Variants of Kurt Gödel’s GoedelNotes, resp. Dana Scott’s ScottNotes, modal ontological argument have previously been analysed, studied and verified on the computer by Benzmüller and Woltzenlogel C55 and Benzmüller and Fuenmayor J52, and even some unknown flaws 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 on this.
In this paper a simplified and improved variant of Gödel’s modal ontological argument is presented. This simplification has been explored in collaboration with the proof assistant Isabelle/HOL [Nipkow et al.2002], while employing Benzmüller’s J41,J23 shallow semantical embedding (SSE) approach as enabling technology. This technology supports the reuse of theorem proving (ATP) systems for classical higher-order logic (HOL) to represent and reason with a wide range of non-classical logics and theories, including higher-order modal logic (HOML) and Gödel’s modal ontological argument, which are in the focus of this paper.
The new, simplified modal argument is as follows. Gödel definition of a Godlike entity remains unchanged ( is an uninterpreted constant denoting positive properties):
A Godlike entity thus possesses all positive properties.
The three single axioms of the new theory are222An alternative to A1’ would be: The universal property () is a positive property, and the empty property () is not. The third-order formalization of A3 given here has been proposed by Anderson and Gettings AndersonGettings, see also Fitting fitting02:_types_tableaus_god. Axiom A3, together with the definition of G, implies that being Godlike is a positive property. Since supporting this inference is the only role this axiom plays in the argument, could be (and has been; cf. Scott ScottNotes) taken as an alternative to our A3.
where the following defined terms are used ( is a possibilist quantifier and is an actualist quantifier for individuals):
Informally we have:
Self-identity is a positive property, self-difference is not.
A property entailed or necessarily entailed by a positive property is positive.
The conjunction of any collection of positive properties is positive. (Technical reading: if is any set of positive properties, then the property obtained by taking the conjunction of the properties in is positive.)
From these premises it follows, in a few argumentations steps in base 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. These observations should render the new theory interesting to formal philosophy and theology.
Compare the above with Gödel’s premises of the modal ontological argument (we give the consistent variant of Scott):
Necessary existence (NE) and essence () are defined as (the other definitions are as above):
Informally: a property is the essence of an entity if (i) has property and (ii) necessarily entails every property of . Moreover, an entity has the property of necessary existence if the essence of is necessarily instantiated.
We also give informal readings of Gödel’s axioms: A1 says that one of a property or its complement is positive. A2 states that a property necessarily entailed by a positive property is positive, and A3 is as before. A4 expresses that any positive property is necessarily so. A5 postulates that necessary existence is a positive property. Axiom B (symmetry of the accessibility relation associated with modal -operator) is added to ensure that we are in modal logic KB instead of just K.333B’s counterpart is implied and could be used instead.
Using Gödel’s premises as stated it can be proved that a Godlike entity possibly and necessarily exists, and this proof can be verified with the computer. However, as is well known [Sobel1987], modal collapse is implied; see also Fitting fitting02:_types_tableaus_god and Sobel sobel2004logic for further background information.
Recent results of Benzmüller and Fuenmayor J52 show that different modal ultrafilter properties can be deduced from Gödel’s premises. These insights are key to the new argument presented in this paper: If Gödel’s premises entail that positive properties form a modal ultrafilter, then why not turning things around, and start with an axiom U1 postulating that positive properties are an ultrafilter? Then use U1 instead of A1 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 the new modal ontological argument presented above, where U1 has been replaced by A1’ and where A2 has been strengthened into A2’ accordingly.
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 and frustration steps in between are dropped.
Paper structure: An SSE of HOML in HOL is introduced in Sect. 2. The foundations outlined there ensure that the paper is sufficiently self-contained; readers familiar with the SSE approach may simply skip it. Modal ultrafilter are defined in Sect. 3. Section 4 recaps the Gödel/Scott variant, and then an ultrafilter-based modal ontological argument is presented in Sect. 5. This new argument is further simplified in Sect. 6, leading to our new proposal based on axioms A1’, A2’ and A3 as presented above. Related work is mentioned in Sect. 7.
