PreprintPDF Available

Computer-supported Analysis of Positiive Properties, Ultrafilters and Modal Collapse in Variants of Gödel's Ontological Argument

Authors:

Abstract and Figures

Three variants of Kurt Gödel's ontological argument, as proposed byDana Scott, C. Anthony Anderson and Melvin Fitting, are encoded and rigorously assessed on the computer. In contrast to Scott's version of Gödel's argument, the two variants contributed by Anderson and Fitting avoid modal collapse. Although they appear quite different on a cursory reading, they are in fact closely related, as our computer-supported formal analysis (conducted in the proof assistant system Isabelle/HOL) reveals. Key to our formal analysis is the utilization of suitably adapted notions of (modal) ultrafilters, and a careful distinction between extensions and intensions of positive properties.
Content may be subject to copyright.
Christoph Benzm¨uller and David Fuenmayor
COMPUTER-SUPPORTED ANALYSIS OF POSITIVE
PROPERTIES, ULTRAFILTERS AND MODAL COLLAPSE
IN VARIANTS OF G ¨
ODEL’S ONTOLOGICAL ARGUMENT
Abstract
Three variants of Kurt odel’s ontological argument, proposed by Dana Scott,
C. Anthony Anderson and Melvin Fitting, are encoded and rigorously assessed
on the computer. In contrast to Scott’s version of odel’s argument the two
variants contributed by Anderson and Fitting avoid modal collapse. Although
they appear quite different on a cursory reading they are in fact closely related.
This has been revealed in the computer-supported formal analysis presented in
this article. Key to our formal analysis is the utilization of suitably adapted
notions of (modal) ultrafilters, and a careful distinction between extensions and
intensions of positive properties.
1. Introduction
The premises of the variant of the modal ontological argument [20] which
was found in Kurt odel’s “Nachlass” are inconsistent; this holds already
in base modal logic K[11, 9]. The premises of Scott’s [28] variant of odel’s
work, in contrast, are consistent [9, 11], but they imply the modal collapse,
ϕ2ϕ, which has by many philosophers been considered an undesirable
side effect; cf. Sobel [30] and the references therein.1
In this article we formally encode and analyze, starting with Scott’s
variant, two prominent further emendations of odel’s work both of which
successfully avoid modal collapse. These two variants have been con-
tributed by C. Anthony Anderson [1, 2] and Melvin Fitting [16], and on a
cursory reading they appear quite different. Our formal analysis, however,
1The modal collapse was already noted by Sobel [29, 30]. One might conclude from
it, that the premises of odel’s argument imply that everything is determined, or alter-
natively, that there is no free will. Sre´cko Kovc [25] argues that modal collapse was
eventually intended by odel.
arXiv:1910.08955v2 [cs.LO] 13 Jan 2020
2
shows that from a certain mathematical perspective they are in fact closely
related.
Two notions are particularly important in our analysis. From set the-
ory, resp. topology, we borrow and suitably adapt, for use in our modal
logic context, the notion of ultrafilter and apply it in two different ver-
sions to the set of positive properties. From the philosophy of language
we adopt the distinction between intensions and extensions of (positive)
properties. Such a distinction has been suggested already by Fitting in his
book “Types, Tableaus and odel’s God” [16], which we take as a starting
point in our formalization work.
Utilizing these notions, and extending Fitting’s analysis, the modi-
fications as introduced by Anderson and Fitting to odel’s concept of
positive properties are formally studied and compared. Our computer-
supported analysis, which is carried out in the proof assistant system Is-
abelle/HOL [27], is technically enabled by the universal logical reason-
ing approach [4], which exploits shallow semantical embeddings (SSEs) of
various logics of interest—such as intensional higher-order modal logics
(IHOML) in the present article—in Church’s simple type theory [5], aka.
classical higher-order logic (HOL). This approach enables the reuse of ex-
isting, interactive and automated, theorem proving technology for HOL to
mechanize also non-classical higher-order reasoning.
Some of the findings reported in this article have, at an abstract level,
already been summarized in the literature before [24, 6, 17], but they have
not been published in full detail yet (for example, the notions of “modal”
ultrafilters, as employed in our analysis, have not been made precise in
these papers). This is the contribution of this article.
In fact, we present and explain in detail the SSE of intensional higher-
order modal logic (IHOML) in HOL (§3.1), the encoding of different types
of modal filters and modal ultrafilters in HOL (§3.2), and finally the en-
coding and analysis of the three mentioned variants of odel’s ontological
argument in HOL utilizing the SSE approach (§4, §5 and §6). We start out
(§2) by pointing to related prior work and by outlining the SSE approach.
2. Prior Work and the SSE Approach
The key ideas of the shallow semantical embedding (SSE) approach, as
relevant for the remainder of this article, are briefly outlined. This section is
intended to make the article sufficiently self-contained and to give references
3
to related prior work. The presentation in this section is taken and adapted
from a recently published related article [24, §1.1]; readers already familiar
with the SSE approach may simply skip it, and those who need further
details may consult further related documents [7, 4].
Earlier papers, cf. [4] and the references therein, focused on the de-
velopment of SSEs. These papers show that the standard translation
from propositional modal logic to first-order logic can be concisely mod-
eled (i.e., embedded) within higher-order theorem provers, so that the
modal operator 2, for example, can be explicitly defined by the λ-term
λϕ.λw.v.(Rwv ϕv), where Rdenotes the accessibility relation asso-
ciated with 2. Then one can construct first-order formulas involving 2ϕ
and use them to represent and proof theorems. Thus, in an SSE, the target
logic is internally represented using higher-order constructs in a proof assis-
tant system such as Isabelle/HOL. The first author, in collaboration with
Paulson [7], developed an SSE that captures quantified extensions of modal
logic (and other non-classical logics). For example, if x.φx is shorthand in
higher-order logic (HOL) for Π(λx.φx), then 2xP 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 described above.
To see how these expressions can be resolved to produce the right rep-
resentation, consider the following series of reductions:
2xP 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 2xPx in HOL. Of course, types
are assigned to each term of the HOL language. More precisely, in the
SSE presented in Fig. 1, we will assign individual terms (such as variable
xabove) the type e, and terms denoting worlds (such as variable wabove)
the type i. From such base choices, all other types in the above presenta-
tion can be inferred. While types have been omitted above, they will often
be given in the remainder of this article.
4
The SSE technique provided a fruitful starting point for a natural en-
coding of odel’s ontological argument in second-order modal logics S5
and KB [9]. Initial studies investigated odel’s and Scott’s variants of
the argument within the higher-order automated theorem prover (hence-
forth ATP) LEO-II [8]. Subsequent work deepened these assessment stud-
ies [11, 12]. Instead of using LEO-II, these studies utilized the higher-order
proof assistant Isabelle/HOL, which is interactive and which also supports
strong proof automation. Some experiments were also conducted with the
proof assistant Coq [10]. Further work (see the references in [24, 4]) con-
tributed a range of similar studies on variants of the modal ontological
argument that have been proposed by Anderson [1], Anderson and Get-
tings [2], ajek [21, 22, 23], Fitting [16], and Lowe [26]. Particularly rele-
vant for this article is some prior formalization work by the authors that
has been presented in [18, 17]. The use of ultrafilters to study the distinc-
tion between extensional and intensional positive properties in the variants
of Scott, Anderson and Fitting has first been mentioned in invited keynotes
presented at the AISSQ-2018 [6] and the FMSPh-2019 [3] conferences.
3. Further Preliminaries
The formal analysis in this article takes Fitting’s book [16] as a starting
point; see also [18, 17]. Fitting suggests to carefully distinguish between
intensions and extensions of positive properties in the context of odel’s
argument, and, in order to do so within a single framework, he introduces
a sufficiently expressive higher-order modal logic enhanced with means for
the explicit representation of intensional terms and their extensions, which
we have termed intensional higher-order modal logic (IHOML) in previous
work [17]. The SSE of IHOML in HOL, that we utilize in the remainder
of this article, is presented in §3.1. Notions of ultrafilters on sets of in-
tensions, resp. extensions, of (positive) properties are then introduced in
§3.2. Since we develop, explain and discuss our formal encodings directly
in Isabelle/HOL [27], some familiarity with this proof assistant and its
background logic HOL [5] is assumed.
3.1. Intensional Higher-Order Modal Logic in HOL
An encoding of IHOML in Isabelle/HOL utilizing the SSE approach, is
presented in Fig. 1. It starts in line 3 with the declaration of two base
5
Fig. 1. Shallow semantical embedding of IHOML in HOL.
types in HOL as mentioned before: type istands for possible worlds and
type efor entities/individuals. To keep the encoding concise some type
synonyms are introduced in lines 4–7, which we explain next.
δand σabbreviate the types of predicates ebool and ibool, re-
spectively. Terms of type δrepresent (extensional) properties of individ-
6
uals. Terms of type σcan be seen to represent world-lifted propositions,
i.e., truth-sets in Kripke’s modal relational semantics [19]. Note that the
explicit transition from modal propositions to terms (truth-sets) of type σ
is a key aspect in SSE approach; see the literature [4] for further details. In
the remainder of this article we make use of phrases such as “world-lifted”
or σ-type” terms to emphasize this conversion in the SSE approach.
τ, which abbreviates the type iibool, stands for the type of ac-
cessibility relations in modal relational semantics, and γ, which stands for
eσ, is the type of world-lifted, intensional properties.
In lines 8–32 in Fig. 1 the modal logic connectives are introduced. For
example, in line 15 we find the definition of the world-lifted -connective
(which is of type σσσ; type information is given here explicitly after the
::-token for mor’, which is the ASCII-denominator for the infix-operator
as introduced in parenthesis shortly after). ϕσψσis defined as abbre-
viation for the truth-set λwiσwiψσwi(i.e., is associated with the
lambda-term λϕσ.λψσ.λwiσwiψσwi). 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 the meta-logic HOL.
The world-lifted ¬-connective is introduced in line 11, and >in lines
9–10, and respective further abbreviations for conjunction, implication and
equivalence are given in lines 14, 16 and 17, respectively. The operators +
and +, introduced in lines 12 and 13, negate properties of types δand γ,
respectively; these operations occur in the premises in the works of Scott,
Anderson and Fitting which govern the definition of positive properties.
As we see in Fig. 1, types can often be omitted in Isabelle/HOL due to
the system’s internal type inference mechanism. This feature is exploited
in our formalization to some extend to improve readability. However, for
all new abbreviations and definitions, we always explicitly declare the types
of the freshly introduced symbols; this not only supports a better intuitive
understanding of these notions but also reduces the number of polymor-
phic terms in the formalization (since polymorphism may generally cause
decreased proof automation performance).
