A (Simplified) Supreme Being Necessarily Exists, says the Computer: Computationally Explored Variants of Gödel's Ontological Argument

01/14/2020 ∙ by Christoph Benzmüller, et al. ∙ Freie Universität Berlin 0

An approach to universal (meta-)logical reasoning in classical higher-order logic is employed to explore and study simplifications of Kurt Gödel'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 K and T. Key to the gained simplifications of Gödel's original theory is the exploitation of a link to the notions of filter and ultrafilter from topology. The paper illustrates how modern knowledge representation and reasoning technology for quantified non-classical logics can contribute new knowledge to other disciplines. The contributed material is also well suited to support teaching of non-trivial logic formalisms in classroom.

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1 Introduction

Variants of Kurt Gödel’s GoedelNotes, resp. Dana Scott’s ScottNotes, modal ontological argument have previously been analysed and verified on the computer by Benzmüller and Woltzenlogel-Paleo C40,C55 and Benzmüller and Fuenmayor J52, and some previously unknown issues were revealed in these works,111E.g., the theorem prover Leo-II detected that Gödel’s GoedelNotes variant of the argument is inconsistent; this inconsistency had, unknowingly, been fixed in the variant of Scott ScottNotes; cf. [Benzmüller and Woltzenlogel Paleo2016] for more details. and it was shown that logic KB, instead of S5, is already sufficient to derive from Gödel’s axioms that a supreme being necessarily exists.

In this paper simplified variants of Gödel’s modal ontological argument are explored and presented. Our simplifications have been developed in interaction with the proof assistant Isabelle/HOL [Nipkow, Paulson, and Wenzel2002] and by employing Benzmüller’s J41,J23 shallow semantical embedding (SSE) approach as enabling technology. This technology supports the reuse of automated theorem proving (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 Gödel’s modal ontological argument, which are in the focus of this paper.

One of the new, simplified modal arguments is as follows. The notion of being Godlike () is exactly as in Gödel’s original work. Thus, a Godlike entity, by definition, possesses all positive properties ( is an uninterpreted constant denoting positive properties):

The three only axioms of the new theory which constrain the interpretation of Gödel’s positive properties () are:

A1’

Self-difference is not a positive property.222An alternative to A1’ would be: The empty property () is not a positive property.

A2’

A property entailed or necessarily entailed by a positive property is positive.

A3

The conjunction of any collection of positive properties is positive.333The third-order formalization of A3 as given here ( is a third-order variable ranging over sets of properties) has been proposed by Anderson and Gettings AndersonGettings, see also Fitting fitting02:_types_tableaus_god. Axiom A3, together with the definition of , implies that being Godlike is a positive property. Since supporting this inference is the only role this axiom plays in the argument, can be taken (and has been taken; cf. Scott ScottNotes) as an alternative to our A3; cf. also Fig.  7.

Technical reading: if is any set of positive properties, then the property obtained by taking the conjunction of the properties in is positive.

In these premises the following defined symbols are used, where is a possibilist second-order quantifier and where is an actualist first-order quantifier for individuals:

From A1’, A2’ and A3 it follows, in a few argumentation 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 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 theoretical philosophy and theology.

Compare the above with Gödel’s premises of the modal ontological argument (in the consistent variant of Scott):

A1

One of a property or its complement is positive.

A2

A property necessarily entailed by a positive property is positive.

A3

The conjunction of any collection of positive properties is positive (or, alternatively, being Godlike is a positive property).

A4

Any positive property is necessarily positive.

A5

Necessary existence () is a positive property.

Furthermore, axiom B is added to ensure that we operate in logic KB instead of just K. (Remember that logic S5 is not needed.)

Symmetry of the accessibility relation r associated with modal -operator can be postulated alternatively in our meta-logical framework.

Essence () and necessary existence () are defined as (all other definitions are as before):

Informally: Property is an essence of an entity if, and only if, (i) holds for and (ii) necessarily entails every property of . Moreover, an entity has the property of necessary existence if, and only if, the essence of is necessarily instantiated.

Using Gödel’s premises as stated it can be proved automatically that a Godlike entity possibly and necessarily exists [Benzmüller and Woltzenlogel Paleo2016]. However, modal collapse is still implied even in the weak logic KB.444For more information on modal collapse (in logic S5) consult Sobel Sobel,sobel2004logic and Fitting fitting02:_types_tableaus_god.

Benzmüller and Fuenmayor J52 recently showed that different modal ultrafilter properties can be deduced from Gödel’s premises. These insights are key to the argument simplifications developed and studied in this paper: If Gödel’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 properties for ? Then use U1 instead of other axioms for proving that a Godlike entity necessarily exists, and on the fly explore what further simplifications of the argument are triggered. 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 interesting 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. J47, ensures that the paper is sufficiently self-contained; readers familiar with the SSE approach may simply skip it. Modal filter and ultrafilter are defined in Sect. 3. Section 4 recaps, in some more detail, the Gödel/Scott variant of the modal ontological 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 encodings directly in Isabelle/HOL, some familiarity with this proof assistant and its underlying logic HOL [Andrews2002, Benzmüller and Andrews2019] is assumed. The entire sources of our formal encodings, which are available at http://to.be.provided.later, are presented and explained in detail in this paper. The reported experiments were conducted on a standard notebook (2,5 GHz Intel Core i7, 16 GB memory).

The contributions of this paper are thus manifold. Besides the novel variants of the modal ontological argument that we contribute to metaphysics and theology, we demonstrate how the SSE technique, in combination with higher-order reasoning tools, can be employed in practical studies to explore such 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. Benzmüller J41 and Kirchner et al. J47 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., embedded) within HOL 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 FO formulas involving and use them to represent and proof theorems. Thus, in an SSE, the target logic is internally represented using higher-order (HO) constructs in a theorem proving system such as Isabelle/HOL. Benzmüller and Paulson J23 developed an SSE that captures quantified extensions of modal logic. For example, if is shorthand in HOL for , then would be represented as , where stands for the -term , and the gets resolved as 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, types are assigned to each (sub-)term of the HOL language. We 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 actually be inferred.

An explicit encoding of HOML in Isabelle/HOL, following the above ideas, is presented in Fig. 1.555In 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 polymorphic terms in the formalization (heavy use of polymorphism may generally lead to decreased proof automation performance). In lines 4–5 the base types i and e are declared (text passages in red are comments). Note that HOL comes with an inbuilt base type bool, the bivalent type of Booleans. No cardinality constraints are associated with types i and e, except that they must be non-empty. To keep our presentation concise, useful type synonyms are introduced in lines 6–9. abbreviates the type ( is the function type constructor in HOL), and terms of type can be seen to represent world-lifted propositions, i.e., truth-sets in Kripke’s modal relational semantics [Garson2018]. The explicit transition from modal propositions to terms (truth-sets) of type is a key aspect of the SSE technique, and in the remainder of this article we use phrases such as “world-lifted” or “-type” terms to emphasize this conversion in the SSE approach. , which stands for , is the type of world-lifted, intensional properties. and , which abbreviate and , are the types associated with unary and binary modal logic connectives.

Figure 1: SSE of HOML in HOL.

The modal logic connectives are defined in lines 12–25. In line 16, for example, the definition of the world-lifted -connective of type is given; explicit type information is presented after the ::-token for ‘c5’, which is the ASCII-denominator for the (right-associative) infix-operator as introduced in parenthesis shortly after. is then defined as abbreviation for the truth-set , 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 , , , , and are introduced analogously. The operator , introduced in lines 22–23, is inverting properties of types ; this operation occurs in Gödel’s axiom A1. and are defined in lines 24–25 as world-independent, syntactical equality.

