of the argument in G¨odel’s Nachlass, whose relation to the presented

simpliﬁed variants should be further investigated in future work. The version No. 2 they reported has meanwhile been

formalised and veriﬁed in Isabelle/HOL, similar to the work presented above. This version No. 2 avoids the notions of

essence and necessary existence and associated deﬁnitions/axioms, just as our simpliﬁed version does. However, this

version, in many respects, also diﬀers from ours, and it assumes a higher-modal modal logic S5.

3 Discussion

Whether the simpliﬁed variant of G¨odel’s ontological argument presented in this paper actually increases or decreases

the argument’s appeal and persuasiveness is a question I would like to pass on to philosophy and theology. As a

logician, I see my role primarily as providing useful input and clarity to promote informed debate.

I have shown how a signiﬁcantly simpliﬁed version of G¨odel’s ontological variant can be explored and veriﬁed in

interaction with modern theorem proving technology. Most importantly, this simpliﬁed variant avoids modal collapse,

and some further issues, which have triggered criticism on G¨odel’s argument in the past. Future work could inves-

tigate the extent to which such theory simpliﬁcation studies could even be fully automated. The resulting rational

reconstructions of argument variants would be very useful in gaining more intuition and understanding of the theory

in question, in this case a theistic theory, which in turn could lead to its demystiﬁcation and also to the identiﬁcation

of ﬂawed discussions in the existing literature.

In future work, I would like to further deepen ongoing studies of Fitting’s (2002) proposal, which works with

intensions rather than extension of (positive) properties.

Acknowledgements: I thank Andrea Vestrucci for valuable comments that helped improve this article.

References

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5In this countermodel, the possible worlds i1 and i2 are reachable from i2, but only world i1 can be reached from i1. Moreover,

there is non-positive property φwhich holds for ein world i2 but not in i1. Apparently, in world i2, modal collapse ∀s(s→2s) is not

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A Simpliﬁed Variant of G¨odel’s Ontological Argument 7

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8 Christoph Benzm¨uller

Appendix: Sources of Conducted Experiments

A Simpliﬁed Variant of G¨odel’s Ontological Argument 9

Figure 1 The universal meta-logical reasoning approach at work: exemplary shallow semantic embedding of modal higher-order logic

K in classical higher-order logic.

A Simpliﬁed Variant of G¨odel’s Ontological Argument 11

Figure 3 Simpliﬁed ontological argument in modal logic K, respectively KT, using possibilist ﬁrst-order and higher-order quantiﬁers.

12 Christoph Benzm¨uller

Figure 4 Simpliﬁed ontological argument in modal logic K, respectively KT, using actualist quantiﬁers ﬁrst-order quantiﬁers and

possibilist higher-order quantiﬁers.