Strategies for stage II of cosmological arguments

Abstract

The following article will examine three argumentative strategies to address a recent topic of debate in the philosophy of religion known as the “Gap Problem.” It aims to study the “Stage II” of cosmological arguments, where the goal is to establish the theistic properties or attributes that identify the first cause or necessary being with the concept of God. The unique contribution of this study lies in the formalized and systematic presentation of the various solutions proposed by authors in the philosophical field, synthesizing their central ideas and presenting them in the form of arguments.

Introduction

Cosmological arguments for the existence of God constitute the most popular and studied family of arguments in favor of philosophical theism. They begin with a phenomenon from the natural order (the beginning of the cosmos, contingency, the hierarchy of causality), and reason to the existence of a first cause, necessary being, or foundation of reality that explains this phenomenon. However, starting with the work of William Rowe (1998), philosophers have been careful to identify the two "stages" present in this type of argument:

Stage I: Establishing the existence of a first cause/necessary entity.

Stage II: Establishing that this first cause/necessary entity is God.

Historically, the development of the cosmological argument has focused on establishing Stage I, while Stage II has not received the same attention.¹ Therefore, in this article, we will focus on the study of Stage II, presenting the different approaches and arguments put forth by contemporary philosophers to "close" the gap between these two stages and solve the so-called "Gap Problem."

Is the “Gap Problem” a real problem?

Conceptually, there is a central issue with the so-called “Gap Problem,” and that is that it doesn't pose a true problem for the theist. This is because, generally, theistic arguments (in this case, cosmological arguments) are presented as pieces of evidence in favor of theism as a metaphysical worldview. The role of the argument in this case is to demonstrate the reasonableness of theism and increase its probability, even if Stage II of the argument has not been analyzed.

To illustrate this point, let's use a generic cosmological argument as an example:

• 1. Something exists.

• 2. If everything is contingent, then there is no external explanation for contingent things (why contingent things exist).

• 3. There is an external explanation for contingent things.

• 4. Therefore, not everything is contingent.

• 5. Therefore, something non-contingent exists.

• 6. Therefore, there exists a necessary entity.²

As we can see, the conclusion of this argument is that a necessary entity exists. This constitutes what is called "Stage I." However, on its own, this argument provides evidence in favor of theism since the existence of a necessary entity is expected and better predicted under the hypothesis of theism than under naturalism. In Bayesian terms, this could be expressed as P(C|T) > P(C|~ T), where C is the existence of the necessary entity, T is the hypothesis of theism, and ~ T is an alternative non-theistic hypothesis. In other words, the existence of a necessary entity is more likely under the hypothesis of theism than under alternative hypotheses. This is because theism is the worldview that posits the existence of God, a being that, among other attributes, possesses necessary existence. Therefore, given theism, the probability of such a being's existence (given the truth of the worldview) is 100%, while with other hypotheses, the probability of the existence of such a being (given the truth of the worldview) is much lower. For example, under naturalism, the existence of a necessary being is neither predicted nor expected; on the contrary, under naturalism, the probability of the existence of a necessary entity is very low, almost zero.³

Therefore, even if the so-called "Stage II" is set aside, the primary conclusions of cosmological arguments provide evidence in favor of theism. However, the "Gap Problem" offers an additional perspective on these types of arguments: if we manage to establish or identify theistic attributes in this necessary entity, the probability of theism will be even higher, as the correct prediction made by the hypothesis will be even more specific. Suppose, for example, we manage to establish that this necessary entity is also a personal and omnipotent being. These attributes are predicted 100% by the hypothesis of theism, which further distances us from alternative hypotheses (non-theistic ones). So, Stage II can provide additional evidence to strengthen the theistic case, even though strictly speaking, it may not be necessary to establish its reasonableness and plausibility as the correct metaphysical worldview per se.

Taking this point into account, we will now analyze the strategies proposed by different authors to reinforce the theistic case through Stage II and address the so-called "Gap Problem" in cosmological arguments.

Abductive strategy

The abductive strategy aims to establish, through various theoretical criteria, that the best explanation for the nature of the necessary entity or first cause implies its identification with God. This approach has been recently advocated by Byerly (2018)⁴ andMiksa (2023),⁵ and we will now delve into each of their proposals before summarizing each one into arguments.

Explaining necessary existence

This argument has been proposed by Byerly (2018) and consists of analyzing what kind of entity could possess necessary existence. The reasoning begins by posing the question: Why does this being or entity (N) have necessary existence? The suggested answer is that this entity is a perfect being, that is, a being that possesses all perfections, with necessary existence being one of them. This theory has the capacity to explain why this entity has necessary existence because its nature includes this property as one of its constitutive perfections.

This explanation makes use of the internal resources or characteristics of this perfect nature in order to provide a foundation for the necessary existence present in this entity. In other words, it succeeds in explaining this property in terms of N's own internal nature, which gives it explanatory power. However, this theory faces alternative rivals that compete to provide a satisfactory explanation for the necessary existence of N. Let's analyze some of them:

(a) Physical Naturalistic Explanations: This type of theory postulates the existence of natural entities, such as the inflationary segment theorized in the Standard Model, common objects in current cosmology. These explanations may initially seem attractive due to the epistemic support of the standard model in physics and inflation as a mechanism introducing contingency. However, they do not successfully explain why entity N possesses necessary existence. This is because nothing in the intrinsic nature of these natural entities provides the expected explanatory relationship that connects them to the property of existing necessarily. In this particular case, neither the high density, temperature, nor rapid expansion that characterizes the inflationary segment provides relevant information to address the question of N's necessary existence. Therefore, this alternative lacks sufficient explanatory power to surpass the perfect being theory.

(b) Exotic-Abstract Naturalistic Explanations: Another category of naturalistic explanations postulates the existence of exotic or abstract entities with the intention of providing an explanation for the necessary existence of N in terms of a specific internal nature. For example, abstract entities like mathematical objects could be candidates to explain necessary existence. In this scenario, if N were constituted as an abstract entity, its causal inaccessibility (a property that characterizes such objects) would be an internal feature of its nature that could explain why it exists in the way it does. However, despite offering some explanatory advantages over physical naturalistic explanations, this approach has the serious problem of eliminating the causal role that entity N has with respect to contingent reality. Since Stage I of cosmological arguments establishes that this necessary entity possesses the causal power to ground the existence of contingent entities, it is unacceptable to postulate that the nature characterizing N is abstract, as it would eliminate its explanatory role regarding contingency. Therefore, these abstract explanations do not make good candidates in the face of the perfect being theory, as they contradict the explanatory function that N has, as established in cosmological arguments.

(c) Non-Theistic Supernatural Explanations: Another type of naturalistic alternative involves postulating supernatural entities that possess an internal nature capable of explaining their necessary existence and grounding the existence of contingency. However, these types of explanations differ from the theory of the perfect being in that the being in question does not possess a perfect nature but one restricted to having some, but not all, perfections. The problem with these proposals is that they tend to be more complex and specific: To draw an analogy, imagine developing an argument for the existence of a black crow. Now, if we ask why this crow is black, the type of explanation we would expect would be in terms of some quality or internal property of the nature that all crows share in common. However, positing that the explanation for this property is that only some or many (but not all) crows are black is an ad-hoc explanation with no independent motivation. Additionally, this explanation generates greater overall complexity because, in this case, nothing explains why there is this restriction in the set of crows. It is simpler to posit a uniform nature characterized by the possession of all perfections. Therefore, non-theistic supernatural explanations do not provide significant explanatory advantages.

Based on this, the theory of the perfect being emerges as the most satisfactory, simple, and explanatory powerful explanation compared to naturalistic rival explanations. Postulating that the nature of N is perfect provides us with sufficient explanatory resources to establish why it has necessary existence (since it possesses all perfections, including existence in its most perfect form), as well as its role as the cause or ultimate foundation of contingency, as perfection includes the necessary causal power to fulfill this explanatory function. Thus, we can establish that, from an abductive standpoint, the best theory about the nature of N is the perfect being theory. Therefore, N is a perfect being. Here is the formalized argument:

• 1. N has necessary existence (Stage I).

• 2. The best explanation for why N has this perfection is that N is a perfect being.

• 3. Therefore, N is a perfect being.

• 4. But (3) implies being God.