Since we develop, explain and discuss our formal encodings directly in Isabelle/HOL [Nipkow et al.2002], some familiarity with this proof assistant and its underlying logic HOL [Andrews2002] is assumed. The entire sources of our formalization are presented and explained.
2 Modeling HOML in HOL
Related work focused on the development of various SSEs, cf. [Benzmüller2019, Kirchner et al.2019] and the references therein. These contributions, among others, show that the standard translation from propositional modal logic to first-order (FO) logic can be concisely modeled (i.e., 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. 1.444In 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 all new abbreviations and definitions, we usually 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. Note that HOL [Andrews2002] 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 , 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 in SSE technique, and in the remainder of this article we use of phrases such “world-lifted” or “-type” terms to emphasize this conversion in the SSE approach. , which stands for ( is the function type constructor in HOL), is the type of world-lifted, intensional properties. and , which abbreviate and , are the types associated with unary and binary modal logic connectives.
The modal logic connectives are defined in lines 11–24. 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, is inverting properties of types ; this operation occurs in some Gödel’s axiom A1. and are defined in lines 23–24 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 27. In line 28, user-friendly binder-notation for is additionally defined. Instead of distinguishing between and as in our illustrating example, these symbols are overloaded here. The introduction of the possibilist -quantifier in lines 29–30 is analogous.
Further actualist quantifiers, and , are introduced in lines 33–37; their definition is guarded by an explicit, possibly empty, existsAt (infix @) predicate, which encodes whether an individual object actually “exists” at a particular possible world, or not. These additional actualist quantifiers are declared non-polymorphic, and they support quantification over individuals only. In the subsequent study of the ontological argument we will indeed apply and for different types in the type hierarchy of HOL, while we employ and for individuals only.
Global validity of a world-lifted formula , denoted as , is introduced in line 40 as an abbreviation for .
Consistency of the introduced concepts is confirmed by the model finder nitpick [Blanchette and Nipkow2010] in line 43. 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 44–47 its 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 counter models for actualist quantification.
The SSE of HOML in HOL is faithful for base modal logic K.
Follows [Benzmüller and Paulson2013]. ∎
Theory HOML thus successfully models base modal logic K in HOL. To arrive at logic KB the symmetry axiom B as shown earlier can be postulated.
3 Modal Ultrafilter
Theory ModalUltrafilter, see Fig. 2, imports theory HOML and adapts the topological notions of filter and ultrafilter to our modal logic setting. For an introduction to filter and ultrafilter see the literature, e.g., [Burris and Sankappanavar1981].
Modal ultrafilter are introduced in lines 18–19 as world-lifted characteristic functions 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. An 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 (line 9).
A Filter , see lines 12–15, is required to
be large: , where U denotes the full set of -type objects we start with,
exclude the empty set: , where is the world-lifted empty set of -type objects,
be closed under supersets: (world-lifted -relation is defined in line 7), and
be closed under intersections: (where is defined in line 8).
Benzmüller and Fuenmayor J52 have studied two different notions of modal ultrafilter (called - 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 -ultrafilter.
4 Gödel’s Modal Ontological Argument
The full formalization of Scott’s variant of Gödel’s argument, which relies on theories ModalUltrafilter and HOML, is presented in Fig. 3. Line 3 starts out with the declaration of the uninterpreted constant symbol , for positive properties, which is of type . is thus an intensional, world-depended concept.
The premises of Gödel’s argument, as already discussed earlier, are stated in lines 5–24.555 Remark: whether we use actualist or possibilist quantifiers for individuals in the definition of or turned out irrelevant in this paper, and we consistently use actualist quantifiers in the remainder.
An abstract level “proof net” for theorem T6, the necessary existence of a Godlike entity, is presented in lines 26–34. 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 six subproofs and their dependencies have been automatically explored 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 smt are trustworthy components of Isabelle/HOL’s, since they internally reconstruct and check each subproofs the proof assistants small and trusted proof kernel. Using the definitions from Sect. 2, one can also reconstruct and formally verify all proofs with pen and paper.666Reconstruction of 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 ongoing work.
The presented theory is consistent, which is confirmed in line 37 by model finder nitpick; nitpick reports a model (not shown here) consisting of one world and one Godlike entity.
