The premises of Kurt Gödel’s original variant of his modal ontological argument , as was found in his “Nachlass”, are inconsistent; this holds already in base modal logic K [11, 9]. The premises of Scott’s  variant of Gödel’s work, in contrast, are consistent [9, 11], but they imply the modal collapse, , which has by many philosophers been considered an undesirable side effect; cf. Sobel  and the references therein.111The modal collapse was already noted by Sobel [29, 30]. One might conclude from it, that the premises of Gödel’s argument imply that everything is determined, or alternatively, that there is no free will. Srécko Kovac  argues that modal collapse was eventually intended by Gödel.
In this article we formally encode and analyze, starting with Scott’s variant, two prominent further emendations of Gödel’s work both of which successfully avoid modal collapse. These two variants have been contributed by C. Anthony Anderson [1, 2] and Melvin Fitting , and on a cursory reading they appear quite different. Our formal analysis, however, shows that, from a certain mathematical perspective, they are in fact closely related.
Two notions are particularly important in our analysis. From set theory, resp. topology, we borrow and suitably adapt, for use in our modal logic context, the notion of ultrafilter and apply it in two different versions 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 textbook “Types, Tableaus and Gödel’s God” , which we take as a starting point in our formalization work.
Utilizing these notions, and extending Fitting’s analysis, the modifications as introduced by Anderson and Fitting to Gödel’s concept of positive properties are formally studied and compared. Our computer-supported analysis, which is carried out in the proof assistant system Isabelle/HOL , is technically enabled by the universal logical reasoning approach , 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 , aka. classical higher-order logic (HOL). This approach enables the reuse of existing, 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 encoding and analysis of the three mentioned variants of Gödel’s ontological argument in HOL utilizing the SSE approach (§4, §5 and §6). We start out (§2) with pointing to related prior work and with outlining the SSE approach.
2 Prior Work and the SSE Approach
We briefly outline the key ideas of the shallow semantical embedding (SSE) approach that is essential for the remainder of this article. This section is intended to make the article sufficiently self-contained and to give references to related prior work. The presentation in this section is taken and adapted from a related article recently published [24, §1.1]; readers already familiar with the SSE approach may simply skip it, and those who need further details may consult further related articles [7, 4].
Earlier papers, cf.  and the references therein, focused on the development of SSEs. These papers show that the standard translation from propositional modal logic to first-order logic can be concisely modeled (i.e., embedded) within higher-order theorem provers, so that the modal operator , for example, can be explicitly defined by the -term , where denotes the accessibility relation associated with . Then one can construct first-order formulas involving 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 assistant system such as Isabelle/HOL. The first author, in collaboration with Paulson , developed an SSE that captures quantified extensions of modal logic (and other non-classical logics). For example, if is shorthand in HOL for , then would be represented as , where stands for the -term , and the gets resolved as described above.
To see how
these expressions can be resolved to produce the right
representation, consider the following series of reductions:
Thus, we end up with a representation of in HOL. Of course, in this setting types are assigned to each term of the language, which have been omitted above, but which will be given in the remainder of this article. More precisely, in the SSE presented in Fig. 1, we will assign individual terms (such as variable above) the type e, and terms denoting worlds (such as variable above) the type i. From such base choices, all other types in the above presentation can be inferred.
The SSE technique was also the starting point for a natural encoding of Gödel’s modern variant of the ontological argument in second-order S5 modal logic. Various computer formalizations and assessments of recent variants of the ontological argument in higher-order theorem provers emerged from earlier work with colleagues and students. Initial studies  investigated Gödel’s and Scott’s variants of the argument within the higher-order automated theorem prover (henceforth ATP) LEO-II . Subsequent work deepened these assessment studies [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 of these experiments were reconstructed in the proof assistant Coq . Additional follow-up work contributed a range of similar studies (see the references in ) and includes a range of variants of the ontological argument proposed by other authors, such as Anderson, Hájek, Fitting, and Lowe [1, 2, 21, 22, 23, 16, 26]; particularly relevant for this article is also some prior work of the authors [18, 17]. The use of ultrafilters to study the distinction between extensional and intensional positive properties in the variants of Scott, Anderson and Fitting has first been mentioned in an invited paper presented in 2018 at AISSQ conference in Bhubaneswar, India , and subsequently at FMSPh-2019 conference in Dubrovnik, Croatia .