• 5. Therefore, N is God.

The aesthetic virtues of theism

This second abductive strategy, proposed by Miksa (2023), involves analyzing the aesthetic virtues of causal theories in cosmological arguments. When we talk about aesthetic virtues, we refer to a type of theoretical criterion that allows us to compare and weigh various theories in the philosophical and scientific fields and infer which one is the best explanation for a certain phenomenon. Specifically, aesthetic virtues consist of the criteria of beauty, simplicity, and unification, as described below:

(B)	Beauty	The theory (T) results in greater aesthetic consistency than its rivals.
(S)	Simplicity	T explains the same facts as rivals but with less theoretical content.
(U)	Unification	T explains more types of facts than rivals with the same amount of theoretical content.

These principles carry some epistemic weight when comparing and deciding among rival theories because they provide reasons to prefer certain virtuous explanations over others. In the case of cosmological arguments, Stage I concludes with the existence of a necessary entity N, which plays a causal explanatory role in relation to the phenomenon of contingency. However, this conclusion opens the door to a wide range of theories about the nature of this entity N. In this context, the use of the previously developed aesthetic virtues allows us to compare the different types of theories and decide which one better aligns with these criteria.

Now, theories about the nature of N can be essentially classified into two categories: the theory of the imperfect cause (IC) and the theory of the perfect cause (PC). The first category includes all explanations that postulate an essentially limited nature in N, whether in terms of power, knowledge, moral status, or another type of perfection, regardless of the non-logical restrictions posited and the degree of theoretical limitation theorized (even if it is slight). On the other hand, the second category consists of postulating a perfect nature in N, and therefore, the existence of a maximally perfect being, without non-logical restrictions on its fundamental properties.

Considering these criteria, we will make a comparative analysis of PC versus IC as follows:

(a) The Beauty of PC versus IC: Beauty is a primitive and intuitive notion that, in this context, involves forming an aesthetic judgment about a theory. Regarding PC, it seems clear that its beauty as a theory is superior to IC due to the way its qualities are formed and grouped in relation to a single concept, perfection. Whereas IC postulates the isolated existence of a series of attributes or properties that are not related or grouped in the same way. However, an objector could argue that beauty is irrelevant as an epistemic criterion when deciding between theories since its nature is fundamentally relative and personal. But even the most ardent relativists make non-relative judgments about other disciplines or activities (such as in music, nature, or certain objects). The problem with this persistent rejection is that it leads to certain absurd conclusions, such as judging random and rudimentary noise as equally beautiful as the music of a privileged artist like Beethoven.⁶ The point is that, with sufficient information and regular cognitive abilities, an individual makes rational judgments about aesthetics, so this criterion can be used in this context to lean towards the PC theory over IC.

(b) The Simplicity of PC versus IC: The criterion of simplicity involves analyzing the parsimony (i.e., how many entities are posited) and elegance (how many basic theoretical principles are required) of a theory. All else being equal, this principle inclines us to prefer simpler theories, i.e., those that postulate the fewest and simplest types of entities and theoretical principles possible. In the case of PC, it posits the existence of a single entity (N) with a single primitive property or characteristic: perfection, from which all other attributes or perfections are linked and grounded. In other words, the theory asserts the existence of only one entity and only one primitive property. On the other hand, IC does not characterize N as a perfect being, so its limited properties (such as power or knowledge, among others) are all primitive properties that cannot be explained, grounded, or reduced to another primary property, as is the case with PC. However, an objector could question whether perfection implies or entails the possession of other properties, which would not give any comparative advantage to PC over IC. But the criterion of simplicity itself leans us toward preferring a PC theory where attributes are grounded in the property of perfection, rather than a PC theory where all perfections are isolated and primitive attributes. Furthermore, even if this were true, PC maintains a higher level of simplicity in individual terms compared to IC: that is, by positing properties to a maximal degree without limitation, PC is considerably simpler than positing limited, finite, and exceptional properties, as done by IC. Scientific disciplines provide a practical example where hypotheses postulating laws without exceptions or maximal degrees of certain properties tend to be preferred and prima facie more probable explanations.

Additionally, elegance is another relevant criterion when conducting a comparative analysis of simplicity: concerning PC, the theory can be described using only one primitive term, perfection. This is also known as ideological qualitative parsimony, an epistemic virtue possessed by theories that postulate the fewest primitive ideological types, i.e., concepts that resist being defined by other concepts. In this case, PC exhibits an ideal case of ideological parsimony, as only one ideological concept is needed to describe the theory, and nothing more. However, IC presents much greater ideological complexity since it multiplies the number of primitive types considerably. Since IC cannot be reduced to a single ideological concept, it necessarily resorts to the use of subconcepts or a longer structure of primitives. Therefore, PC proves to be a much simpler theory (in terms of parsimony and elegance) than IC.

(c) Unification: This final criterion analyzes the explanatory simplicity of a theory, that is, how many types of facts it can explain with a given amount of theoretical content. Naturally, the more types of facts a theory can explain with the same theoretical content, the greater its unification. In the case of PC, perfection has the advantage (as seen in the previous argument) of being able to explain to some extent why N is a necessary being, in addition to explaining the existence of contingent reality. However, IC does not offer an internal explanation in terms of the nature of N for the necessity of its existence. Therefore, PC can explain more types of facts with the same theoretical content. Additionally, PC offers great explanatory simplicity when addressing other types of phenomena. For example, if we consider the fine-tuning of the universe, the existence of objective morality, or other aspects of reality, this theory possesses the flexibility and the necessary qualities to explain these facts, appealing solely to the perfect nature of N. However, IC is limited to explaining a very narrow range of facts, and when used as an explanation for other phenomena, its theoretical content is insufficient to address them satisfactorily and non-ad-hoc. Therefore, PC proves to be a theory with greater explanatory unification than IC.

In conclusion, the theory that postulates the nature of N as perfect better aligns with the described aesthetic virtues: it possesses greater beauty, simplicity, and unification than rival theories that postulate a limited, imperfect, or mixed nature. Thus, PC emerges as the best explanation of the nature of N in terms of theoretical superiority. Therefore, N is a perfect being. Below is the formal argument:

• 1. N has necessary existence (Stage I).

• 2. The theory of the perfect being is the most beautiful, simple, and unified explanation of the nature of N.

• 3. Therefore, the best theory about the nature of N is the theory of the perfect being.

• 4. But (3) implies being God.

• 5. Therefore, N is God.

Deductive strategy

The deductive strategy aims to establish, in a strictly metaphysical and logically deductive manner, that the necessary entity or first cause of Stage I possesses certain theistic attributes that identify it with God. This approach has been historically developed by medieval philosophers such as Thomas Aquinas, John Duns Scotus, or Avicenna,⁷ and more recently by philosophers like Rasmussen (2009), Pruss (2009), or Gellman (2000). Below, we will present some of the contemporary proposals that have been developed in philosophical literature.

Personal agency

Argument I

One of the essential attributes that can be established in Stage II is the agency of the necessary entity, i.e., its volitional capacity to act freely. To establish this characteristic, Rasmussen (2009) argues that the causal connection between contingent substances and the necessary entity (let's call it "N") suggests the presence of a non-deterministic volitional act that would establish the personality of the necessary being. To establish this attribute, Rasmussen makes use of the following principle of causality:

(C1)	Every set of contingent intrinsic properties or relations can be causally explained.8

This principle (similar to those proposed in Stage I) states that for every contingent property or contingent relation present in a concrete object/substance, it is possible that this exemplification can be causally explained.

The motivation behind C1 is highly intuitive and justified, and to illustrate it, we will use the following scenario: suppose there are two spherical objects, one red and the other blue. Both objects exist contingently and exemplify their colors contingently, meaning that neither their existence nor their colors are necessary, as they could have not existed or exemplified a different color. Naturally, the following question arises: why do these objects exist, and why do they have the colors they have? An explanation in causal terms for the contingent existence, properties, and relations of both objects should be metaphysically possible. This is what C1 claims.

The second principle that will be used states that:

(D1)	For every finite attribute A, where A consists of possessing a certain property D to the degree μ, and any concrete object x that has A, there is a degree such that it is possible for x to have D to the degree μ – e or μ + e. (see Koons (1997)).

What D1 states is that for any finite attribute possessed by a concrete object/substance, that attribute will be contingent since it could have been possessed to a greater or lesser degree. To illustrate this, we could imagine a certain object with a measurable finite power. Suppose this object possesses 100 units of power x. Therefore, this degree of power is contingent because it could have been possessed to a degree of 100 + e or 100 – e (for example, 99.999 or 100.001). Therefore, the power possessed by this object will be contingent as it could have been exemplified to a different degree.

Based on these principles, Rasmussen presents the following strategy to demonstrate that N is a personal agent: From Stage I of the cosmological argument, we know that N is a necessary entity that explains the existence of all contingent concrete things/objects that exist (let's call this set of things "L"). But if N does not possess the ability to act freely (i.e., if it were not a personal agent), then the mere existence of N would be a sufficient condition for the existence of L. Since N is a necessary entity, the existence of L would also be necessary. However, this is contradictory because L is, by definition, contingent. Therefore, N cannot be an impersonal entity.

However, one could propose an impersonal non-deterministic alternative, meaning that even though N is not a personal agent, the explanation it provides for the existence of L is indeterministic. This would imply that N has a fixed degree of probability, denoted as H, of causing the existence of L. However, due to principle (D1), this probabilistic property is contingent because it could have been possessed to a different degree. But according to (C1), this probabilistic property can be causally explained. This leads to a circular problem since N would need to possess a probabilistic property to causally explain the contingent properties and relations, including the probabilistic properties that N itself possesses in the first place.⁹ This problem arises from the assumption that N is an impersonal entity that explains the existence of L in a non-deterministic way. Therefore, N cannot be an impersonal entity with probabilistic properties to causally explain L.

Because of this, the only remaining option is that N is a personal agent. Since its causal activity is not impersonal, it follows that N is a being with volition and free will. Thus, the explanation for the existence of L will be in terms of a personal agent. Below is the formal argument:

• 1. N is a necessary entity that explains the existence of a specific L in w (Stage I).

• 2. N is a (i) personal or (ii) impersonal entity.

• 3. Suppose that (ii) N is an impersonal entity (reductio).

• 4. If (ii), then the explanation for the existence of L can be (a) impersonal deterministic or (b) impersonal non-deterministic.

• 5. Suppose (a) the explanation is impersonal deterministic.

• 6. Therefore, the mere existence of N is a sufficient condition for the existence of L in w.

• 7. Therefore, the existence of L is metaphysically necessary since N is a necessary being.

• 8. But the existence of L is contingent (by definition of "set L").

• 9. But (8) contradicts (7).

• 10. Therefore, the explanation is (b) impersonal non-deterministic.

• 11. If the explanation is impersonal non-deterministic, then N explains the existence of L under a fixed probability degree.

• 12. But if N has the property of causing the existence of L under a fixed probability degree H, it is metaphysically possible that N could have that property to the degree H + e or H–e (principle (D1)).

• 13. Therefore, that property of N is contingent.

• 14. Therefore, that property can be causally explained.

• 15. But N is the cause of all contingent things and properties in w (Stage I and principle (C1)).

• 16. But then the probabilistic property of N in w is ultimately explained by the causal activity of N.

• 17. But (16) is a circular explanation since the probabilistic property of N is explained by the causal activity of N, which requires a probabilistic property in the first place.

• 18. Therefore, N cannot have a fixed probabilistic property of causing L.

• 19. But (18) contradicts (11).

• 20. Therefore, the explanation is neither (a) impersonal deterministic nor (b) impersonal non-deterministic.

• 21. But (20) contradicts (4).

• 22. Therefore, the supposition (3) is false.

• 23. Therefore, N is not an impersonal entity.

• 24. Therefore, (i) N is a personal entity.

Argument II

The following argument developed by Pruss (2009) to establish agency involves analyzing the type of explanation that the necessary being provides for the existence of contingent reality, specifically, what kind of causal activity allowed the existence of the contingent. When we examine the possible types of explanations, we encounter three distinct categories: scientific explanations in terms of contingent laws, conditions, and causes; personal explanations in terms of agents and their volition; and conceptual explanations.

Now, what type of explanation does the necessary being provide for the existence of contingent reality? Pruss argues that a conceptual explanation is not viable because the existence of contingent substances in reality cannot be conceptually explained by something other than the substances themselves. This is because they are self-sufficient, and even if we allow that they can be conceptually explained by their constituent parts, those parts themselves are substantial, leading to the same problem. Therefore, this option is ruled out. With respect to scientific explanations, the entities involved in such explanations are laws and contingent conditions. However, this is not possible in this scenario because, by definition, the necessary being does not exist contingently. Therefore, given this triple disjunction, the only remaining option is an explanation in personal terms. This implies that the necessary being explains the existence of contingent reality through its volitional activity, making it an intentional agent. Here is the formal argument:

• 1. N has necessary existence (Stage I).

• 2. The explanation for L is (a) scientific, (b) conceptual, or (c) personal.

• 3. Suppose (a) the explanation for L is scientific.

• 4. Scientific explanations occur in terms of contingent causes.

• 5. Therefore, the explanation for L is a contingent cause or causes.

• 6. But (5) contradicts (1) since N has necessary existence.

• 7. Therefore, (b) the explanation for L is conceptual.

• 8. However, the substances in L cannot be conceptually explained by something other than the substances themselves.

• 9. Therefore, (c) the explanation for L is personal.

• 10. Therefore, N is a personal being.

Argument III

Another argument to establish personal agency has been proposed by Craig (2008), and it consists of analyzing the type of entity that possesses or fulfills the characteristics associated with the necessary entity.

Particularly, Stage I establishes the existence of a necessary entity N with certain causal powers, from which the existence of contingent reality is grounded. However, given its nature, this entity must be essentially immaterial because material composition implies a certain level of contingency in terms of spatial location, composition, and quantity of matter, contrary to its necessary essence. But given these properties (necessity and immateriality), the types of entities that fit this description are reduced to only two: abstract objects or minds.

The first type, abstract objects, are immaterial and necessary entities that exist outside the concrete realm, understood as the causal network among objects that have the capacity to affect each other in some of their properties or relations. On the other hand, mental entities are those that possess intentionality and causal capacity.

Now, since N is the cause or ultimate foundation of contingent reality, it follows that it must possess some kind of causal power or explanatory capacity concerning contingent objects; otherwise, it could not be the ultimate causal explanation of them, contrary to the conclusion of Stage I. But if this is the case, N cannot be an abstract entity, as by definition abstract objects are outside the concrete realm and, therefore, causally inert. Due to this, the type of entity that corresponds to the nature of N is mental entities. Therefore, N is an intentional agent with mental capacities. Here is the argument in formal notation:

• 1. N is a necessary entity that explains the existence of contingent reality (Stage I).

• 2. If (1), then N is immaterial (since materiality implies contingency).

• 3. If (2), then N is (a) an abstract entity or (b) a mind.

• 4. Suppose N is (a) an abstract entity.

• 5. But abstract objects are causally inert.

• 6. But (5) contradicts (1), as in that case, N could not be the ultimate cause of contingency.

• 7. Therefore, N is (b) a mind.

Unity

Argument IV

Unity or uniqueness is the status or attribute of existing in a unique and irreplaceable manner, one of the distinctive features of monotheism in which this quality is attributed to God. However, theistic cosmological arguments in Stage I establish the existence of a necessary entity and often appeal to the principle of Occam's Razor or similar principles to argue that the hypothesis of a single necessary being is the simplest and most parsimonious way to explain contingent reality. They argue that postulating more entities of this kind diminishes the plausibility and simplicity of the hypothesis, making the singular existence of the necessary being the most reasonable position.

However, there are other arguments that go further and attempt to establish logically that the existence of more than one necessary being (a se) is impossible. The present argument that we will develop has been formulated by Gellman (2000), and it aims to demonstrate the contradiction that would arise from postulating two "creators." Suppose there are two necessary beings called N₁ and N₂. Each of them can be called a "creator" under the following definition:

(CR)	N is a creator being in w = def N is a necessary being whose causal power explains the existence of all contingent beings in w.

This characterization arises from the conclusion of the Stage I of cosmological arguments, in which such a necessary being is the ultimate explanation of contingency in a possible world. Therefore, we would have the creator being N₁, responsible for the contingent reality in w₁, and the creator being N₂, responsible for the contingent reality of w₂. Advancing with the argument, Gellman argues that the causal powers of both beings will be essential, that is, not contingent, so they will possess them in every possible world. To establish this point, he uses the following iterative principle:

(PI)	If x possesses the power to obtain the power to do A, then x already possesses that power to do A.

This principle seems self-evident, and it states that if a being has the capacity to exercise a certain power P, even if such exercise requires an intermediate instance, the being in question already possesses power P. Now, suppose that a creator being N possesses its causal powers contingently. Let's call the set of all its contingent powers P. Due to the contingency of P, its existence or instantiation in N will require an explanation. But, since the creator being N explains the existence of everything and every contingent property, it follows that N is the ultimate explanation of P, its contingent powers. Therefore, N must possess some power distinct from P, that is, a power P' through which it obtained P, its contingent powers. But since P encompasses all of its contingent powers, it follows that P' must be an essential power, and due to the iterative principle (PI), that power will include or contain all its contingent powers. But this contradicts the initial hypothesis that N possesses its powers contingently. Therefore, N possesses all its powers essentially.¹⁰

Once the essential nature of the powers of N₁ and N₂ is established, Gellman proceeds to construct a dilemma based on the concept of creative power:

(R)	x possesses creative power over y in w = def x can prevent y from creating certain contingent beings in w.

And the dilemma is as follows: Does N₂ have creative power over N₁ in w₁? In other words, can N₂ prevent N₁ from creating the contingent beings that it creates in w₁? This dilemma can only be answered in two ways: yes or no. Let's see what happens in each of these options:

Affirmative case: Suppose that N₂ indeed has creative power over N₁ in w₁. In that case, N₂, contingently, chooses not to exercise that power over N₁ since N₁ is the explanation for all the contingent beings in w₁, and for this reason, its causal activity should not have been interrupted. Now, if N₁ is indeed responsible for the existence of all contingent beings in w₁, then N₁ possesses the essential power to determine the contingent reality that will exist in w₁, and this implies that N₁ can prevent a different contingent reality from being actualized than the one it creatively decided. Therefore, N₁ has creative power over N₂ in w₁. But this contradicts the initial assumption, so N₂ has and does not have creative power over N₁. Contradiction.

Negative case: Now suppose that N₂ does not have creative power over N₁ in w₁. If that is the case, then the absence of that power is not contingent but essential: due to the principle (PI) stated earlier, neither N₁ nor N₂ can possess contingent powers, only essential ones. Therefore, if some power is absent in N₁ or N₂, that absence is essential; that power is absent in every possible world. But if that is the case, N₂ does not have creative power over N₁ even in w₂. However, N₂ is responsible for the existence of the contingent reality in w₂, so it must have creative power over N₁ in w₂ because it has the power to determine the contingent reality that will exist in w₂ and can prevent a different contingent reality from being actualized than the one it creatively decided. But this means that N₂ actually has creative power over N₁, and therefore, it must have it essentially in every possible world. But this contradicts the initial assumption, so N₂ has and does not have creative power over N₁. Contradiction.

In this way, by leading the disjunction to absurdity, Gellman manages to demonstrate that if we were to admit the existence of two creator beings N₁ and N₂, it would lead us to a contradictory scenario where N₁ has creative power over N₂, and N₂ has creative power over N₁ essentially. Therefore, the existence of two necessary beings (a se) with essential causal powers (as established in Stage I) is impossible; there can only be one.¹¹ Here is the formal argument:

• Set L = L is the totality of contingent things and their properties (attributes) in a possible world w.

• Creator being N = N is a necessary being (a se) that explains the existence of L in w.

• Creative power = x possesses creative power over y when x can prevent y from creating a certain L in w.

• 1. Suppose there are two creator beings, N₁ and N₂, which explain the existence of L₁ and L₂ in possible worlds w₁ and w₂, respectively (reductio).

• 2. N₁ and N₂ possess their powers essentially (if they were contingent, their own causal activity would explain their powers, which is circular).

• 3. If (2) then N₂ (i) has creative power over N₁ in w₁ or (ii) does not have creative power over N₁ in w₁.

• 4. Suppose that (i) N₂ has creative power over N₁ in w₁.

• 5. Since N₁ is, by definition, a creator being in w₁, the existence of L in w₁ depends on its causal activity.

• 6. Therefore, N₁ has creative power over N₂ in w₁ since N₁ can prevent N₂ from actualizing a different L in w₁.

• 7. But (6) contradicts (4).

• 8. Therefore, (ii) N₂ does not have creative power over N₁ in w₁.

• 9. But given (2), N₂ does not possess creative power over N₁ essentially (not contingently).

• 10. Therefore, N₂ does not have creative power over N₁ in any possible world (including w₂).

• 11. Since N₂ is, by definition, a creator being in w₂, the existence of L in w₂ depends on its causal activity.

• 12. Therefore, N₂ has creative power over N₁ in w₂ since N₂ can prevent N₁ from actualizing a different L in w₂.

• 13. But (12) contradicts (10).

• 14. Therefore, from premise (3) arises a contradiction (since N₂ has and does not have a creative power over N₁ in w₁, which is absurd).

• 15. Therefore, the assumption (1) is false.

• 16. Therefore, there cannot be more than one creator being N.

Argument V

This argument originates from the work of the great Thomas Aquinas in Summa Contra Gentiles (Book 1, Chapter 42), which has been analyzed and expanded upon in a contemporary context by Kretzmann (1997). It is an argument that examines the concept of necessary existence and establishes that the multiplicity of beings with this quality is impossible. This argument is very useful because it allows us to rule out the possibility of a plurality of necessary beings (a se) without reference to any additional attribute or characteristic.

The argument begins by considering a scenario in which two necessary beings, N₁ and N₂, exist. Since they are two distinct and differentiated beings, their "individuation" is due to a property or characteristic that distinguishes them, such that N₁ ≠ N₂.¹² Now, this property can either be (a) accidental or (b) a property of the necessity of their being: both options are logically exhaustive, meaning there is no third available option. Therefore, the argument starts with a main disjunction that will branch out into other options, depending on the path we take.

Let's begin by analyzing option (a):

(a) Difference by an accidental property: If what constitutes the difference between N₁ and N₂ is an accidental property D, then this property must have some explanation for its existence or instantiation. This follows from applying the principle of sufficient reason, as used in Stage I or as previously explained in other arguments like (C1). Now, the causal explanation of the accidental property D could either be (i) due to the essence of N₁ or N₂ or (ii) due to an entity external to N₁ or N₂. Suppose it's due to (i). But in that case, and since they share the same essence of necessary existence, this property will be possessed by both N₁ and N₂, so D does not serve as a distinguishing property between them. Therefore, D is explained by (ii) an entity external to N₁ or N₂. However, in that case, neither N₁ nor N₂ will be beings with necessary existence (a se) because in this case, their existence as two different beings depends on an external cause that causally explains this distinguishing property D, which contradicts their ontological independence. Therefore, the part (a) of the disjunction is false.

Now, let's analyze option (b):

(b) Difference by a property of the necessity of their being: If what differentiates N₁ from N₂ is an essential property D, then this property could either be (i) common to the essence of necessary existence or (ii) something that distinguishes them into two distinct species. If (i), then this property cannot serve as the difference between N₁ and N₂, as both share the same essence of necessary existence, so they both possess the same property D. Kretzmann uses the example of the property of being "animated." This property is present in all beings that share an animal essence, so it is of no use in distinguishing (for example) between a tiger and a worm. Therefore, the essential property D is (ii) something that differentiates N₁ and N₂ into two distinct species. In this case, D₁ would be the property that defines species E₁, and D₂ would define species E₂. Just as animals can be divided into species (tigers or worms, to use the previous example), necessary beings could also be divided into two species, in this case, E₁ and E₂.

In this scenario, a necessary being does not exist simply and directly as such (necessarily), but it must exist as E₁ or E₂, with a distinguishing property (D₁ or D₂) that defines it. This creates the problem that such a necessary being depends on something external to the essence of necessary existence (i.e., a distinguishing property) that is neither derived from that essence nor grounded in it but is a property related in a merely contingent way to its necessary essence. However, this contradicts the very notion of necessary existence because in this case, such a being depends on a property disconnected from necessary existence to exist individually. In other words, the essence of necessary existence would not be sufficient for the existence of a necessary being, as something additional and distinct from it is required for the individuation and existence of a necessary being, which contradicts its independence. Therefore, the part (b) of the disjunction is false.

From this argument, it is demonstrated that the assumption that there is more than one necessary being (a se) is false, as it generates contradictions with the concept of necessary existence in all possible cases.¹³ Below is the formal representation of the argument:

• 1. Suppose that there are two necessary beings, N₁ and N₂ (reductio).

• 2. Therefore, N₁ and N₂ differ either (a) by some property of the necessity of their being or (b) by some accidental property.

• 3. Suppose that N₁ and N₂ differ by (b) some accidental property.

• 4. If (3), then the cause of this accidental property is either (i) the necessary essence or (ii) something external.

• 5. Suppose that the cause of the distinguishing accidental property is (i) the necessary essence.

• 6. But if (i), then this accidental property will be present in both N₁ and N₂, as they both share the same necessary essence.

• 7. But (6) contradicts (5).

• 8. Therefore, the accidental property is caused by (ii) something external.

• 9. But if (ii), then the existence of N₁ and N₂ depends on an external cause that distinguishes them, contradicting their necessary existence.

• 10. Therefore, (i) and (ii) are false.

• 11. Therefore, the assumption (3) is false.

• 12. Therefore, N₁ and N₂ differ by (a) some property of the necessity of their being.

• 13. If (12), then this distinguishing property will be either (iii) something included in the common necessary nature or (iv) something that distinguishes the two natures into two species.

• 14. Suppose that the distinguishing property is (iii) something included in the common necessary nature.

• 15. But if (iii), then this property will be common to both beings with necessary existence.

• 16. But (15) contradicts (14).

• 17. Therefore, the distinguishing property is (iv) something that distinguishes the two natures into two species.

• 18. But if (iv), then N₁ will possess a property D₁ that distinguishes it from N₂, and N₂ will possess a property D₂ that distinguishes it from N₁.

• 19. But then, N₁ and N₂ will depend on these external properties for their differentiation, contradicting their necessary existence.

• 20. Therefore, (i), (ii), (iii), and (iv) are false.

• 21. Therefore, the assumption (1) is false.

• 22. Therefore, there cannot be more than one necessary being N.

Argument VI

The following argument is inspired by the medieval Christian philosopher John Duns Scotus, particularly from his work De Primo Principio (Treatise on the First Principle), and it has been recently studied by O'Connor (1996). It attempts to analyze the consequences of what would happen if a kind of necessary existence allowed for multiple instances, that is, the possibility of more than one exemplification of that nature in various individuals.

To begin, the argument starts with the assumption that indeed necessary existence, as a kind, admits multiplicity. Now, if that is the case, then concerning the kind itself, there can be nothing intrinsic that limits the possibility of instances to a particular finite number of beings. Clearly, this applies to all kinds, regardless of their type. It seems that, concerning kinds, there is nothing inherent in them that makes it impossible for there to be a limited number x of instances. Of course, given various external factors, the existence of an infinite number of individuals of a particular kind could be causally impossible, but what this argument is examining are not these extrinsic conditions but the kind itself in isolation and its potential to admit instances indefinitely.

Now, it follows from this that if a kind of necessary existence allows for an infinite number of instances, then there effectively exists an actual infinite number of necessary beings. This is not controversial and indeed follows from the very definition of necessary beings: they exist in all possible worlds, not just in some. Due to this, and unlike any contingent kind, if a kind of necessary beings admits the metaphysical possibility of its existence in an infinite form, then (and using the S5 axiom of modal logic) an infinite number of necessary beings (i.e., particular instances of the kind) exist in all possible worlds.

But here arises an obvious problem, namely, the existence of an infinite actual number of concrete objects is metaphysically impossible. This problem has been well-known from medieval thinkers to our times, and it has been illustrated in many different ways through paradoxes and various scenarios that, while allowing for the possibility of an infinite actual number of concrete things, would lead to a metaphysical absurdity. Below, I will present a simple way to illustrate the issue and justify this point, following the argument presented in Loke (2012):

The impossibility of an actual concrete infinite

Suppose we have an infinite group of people, each with a Christmas gift. Let's also assume that all the gifts are essentially the same (same size, content, shape, etc.). Now, each person places their gift in front of them on the ground, and then they take another gift as follows: person 1 takes the gift of person 2, person 2 takes the gift of person 4, person 4 takes the gift of person 8, and so on, with person n taking the gift of person 2n. What will happen in this scenario is that all the people will end up with a gift, and yet there will still be an infinite number of gifts left on the ground. However, now, if all the people put the gifts back on the ground and person 1 takes back the gift that was originally theirs, and so on, with person n taking back the gift that belonged to person n, we will see that in this case, there will be no gifts left on the ground, and everyone will have a gift in their hands.

But this is absurd: in one scenario, an infinite number of remaining gifts are left on the ground, and all people have a gift in their hands, and in the other scenario, no gifts are left on the ground, and everyone has a gift in their hands. But how? If all people took just one gift for themselves, how can it be that by performing the same action of taking back a gift (but in a different arrangement), a different number of gifts remains on the ground? What is happening here is a violation of a basic metaphysical principle, which is that numbers do not have causal power: simply modifying the quantity of objects in a set does not change the causal capacities of that set.

For example, suppose a certain thing x has zero mass. In this case, the quantity of objects x will be irrelevant regarding the measurement of its weight, as whether there is one, two, or a hundred objects x with zero mass, it will not change the fact that the set of objects has no mass, nor will it add new properties beyond the qualities inherent to the object itself. But in this scenario of infinite Christmas gifts, it seems that the presence of an infinite number of objects actually affects the causal capacity of the set of objects itself. However, this is impossible since the causal capacities of a set of objects depend on its qualities and characteristics (of the object itself), not on the number of objects. Therefore, the existence of an actual infinite number of concrete objects is metaphysically impossible.

Therefore, since the kind of necessary existence, if it were to allow multiplicity, would generate the actual existence of infinite necessary beings, it follows that the initial assumption is false: there cannot be more than one instance of the kind of necessary existence, and therefore, there can only be a single necessary being (a se). However, a possible objection to this argument could be based on the assumption that the multiplicity of a kind can potentially allow for infinite instances.

An objector could argue that perhaps only the instantiation of the kind in two individuals is possible. Perhaps only two necessary beings can exist, neither more nor less. The problem with this type of suggestion (that perhaps only a limited number of instances of a kind are possible) is that it would violate the Principle of Sufficient Reason. To illustrate the issue, imagine that there are only two necessary beings, N₁ and N₂. Then the question arises: Why has this common nature (kind) been particularized into two instances, and not three, or four, etc.? It seems that admitting a multiplicity of instances implies contingency. And because of this, there must be an explanation for this existential fact. But as we have seen, nothing in their common nature dictates or explains why this is the case, nor can an external explanation to N₁ and N₂ explain this fact, as in that case, neither of them would be necessary beings (a se), but their existence would be derived or dependent on something external.

Therefore, the argument concludes that necessary existence (a se) does not allow for multiplicity of individuated instances but can only have a single necessary being. Here is the formal representation of the argument:

• 1. Suppose that more than one necessary being N can exist (reductio).

• 2. If a kind is capable of existing in more than one individual, then concerning the kind itself, it is capable of existing in a potential infinite number of individuals.

• 3. Therefore, there can potentially be an infinite number of necessary beings.

• 4. But what is necessary must exist in every possible world (S5).

• 5. Therefore, there are infinite necessary beings.

• 6. However, the actual existence of infinite concrete objects is impossible.

• 7. But (6) contradicts (5).

• 8. Therefore, the assumption (1) is false.

• 9. Therefore, there cannot be more than one necessary being N.

Omnipotence

Argument VII

The attribute of omnipotence is another relevant aspect of philosophical theism, common to all models concerning the nature of God. The following argument is inspired by Duns Scotus and aims to establish, based on the creative act of the necessary being, that its causal power is infinite or maximal. Rasmussen (2009) has extended the argument using modal logic and possible worlds semantics, allowing for a formulation of the argument in light of contemporary metaphysics.

The argument begins with an initial reasoning about contingent reality, which states that for any set of contingent entities L, there will always be another set L' that requires a greater difficulty or causal power to actualize. This first premise is not controversial because there will naturally be a metaphysical possibility that a certain set of contingent concrete objects could include a greater number of objects. Extending this idea, it concludes that there is no limit to the difficulty of actualizing a certain contingent set:

(M)	For every set of contingent concrete objects L, there is a set L' that requires greater power to actualize.

Since Stage I establishes that the necessary being N is the ultimate cause or foundation of contingent reality, it follows that N is responsible for the existence of the contingent set L in the actual world. However, for every contingent reality in every possible world, given the necessary existence of N and its causal connection with contingency, every contingent set L' in every possible world w' will ultimately be a product of N's creative act. In conjunction with the principle (M), it follows that N's causal power cannot be limited. To claim that eventually there will be a set L that N cannot causally explain would conflict with the Principle of Sufficient Reason used in Stage I, which states that contingent reality (regardless of its specific form, characteristics, or particular arrangement) requires an ultimate causal explanation in terms of a necessary entity external to that set.

Therefore, the necessary being N that explains contingent reality possesses unlimited or maximal power, corresponding to the concept of omnipotence.¹⁴ Here is the formal representation of the argument:

• 1. For every L, there is always an L' that requires greater power to actualize.

• 2. N possesses the power to actualize a particular L in w (by definition of "creator being").

• 3. Suppose that N does not possess the power to actualize a particular L' (reductio).

• 4. Therefore, there exists an L' that cannot be causally explained.

• 5. But there cannot be a contingent set L that cannot be causally explained (since every contingent set can have an external explanation for its existence).

• 6. But (5) contradicts (4).

• 7. Therefore, the assumption (3) is false.

• 8. Therefore, N possesses the power to actualize every possible L.

• 9. Due to (1), there is no limit to N's causal power.

• 10. Therefore, the causal power of N is unlimited.

• 11. Therefore, N is omnipotent.

Argument VIII

This second argument is based on the contingency of finite attributes and their possibility to be caused to arrive at the conclusion that the necessary being does not possess power in a limited way.

To reach this conclusion, Rasmussen (2009) performs a reductio ad absurdum by assuming that the necessary being N possesses a limited degree of power. Using the principles (C1) and (D1) that we presented earlier, the argument analyzes the consequences of applying them to this scenario.

(C1) establishes that every contingent property or relation can be causally explained. In conjunction with (D1), which states that any finite attribute possessed to a particular degree μ could have been instantiated to a degree μ – e or μ + e, it follows that any finite attribute is contingent and admits an explanation in causal terms, meaning that something external can explain the possession of that attribute by the entity that possesses it.

Now, the problem with assuming that N possesses power to a limited degree is that it would lead us to the conclusion that this property of N admits an explanation in causal terms. But this implies the absurd conclusion that N itself explains its contingent properties. Since N explains contingent reality (i.e., contingent objects and properties), it follows that N must possess some kind of power to explain its own contingent powers, which is circular. But this absurd scenario arises from the assumption that N possesses power to a limited degree. Therefore, the assumption is false, and consequently, N possesses power in an unlimited way.

A simpler way to see the problem is as follows: a limited degree of power in N, and therefore contingent, would generate a certain level of dependence in the fundamental nature of N. That is, assuming that a necessary being possesses a limited degree of power implies that N possesses an intrinsic property contingently. But this is contradictory to the very notion of necessary existence, as the intrinsic properties (i.e., essence) of a necessary being are instantiated necessarily in all possible worlds. Asserting that an intrinsic property is instantiated contingently means that it does not exist in all possible worlds, and consequently, it would compromise the necessary existence (a se) of N. Therefore, N does not possess a limited degree of power but is omnipotent. Here is the formal argument:

• 1. N possesses a certain causal power (by definition of "creator being").

• 2. Suppose that N possesses a certain degree of limited power P (reductio).

• 3. Therefore, N could possibly have power to the degree of P + e or P–e.

• 4. Therefore, the degree of power P of N is contingent.

• 5. Therefore, this property can be causally explained.

• 6. But N is the cause of all contingent things and properties in w (Stage I and principle (C1)).

• 7. Therefore, the contingent power of N in w is ultimately explained by the causal activity of N.

• 8. But (7) is a circular explanation since the power of N is explained by the causal activity of N, which requires power in the first place.

• 9. Therefore, the power of N cannot be contingent.

• 10. But (9) contradicts (4).

• 11. Therefore, the assumption (2) is false.

• 12. Therefore, the power of N is unlimited.

• 13. Therefore, N is omnipotent.

Argument IX

The third argument we will present is based on Thomas Aquinas' idea of creation ex nihilo and the absence of passive potentiality to demonstrate the omnipotence of the first cause. Madden and Mancha Jr. (2005) use Aquinas' central argument in the Summa Theologiae (Book 1, Chapter 45) to construct a new version of it, which deduces that the being that explains the existence of contingent reality possesses unlimited power.

The argument begins with the assumption (established in Stage I) that there exists a certain agent N that creates a world ex nihilo at time t. When speaking of a "world," we refer to a maximally compossible set of states of affairs that describe reality. Therefore, this creative act C carried out by N assigns truth values to these states of affairs.¹⁵ Additionally, this creative act C is performed ex nihilo, which in this context means that no substance or prior causal event intervenes in the instantiation of this world.¹⁶

Given the nature of this creative act, it is deduced that N acts without any extrinsic limitation to its causal power. This is because any substance or event distinct from N is posterior to the creative act C in question and does not participate either wholly or partially in this action. What this means is that no logically possible state of affairs could have counterfactually prevented N from creating, given its explanatory and causal priority.

At this point, and given N's unrestricted act, it can be maintained that any other creative act C' of the same type could be performed by N at time t. This follows from N's own power, free from extrinsic restrictions: that any other instance of creative act would also be free from any external condition or logically possible state of affairs that could block it. To illustrate this point, Madden and Mancha Jr. provide the following example: imagine a person who is 5′8" tall and can climb stairs with small steps. Now, imagine another staircase with steps that are 6′5" apart. Does the fact that this agent can climb the first staircase imply that they can also perform another instance of stair climbing, such as the second one? It seems not, and this is because the agent is related to other entities in such a way that they have extrinsic limitations on their power that prevent them from performing other instances of stair climbing. However, if the agent in question could create ex nihilo, it follows that no extrinsic limitation (i.e., no state of affairs) could counterfactually prevent them from climbing other types of stairs. In this context, the staircase with a 6′5" distance between steps would not provide an extrinsic limitation against the agent (since no logically possible state of affairs could do so), so he could climb it. This is the case with N, the being that creates ex nihilo. Therefore, N has the power to actualize any creative act of type-C at time t.

Now, given that the creative act C, which involves the ability to actualize a maximal compossible set of states of affairs (a "world," as we have defined it), also implies in some way attributing truth value to them, and therefore, this act in time t will also affect subsequent states of affairs after t. In other words, its creative capacity arranges compossible states of affairs that occur later or following this creative moment, so N's capacity is not limited solely to that initial moment t.

But this implies that ultimately, N has the ability to actualize any compossible set of states of affairs, that is, to make or produce any logically possible description. But this is what omnipotence fundamentally means. Therefore, if N is the necessary being that through its causal activity explains reality or our "world" ex nihilo, then N is necessarily omnipotent. Here is the argument in formal form:

• World = a world is a maximally compossible set of states of affairs that exhausts all of reality.

• Creative act C = an action in which an agent actualizes a world without any prior causal substance or event.

• 1. If N explains the existence of a world at time t, then there is no extrinsic limitation that could have counterfactually prevented N's creative act C.

• 2. There is no logically possible state of affairs that could have prevented N from actualizing C at time t.

• 3. If there is no logically possible state of affairs that could have prevented N from actualizing C at time t, then at time t, N could have actualized any creative act of type-C that is logically possible.

• 4. Therefore, N could have actualized any type-C action at time t.

• 5. For every x, if there is a time t at which x could actualize any type-C action, then x could arrange any compossible state of affairs for any time subsequent to t.

• 6. N could have arranged to actualize any compossible state of affairs.

• 7. But if x has the power to arrange any compossible state of affairs, then x is omnipotent.

• 8. Therefore, N is omnipotent.

Argument X

A more direct version of the Thomistic argument presented earlier has been elaborated by Kretzmann (1997), using the notion of active power and its relationship with passive potentiality for the production of a certain effect.

The argument begins by defining how active power is measured: an agent x, in the production of a certain effect or in actualizing a certain state of affairs, utilizes a certain amount of passive potentiality. This means that the agent depends on certain external circumstances or entities that must be present and contribute to the total production of the effect in question. Therefore, the presence of passive potentiality in an agent indicates its dependence on certain circumstances outside itself for the realization of a certain effect. Naturally, the greater the amount of passive potentiality required by the agent to bring about a certain effect, the lower the degree of active power it possesses.

To illustrate this concept, Kretzmann provides the following example: suppose agent A draws a picture of a house on a sheet of paper, and agent B traces the already completed drawing. In this case, B has less active power because it used more passive potentiality than A to perform the action: B required the sheet of paper and additionally the completed drawing of the house to make the trace, while A only needed the sheet of paper to make the drawing. Therefore, since B uses more passive potentiality than A to bring about its respective effect, it follows that B has less active power than A. In summary, the less passive potentiality is required (i.e., the less dependence on external events or entities contributing to the effect), the greater the active power exhibited by the agent.

Expanding this analysis to the necessary being N, who is the ultimate explanation or causal foundation of all contingent reality, it follows that every event, circumstance, or entity outside of N is logically posterior to him and therefore dependent on his causal activity. Therefore, at the logical moment when N creates, nothing other than N could have possibly contributed to the production of that creative effect. Therefore, N is absent of all passive potentiality. But since less potentiality used implies greater active power, and since the least possible potentiality implies the maximum possible active power, it follows that if an agent produces an effect in the total absence of passive potentiality, its active power is infinite. Mathematically, we can represent active power A as the quotient of the produced effect E and the used passive potentiality P. Since P = 0, if N brings about a certain effect, then A = 1/0, resulting in A = ∞. Therefore, given the absence of passive potentiality in the creative act, it follows that N is omnipotent. Here is the formal argument:

• 1. The degree of active power of an agent x varies inversely with the amount of passive power used to actualize a certain effect E.

• 2. N is the ultimate explanation of L in w (Stage I).

• 3. Since N is explanatorily primary in w, every substance or event external to N is causally dependent on its creative act C.

• 4. Therefore, the creative act C does not presuppose any passive power.

• 5. But (4) implies the actualization of a certain effect E without any passive power.

• 6. Therefore, and given (1), the active power of N is infinite.

• 7. Therefore, N is omnipotent.

Omniscience

Argument XI

The attribute of omniscience is another classical aspect of philosophical theism, implying the possession of unlimited or maximal knowledge. This first argument uses the same strategy as previously employed to demonstrate that possessing finite knowledge implies contingency and that, therefore, the necessary being N cannot possess this quality to a limited degree.

By utilizing the principles (C1) and (D1), Rasmussen (2009) conducts a reductio ad absurdum as follows: Suppose that N possesses a limited degree of knowledge K. This initial assumption naturally follows since N, as we have already established, is an agent with volition and therefore capable of having knowledge. However, due to (C1), it follows that any finite attribute, instantiated to a limited degree, implies contingency, and this allows for the possibility of an external causal explanation, as per (D1). In other words, the degree of knowledge K possessed by N is contingent and thus could admit an explanation in terms of something external that is the cause of the instantiation of this attribute.

Again, as argued previously, this scenario leads us to the absurd conclusion that N itself explains its contingent properties. This is because N, being the ultimate explanation of contingent reality (i.e., contingent objects and properties), must possess some form of knowledge to cause that contingency, including its own finite knowledge, which is circular, as it would be using its knowledge to explain the possession of that knowledge itself. Alternatively, we can say that if an intrinsic property is instantiated contingently, it means that it does not exist in all possible worlds, and consequently, it would compromise the necessary existence (a se) of N. Therefore, the assumption that N possesses limited knowledge is false, and it follows that N is omniscient.¹⁷ Here is the formal argument:

• 1. N possesses certain knowledge due to its volitional agency.

• 2. Suppose N possesses a certain degree of limited knowledge K (reductio).

• 3. Therefore, N could possibly have knowledge to the degree K + e or K–e.

• 4. Therefore, the degree of knowledge K of N is contingent.

• 5. Therefore, this property can be causally explained.

• 6. But N is the cause of all contingent things and properties in the world w (Stage I and Principle (C1)).

• 7. Therefore, the contingent knowledge of N in w is ultimately explained by N's causal activity.

• 8. But (7) is a circular explanation, as N's knowledge is explained by N's causal activity, which requires knowledge in the first place.

• 9. Therefore, N's knowledge cannot be contingent.

• 10. But (9) contradicts (4).

• 11. Therefore, the assumption (2) is false.

• 12. Therefore, N's knowledge is unlimited.

• 13. Therefore, N is omniscient.

Argument XII

A second argument proposed by Hoffman and Rosenkrantz (2002) focuses on the analysis of omnipotence and what this attribute implies in relation to omniscience. The argument is based on the idea that the possession of power implies not only ability but also opportunity. An omnipotent being is one that has efficacy in its will: if certain circumstances or external obstacles can prevent this being from exercising its abilities, then its power is not maximal but is restricted to some extent.

Now, for an omnipotent being to be able to exercise its abilities in every logically possible context, it must possess the necessary information to act with perfect efficacy and without extrinsic restrictions. But if omnipotence implies the power to actualize any logically compossible state of affairs, then its knowledge must also extend to relevant information in all possible scenarios. But any state of knowledge less than omniscience would imply a restriction in the opportunity for action of an omnipotent being in some logically possible scenarios. Therefore, since the necessary being (as we have seen) is omnipotent, it follows that it is also omniscient. Below is the argument in formal terms:

• 1. N is omnipotent (arguments (VII)-(X)).

• 2. Power implies both ability and opportunity.

• 3. If (2), then an omnipotent being cannot be restricted by external circumstances from exercising its abilities.

• 4. Suppose N has finite knowledge (reductio).

• 5. If (4), then there will be logically possible scenarios in which N does not have the opportunity to exercise its abilities.

• 6. But (5) contradicts (3).

• 7. Therefore, the assumption (4) is false.

• 8. Therefore, N has infinite knowledge.

• 9. Therefore, N is omniscient.

Omnibenevolence

Argument XIII

Omnibenevolence or moral perfection is another of the central attributes of classical models of theism and constitutes the possession of a maximally good character and behavior with respect to morality. The following argument, primarily proposed by Swinburne (2016) and Weaver (2015), is based on certain metaethical theses that, in conjunction with other descriptive attributes we have established, will seek to demonstrate how omnibenevolence is deduced and linked from omnipotence and omniscience.

The argument begins by analyzing the concept of action: when we say that an agent performs an action, that action is mobilized by the existence of a purpose or reason that the agent holds, even if it is minimal. For an agent to have a reason to act means that they consider certain achievable states of affairs as good through that action, either indirectly (valuing a subsequent state of affairs) or directly (valuing the action itself).

Now, if an agent has decisive reasons not to take a certain action A, and yet performs A, the explanation for such behavior goes beyond mere reasons for action, so non-rational factors come into play. It becomes unintelligible to claim that a certain agent performs a specific action while also having decisive reasons not to perform it, unless external factors beyond their control influence their behavior, which is known as a constraint of the will.

Based on these concepts, we can now introduce the two main metaethical theses of this argument: moral realism and moral rationalism. With respect to the former thesis, there is not much to say since it is the starting point for any significant analysis of moral goodness. What moral realism maintains is that moral truths exist: that certain things are objectively good and bad, independently of the desires or motivations of agents.¹⁸ On the other hand, moral rationalism is a metaethical thesis that asserts:

(MR)	If a moral agent x determines that action A in situation S is good (or obligatory), then in S, x will be motivated to do A, or it will be practically irrational

What this thesis means is that if there is a moral agent capable of deliberating on moral issues, who knows that a certain action is morally correct in a certain context, then in that context, the agent will be motivated to perform that action. Otherwise, the agent will be subject to non-rational influences (what we have called a constraint of the will).

Taking this thesis into account, we can now see how omnibenevolence can be deduced from omnipotence and omniscience: as we have seen, the necessary being N is omnipotent and omniscient. Being omnipotent, no external factor or event can causally influence its efficacy of the will: this means that nothing other than N can ultimately determine how N will act. This implies that its actions only arise based on objective reasons for action: no irrational external influence can generate constraints on its will. To suggest otherwise would compromise its omnipotence because we would be postulating that certain external factors, beyond N's control, can ultimately determine its actions, which implies a limitation in its power. Therefore, we can say that N is also perfectly rational (or free), in the sense that its actions are not influenced by any external causal factor, and therefore, it is guided solely by rational considerations (since every agent is motivated by some reason to act). N is also an omniscient being, which means it knows all true propositions. Consequently, and given moral realism, propositions about morality (i.e., what is good and bad in every logically possible situation) are known by N. Finally, if we introduce the thesis (MR) into the equation, it follows that the necessary being N will always act perfectly good: since N is perfectly rational, its actions will be motivated solely by objective reasons for action. Given its omniscience, N knows all moral truths, i.e., all moral reasons for action in every logically possible situation. And according to moral rationalism (MR), N will be motivated to act based on these reasons. Therefore, N is perfectly good or omnibenevolent. The following is the formal representation of the argument.

• 1. N is omnipotent and omniscient (arguments (III)-(VI)).

• 2. Due to its omnipotence, N's will cannot be limited by external influences.

• 3. Therefore, N's will is not restricted by non-rational influences.

• 4. Therefore, N is perfectly rational.

• 5. Due to its omniscience, N knows all moral truths.

• 6. Due to its perfect rationality, N always acts in accordance with moral truths (thesis (MR)).

• 7. Therefore, N always acts in a perfectly moral way.

• 8. Therefore, N is omnibenevolent.

Eternity

Argument XIV

Eternity can be ascribed to entities that neither begin nor cease to exist, and therefore possess a mode of permanent existence. In the case of N, the reason for its eternal existence is evident: anything that begins or ceases to exist is characterized as a contingent entity (meaning its non-existence is possible) and therefore cannot be characterized as a necessary entity. But N is a being that exists in a metaphysically necessary way, as established in Stage I. Therefore, N is eternal. This leaves open the question of whether N exists in a timeless or temporally eternal manner. The essential point is that its mode of existence is permanent, whether outside or within the temporal dimension. Below is the argument in formal terms:

• 1. Everything that begins or ceases to exist is contingent.

• 2. N is a necessary being (Stage I).

• 3. Therefore, N is not contingent.

• 4. Therefore, N neither begins nor ceases to exist.

• 5. Therefore, N is eternal.

Immateriality

Argument XV

Immateriality is the property of being an entity devoid of material composition, spatial location, or extension, and thus outside the realm of the fundamental laws of matter. In the case of the necessary being N, being metaphysically necessary, it is quite straightforward to elucidate why it must be essentially immaterial.

Material entities possess a series of finite and quantitative characteristics, such as mass, charge, velocity, energy, etc., and each of these properties constitutes limited and graded attributes. Now, given the nature of material composition, it is evident that any entity with such a nature will exist contingently since these quantitative characteristics could be possessed to a slightly higher or lower degree. Causal principles like (D1) that we have developed earlier allow us to conclude that every finite and gradable property is instantiated contingently, opening the possibility that its possessor could have had that property to a different degree. But since N is a metaphysically necessary being, it follows that its fundamental nature is instantiated necessarily, and therefore, we cannot characterize it as essentially material because that would contradict its necessary nature. Therefore, N is an immaterial entity, devoid of all physical and spatial composition or limitation. Here is the formal argument:

• 1. N is a necessary being (Stage I).

• 2. Suppose that N is essentially material (reductio).

• 3. Materiality implies instantiating finite properties to a degree μ.

• 4. If (3), then materiality implies contingency.

• 5. But (4) contradicts (1).

• 6. Therefore, the assumption (2) is false.

• 7. Therefore, N is an immaterial being.

Perfection

Argument XVI

Perfection is the fundamental definition or attribute of God from which all other omni-attributes stem. When we speak of perfection, we refer to the quality of being maximal with respect to positive properties (perfections) and the absence of any limit or imperfection in that fundamental nature. A positive aspect or property is one that contributes to increasing the intrinsic value or greatness of a being, such as knowledge, power, or goodness. Therefore, a supreme or perfect being is one that possesses a maximal nature without limits with respect to its positive properties.¹⁹ The following argument developed by Rasmussen (2024) uses the notion of necessary existence or fundamentality to show that the necessary being N must have a perfect nature that excludes any limited or imperfect property.

First, the argument analyzes the nature of limited properties: When we talk about limited entities, we refer to anything that possesses a finite or non-maximal fundamental nature.²⁰ For example, having a finite amount or degree of mass, size, power, etc., constitutes a limit in the nature of a being. As we have seen previously through principles like (D1), this type of property brings us into the domain of contingency, meaning the possibility that such a limited property could have been instantiated to a different degree. Based on this metaphysical principle, the argument establishes that anything that is limited can have an external explanation.

To illustrate this point, consider the following example: a human being is dependent, meaning it is not a being that exists out of necessity of its own nature. This means that the existence of a human being has an explanation external to itself (for example, their parents giving birth to them). Following this reasoning, we can think that every human being (regardless of their age, size, height, intelligence, etc.) is dependent, meaning their existence is explained externally.

A second illustrative example could be the existence of a mountain. Imagine a mountain with two peaks. This mountain is dependent: its shape is due to external factors, such as erosion and the pattern in which wind and water affected it. But now, imagine a mountain with two thousand peaks: in this case, is there an external explanation for the shape and number of peaks of the mountain? Certainly, there is, as the mere difference in the shape or size of a mountain is irrelevant to the need for an external explanation.

Therefore, based on examples of the same nature, a fundamental principle explaining this need for an external explanation can be deduced:

(L)	All limits are categorically uniform with respect to their dependence on an external explanation.21

This means that mere differences in limits do not change the fact that they are dependent or explainable by something external. Regardless of the limit we are talking about (whether it's size, shape, power, etc.), a change in the quantity or degree of that limit does not eliminate the need for an external explanation: it will still be dependent (in the absence of a reason to the contrary).

Now, analyzing the case of N, this being is constituted as a metaphysically necessary or fundamental entity, which means its existence and nature have ontological independence: no external entity or explanation can explain or cause its existence or essential properties, as that would create a level of dependence in N, contradicting its fundamentality. But if this is the case, then any imperfection or limit in its positive properties would generate this tie of dependence or need for an external explanation that grounds these limits. Therefore, its nature excludes any instances of limited properties that imply the possibility of dependence or external explanation. Ultimately, this reveals that the fundamental nature of N is perfect, as any lower term or limited characterization of its essential properties would generate the need for an external explanation of that particular instantiation, contrary to the fundamentality of N.²²

Alternatively, this perfect nature can be seen in the light of the intrinsic value of N. When we talk about intrinsic value, we refer to those qualities that have value in themselves, in an objective sense. These qualities are valuable insofar as they enhance their possessor and are intrinsically better to have than not to have. Now, N is the foundation through which value flows, and it is through its creative acts and nature that the existence of value in reality is possible: without it, nothing could have existed, and consequently, no value or positive quality could exist, given that N is the ultimate explanation or cause of contingent reality, as established in Stage I. Therefore, N possesses the power to produce value, which is an intrinsically valuable aspect of its nature.

Furthermore, N possesses certain great-making attributes, such as causal powers, necessary existence/fundamentality, or self-sufficiency. Therefore, its nature is inherently valuable and positive. But how much value does N possess? As we have established, N has a nature without limits: that is, its essential properties are maximal. Due to this, the value of N cannot be limited; it is perfect. This leads us to elucidate the deepest and most foundational aspect of N: its complete, purely positive nature, devoid of any imperfection that implies a limit in its intrinsic value. Therefore, N is a perfect being.²³ Below, the argument in a more formal manner:

• 1. Everything that is limited can have an external explanation.

• 2. N is a necessary or fundamental being (Stage I).

• 3. If (2), then N cannot have an external explanation.

• 4. Therefore, the fundamental nature of N is not limited.

• 5. Everything that is not perfect has limits in its fundamental nature.

• 6. N has no limits in its fundamental nature (by (4)).

• 7. