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Abstract

A simplified variant of Gödel'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 Gödel. 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 Gödel'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. (Article to appear in: A. Vestrucci (ed.), Beyond Babel: Religions in a Linguistic Pluralism. Cham: Springer Nature, Series: Sophia Studies in Cross- Cultural Philosophy of Traditions and Cultures, 2023)
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
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(st)(2s2t) (if simplies tholds necessarily, then the
arXiv:2202.06264v1 [cs.LO] 13 Feb 2022
2 Christoph Benzm¨uller
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(s2s), 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)2y(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 2x 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.
φψ((P(φ)(2x(φ(x)ψ(x)))) P(ψ))
necessity of simplies the necessity of t). Modal logic KT additionally provides the T axiom: 2ss(if sholds necessarily, then s),
respectively its dual s3s(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(φ23φ), or in words, if φthen φis necessarily possible.
THEOREM 1 Positive properties are possibly exemplified.
φ(P(φ)3x φ(x))
Follows from AXIOM 1 and AXIOM 2.
CORO Possibly there exists a God-like being.
3xG(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.
2xG(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(XdZPX)), where Pos Zstands for X(ZXPX) and XdZis shorthand for 2u.(X u (Y. ZYY 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.
φψ((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.
φψ((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.
2xG(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:3xG(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. 3xG(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(2ss) is postulated, that is, a property holds if it necessarily holds. This postulate
appears uncontroversial. AXIOM T is equivalent to AXIOM T’:s(s3s), 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)3x φ(x))
Follows from CORO 1,CORO 2 and AXIOM T’.
CORO Possibly there exists a God-like being.
3xG(x)
Follows from THEOREM 1 and AXIOM 3.
THEOREM 2 The possible existence of a God-like being implies its necessary existence.
3xG(x)2xG(x)
Follows from AXIOM 3,CORO 1 and CORO 2.
THEOREM 3 Necessarily, there exists a God-like being.
2xG(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.
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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,
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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.
10 Christoph Benzm¨uller
Figure 2 Verification of Scott’s variant of G¨odel’s ontological argument in modal higher-order logic KB, using first-order and higher
-order possibilistic quantifiers; the theory HOML from Fig. 1 is imported.
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.
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