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A (Simplified) Supreme Being Necessarily Exists, says the Computer:
Computationally Explored Variants of G¨
odel’s Ontological Argument
Christoph Benzm¨
uller∗
Freie Universit¨
at Berlin
c.benzmueller@fu-berlin.de
Abstract
An approach to universal (meta-)logical reasoning in classical
higher-order logic is employed to explore and study simpli-
fications of Kurt G¨
odel’s modal ontological argument. Some
argument premises are modified, others are dropped, modal
collapse is avoided and validity is shown already in weak
modal logics Kand T. Key to the gained simplifications of
G¨
odel’s original theory is the exploitation of a link to the no-
tions of filter and ultrafilter in topology.
The paper illustrates how modern knowledge representation
and reasoning technology for quantified non-classical logics
can contribute new knowledge to other disciplines. The con-
tributed material is also well suited to support teaching of
non-trivial logic formalisms in classroom.
1 Introduction
Variants of Kurt G ¨
odel’s (1970), resp. Dana Scott’s (1972),
modal ontological argument have previously been studied
and verified on the computer by Benzm¨
uller and Woltzen-
logel Paleo (2014; 2016) and Benzm¨
uller and Fuenmayor
(2020), and some previously unknown issues were revealed
in these works,1and it was shown that logic KB, instead of
S5, is already sufficient to derive from G¨
odel’s axioms that
a supreme being necessarily exists.
In this paper simplified variants of G¨
odel’s modal onto-
logical argument are explored and assessed. These simpli-
fications have been developed in interaction with the proof
assistant Isabelle/HOL (Nipkow, Paulson, and Wenzel 2002)
and by employing the shallow semantical embedding (SSE)
approach (Benzm¨
uller 2019) as enabling technology. This
technology supports the reuse of automated theorem prov-
ing (ATP) and model finding tools for classical higher-order
logic (HOL) to represent and reason with a wide range
of non-classical logics and theories, including higher-order
modal logics (HOMLs) and G¨
odel’s modal ontological ar-
gument, which are in the focus of this paper.
One of the new, simplified modal arguments is as follows.
The notion of being Godlike (G) is exactly as in G¨
odel’s
∗This paper (without appendix) was accepted for KR 2020.
1E.g., the theorem prover LE O-II detected that G¨
odel’s
(1970) variant of the argument is inconsistent; this inconsistency
had, unknowingly, been fixed in the variant of Scott (1972);
cf. (Benzm¨
uller and Woltzenlogel Paleo 2016) for more details.
original work. Thus, a Godlike entity, by definition, pos-
sesses all positive properties (Pis an uninterpreted constant
denoting positive properties):
Gx≡∀Y.(PY→Y x)
The three only axioms of the new theory which constrain
the interpretation of G¨
odel’s positive properties (P) are:
A1’ Self-difference is not a positive property.2
¬P(λx.x 6=x)
A2’ A property entailed or necessarily entailed by a positive
property is positive.
∀X Y. ((PX∧(XvY∨XVY)) →PY)
A3 The conjunction of any collection of positive properties
is positive.3
∀Z.(Pos Z→ ∀X.(XdZ→PX))
(Technical reading: if Zis any set of positive properties,
then the property Xobtained by taking the conjunction of
the properties in Zis positive.)
In these premises the following defined symbols are used,
where ∀is a possibilist second-order quantifier and where
∀Eis an actualist first-order quantifier for individuals:
XvY≡∀Ez.(X z →Y z)
XVY≡2(XvY)
Pos Z ≡ ∀X.(ZX→PX)
XdZ ≡ 2∀Eu.(X u ↔(∀Y. ZY→Y u))
From A1’,A2’ and A3 it follows, in a few argumenta-
tion steps in modal logic K, that a Godlike entity possibly
and necessarily exists. Modal collapse, which expresses that
there are no contingent truths and which thus eliminates the
2An alternative to A1’ would be: The empty property (λx.⊥) is
not a positive property.
3The third-order formalization of A3 as given here (Zis a third-
order variable ranging over sets of properties), together with the
definition of G, implies that being Godlike is a positive property.
Since supporting this inference is the only role this axiom plays in
the argument, (P G)can be taken—and has been taken, cf. Scott
(1972)—as an alternative to our A3; cf. also Fig. 7.
possibility of alternative possible worlds, does not follow
from these axioms. Also monotheism is not implied. These
observations should render the new theory interesting to the-
oretical philosophy and theology.
Compare the above with G¨
odel’s premises of the modal
ontological argument (in the consistent variant of Scott):
A1 One of a property or its complement is positive.
∀X.(¬(PX)↔P(+X))
A2 A property necessarily entailed by a positive property is
positive.