The already discussed, world-lifted modal -operator is introduced in lines 19–20; accessibility relation is now named r. The definition of in line 21 is 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 as in our illustrating example, -symbols are overloaded here. The introduction of the possibilist -quantifier in lines 31–33 is analogous.

Additional actualist quantifiers, and , are introduced in lines 36–42. Their definition is guarded by an explicit, possibly empty, existsAt (@) predicate, which encodes whether an individual object actually “exists” at a particular world, or not. The actualist quantifiers are declared non-polymorphic, and they support quantification over individuals only. In the remainder we will indeed apply and for different types in the type hierarchy of HOL, while and exclusively quantify over individuals only.

Global validity of a world-lifted formula , denoted as , is introduced in line 45 as an abbreviation for .

Consistency of the introduced concepts is confirmed by the model finder nitpick [Blanchette and Nipkow2010] 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 paramater settings for some of the reasoning tools that we employ.

Theorem 1.

The SSE of HOML in HOL is faithful for base logic K.

Proof.

Analagous to Benzmüller and Paulson J23. ∎

Theory HOML thus models base logic K in HOL. Axiom B, as mentioned before, can postulated to arrive at logic KB.

3 Modal Filter and Ultrafilter

Theory file MFilter, for “modal filter”, see Fig. 2, imports theory the HOML and adapts the topological notions of filter and ultrafilter to our modal logic setting. For an introduction to the notions of filter and ultrafilter see the literature, e.g., [Burris and Sankappanavar1981].

Figure 2: Definition of filter and ultrafilter (for possible worlds).

Our notion of modal ultrafilter is introduced in lines 20–21 as a world-lifted characteristic function of type

. A modal ultrafilter is thus a world-dependent set of intensions of -type properties; in other words, a -subset of the -powerset of -type property extensions. A modal ultrafilter is defined as a modal filter satisfying an additional maximality condition: , where is elementhood of -type objects in -sets of -type objects (see lines 3–4), and where is the relative set complement operation on sets of entities (line 11).

A modal filter , see lines 14–17, is required to

  1. be large: , where U denotes the full set of -type objects we start with,

  2. exclude the empty set: , where is the world-lifted empty set of -type objects,

  3. be closed under supersets: (the world-lifted -relation is defined in lines 7–8), and

  4. be closed under intersections: (where is defined in lines 9–10).

Benzmüller and Fuenmayor J52 have studied two different notions of modal ultrafilter (termed - and -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 their notion of -ultrafilter.

4 Gödel/Scott Variant

We start out in Fig. 3 with the introduction of some basic abbreviations and definitions for the Gödel/Scott variant of the modal ontological argument. This theory file, which is termed BaseDefs and which imports HOML, is reused without modification also in all other variants as explored in this paper later on.

Figure 3: Definitions for all variants discussed in the remainder.

In line 4 the uninterpreted constant symbol , for “positive properties”, is declared; it has type . thus denotes an intensional, world-depended concept. In lines 7–11 abbreviations for the previously discussed relations and predicates , , and are introduced. In line 14–15, Gödel’s notion of “being Godlike” () is defined, and in lines 18–21 the previously discussed definitions for Essence () and Necessary Existence () are given.

The full formalization of Scott’s variant of Gödel’s argument is presented as theory ScottVariant in Fig. 4. This theory imports and relies on the previously introduced notions from theory files HOML, MFilter and BaseDefs.

The premises of Gödel’s argument, as already discussed earlier, are stated in lines 4–10. In line 12 a semantical counterpart B’ (symmetry of the accessibility relation r associated with the operator) of the B axiom is proved.

Figure 4: Gödel’s modal ontological argument; Scott’s variant.

An abstract level “proof net” for theorem T6, the necessary 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 exemplified. From A3 and the defn. of obtain 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 and to infer T4: being Godlike is an essential property of any Godlike entity. From this, A5, B’ and the defns. of , and 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 state-of-the-art ATP system integrated with Isabelle/HOL via its sledgehammer tool; sledgehammer then 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, simp and 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 modulo solver (CVC4 in our case), is less trusted, but we nevertheless 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 restrictions). Using the definitions from Sect. 2, one can generally reconstruct and formally verify all presented proofs with pen and paper 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 Kanckos2017VariantsOG or Fitting fitting02:_types_tableaus_god is also possible; in this paper we leave this as an exercise for the reader.