3 Further Preliminaries
The formal analysis in this article takes Fitting’s textbook  as a starting point; see also the related prior work [18, 17]. Fitting suggests to carefully distinguish between intensions and extensions of positive properties in the context of Gödel’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 in previous work  intensional higher-order modal logic (IHOML). 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 intensions, resp. extensions, of (positive) properties are then introduced in §3.2. Since we develop, explain and discuss our formal encodings directly in Isabelle/HOL , some familiarity with this proof assistant and its background logic HOL  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 types in HOL as mentioned before: type i stands for possible worlds and type e for entities/individuals. To keep the encoding concise some type synonyms are introduced in lines 4–7, which we explain next.
and abbreviate the types of predicates and , respectively. Terms of type represent (extensional) properties of individuals. Terms of type can be seen to represent world-lifted propositions, i.e., truth-sets in Kripke’s modal relational semantics . Note that the explicit transition from modal propositions to terms (truth-sets) of type is a key aspect in SSE approach; see the literature  for further details. In the remainder of this article we make use of phrases such “world-lifted” or “-type” terms to emphasize this conversion in the SSE approach.
, which abbreviates type , is accordingly the type of accessibility relations in modal relational semantics, and , which stands for , 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 ; 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). is defined as abbreviation for the truth-set , respectively, is associated with the lambda-term . 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.
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, are inverting properties of types and , respectively; these operations will occur in 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 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, it also reduces the amount of polymorphic constructs in the formalization, which may cause a decreased practical automation performance.
The world-lifted modal -operator and the (polymorphic)222In line 19, ’a represents a type variable. world-lifted universal quantifier , as already discussed in §2, are introduced in lines 31 and 19, respectively. In line 20, user-friendly binder-notation for is additionally defined. In addition to the (polymorphic) possibilist quantifiers, and , defined this way in lines 19–22, further actualist quantifiers, and , 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 additional actualist quantifiers are declared non-polymorphic, so that they support quantification over individuals only. In the subsequent analysis of the variants of Gödel’s argument as contributed by Scott, Anderson and Fitting, we will indeed apply and for different types in the type hierarchy of HOL, while we need and for individuals only.
The notion of global validity of a world-lifted formula , denoted as , is introduced in line 34 as an abbreviation for .
Note that an (intensional) base modal logic K is introduced in theory IHOML. 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 and respectively, i.e., they are technically handled in HOL as functions over worlds whose extensions are obtained by applying them to a given world in context. The operation in line 37 is trivially converting a world-independent proposition of Boolean type into a rigid world-lifted proposition of type ; 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 , is slightly more involved. It evaluates its second argument, which is a property of type , for a given world , and it then rigidly intensionalizes the obtained extension of in . For technical reasons, however, is introduced as a binary operator, with its first argument being a world-lifted predicate that is being applied to the rigidly intensionalized ; 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 are generally different from their rigidly intensionalized counterparts : Isabelle/HOL’s model finder Nitpick  generates a countermodel to the claim that they are (Leibniz-)equal.
A related (non-bold) binary operator , of type , is introduced in line 44. Its first argument is a predicate applicable to extensions of properties, and its second argument is an intensional property. The -operator evaluates its second argument in a given world , thereby obtaining an extension of type , and then it applies its first argument to this extension. The -operator is analogous, but its first argument is now of type , which can be understood as world-lifted binary predicate whose first argument is of type and its second argument of type e. The -operator evaluates the intensional argument , given to it in second position, in a given world , and it then applies to the result of this operation and subsequently to its (unmodified) second argument .