∀X Y.((PX∧(XVY)) →PY)
A3 The conjunction of any collection of positive properties
is positive (or, alternatively, being Godlike is a positive
property).
∀Z.(Pos Z→ ∀X.(XdZ→PX))
A4 Any positive property is necessarily positive.
∀X.(PX→2(PX))
A5 Necessary existence (N E) is a positive property.
P N E
Furthermore, axiom Bis added to ensure that we operate
in logic KB instead of just K4(remember that logic S5 is not
needed):
∀ϕ.(ϕ→23ϕ)
Essence (E) and necessary existence (N E) are defined as
(other definitions are as before):
EY x ≡Y x ∧ ∀Z.(Z x →(YVZ))
N E x≡∀Y.(EY x →2∃EY)
Informally: Property Yis an essence Eof an entity xif, and
only if, (i) Yholds for xand (ii) Ynecessarily entails every
property Zof x. Moreover, an entity xhas the property
of necessary existence if, and only if, the essence of xis
necessarily instantiated.
Using G¨
odel’s premises as stated it can be proved auto-
matically that a Godlike entity possibly and necessarily ex-
ists (Benzm¨
uller and Woltzenlogel Paleo 2016). However,
modal collapse is still implied even in the weak logic KB.5
Own prior work recently showed that different modal ul-
trafilter properties can be deduced from G¨
odel’s premises
(Benzm¨
uller and Fuenmayor 2020). These insights are key
to the argument simplifications developed and studied in this
paper: If G¨
odel’s premises entail that positive properties
form a modal ultrafilter, then why not turning things around,
and start out with an axiom U1 postulating ultrafilter proper-
ties for P? Then use U1 instead of other axioms for proving
4Symmetry of the accessibility relation rassociated with the
modal 2-operator can be postulated alternatively in our meta-
logical framework.
5For more information on modal collapse (in logic S5) consult
Sobel (1987; 2004), Fitting (2002) and Kovaˇ
c (2012); see also the
references therein.
that a Godlike entity necessarily exists, and on the fly ex-
plore what further simplifications of the argument are trig-
gered. This research plan worked out and it led to simplified
argument variants as presented above and in the remainder.
The proof assistant Isabelle/HOL and its integrated ATP
systems have supported our exploration work surprisingly
well, despite the undecidability and high complexity of the
underlying logic setting. As usual, we here only present the
main steps of the exploration process, and various interest-
ing eureka or frustration steps in between are dropped.
The structure of this paper is as follows: An SSE of
HOML in HOL is introduced in Sect. 2. This section, parts
of which have been adapted from Kirchner et al. (2019), en-
sures that the paper is sufficiently self-contained; readers fa-
miliar with the SSE approach may simply skip it. Modal fil-
ter and ultrafilter are defined in Sect. 3. Section 4 recaps, in
some more detail, the G¨
odel/Scott variant of the modal on-
tological argument from above. Subsequently, an ultrafilter-
based modal ontological argument is presented in Sect. 5.
This new argument is further simplified in Sect. 6, leading
to our proposal based on axioms A1’,A2’ and A3 as presented
before. Further simplifications and modifications are studied
in Sect. 7, and related work is discussed in Sect. 8.
Since we develop, explain and discuss our formal encod-
ings directly in Isabelle/HOL, some familiarity with this
proof assistant and its underlying logic HOL (Andrews
2002; Benzm¨
uller and Andrews 2019) is assumed. The en-
tire sources6of our formal encodings are presented and ex-
plained in detail in this paper.
The contributions of this paper are thus manifold. Be-
sides the novel variants of the modal ontological argument
that we contribute to metaphysics and theology, we demon-
strate how the SSE technique, in combination with higher-
order reasoning tools, can be employed in practical studies
to explore new knowledge. Moreover, we contribute useful
source encodings that can be reused and adapted to teach
quantified modal logics in interdisciplinary lecture courses.
2 Modeling HOML in HOL
Various SSEs of quantified non-classical logics in HOL
have been developed, studied and applied in related work,
cf. Benzm¨
uller (2019) and Kirchner et al. (2019) and the
references therein. These contributions, among others, show
that the standard translation from propositional modal logic
to first-order (FO) logic can be concisely modeled (i.e., em-
bedded) within HOL theorem provers, so that the modal op-
erator 2, for example, can be explicitly defined by the λ-
term λϕ.λw.∀v.(Rwv →ϕv), where Rdenotes the acces-
sibility relation associated with 2. Then one can construct
FO formulas involving 2ϕand use them to represent and
prove theorems. Thus, in an SSE, the target logic is inter-
nally represented using higher-order (HO) constructs in a
theorem proving system such as Isabelle/HOL. Own prior
work developed an SSE that captures quantified extensions
6See logikey.org →Computational-Metaphysics. The experi-
ments reported in this paper were conducted on a standard note-
book (2,5 GHz Intel Core i7, 16 GB memory).
of modal logic (Benzm¨
uller and Paulson 2013). For ex-
ample, if ∀x.φx is shorthand in HOL for Π(λx.φx), then
2∀xP x would be represented as 2Π0(λx.λw.P xw), where
Π0stands for the λ-term λΦ.λw.Π(λx.Φxw), and the 2gets
resolved as above.