The presented theory is consistent, which is confirmed in line 31 by model finder nitpick; nitpick reports a model consisting of one world and one Godlike entity.

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 Gödel’s notion of positive properties constitutes a modal ultrafilter, cf. lines 45–55. This was key to the idea of taking the modal ultrafilter property of as 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 of all supersets of ; it follows from Gödel’s theory that this set is a modal filter (line 66) resp. modal ultrafilter (line 67), i.e., is the modal Hauptfilter of . The necessary existence of a Godlike entity now becomes a simple corollary of this result. In Sect.7.0.4 we will later present an argument variant that is based on this observation.666It was actually Manfred Droste at the University of Leipzig who pointed me to this proof argument alternative in an email communication.

5 Ultrafilter Variant

Figure 5: Ultrafilter variant.

Taking U1 as an axiom for Gödel’s theory 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 Gödel’s axiom A1 can be dropped, but also axioms A4 and A5, together with defns.  and . 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 have 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 have T5 (possible existence of a Godlike entity implies its necessary existence). Use T5 and T3 to conclude T6 (necessary existence of a Godlike entity).

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 countermodel consists of a single entity and two worlds and with accessibility relation . Trivially, formula is such that holds in but not in , which invalidates MC at world . is the Godlike entity in both worlds, i.e., is the property that holds for in and , which we may denote as . Using tuple notation we may write .

Remember that , which is of type , is an intensional, world-depended concept. In our countermodel for MC in line 23 the extension of for world has the above and as its elements, while in world we have and . Using tuple notation we may note as
         
         
In order to verify that is a modal ultrafilter we have to check whether the respective modal ultrafilter conditions are satisfied in both worlds. in and also in , since both and are in ; in and also in , since both and are not in . It is also easy to verify that is closed under supersets and intersection in both worlds.

Note that in our countermodel for MC, also Gödel’s axiom A4 is invalidated. Consider , i.e., is true for in , but false for in . We have in , but we do not have in , since does not hold in , which is reachable in r from .

nitpick is capable of computing all partial modal ultrafilters as part of its countermodel exploration: out of 512 candidates, nitpick identifies 32 structures of form , for , in which satisfies the ultrafilter conditions in the specified world . An example for such an is
          ,
         
is not a proper modal ultrafilter, since fails to be an ultrafilter in world .

6 Simplified Variant

Figure 6: Simplified variant.

What modal ultrafilters properties of are actually needed to support T6? Which ones can be dropped? Experiments with our framework, as displayed in theory file SimpleVariant in Fig. 6, confirm that only the filter conditions from Sect. 3 must be upheld for ; maximality can be dropped. However, it is possible to merge filter condition 3 (closed under supersets) for with Gödel’s A2 into axiom A2’ as shown in line 6 of Fig. 6. Moreover, instead of requiring that the empty set must not be a positive property, we postulate that self-difference is 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 motivation for this switch. As intended, filter condition 4 is now implied by the theory (see theorem F1 proved in line 34), as well as positiveness of self-identity (line 18). The essential idea of the theory SimpleVariant in Fig. 6 is to show that it actually suffices, in combination with A3, to postulate that is a modal filter, and this is what our simplified axioms do.

From the definition of and the axioms A1’, A2’ and A3 (lines 5–7) theorem T6 immediately follows: in line 10 several theorem provers integrated with sledgehammer quickly report a proof ( 1sec). Moreover, a more detailed and more intuitive “proof net” is presented in lines 11–16; the proof argument is analogous to what has been discussed before.

In lines 21–26, consistency is confirmed by nitpick, and countermodels for modal collapse MC (similar to the one discussed before) and for monotheism MT are reported.

Further questions are answered experimentally (lines 29–31): neither A1, nor A4 or A5, of the premises we dropped from Gödel’s theory are implied anymore, all have countermodels. In lines 34–35 we see that is still a filter, but not an ultrafilter. Since some of these axioms, e.g. Gödel’s strong A1, have been discussed controversially in the history of Gödel’s argument, and since MC and MT are independent, we have arrived at a philosophically and theologically potentially relevant simplification of Gödel’s work.

7 Further Simplified Variants

7.0.1 Postulating T2 instead of A3

Instead of working with third-order axiom A3 to infer T2 as in theory SimpleVariant, we directly postulate T2 as an axiom in theory SimpleVariantPG; 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 21–23).

Figure 7: Simplified variant with axiom T2.

7.0.2 Simple Entailment in Axiom A2’

Figure 8: Simplified variant with simple entailment in logic K.

Instead of using a disjunction 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 SimpleVariantSE displayed in Fig. 8. Proofs for T6, the necessary existence 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 existence of a Godlike entity) is no longer implied; see line 14. As nitpick informs us, T3 now has an undesired countermodel 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).

7.0.3 Simple Entailment in Logic T

The above discussion motivates a further alternative of the simplified modal ontological argument; see theory file SimpleVariantSEinT in Fig. 9. This argument is assuming modal logic T (which comes with axiom T as discussed above), and as before it postulates 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.

Figure 9: Simplified variant with simple entailment in logic T.

A 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 some more detail (note that this simpler proof argument is obviously applicable to previous variants as well):777The existence of such an eventually unintended derivation was already hinted at by Fuenmayor and Benzmüller [Footn. 11]C65.

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.
— This follows from L1 and axiom A1’.

T1’

Positive properties are exemplified.
— A reformulation of L2.

T3’

A Godlike entity exists.
— Simple consequence of T1’ and axiom T2.

T3

A Godlike entity possibly exists.
— Follows from T3’ and T’ (the contrapositive of axiom T); note that T3 is in fact not needed to obtain T6 in the next step; generally, axiom T (resp. its contrapositive T’) is only needed for deriving T3 and not for T6.

T6

A Godlike entity necessarily exists.
— Follows from T3’ by necessitation.

7.0.4 Hauptfiltervariant

Figure 10: Hauptfiltervariant.

Another drastically simplified variant of the modal ontological argument is related to the observations discussed earlier at the end of Sec. 4. There it was shown that the set , consisting of all supersets of , is a modal filter (so that is the Hauptfilter of ); this then directly implies the necessary existence of a Godlike entity (in fact, the only essential requirement that is enforced here is that is not in , hence cannot be identical to ). A new variant based on this observation is presented in theory file SimpleVariantHF in Fig. 10. Here the set of all supersets of property (being Godlike) is postulated to be a modal filter (axiom F1 in line 7). 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 19–25). Moreover, the theory is consistent (line 28) and neither modal collapse nor monotheism are implied (lines 31–33).

8 Related Work

Fitting fitting02:_types_tableaus_god suggested to carefully distinguish between intensions and extensions of positive properties in the context of Gödel’s modal ontological argument, and, in order to do so within a single framework, he introduced a sufficiently expressive HOML enhanced with means for the explicit representation of intensional terms and their extensions; cf. also the intensional operations used by Fuenmayor and Benzmüller C65,J52 in their verification of the works of Fitting and Anderson Anderson90,AndersonGettings.

The application of computational methods to philosophical problems was initially limited to first-order theorem provers. Fitelson and Zalta FitelsonZalta used Prover9 to find a proof of the theorems about situation and world theory in [Zalta1993] and they found an error in a theorem about Plato’s Forms that was left as an exercise in [Pelletier and Zalta2000]. Oppenheimer and Zalta OppenheimerZalta2011 discovered, using Prover9, that one of the three premises used in their reconstruction of Anselm’s ontological argument [Oppenheimer and Zalta1991] was sufficient to derive the conclusion. The first-order conversion techniques that were developed and applied in these works are outlined in some detail in related work by Alama, Oppenheimer and Zalta AlamaZalta2015.

More recent related work makes use of higher-order proof assistants. Besides the already mentioned work of Benzmüller and colleagues, this includes Rushby’s Rushby study on Anselm’s ontological argument in the PVS proof assistant and Blumson’s Blumson related study in Isabelle/HOL.

The development of Gödel’s ontological argument has recently been addressed also in an article by Kanckos and Lethen KanckosLethen19; they present some previously unknown insights about Gödel’s work.

9 Discussion

The simplifications of Gödel’s theory as presented are partly quite far reaching. In fact, one may be tempted to ask “Is this really what Gödel 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 entailment () really adequate in the context of the modal ontological argument, or shouldn’t this definition be replaced by concept containment, so that the property of being self-different, resp. the empty property, would no longer -entail any other property?888See also the discussion of concept containment in [Alama, Oppenheimer, and Zalta2015] and [Fuenmayor and Benzmüller2017, Footnote 11].

Moreover, assuming the simplified theory SimpleVariant from Fig. 6 is indeed still in line with what Gödel 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”):

An entity x is maximally-rational () iff it has all rational/consistent properties.

A1’

Self-difference is not a rational/consistent property.

A2’

A property entailed or necessarily entailed by a rational/consistent property is rational/consistent.

T2

Maximal-rationality is a rational/consistent property.

It follows:

A maximally-rational entity necessarily exists.

It would be still be possible, but not mandatory, to understand a maximally-rational being also as a Godlike being.

Independent of this discussion we expect the Isabelle/HOL theory files we contributed to be useful for teaching quantified modal logics in classroom, as previously demonstrated by Wisniewski et al. C58 in their lecture course on “Computational Metaphysics”. The developed corpus of example problems is furthermore suited as a benchmark for other ambitious knowledge representation and reasoning projects in the KR community: Can the alternative approaches encode such as arguments from metaphysics as well? How about their proof automation capabilities and how about modal finding? Description logics, for example, due to their expressivity restrictions, are most likely not well suited for such kind of applications in metaphysics; cf. also the work by Kirchner et al. J50. Moreover, our problem set constitutes an interesting small benchmark set for other HOL automated theorem provers and should therefore be converted into TPTP THF [Sutcliffe and Benzmüller2010] representation.

10 Conclusion

Gödel’s modal ontological argument stands in prominent tradition of western philosophy. It has its roots in the Proslogion 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 Gödel.

In this paper we have linked Gödel’s theory to a suitably adapted mathematical theory (modal filter and modal ultrafilter), and subsequently we have developed simplified modal ontological arguments which avoid some of Gödel’s axioms and consequences, including modal collapse, that have led to criticism in the past. At the same time the offered simplifications are very far reaching, eventually even too far. Anyhow, the insights that were presented in this paper appear relevant to inform ongoing studies of the modal ontological argument in theoretical philosophy and theology.

Regarding the ongoing debate on the relevance and difference of subsymbolic and symbolic AI technologies we may comment: While data scientists apply subsymbolic AI techniques to obtain approximating and rather opaque models in their application domains of interest, we have in this paper applied modern symbolic AI techniques to arrive at deep, explainable and verifiable models of the metaphysical concepts we are interested in. In particular, we have illustrated how state of the art theorem proving systems, in combination with latest knowledge representation and reasoning technology, can fruitfully be employed to explore and contribute deep new knowledge to other disciplines.

Acknowledgements

I am grateful to friends and colleagues who have supported this line of research in the past, including (in alphabetical order) C. Brown, M. Droste, D. Fuenmayor, T. Gleißner, D. Kirchner, X. Parent, R. Rojas, D. Scott, A. Steen, L. van der Torre, E. Weydert, D. Streit, B. Woltzenlogel-Paleo, E. Zalta.

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