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. ).
-Ultrafilters are introduced in line 26 as world-lifted characteristic functions of type. They thus denote -sets of -sets of objects of type . In other words, a -Ultrafilter is a -subset of the -powerset of -type property extensions.
A -Ultrafilter is defined as a -Filter satisfying an additional maximality condition: , where is elementhood of -type objects in -sets of -type objects (see line 4), and where is the relative set complement operation on sets of entities (see line 14).
The notion of -Filter is introduced in lines 17 and 18. A -Filter is required to
be large: , where denotes the full set of -type objects we start with (see line 8),
exclude the empty set: , where is the world-lifted empty set of -type objects (see line 6),
be closed under supersets: (the world-lifted subset relation is defined in line 10), and
be closed under intersections: (the intersection operation is defined in line 12).
-Ultrafilters, which are of type , are analogously defined as a -subset of the -powerset of -type property extensions.
The distinction of both notions of ultrafilters is needed in our subsequent 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, “duplications” of abbreviations in theory ModalUltrafilter could actually be avoided. To support a more precise understanding of - and -Ultrafilters, and their differences, however, we have decided to be very transparent and explicit regarding type information in the provided definitions.
4 Scott’s Variant of Gödel’s Argument
Scott’s variant of Gödel’s argument has been reproduced by Fitting in his textbook . 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.
First, some auxiliary definitions are introduced in lines 5–7. Then Part I of the argument is reconstructed and verified with automated reasoning tools.333The 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 , 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. In this part we conclude from the premises and definitions (lines 4–12) in theorem T3 that a Godlike being possibly exists (line 15): ; this follows from theorems T1 and T2 that are proved in lines 13 and 14. Note that, using binder notation, can be more intuitively presented as . The most essential definition, the definition of property G, which is of type and which defines a Godlike being to possess all (intensional!) positive properties , is given in line 9. Premises that govern the notion of (intensional) positive properties are A1 (which is split into A1a and A1b), A2 and A3; see lines 13–15.
Part II of the argument is presented in lines 16–24. In line 17 we switch from base modal logic K to 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 24, the main theorem T6, which states that a Godlike being necessarily exists: , resp. .
Consistency of the Isabelle/HOL theory ScottVariant, as introduced up to here, is confirmed by Nitpick in line 25 (which constructs a model with one world and one entity—God)
In lines 27–33 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 ) still fails. A little interactive help is needed here to show that modal collapse indeed follows from the premises in Scott’s variant of Gödel’s argument.
For more background information and details on the formalization of Scott’s argument, and also on the arguments by Anderson and Fitting as presented in the following sections, we refer to Fitting’s textbook [16, §11].
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 36–41, states that the set of positive properties in Scott’s variant constitutes a -Ultrafilter.
In line 42, a modified notion of positive properties is defined as the set of properties whose rigidly intensionalized extensions are in . It is then shown in theorem U2 (lines 44–48), that also constitutes a -Ultrafilter. And theorem U3 in line 49 shows that these two sets, and , are in fact equal.
In line 51 we switch from logic KB to logic S5 by postulating reflexivity and transitivity of the accessibility relation r in addition to symmetry (line 17 in Fig. 3); and we show consistency again (line 52). In the remaining lines 53–57 in Fig. 4 we show that the Barcan and the converse Barcan formulas are valid for types e and ; we use for the former type actualist quantifiers (as in the argument) and for the latter type possibilist quantifiers.
5 Andersons’s Variant of Gödel’s Argument
Anderson’s variant of Gödel’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 modification, 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 further emendations: it strengthens the notions of Godlikeness (in line 5) and essence (in line 14). The emended notions, referred to by and , are as follows:
An individual is Godlike if and only if all and only the necessary/essential properties of are positive, i.e., .
A property is an essence of an individual if and only if all of ’s necessary/essential properties are entailed by and (conversely) all properties entailed by are necessary/essential properties of .
As is shown in lines 3–19, no further modifications are required to ensure that the intended theorem T6, the necessary existence of a -like being, can (again) be proved.444In 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.
In line 20, the model finder Nitpick confirms that modal collapse is indeed countersatisfiable in Anderson’s variant of Gödel’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; here only one world and one entity is needed.
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 in Anderson’s variant does not constitute a -Ultrafilter; Nitpick finds a countermodel to statement U1 in line 23 that consists of two worlds and one entity. However, the modified notion , i.e., the set of all properties , whose rigidly intensionalized extensions are in (line 24), still is a -Ultrafilter; see theorem U2, which is proved in lines 25–30. Consequently, the sets and are 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 e and . 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 Gödel’s Argument
In Fitting’s variant of Gödel’s Argument, see Fig. 6, the notion of positive properties in the definition of Godlikeness G ranges over extensions of properties, i.e., over terms of type , and not over -type intensional properties as in Scott’s and Anderson’s variants. In Fitting’s understanding, 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. Gödel) defines G as the property (line 9 in Fig. 3), whereas Fitting modifies this into (line 5 in Fig. 6). In an analogous way, the notion of essence is emended by Fitting: while in Scott’s variant, see line 19 in Fig. 3, is defined as , it becomes in Fitting’s variant (see line 15 in Fig. 6; also note in line 4 that is now defined on type and not on type as in Scott’s variant).
The definition of necessary existence NE in line 17 is adapted accordingly, and in several other places of Fitting’s variant respective emendations are required to suitably address his alternative interpretation of Gödel’s notion of positive properties (see, e.g., theorem T2 in line 9 or axiom A5 in line 18). Fitting’s expressive logical system (IHOML) also allows us to distinguish 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 introduced up to here, is confirmed by Nitpick in line 26 (one world, one entity).
6.1 Positive Properties and Ultrafilters: Fitting
The type of has changed in Fitting’s variant from the prior to . Hence, in our ultrafilter analysis, the notion of a -Ultrafilter no longer applies and we must consult the corresponding notion of a -Ultrafilter. Theorem U1, which is proved in lines 28–33 of Fig. 6, confirms that Fitting’s emended notion of indeed constitutes a -Ultrafilter.
In line 35 we again switch from modal logic KB to logic S5. Consistency 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 .
Anderson and Fitting both succeed in altering Gödel’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 article [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 -Ultrafilter, which is of type , is defined on the powerset of individuals, i.e., on the set of rigid properties, and a -Ultrafilter, which is of type , 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 that applies to (intensional) properties and a restricted notion that applies to properties whose rigidified extensions are -positive. Using these definitions the following results were proved computationally:
In Scott’s variant both and coincide, and both are -Ultrafilters.
In Anderson’s variant and do not coincide, and only , but not , is a -Ultrafilter.
In Fitting’s variant, the in the sense of Scott and Anderson is not considered an appropriate notion. Fitting’s emended notion of positive properties , however, which applies to extensions of properties, corresponds to our definition of in Scott and Anderson. And, as was to be expected, Fitting’s emended notion of constitutes a -Ultrafilter.
The presented computational experiments thus reveal an intriguing correspondence between the variants of the ontological argument by Anderson and Fitting, which otherwise seem quite different. The variants of Anderson and Fitting require that only the restricted notion of a positive property is an ultrafilter.
The notion of positive properties in Gödel’s ontological argument is thus aligned with the mathematical notion of a (principal) modal ultrafilter 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 “” of Gödel is thus in close correspondence to the -object in a principal modal ultrafilter “” of positive properties. 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 Gödel’s ontological argument. We conjecture that different notions of equality will eventually support both views, and a respective formal exploration study could take Kordula Świȩtorzecka’s related work  as a starting point.
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.
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