To see how these expressions can be resolved to produce
the right representation, consider the following series of re-
ductions:
2∀xP x
≡2Π0(λx.λw.P xw)
≡2((λΦ.λw.Π(λx.Φxw))(λx.λw .P xw))
≡2(λw.Π(λx.(λx.λw.P xw )xw))
≡2(λw.Π(λx.P xw))
≡(λϕ.λw.∀v.(Rwv →ϕv))(λw.Π(λx.P xw))
≡(λϕ.λw.Π(λv.Rwv →ϕv))(λw.Π(λx.P xw))
≡(λw.Π(λv.Rwv →(λw.Π(λx.P xw))v))
≡(λw.Π(λv.Rwv →Π(λx.P xv)))
≡(λw.∀v.Rwv → ∀x.P xv)
≡(λw.∀vx.Rwv →P xv)
Thus, we end up with a representation of 2∀xP x in HOL.
Of course, types are assigned to each (sub-)term of the HOL
language. We assign individual terms (such as variable x
above) the type e, and terms denoting worlds (such as vari-
able wabove) the type i. From such base choices, all other
types in the above presentation can actually be inferred.
An explicit encoding of HOML in Isabelle/HOL, follow-
ing the above ideas, is presented in Fig. 1.7In lines 4–5 the
base types iand eare declared (text passages in red are com-
ments). Note that HOL comes with an inbuilt base type bool,
the bivalent type of Booleans. No cardinality constraints
are associated with types iand e, except that they must be
non-empty. To keep our presentation concise, useful type
synonyms are introduced in lines 6–9. σabbreviates the
type i⇒bool (⇒is the function type constructor in HOL),
and terms of type σcan be seen to represent world-lifted
propositions, i.e., truth-sets in Kripke’s modal relational se-
mantics (Garson 2018). The explicit transition from modal
propositions to terms (truth-sets) of type σis a key aspect
of the SSE technique, and in the remainder of this article
we use phrases such as “world-lifted” or “σ-type” terms to
emphasize this conversion in the SSE approach. γ, which
stands for e⇒σ, is the type of world-lifted, intensional prop-
erties. µand ν, which abbreviate σ⇒σand σ⇒σ⇒σ, are
the types associated with unary and binary modal logic con-
nectives.
The modal logic connectives are defined in lines 12–25.
In line 16, for example, the definition of the world-lifted
∨-connective of type νis given; explicit type information
is presented after the ::-token for ‘c5’, which is the ASCII-
denominator for the (right-associative) infix-operator ∨as
7In Isabelle/HOL explicit type information can often be omitted
due the system’s internal type inference mechanism. This feature
is exploited in our formalization to improve readability. However,
for new abbreviations and definitions, we often explicitly declare
the types of the freshly introduced symbols. This supports a better
intuitive understanding, and it also reduces the number of polymor-
phic terms in the formalization (heavy use of polymorphism may
generally lead to decreased proof automation performance).
Figure 1: SSE of HOML in HOL.
introduced in parenthesis shortly after. ϕσ∨ψσis then de-
fined as abbreviation for the truth-set λw.(ϕσw)∨(ψσw),
respectively. In the remainder we generally use bold-face
symbols for world-lifted connectives (such as ∨) in order to
rigorously distinguish them from their ordinary counterparts
(such as ∨) in meta-logic HOL.
Further modal logic connectives, such as ⊥,>,¬,→,
↔, are introduced analogously. The operator +, introduced
in lines 22–23, is inverting properties of types γ; this oper-
ation occurs in G¨
odel’s axiom A1.=and 6=are defined in
lines 24–25 as world-independent, syntactical (in-)equality.
The world-lifted modal 2-operator is introduced in lines
19–20; accessibility relation Ris now synonymously named
rin infix notation. The definition of 3is analogous.
The world-lifted (polymorphic) possibilist quantifier ∀as
discussed before is introduced in line 28–29. In line 30,
user-friendly binder-notation for ∀is additionally defined.
Instead of distinguishing between ∀and Π0as in our illus-
trating example, ∀-symbols are overloaded here. The intro-
duction of the possibilist ∃-quantifier is analogous.
Additional actualist quantifiers, ∀Eand ∃E, are intro-
duced in lines 36–42. Their definition is guarded by an
explicit, possibly empty, existsAt (@) predicate, which en-
codes whether an individual object actually “exists” at a par-
ticular world, or not. The actualist quantifiers are declared
non-polymorphic, and they support quantification over indi-
viduals only. In the remainder we will indeed apply ∀and ∃
for different types in the type hierarchy of HOL, while ∀E
and ∃Eexclusively quantify over individuals only.
Global validity of a world-lifted formula ψσ, denoted as
bψc, is introduced in line 45 as an abbreviation for ∀wi.ψw.
Consistency of the introduced concepts is confirmed by
the model finder nitpick (Blanchette and Nipkow 2010) in
line 48. Since only abbreviations and no axioms have been
introduced so far, the consistency of the Isabelle/HOL theory
HOML as displayed in Fig. 1 is actually evident.
In line 49–52 it is studied whether instances of the Barcan
and the converse Barcan formulas are implied. As expected,
both principles are valid only for possibilist quantification,
while they have countermodels for actualist quantification.
Lines 54–55 declare some specific parameter settings for
some of the reasoning tools that we employ.
Theorem 1. The SSE of HOML in HOL is faithful (for K).
Proof. Analogous to (Benzm¨
uller and Paulson 2013).
Theory HOML thus models base logic Kin Isabelle/HOL.
Axiom B, see above, can be postulated to arrive at logic KB.
3 Modal Filter and Ultrafilter
Theory MFilter, for “modal filter”, see Fig. 2, imports theory
HOML and adapts the topological notions of filter and ultra-
filter to our modal logic setting. For an introduction to the
notions of filter and ultrafilter see the literature, e.g., (Burris
and Sankappanavar 1981) or also (Odifreddi 2000).
Our notion of modal ultrafilter is introduced in lines
20–21 as a world-lifted characteristic function of type
(γ⇒σ)⇒σ. A modal ultrafilter is thus a world-dependent
set of intensions of γ-type properties; in other words, a σ-
subset of the σ-powerset of γ-type property extensions. A
modal ultrafilter φis defined as a modal filter satisfying an
additional maximality condition: ∀ϕ.ϕ ∈φ∨(−1ϕ)∈φ,
where ∈is elementhood of γ-type objects in σ-sets of γ-
type objects (see lines 3–4), and where −1is the relative set
complement operation on sets of entities (line 11).
A modal filter φ, see lines 14–17, is required to
1. be large: U∈φ, where Udenotes the full set of γ-type
objects we start with,
Figure 2: Definition of filter and ultrafilter (for possible worlds).
2. exclude the empty set: ∅ 6∈ φ, where ∅is the world-lifted
empty set of γ-type objects,
3. be closed under supersets: ∀ϕ ψ.(ϕ∈φ∧ϕ⊆ψ)→
ψ∈φ(the world-lifted ⊆-relation is defined in lines 7–
8), and
4. be closed under intersections: ∀ϕ ψ .(ϕ∈φ∧ψ∈φ)→
(ϕuψ)⊆φ(where uis defined in lines 9–10).
Own prior work (Benzm¨
uller and Fuenmayor 2020) stud-
ied two different notions of modal ultrafilter (termed γ- and
δ-ultrafilter), which are defined on intensions and extensions
of properties, respectively. This distinction is not needed in
this paper; what we call modal ultrafilter here corresponds
to our prior notion of γ-ultrafilter.
4 G¨
odel/Scott Variant
We start out in Fig. 3 with the introduction of some basic
abbreviations and definitions for the G¨
odel/Scott variant of
the modal ontological argument. This theory file, which is
termed BaseDefs and which imports HOML, is reused with-
out modification also in all other variants as explored in this
paper later on. In line 4 the uninterpreted constant symbol
P, for “positive properties”, is declared; it has type γ⇒σ.
Pthus denotes an intensional, world-depended concept. In
lines 7–11 abbreviations for the previously discussed rela-
tions and predicates v,V,dand Pos are introduced. In
lines 14–15, G¨
odel’s notion of “being Godlike” (G) is de-
fined, and in lines 18–21 the previously discussed definitions
for Essence (E) and Necessary Existence (N E ) are given.
The full formalization of Scott’s variant of G¨
odel’s argu-
ment is presented as theory ScottVariant in Fig. 4. This the-
ory imports and relies on the previously introduced notions
from theory files HOML,MFilter and BaseDefs.
The premises of G¨
odel’s argument, as already discussed
Figure 3: Definitions for all variants discussed in the remainder.
earlier, are stated in lines 4–10. In line 12 a semantical coun-
terpart B’ (symmetry of the accessibility relation rassociated
with the 2-operator) of the Baxiom is proved.