Validity of modal collapse (MC) is confirmed in lines 40–47; 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 50–57. This was key to the idea of taking the modal ultrafilter property of as an axiom U1; see the next section.
5 Ultrafilter Modal Ontological Argument
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 16–28 in Fig. 4: 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 31; again a model with one world and one Godlike entity is reported.
Most interestingly, modal collapse MC now has a simple counter model as nitpick informs us. This counter model consists of a single entity and two worlds and with Trivially, formula is such that holds in but not in , 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
an intensional, world-depended concept. In our counter model for MC the
extension of for world has the above and
as its elements, while
in world we have and
tuple notation we note
In order to verify that is a modal ultrafilter we have to verify that 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 . Is also easy to verify that is closed under supersets and intersection in both worlds.
Note that in this counter model 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 from .
nitpick is e.g. capable of computing and displaying all partial
modal ultrafilters in this counter model: 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 Modal Ontological Argument
What modal ultrafilters properties of are actually needed to support T6? Which ones can be dropped? Experiments with the computer confirm that, in modal logic K, the ultrafilter conditions 1-3 from Sect. 3 must be upheld for , while 4 can be dropped. However, it is possible to merge condition 3 (closed under supersets) for with Gödel’s A2 into A2’ as shown in line 18 of Fig. 5. Moreover, instead of requiring the universal set/property to be a positive property, we postulate that self-identity , which is extensionally equal to U, is in . Analogously, we replace in ultrafilter condition 2 for by self-difference . Self-identity and self-difference have used frequently in the history of the ontological argument, which is part of the motivation for this switch.
Now, from the definition of (line 13) and the axioms A1’, A2’ and A3 (lines 17–19) theorem T6 immediately follows: in line 22 several theorem provers integrated with sledgehammer report a proof in about one second when running the experiments on a standard notebook. Moreover, a more detailed “proof net” is presented in lines 23–28; the proof argument is analogous to what has been discussed before.
Consistency is confirmed by nitpick in line 31, and a counter model (similar to the one discussed in the previous section) is reported to MC in line 34.
In lines 37–43 further questions are answered experimentally: neither A1, nor A4 or A5, of the premises we dropped from Gödel’s theory are implied anymore. Also Monotheism is not implied (line 44); by postulating A1 it can be enforced. Since some of these axioms, e.g. the strong A1, have been discussed controversially in the history of Gödel’s argument and since also MC is independent, it is justified to claim that we have arrived at a philosophically and theologically potentially relevant simplification of the modal ontological argument.
7 Related Work
Fitting fitting02:_types_tableaus_god has 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 introduces a sufficiently expressive HOML enhanced with means for the explicit representation of intensional terms and their extensions; see also the intensional operations formalized by Fuenmayor and Benzmüller C65,J52 in the course of their analysis and verification of the variants of Fitting and Anderson Anderson90,AndersonGettings contributions.
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 1 of the 3 premises used in their reconstruction of Anselm’s ontological argument (in [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 [Alama et al.2015].
More recent related work makes use of higher-order proof assistants. Besides some already given references to the work of Benzmüller and colleagues, this includes John Rushby’s Rushby study on the Anselm’s ontological argument in the PVS proof assistant and Blumson’s Blumson related study in Isabelle/HOL.
Gödel’s modal ontological argument stands in prominent tradition of western philosophy. The ontological argument has its roots in the Proslogion (1078) of Anselm of Canterbury and it has been picked up in the Fifth Meditation (1637) of Descartes 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 ultrafilter), and subsequently we have developed a significantly simplified modal ontological argument that avoids some axioms and consequences in the new theory, including modal collapse, that have led to criticism in the past.
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 sharp and explainable models of the metaphysical concepts we have studied. In particular, we have illustrated how state of the art theorem proving systems, in combination with latest knowledge representation and reasoning technology, can be fruitfully utilized to explore and contribute new knowledge to theoretical philosophy and theology.
For comments and/or support of this work and prior papers I am grateful to, among others, D. Fuenmayor, B. Woltzenlogel-Paleo, D. Scott, E. Zalta, R. Rojas, A. Steen, D. Kirchner, L. van der Torre, X. Parent, E. Weydert, D. Streit.
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