The world-lifted modal 2-operator and the polymorphic, world-lifted
universal quantifier , as already discussed in §2, are introduced in lines
31 and 19, respectively (the ’a in the type declaration for represents a
type variable). In line 20, user-friendly binder-notation for is additionally
defined. In addition to the (polymorphic) possibilist quantifiers, and ,
7
defined this way in lines 19–22, further actualist quantifiers, Eand E,
are introduced in lines 24–28; their definition is guarded by an explicit,
possibly empty, existsAt predicate, which encodes whether an individual
object actually “exists” at a particular given world, or not. These ad-
ditional actualist quantifiers are declared non-polymorphic, so that they
support quantification over individuals only. In the subsequent analysis of
the variants of odel’s argument, as contributed by Scott, Anderson and
Fitting, we will indeed use and for different types in the type hierarchy
of HOL, while keeping Eand Efor quantification over individuals only.
The notion of global validity of a world-lifted formula ψσ, denoted as
bψc, is introduced in line 34 as an abbreviation for wi.ψw.
Note that an (intensional) base modal logic Kis introduced in the
theory IHOML (Fig. 1). In later sections we will switch to logics KB and S5
by postulating respective conditions (symmetry, and additionally reflexivity
and transitivity) on the accessibility relation r.
In lines 35–46 some further abbreviations are declared, which address
the mediation between intensions and extensions of properties. World-lifted
propositions and intensional properties are modeled as terms of types σ
and γrespectively, i.e., they are technically handled in HOL as functions
over worlds whose extensions are obtained by applying them to a given
world win context. The operation LϕMin line 37 is trivially converting
a world-independent proposition of Boolean type into a rigid world-lifted
proposition of type σ; the rigid world-lifted propositions obtained from this
trivial conversion have identical evaluations in all worlds.
The -operator in line 40, which is of type (γσ)γσ, is slightly
more involved. It evaluates its second argument, which is a property Pof
type γ, for a given world w, and it then rigidly intensionalizes the obtained
extension of Pin w. For technical reasons, however, is introduced as
a binary operator, with its first argument being a world-lifted predicate
ϕγσthat is being applied to the rigidly intensionalized Pγ; in fact, all
occurrences of the -operator in our subsequent sections will have this
binary pattern.
The lemma statement in line 41 confirms that intensional properties Pγ
are generally different from their rigidly intensionalized counterparts Pγ:
Isabelle/HOL’s model finder Nitpick [14] generates a countermodel to the
claim that they are (Leibniz-)equal.
A related (non-bold) binary operator , of type (δσ)γσ, is in-
troduced in line 44. Its first argument is a predicate ϕδσapplicable to
8
Fig. 2. Definition of δ-Filters and δ/γ-Ultrafilters.
extensions of properties, and its second argument is an intensional prop-
erty. The -operator evaluates its second argument Pγin a given world w,
thereby obtaining an extension Pγof type δ, and then it applies its first ar-
gument ϕδσto this extension. The 1-operator is analogous, but its first
argument ϕis now of type δγ, which can be understood as world-lifted
binary predicate whose first argument is of type δand its second argument
of type e. The 1-operator evaluates the intensional argument Pγ, given
to it in second position, in a given world w, and it then applies ϕδ(eσ)
to the result of this operation and subsequently to its (unmodified) second
argument ze.
In line 48, consistency of the introduced concepts is confirmed by the
model finder Nitpick [14]. Since only abbreviations and no axioms have
been introduced so far, the consistency of the Isabelle/HOL theory IHOML
in Fig. 3.1 is actually evident.
9
3.2. Filters and Ultrafilters
Two related world-lifted notions of modal filters and modal ultrafilters are
defined in Fig. 2; for a general introduction to filters and ultrafilters we
refer to the corresponding mathematical literature (e.g. [15]).
δ-Ultrafilters are introduced in line 26 as world-lifted characteristic
functions of type (δσ)σ. They thus denote σ-sets of σ-sets of objects
of type δ. In other words, a δ-Ultrafilter is a σ-subset of the σ-powerset of
δ-type property extensions.
Aδ-Ultrafilter φis defined as a δ-Filter satisfying an additional max-
imality condition: ϕ.ϕ δφ(1δϕ)δφ, where δis elementhood of
δ-type objects in σ-sets of δ-type objects (see line 4), and where 1δis the
relative set complement operation on sets of entities (see line 14).
The notion of δ-Filter is introduced in lines 17 and 18. A δ-Filter φis
required to
be large: Uδδφ, where Uδdenotes the full set of δ-type objects
we start with (see line 8),
exclude the empty set: δ6∈δφ, where δis the world-lifted empty
set of δ-type objects (see line 6),
be closed under supersets: ϕ ψ.(ϕδφϕδψ)ψδφ(the
world-lifted subset relation δis defined in line 10), and
be closed under intersections: ϕ ψ.(ϕδφψδφ)(ϕuδψ)δ
φ(the intersection operation uδis defined in line 12).
γ-Ultrafilters, which are of type (γσ)σ, are analogously defined as
aσ-subset of the σ-powerset of γ-type property extensions.