An abstract level “proof net” for theorem T6, the neces-
sary existence of a Godlike entity, is presented in lines 15–
25. Following the literature, the proof goes as follows: From
A1 and A2 infer T1: positive properties are possibly exem-
plified. From A3 and the defn. of Gobtain T2: being Godlike
is 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 Gand Eto infer T4: be-
ing Godlike is an essential property of any Godlike entity.
From this, A5,B’ and the defns. of G, and N E have T5: the
possible existence of a Godlike entity implies its necessary
existence. T5 and T3 then imply T6.
The five subproofs and their dependencies have been
automatically proved using ATP systems integrated with
Isabelle/HOL via its sledgehammer tool (Blanchette et
al. 2013); sledgehammer identified and returned the abstract
level proof justifications as displayed here, e.g. “using T1 T2
by simp”. The mentioned proof engines/tactics blast,metis,
and simp are trustworthy components of Isabelle/HOL’s,
since they internally reconstruct and check each (sub-)proof
in the proof assistants small and trusted proof kernel. The
smt method, which relies on an external satisfiability mod-
ulo solver (CVC4 in our case), is less trusted, but we never-
theless use it here since it was the only Isabelle/HOL method
that was able to close this subproof in a single step (we want
to avoid displaying longer interactive proofs due to space re-
strictions). Using the defns. from Sect. 2, one can generally
reconstruct and verify all presented proofs with pen and pa-
per directly in meta-logic HOL. Moreover, reconstruction
of modal logic proofs from such proof nets within direct
proof calculi for quantified modal logics, cf. Kanckos and
Woltzenlogel Paleo (2017) or Fitting (2002) is also possible.
The presented theory is consistent, which is confirmed in
line 31 by model finder nitpick;nitpick reports a model con-
sisting of one world and one Godlike entity. Figure 4: G ¨
odel’s modal ontological argument; Scott’s variant.
Figure 5: Ultrafilter variant.
Validity of modal collapse (MC) is confirmed in lines 34–
42; a proof net displaying the proofs main idea is shown.
Most relevant for this paper is that the ATP systems were
able to quickly prove that G¨
odel’s notion of positive proper-
ties Pconstitutes a modal ultrafilter, cf. lines 45–55. This
was key to the idea of taking the modal ultrafilter property
of Pas an axiom U1 of the theory; see the next section.
In lines 57–60 some further relevant lemmata are proved.
And in line 62–70 we hint at a much simpler, alternative
proof argument: Take the set HF Gof all supersets of G;
it follows from G¨
odel’s theory that this set is a modal fil-
ter (line 66) resp. modal ultrafilter (line 67), i.e., HF Gis
the modal Hauptfilter of G. The necessary existence of a
Godlike entity now becomes a simple corollary of this result
(see line 70, where T6Again is proved exclusively from F1).
In Sect. 7 we will later present an argument variant that is
based on this observation.8
5 Ultrafilter Variant
Taking U1 (Pis an ultrafilter) as an axiom for G¨
odel’s the-
ory in fact leads to a significant simplification of the modal
ontological argument; this is shown in lines 10–17 of the
theory file UFilterVariant in Fig. 5: not only G ¨
odel’s axiom
A1 can be dropped, but also axioms A4 and A5, together with
defns. Eand N E . Even logic KB can be given up, since K
is now 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 and
defn. of Ghave T2 (being Godlike is a positive property).
T1 and T2 imply T3 (a Godlike entity possibly exists). From
U1,A2,T2 and the defn. of Ghave T5 (possible existence of
a Godlike entity implies its necessary existence). Use T5 and
T3 to conclude T6 (necessary existence of a Godlike entity).
8Manfred Droste from the University of Leipzig pointed me to
this proof argument alternative in an email communication.
Consistency of the theory is confirmed in line 20; again a
model with one world and one Godlike entity is reported.
Most interestingly, modal collapse MC now has a simple
countermodel as nitpick informs us in line 23. This counter-
model consists of a single entity e1and two worlds i1and
i2with accessibility relation r={hi1, i1i,hi2, i1i,hi2, i2i}.
Trivially, formula Φis such that Φholds in i2but not in i1,
which invalidates MC at world i2.e1is the Godlike entity in
both worlds, i.e., Gis the property that holds for e1in i1and
i2, which we may denote as λe.λw.e=e1∧(w=i1∨w=i2).
Using tuple notation we may write G={he1, i1i,he1, i2i}.
Remember that P, which is of type γ⇒σ, is an inten-
sional, world-depended concept. In our countermodel for
MC in line 23 the extension of Pfor world i1has the above
Gand λe.λw.e=e1∧w=i1as its elements, while in world
i2we have Gand λe.λw.e=e1∧w=i2. Using tuple notation
we may note Pas
{h{he1, i1i,he1, i2i}, i1i,h{he1, i1i}, i1i,
h{he1, i1i,he1, i2i}, i2i,h{he1, i2i}, i2i}
In order to verify that Pis a modal ultrafilter we have to
check whether the respective modal ultrafilter conditions are
satisfied in both worlds. U∈Pin i1and also in i2, since
both h{he1, i1i,he1, i2i}, i1iand h{he1, i1i,he1, i2i}, i2iare
in P;∅ 6∈ Pin i1and also in i2, since both h{}, i1iand
h{}, i2iare not in P. It is also easy to verify that Pis closed
under supersets and intersection in both worlds.
Note that in our countermodel for MC, also G¨
odel’s axiom
A4 is invalidated. Consider X=λe.λw.e=e1∧w=i2, i.e.,
Xis true for e1in i2, but false for e1in i1. We have PX
in i2, but we do not have 2(PX)in i1, since PXdoes not
hold in i1, which is reachable in rfrom i2.
nitpick is capable of computing different partial modal ul-
trafilters as part of its countermodel exploration: out of 512
candidates, nitpick identifies 32 structures of form hF, ii, for
i∈ {i1, i2}, in which Fsatisfies the ultrafilter conditions in
the specified world i. An example for such an hF, iiis
h{h{he1, i1i}, i1i,h{}, i1i,
h{he1, i1i,he1, i2i}, i2i,h{he1, i2i}, i2i}, i2i
Fis not a proper modal ultrafilter, since Ffails to be an
ultrafilter in world i1.
6 Simplified Variant
What modal ultrafilters properties of Pare actually needed
to support T6? Which ones can be dropped? Experiments
with our framework, as displayed in theory file SimpleVari-
ant in Fig. 6, confirm that only the filter conditions from
Sect. 3 must be upheld for P; maximality can be dropped.
However, it is possible to merge filter condition 3 (closed
under supersets) for Pwith G¨
odel’s A2 into axiom A2’ as
shown in line 6 of Fig. 6. Moreover, instead of requiring
that the empty set ∅=λx.⊥must not be a positive prop-
erty, we postulate that self-difference λx.x6=xis not (line
5); note that self-difference is extensionally equal to ∅. Self-
identity and self-difference have been used frequently in the
history of the ontological argument, which is part of the mo-
Figure 6: Simplified variant.
tivation for this switch. As intended, filter condition 4 is now
implied by the theory (see theorem F1 proved in line 31), as
well as positiveness of self-identity (line 10). The essential
idea of the theory SimpleVariant in Fig. 6 is to show that it
actually suffices, in combination with A3, to postulate that P
is a modal filter, and this is what our simplified axioms do.
From the definition of Gand the axioms A1’,A2’ and A3
(lines 5–7) theorem T6 immediately follows: in line 13 sev-
eral theorem provers integrated with sledgehammer quickly
report a proof (≤1sec). Moreover, a more detailed and more
intuitive “proof net” is presented in lines 14–18; the proof
argument is analogous to what has been discussed before.
In lines 21–23, countermodels for modal collapse MC
(similar to the one discussed before) and for monotheism
MT are reported.
Further questions are answered experimentally (lines 26–
38): neither A1, nor A4 or A5, of the premises we dropped
from G¨
odel’s theory are implied anymore, all have counter-
models. In lines 31–32 we see that Pis still a filter, but
not an ultrafilter. Since some of these axioms, e.g. G¨
odel’s
strong A1, have been discussed controversially in the history
of G¨
odel’s argument, and since MC and MT are independent,
we have arrived at a philosophically and theologically po-
tentially relevant simplification of G¨
odel’s work.
7 Further Simplified Variants
Postulating T2 instead of A3 Instead of working with
third-order axiom A3 to infer T2 as in theory SimpleVariant,
Figure 7: Simplified variant with axiom T2.
Figure 8: Simplified variant with simple entailment in logic K.
we directly postulate T2 as an axiom in theory SimpleVari-
antPG; cf. the new axiom T2 in line 7 of in Fig. 7.
Theorem T6 can be proved essentially as before (lines 10–
15), and MC and MT still have countermodels (lines 20–22).
Simple Entailment in Axiom A2’ Instead of using a dis-
junction of simple entailment and necessary entailment in
axiom A2’ we may in fact only require simple entailment
in A2’; see axiom A2” (line 6) of the theory file SimpleVari-
antSE displayed in Fig. 8. Proofs for T6, the necessary exis-
tence of a Godlike entity, and also T7, existence of a Godlike
entity, can still be quickly found (lines 10–11).
However, after replacing A2’ by A2”,T3 (the possible ex-
istence of a Godlike entity) is no longer implied; see line 14.
As nitpick informs us, T3 now has an undesired counter-
model consisting of one single world that is not connected to
itself. By assuming modal axiom T(what is necessary true is
true in the given world) this countermodel can be eliminated
so that T3 is implied as desired (lines 16–18).
Figure 9: Simplified variant with simple entailment in logic T.
Simple Entailment in Logic T The above discussion mo-
tivates a further alternative of the simplified modal ontolog-
ical argument; see theory file SimpleVariantSEinT in Fig. 9.
This argument is assuming modal logic T(which comes
with axiom Tas discussed above), and, as before, it pos-
tulates axioms A1’ A2” and T2 (lines 5–7):
A1’ Self-difference is not a positive property.
A2” A property entailed by a positive property is positive.
T2 Being Godlike is a positive property.
One possible proof argument for T6 is as before; see lines
15–19. However, there is also a much simpler proof, see
lines 23–30, which we explain in more detail (this simple
proof is applicable to previous variants as well; it is also
key to proving T5 from A1’,A2’/A2” and T2/A3 in previous
variants, including the simplified variant in Fig. 6):9
L1 The existence of a non-exemplified positive property
implies that self-difference is a positive property—This
follows from axiom A2”.
L2 There is no non-exemplified positive property—From
L1 and axiom A1’.
T1’ Positive properties are exemplified—Equivalent to L2.
T3’ A Godlike entity exists—From T1’ and axiom T2.
L3 A Godlike entity possibly exists—From T3’ and T’ (con-
trapositive of axiom T); note that L3 is not needed to ob-
tain T6 in the next step; generally, axiom T(resp. its con-
trapositive T’) is only needed for deriving T3/L3.
9The existence of such an unintended derivation was already
hinted at by Fuenmayor and Benzm¨
uller (2017, Footn. 11).
Figure 10: Hauptfiltervariant.
T6 A Godlike entity necessarily exists—From T3’ by neces-
sitation.
Hauptfiltervariant Another drastically simplified variant
of the modal ontological argument is related to the obser-
vations discussed earlier at the end of Sec. 4. There it has
been shown that the set HF G, consisting of all supersets of
G, is a modal filter (so that HF Gis the Hauptfilter of G); this
then directly implies the necessary existence of a Godlike
entity. A new variant based on this observation is presented
in theory file SimpleVariantHF in Fig. 10. Here the set of all
supersets HF Gof property G(being Godlike) is postulated
to be a modal filter (axiom F1 in line 7).10 The existence and
necessary existence of a Godlike entity then directly follows
from F1, see lines 10–16. And as already observed before,
the possible existence of a Godlike entity is independent, but
can be enforced by postulating modal axiom T(lines 19–
24). Moreover, the theory is consistent (line 26) and neither
modal collapse nor monotheism are implied (lines 29–31).
8 Related Work
Fitting (2002) suggested to carefully distinguish between in-
tensions and extensions of positive properties in the con-
text of G¨
odel’s modal ontological argument, and, in order
to do so within a single framework, he introduced a suf-
ficiently expressive HOML enhanced with means for the
explicit representation of intensional terms and their exten-
sions; cf. also the intensional operations used by Fuenmayor
10In fact, the only essential requirement that is enforced here is
that ∅is not in HF G, hence Gcannot be identical to ∅.
and Benzm¨
uller (2017; 2020) in their formal study of the
works of Fitting and Anderson (1990; 1996).
The application of computational methods to philosoph-
ical problems was initially limited to first-order theorem
provers. Fitelson and Zalta (2007) used PROVE R9 to find
a proof of the theorems about situation and world theory
in (Zalta 1993) and they found an error in a theorem about
Plato’s Forms that was left as an exercise in (Pelletier and
Zalta 2000). Oppenheimer and Zalta (2011) discovered, us-
ing Prover9, that one of the three premises used in their
reconstruction of Anselm’s ontological argument (Oppen-
heimer and Zalta 1991) was sufficient to derive the conclu-
sion. The first-order conversion techniques that were devel-
oped and applied in these works are outlined in some detail
in related work by Alama, Oppenheimer and Zalta (2015).
More recent related work makes use of higher-order
proof assistants. Besides the already mentioned own prior
work with colleagues, this includes Rushby’s (2018) study
on Anselm’s ontological argument in the PVS system and
Blumson’s (2017) related study in Isabelle/HOL. Related is
also includes Odifreddi’s (2000) discussion on ultrafilters,
dictators and God, which I was pointed to by a reviewer.
The development of G¨
odel’s ontological argument has re-
cently 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.11
9 Discussion
The simplifications of G¨
odel’s theory as presented are far-
reaching. In fact, one may ask “Is this really what G¨
odel had
in mind?”, or are there some technical issues, such as the
alternative proof in lines 22–30 from Fig. 9, that have been
ignored so far? Moreover, is the definition of property entail-
ment (v) really adequate in the context of the modal onto-
logical argument, or shouldn’t this definition be replaced by
concept containment, so that self-difference, resp. the empty
property, would no longer v-entail any other property?12
Moreover, assuming that the simplified theory SimpleVari-
ant from Fig. 6 is indeed still in line with what G¨
odel had in
mind, why not presenting the definitions and axioms using
an alternative wording, for example, as follows (where we
replace “positive property” by “rational/consistent property”
and “Godlike entity” by “maximally-rational entity”):
GAn entity x is maximally-rational (G) iff it has all ratio-
nal/consistent properties.
11An email discussion (March, 2020) with Tim Lethen revealed
the following: In particular version No. 2 of G¨
odel’s argument as
presented in (Kanckos and Lethen 2019) appears related, though
not equivalent, to our simplified versions. Version No. 2 –which we
have meanwhile formalized and verified in Isabelle/HOL – avoids
the notions of essence and necessary existence and associated def-
initions/axioms, just as our simplified versions do. Moreover, their
findings also suggest that instead of axiom A1’ we may just postu-
late that a non-positive property exists (and experiments confirm
this claim; A1’ is then implied). However, I prefer axiom A1’.
12See also (Alama, Oppenheimer, and Zalta 2015) and (Fuen-
mayor and Benzm¨
uller 2017, Footn. 11).
A1’ Self-difference is not a rational/consistent property.
A2’ A property entailed or necessarily entailed by a ratio-
nal/consistent property is rational/consistent.
T2 Maximal-rationality is a rational/consistent property.
It follows: A maximally-rational entity necessarily exists.
It would still be possible, but not mandatory, to under-
stand a maximally-rational being also as a Godlike being.
Independent of this discussion we expect the Is-
abelle/HOL theory files we contributed to be useful for
teaching quantified modal logics in classroom, as previously
demonstrated in our awarded lecture course on “Compu-
tational Metaphysics” (Wisniewski et al. 2016). The de-
veloped corpus of example problems is furthermore suited
as a benchmark for other ambitious knowledge represen-
tation and reasoning projects in the KR community: Can
the alternative approaches encode such metaphysical argu-
ments as well? How about their proof automation capa-
bilities and how about model finding? Description logics
or argumentation theory, for example, due to their limited
expressivity, appear unsuited to support such ambitious ap-
plications, while the techniques presented here demonstra-
bly scale for the encoding and assessment of other ambi-
tious theories in metaphysics (Kirchner, Benzm¨
uller, and
Zalta 2020) and mathematics (Benzm¨
uller and Scott 2020;
Tiemens et al. 2020). Moreover, our problem set constitutes
an interesting benchmark for other HOL automated theorem
provers and should therefore be converted into TPTP THF
representation (Sutcliffe and Benzm¨
uller 2010) and be used
in theorem prover competitions.
10 Conclusion
G¨
odel’s modal ontological argument stands in prominent
tradition of western philosophy. It has its roots in the Proslo-
gion of Anselm of Canterbury and it has been picked up
in Descartes’ Fifth Meditation and in the works of Leibniz,
which in turn inspired and informed the work of G¨
odel.
In this paper we have linked G¨
odel’s theory to a suit-
ably adapted mathematical theory (modal filter and modal
ultrafilter), and subsequently we have developed simplified
modal ontological arguments which avoid some of G¨
odel’s
axioms and consequences, including modal collapse, that
have led to criticism in the past. At the same time the of-
fered simplifications are very far reaching, eventually too
far. Anyhow, the insights that were presented in this paper
appear relevant to inform ongoing studies of the modal on-
tological argument in theoretical philosophy and theology.
We have in this paper applied modern symbolic AI tech-
niques to arrive at deep, explainable and verifiable models of
the metaphysical concepts we are interested in. In particu-
lar, we have illustrated how state of the art theorem proving
systems, in combination with latest knowledge representa-
tion and reasoning technology, can fruitfully be employed to
explore and contribute deep new knowledge to other disci-
plines.
Acknowledgements
I am grateful to friends and colleagues who have supported
this line of research in the past and/or directly contributed
with comments and suggestions, including (in alphabetical
order) C. Brown, M. Droste, D. Fuenmayor, T. Gleißner, D.
Kirchner, S. Kovaˇ
c, T. Lethen, X. Parent, R. Rojas, D. Scott,
A. Steen, L. van der Torre, E. Weydert, D. Streit, B. Woltzen-
logel Paleo, E. Zalta. Moreover, I am grateful to the review-
ers of this paper for useful comments and for the pointer to
P. Odifreddi’s paper.
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Figure 11: Formal study of “version No.2” of G ¨
odel’s argument as reported by Kanckos and Lethen (2019).