The distinction of both notions of ultrafilters is needed in our subse-
quent investigation. This is because we will rigorously distinguish between
positive property intensions (as used by Scott and Anderson) and positive
property extensions (as utilized by Fitting).
By using polymorphic definitions, several “duplications” of abbrevi-
ations in the theory ModalUltrafilter (Fig. 2) could be avoided. To
support a more precise understanding of δ- and γ-Ultrafilters, and their
differences, however, we have decided to be very transparent and explicit
regarding type information in the provided definitions.
10
4. Scott’s Variant of odel’s Argument
Scott’s variant of odel’s argument has been reproduced by Fitting in his
book [16]. It is Fitting’s formalization of Scott’s variant that we have
encoded and verified first in our computer-supported analysis of positive
properties, ultrafilters and modal collapse. This encoding of Scott’s variant
is presented in Fig. 3 and its presentation is continued in Fig. 4.
Part I of the argument is reconstructed in lines 4-11 of Fig. 3 and ver-
ified with automated reasoning tools.2In this part we conclude from the
premises and definitions (lines 5–8) that a Godlike being possibly exists
(theorem T3 in line 11): b3EGc; this follows from theorems T1 and T2
that are proved in lines 9 and 10. Note that, using binder notation, b3EGc
can be more intuitively presented as b3Ex.Gxc. The most essential def-
inition, the definition of property G, which is of type γand which defines
a Godlike being xeto possess all (intensional!) positive properties P, is
given in line 5. Premises that govern the notion of (intensional) positive
properties Pare A1 (which is split into A1a and A1b), A2 and A3; see lines
6–8. Scott [28] actually avoids axiom A3 and instead directly postulates
T2 (the sole purpose of A3 is to support T2). Although we here explicitly
include the inference from A3 to T2, it could also be left out without any
implications for the rest of the proof.
Part II of the argument is presented in lines 12–20. In line 13 we
switch from base modal logic Kto logic KB by postulating symmetry
of the accessibility relation r. Utilizing the same tools as before, and by
exploiting theorems T3, T4 and T5, we finally prove, in line 20, the main
theorem T6, which states that a Godlike being necessarily exists: b2EGc,
resp. b2Ex.Gxcusing binder notation.
Consistency of the Isabelle/HOL theory ScottVariant, as introduced
up to here, is confirmed by the model finder Nitpick [14] in line 21 (which
constructs a model with one world and one Godlike entity).
2The automated reasoning tools that are integrated with Isabelle/HOL, and which
we utilize in this article, include metis,smt,simp,blast,force, and auto. In fact,
in each case where those occur in the presented Isabelle/HOL formalizations, we have
actually first used a generic hammer-tool, called sledgehammer [13], which calls state-
of-the-art ATPs to prove the statements in question fully automatically and without
the need for specifying the particularly required premises; sledgehammer, in case of
success, subsequently attempts to reconstruct the external proofs reported by the ATPs
in Isabelle/HOL’s trusted kernel by applying the mentioned automated reasoning tools.
11
Scott’s Axioms and Definitions
(df.G)Gx Yγ.PYY x
(A1a) X.P(+X ) ¬(PX) where +is set/predicate negation
(A1b) X.¬(PX) P(+X)
(A2) XY .(PX2(Ez.Xz Y z )) P Y
(A3) ZX.((Y .Z Y P Y)2(x.Xx (Y .Z Y Y x))) P X
(A4) X.PX2(PX)
(df.E)EY x Y x (Z.Z x 2(Ez.Y z Zz))
(df.NE) NE x Y.EY x 2EY
(A5) PNE
Fig. 3. Scott’s variant of odel’s argument, following Fitting [16].
In lines 23–29 modal collapse is proved. This is one of the rare cases
in our experiments where direct proof automation with Isabelle/HOL’s
integrated automated reasoning tools (incl. sledgehammer [13]) still fails.
A little interactive help is needed here to show that modal collapse indeed
follows from the premises in Scott’s variant of odel’s argument.
For more background information and details on the formalization of
12
Fig. 4. Ultrafilter-analysis of Scott’s variant (continued from Fig. 3).
Scott’s argument, and also on the arguments by Anderson and Fitting as
presented in the following sections, we refer to Fitting’s book [16, §11] and
our previous work [17].
4.1. Positive Properties and Ultrafilters: Scott
Interesting findings regarding positive properties and ultrafilters in Scott’s
variant are revealed in Fig. 4.
Theorem U1, which is proved in lines 32–37, states that the set of
positive properties Pin Scott’s variant constitutes a γ-Ultrafilter.
In line 38, a modified notion of positive properties P0is defined as the
set of properties ϕwhose rigidly intensionalized extensions ϕare in P.
It is then shown in theorem U2 (lines 39–44), that also P0constitutes a
γ-Ultrafilter. And theorem U3 in line 45 shows that these two sets, Pand
P0, are in fact equal.
In line 47 we switch from logic KB to logic S5 by postulating reflexiv-
ity and transitivity of the accessibility relation rin addition to symmetry
(line 13 in Fig. 3); and we show consistency again (line 48). In the re-
maining lines 49–53 in Fig. 4 we show that the Barcan and the converse
13
Barcan formulas are valid for types eand γ; we use for the former type
actualist quantifiers (as in the argument) and for the latter type possibilist
quantifiers.
5. Anderson’s Variant of odel’s Argument
Anderson’s variant of odel’s argument is presented in Fig. 5.
A central change in comparison to Scott’s variant concerns Scott’s
premises A1a and A1b. Anderson drops A1b and only keeps A1a: “If
a property is positive, then its negation is not positive”. This modifi-
cation, however, has the effect that the necessary existence of a Godlike
being would no longer follow (and the reasoning tools in Isabelle/HOL can
confirm this; not shown here). Anderson’s variant therefore introduces fur-
ther emendations: it strengthens the notions of Godlikeness (in line 5) and
essence (in line 14). The emended notions, referred to by GAand EA, are
as follows:
GAAn individual xis Godlike GAif and only if all and only the neces-
sary/essential properties of xare positive, i.e., GAx Y(PY
2(Y x)).
EAA property Yis an essence EAof an individual xif and only if all of
x’s necessary/essential properties are entailed by Yand (conversely)
all properties entailed by Yare necessary/essential properties of x.
As is shown in lines 3–19, no further modifications are required to ensure
that the intended theorem T6, the necessary existence of a GA-like being,
can (again) be proved.3
In line 20, the model finder Nitpick confirms that modal collapse is
indeed countersatisfiable in Anderson’s variant of odel’s argument. As
expected, the reported countermodel consists of two worlds and one entity.
Consistency of theory AndersonVariant is confirmed by Nitpick in line
21, by finding a model with only one world and one entity (not shown).
3In a very stringent interpretation this statement is not entirely true: Theorem T2 in
Scott’s argument, which was derived in Fig. 3 from axiom A3 and the definition of G, is
now directly postulated here (for simplicity reasons) and axiom A3, which had no other
purpose besides supporting T2, is dropped. This simplification, however, is obviously
independent from the aspects as discussed.
14
Anderson’s Axioms and Definitions
(df.GA)GAx Yγ.PY2(Y x)
(A1a) X.P(+X ) ¬(PX) where +is set/predicate negation
(A2) XY .(PX2(Ez.Xz Y z )) P Y
(T2) PGA
(A4) X.PX2(PX)
(df.EA)EAY x Z.2(Z x)2(Ez.Y z Zz)
(df.NEA) NEAx Y.EAY x 2EY
(A5) PNEA
Fig. 5. Anderson’s variant of odel’s argument, following Fitting [16].
15
5.1. Positive Properties and Ultrafilters: Anderson
Regarding positive properties and ultrafilters an interesting difference to
our prior observations for Scott’s version is revealed by the automated
reasoning tools: the set of positive properties Pin Anderson’s variant does
not constitute a γ-Ultrafilter; Nitpick finds a countermodel to statement U1
in line 23 that consists of two worlds and one entity. However, the modified
notion P0, i.e., the set of all properties ϕ, whose rigidly intensionalized
extensions are in P(line 24), still is a γ-Ultrafilter; see theorem U2, which
is proved in lines 25–30. Consequently, the sets Pand P0are not generally
equal anymore and Nitpick reports a countermodel for statement U3 in line
31.
In lines 32–40, we once again switch from logic KB to logic S5, we
again show consistency, and we again analyze the Barcan and the converse
Barcan formulas for types eand γ. In contrast to before, the Barcan
and converse Barcan formulas for type e, when formulated with actualist
quantifiers, are not valid anymore; Nitpick presents countermodels with
two worlds and two entities.
6. Fitting’s Variant of odel’s Argument
In Fitting’s variant of odel’s Argument, see Fig. 6, the notion of posi-
tive properties Pin the definition of Godlikeness Granges over extensions
of properties, i.e., over terms of type δ, and not over γ-type intensional
properties as in Scott’s and Anderson’s variants. In Fitting’s understand-
ing, positive properties are thus fixed from world to world, while they
are world-dependent in Scott’s and Anderson’s. In technical terms, Scott
(resp. odel) defines Gxas Yγ.PYY x (line 5 in Fig. 3), whereas Fit-
ting modifies this into Yδ.PYLY xM(line 5 in Fig. 6). In an analogous
way, the notion of essence is emended by Fitting: in Scott’s variant, see line
15 in Fig. 3, EY x is defined as Y x (Z.Zx 2(Ez.Y z Zz)), while
it becomes LY xM(Z.LZxM2(Ez.LY zMLZ zM) in Fitting’s variant
(see line 15 in Fig. 6).
The definition of necessary existence NE in line 17 is adapted accord-
ingly, and in several other places of Fitting’s variant respective emendations
are required to suitably address his alternative interpretation of odel’s no-
tion of positive properties (see, e.g., theorem T2 in line 9 or axiom A5 in
16
Fitting’s Axioms and Definitions
(df.G)Gx Yδ.PYLY xM
(A1a) X.P(+X ) ¬(PX) where +is set/predicate negation
(A1b) X.¬(PX) P(+X)
(A2) XY .(PX2(Ez.LXz MLY zM)) P Y
(T2) P↓G
(df.E)EY x LY xM(Z.LZ xM2(Ez.LY zMLZ zM)
(df.NE) NE x Y.EY x 2(Ez.LY zM)
(A5) P↓NE
Fig. 6. Fitting’s variant of odel’s argument.
17
line 18). Fitting’s expressive logical system (IHOML) also allows us to dis-
tinguish between de dicto and de re readings of theorems T3, T5, and T6.
Except for the de dicto reading of T3, which has a countermodel with two
worlds and two entities, all of these statements are proved automatically
by the reasoning tools integrated with Isabelle/HOL.
As intended by Fitting, modal collapse is not provable anymore, which
can be seen in line 25, where Nitpick reports a countermodel with two
worlds and one entity.
Consistency of the Isabelle/HOL theory FittingVariant, as intro-
duced up to here, is confirmed by Nitpick in line 26 (one world, one entity).
6.1. Positive Properties and Ultrafilters: Fitting
The type of Phas changed in Fitting’s variant from the prior γσto δσ.
Hence, in our ultrafilter analysis, the notion of a γ-Ultrafilter no longer
applies and we must consult the corresponding notion of a δ-Ultrafilter.
Theorem U1, which is proved in lines 28–33 of Fig. 6, confirms that Fitting’s
emended notion of Pindeed constitutes a δ-Ultrafilter.
In line 35 we again switch from modal logic KB to logic S5. Consis-
tency of the Isabelle/HOL theory FittingVariant in S5 is confirmed in
line 36, and countersatisfiability of modal collapse is reconfirmed in line 37.
Moreover, like for Anderson’s variant before, we get a countermodel
for the Barcan formula and the converse Barcan formula on type e, when
formulated with actualist quantifiers. The Barcan formula and its converse
are proved valid for type γ.
7. Conclusion
Anderson and Fitting both succeed in altering odel’s modal ontological
argument in such a way that the intended result, the necessary existence of
a Godlike being, is maintained while modal collapse is avoided. And both
solutions, from a cursory reading, are quite different.
We conclude by rephrasing in more precise, technical terms what has
been mentioned at abstract level already in the mentioned related arti-
cle [24, §2.3]:
In order to compare the argument variants by Scott, Anderson, and
Fitting, two notions of ultrafilters were formalized in Isabelle/HOL: A δ-
Ultrafilter, of type (δσ)σ, is defined on the powerset of individuals,
18
i.e., on the set of rigid properties, and a γ-Ultrafilter, which is of type
(δσ)σ, is defined on the powerset of concepts, i.e., on the set of non-
rigid, world-dependent properties. In our formalizations of the variants,
a careful distinction was made between the original notion of a positive
property Pthat applies to (intensional) properties and a restricted notion
P0that applies to properties whose rigidified extensions are P-positive.
Using these definitions the following results were proved computationally:
In Scott’s variant both Pand P0coincide, and both are γ-Ultrafilters.
In Anderson’s variant Pand P0do not coincide, and only P0, but not
P, is a γ-Ultrafilter.
In Fitting’s variant, the Pin the sense of Scott and Anderson is not
considered an appropriate notion. However, Fitting’s emended notion
of a positive property P, which applies to extensions of properties,
corresponds to our definition of P0in Scott’s and Anderson’s variants;
and, as was to be expected, Fitting’s emended notion of Pconstitutes
aδ-Ultrafilter.
The presented computational experiments thus reveal an intriguing corre-
spondence between the variants of the ontological argument by Anderson
and Fitting, which otherwise seem quite different. The variants of Ander-
son and Fitting require that only the restricted notion of a positive property
is an ultrafilter.
The notion of positive properties in odel’s ontological argument is
thus aligned with the mathematical notion of a (principal) modal ultrafil-
ter on intensional properties, and to avoid modal collapse it is sufficient to
restrict the modal ultrafilter-criterion to property extensions. In a sense,
the notion of Godlike being Gx of odel is thus in close correspondence
to the x-object in a principal modal ultrafilter Fx of positive proper-
ties. This appears interesting and relevant, since metaphysical existence
of a Godlike being is now linked to existence of an abstract object in a
mathematical theory.
Further research could look into a formal analysis of monotheism and
polytheism for the studied variants of odel’s ontological argument. We
conjecture that different notions of equality will eventually support both
views, and a respective formal exploration study could take Kordula ´
Swi¸etor-
zecka’s related work [31] as a starting point.
19
Acknowledgements. This work was supported by VolkswagenStiftung
under grant CRAP (Consistent Rational Argumentation in Politics). As
already mentioned, the technical results presented in this article have been
summarized at abstract level in a joint article with Daniel Kirchner and
Ed Zalta. We are also grateful to the anonymous reviewer.
References
[1] C. Anthony Anderson. Some emendations of odel’s ontological proof. Faith
and Philosophy, 7(3):291–303, 1990.
[2] C. Anthony Anderson and M. Gettings. odel’s ontological proof revisited.
In odel’96: Logical Foundations of Mathematics, Computer Science, and
Physics: Lecture Notes in Logic 6, pages 167–172. Springer, 1996.
[3] Christoph Benzm¨uller. Computational metaphysics: New insights on
odel’s ontological argument and modal collapse. In Sre´cko Kovaˇc and
Kordula ´
Swi¸etorzecka, editors, Formal Methods and Science in Philosophy
III, Informal Proceedings, pages 3–4, Dubrovnik, Croatia, 2019.
[4] Christoph Benzm¨uller. Universal (meta-)logical reasoning: Recent successes.
Science of Computer Programming, 172:48–62, March 2019.
[5] Christoph Benzm¨uller and Peter Andrews. Church’s type theory. In Ed-
ward N. Zalta, editor, The Stanford Encyclopedia of Philosophy, pages pp. 1–
62 (in pdf version). Metaphysics Research Lab, Stanford University, summer
2019 edition, 2019.
[6] Christoph Benzm¨uller and David Fuenmayor. Can computers help to
sharpen our understanding of ontological arguments? In Sudipto Gosh,
Ramgopal Uppalari, K. Vasudeva Rao, Varun Agarwal, and Sushant
Sharma, editors, Mathematics and Reality, Proceedings of the 11th All India
Students’ Conference on Science & Spiritual Quest (AISSQ), 6-7 October,
2018, IIT Bhubaneswar, Bhubaneswar, India, pages 195–226. The Bhak-
tivedanta Institute, Kolkata, www.binstitute.org, 2018.
[7] Christoph Benzm¨uller and Lawrence Paulson. Quantified multimodal logics
in simple type theory. Logica Universalis, 7(1):7–20, 2013.
[8] Christoph Benzm¨uller, Nik Sultana, Lawrence C. Paulson, and Frank
Theiß. The higher-order prover LEO-II. Journal of Automated Reasoning,
55(4):389–404, 2015.
[9] Christoph Benzm¨uller and Bruno Woltzenlogel Paleo. Automating odel’s
ontological proof of God’s existence with higher-order automated theorem
20
provers. In Torsten Schaub, Gerhard Friedrich, and Barry O’Sullivan, edi-
tors, ECAI 2014, volume 263 of Frontiers in Artificial Intelligence and Ap-
plications, pages 93 98. IOS Press, 2014.
[10] Christoph Benzm¨uller and Bruno Woltzenlogel Paleo. Interacting with
modal logics in the Coq proof assistant. In Lev D. Beklemishev and Daniil V.
Musatov, editors, Computer Science Theory and Applications 10th In-
ternational Computer Science Symposium in Russia, CSR 2015, Listvyanka,
Russia, July 13-17, 2015, Proceedings, volume 9139 of Lecture Notes in Com-
puter Science, pages 398–411. Springer, 2015.
[11] Christoph Benzm¨uller and Bruno Woltzenlogel Paleo. The inconsistency in
odel’s ontological argument: A success story for AI in metaphysics. In
Subbarao Kambhampati, editor, IJCAI 2016, volume 1-3, pages 936–942.
AAAI Press, 2016.
[12] Christoph Benzm¨uller and Bruno Woltzenlogel Paleo. An object-logic expla-
nation for the inconsistency in odel’s ontological theory (extended abstract,
sister conferences). In Malte Helmert and Franz Wotawa, editors, KI 2016:
Advances in Artificial Intelligence, Proceedings, Lecture Notes in Computer
Science, pages 244–250, Berlin, Germany, 2016. Springer.
[13] Jasmin C. Blanchette, Sascha ohme, and Lawrence C. Paulson. Extend-
ing Sledgehammer with SMT solvers. Journal of Automated Reasoning,
51(1):109–128, 2013.
[14] Jasmin Christian Blanchette and Tobias Nipkow. Nitpick: A counterexample
generator for higher-order logic based on a relational model finder. In Matt
Kaufmann and Lawrence C. Paulson, editors, Interactive Theorem Proving,
First International Conference, ITP 2010, Edinburgh, UK, July 11-14, 2010.
Proceedings, volume 6172 of Lecture Notes in Computer Science, pages 131–
146. Springer, 2010.
[15] Burris, Stanley and Sankappanavar, Hanamantagouda P. A course in uni-
versal algebra. Millenium Edition, 2012.
[16] Melvin Fitting. Types, Tableaus, and odel’s God. Kluwer, 2002.
[17] David Fuenmayor and Christoph Benzm¨uller. Automating emendations of
the ontological argument in intensional higher-order modal logic. In KI 2017:
Advances in Artificial Intelligence 40th Annual German Conference on AI,
Dortmund, Germany, September 25-29, 2017, Proceedings, volume 10505 of
LNAI, pages 114–127. Springer, 2017.
[18] David Fuenmayor and Christoph Benzm¨uller. Types, Tableaus and odel’s
God in Isabelle/HOL. Archive of Formal Proofs, pages 1–34, 2017. Note:
verified data publication.
21
[19] James Garson. Modal logic. In Edward N. Zalta, editor, The Stanford
Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University,
fall 2018 edition, 2018.
[20] Kurt odel. Appendix A. Notes in Kurt odel’s Hand. In J.H. Sobel,
editor, Logic and Theism: Arguments for and Against Beliefs in God, pages
144–145. Cambridge University Press, 1970.
[21] Petr ajek. Magari and others on odel’s ontological proof. In A. Ursini
and P. Agliano, editors, Logic and algebra, page 125135. Dekker, New York
etc., 1996.
[22] Petr ajek. Der Mathematiker und die Frage der Existenz Gottes. In
B. Buldt et al., editor, Kurt odel. Wahrheit und Beweisbarkeit, pages 325–
336. ¨obv & hpt Verlagsgesellschaft mbH, Wien, 2001. ISBN 3-209-03835-X.
[23] Petr ajek. A new small emendation of odel’s ontological proof. Studia
Logica, 71(2):149–164, 2002.
[24] Daniel Kirchner, Christoph Benzm¨uller, and Edward N. Zalta. Computer
science and metaphysics: A cross-fertilization. Open Philosophy, 2:230251,
2019.
[25] Sre´cko Kovc. Modal collapse in odel’s ontological proof. In Miroslaw
Szatkowski, editor, Ontological Proofs Today, pages 50–323. Ontos Verlag,
2012.
[26] Edward J. Lowe. A modal version of the ontological argument. In J. P. More-
land, K. A. Sweis, and C. V. Meister, editors, Debating Christian Theism,
chapter 4, pages 61–71. Oxford University Press, 2013.
[27] Tobias Nipkow, Lawrence C. Paulson, and Makarius Wenzel. Isabelle/HOL:
A Proof Assistant for Higher-Order Logic. Number 2283 in Lecture Notes
in Computer Science. Springer, 2002.
[28] Dana S. Scott. Appendix B: Notes in Dana Scott’s Hand. In J.H. Sobel,
editor, Logic and Theism: Arguments for and Against Beliefs in God, pages
145–146. Cambridge U. Press, 1972.
[29] Jordan H. Sobel. odel’s ontological proof. In Judith Jarvis Tomson, editor,
On Being and Saying. Essays for Richard Cartwright, pages 241–261. MIT
Press, 1987.
[30] Jordan H. Sobel. Logic and Theism: Arguments for and Against Beliefs in
God. Cambridge University Press, 2004.
[31] Kordula ´
Swi¸etorzecka. Identity or equality of odelian God. Draft paper,
private communication, May 2019.
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
Computational philosophy is the use of mechanized computational techniques to unearth philosophical insights that are either difficult or impossible to find using traditional philosophical methods. Computational metaphysics is computational philosophy with a focus on metaphysics. In this paper, we (a) develop results in modal metaphysics whose discovery was computer assisted, and (b) conclude that these results work not only to the obvious benefit of philosophy but also, less obviously, to the benefit of computer science, since the new computational techniques that led to these results may be more broadly applicable within computer science. The paper includes a description of our background methodology and how it evolved, and a discussion of our new results.
Article
Full-text available
Classical higher-order logic, when utilized as a meta-logic in which various other (classical and non-classical) logics can be shallowly embedded, is suitable as a foundation for the development of a universal logical reasoning engine. Such an engine may be employed, as already envisioned by Leibniz, to support the rigorous formalisation and deep logical analysis of rational arguments on the computer. A respective universal logical reasoning framework is described in this article and a range of successful first applications in philosophy, artificial intelligence and mathematics are surveyed.
Preprint
Full-text available
Classical higher-order logic, when utilized as a meta-logic in which various other (classical and non-classical) logics can be shallowly embedded, is suitable as a foundation for the development of a universal logical reasoning engine. Such an engine may be employed, as already envisioned by Leibniz, to support the rigorous formalisation and deep logical analysis of rational arguments on the computer. A respective universal logical reasoning framework is described in this article and a range of successful first applications in philosophy, artificial intelligence and mathematics are surveyed. DOI: 10.1016/j.scico.2018.10.008
Conference Paper
Full-text available
A shallow semantic embedding of an intensional higher-order modal logic (IHOML) in Isabelle/HOL is presented. IHOML draws on Montague/Gallin intensional logics and has been introduced by Melvin Fitting in his textbook Types, Tableaus and Gödel's God in order to discuss his emendation of Gödel's ontological argument for the existence of God. Utilizing IHOML, the most interesting parts of Fitting's textbook are formalized, automated and verified in the Isabelle/HOL proof assistant. A particular focus thereby is on three variants of the ontological argument which avoid the modal collapse, which is a strongly criticized side-effect in Gödel's resp. Scott's original work.
Article
Full-text available
A computer-formalisation of the essential parts of Fitting’s text- book Types, Tableaus and Gödel’s God in Isabelle/HOL is presented. In particular, Fitting’s (and Anderson’s) variant of the ontological argument is verified and confirmed. This variant avoids the modal collapse, which has been criticised as an undesirable side-effect of Kurt Gödel’s (and Dana Scott’s) versions of the ontological argument. Fitting’s work is employing an intensional higher-order modal logic, which we shallowly embed here in classical higher-order logic. We then utilize the embedded logic for the formalisation of Fitting’s argument.
Conference Paper
Full-text available
This paper discusses the inconsistency in Gödel's ontological argument. Despite the popularity of Gödel's argument, this inconsistency remained unnoticed until 2013, when it was detected automatically by the higher-order theorem prover Leo-II. Complementing the meta-logic explanation for the inconsistency available in our IJCAI 2016 paper [6], we present here a new purely object-logic explanation that does not rely on semantic argumentation.
Conference Paper
Full-text available
This paper discusses the discovery of the inconsistency in Gödel's ontological argument as a success story for artificial intelligence. Despite the popularity of the argument since the appearance of Gödel's manuscript in the early 1970's, the inconsistency of the axioms used in the argument remained unnoticed until 2013, when it was detected automatically by the higher-order theorem prover Leo-II. Understanding and verifying the refutation generated by the prover turned out to be a time-consuming task. Its completion, as reported here, required the reconstruction of the refutation in the Isabelle proof assistant, and it also led to a novel and more efficient way of automating higher-order modal logic S5 with a universal accessibility relation. Furthermore, the development of an improved syntactical hiding for the utilized logic embedding technique allows the refutation to be presented in a human-friendly way, suitable for non-experts in the technicalities of higher-order theorem proving. This brings us a step closer to wider adoption of logic-based artificial intelligence tools by philosophers .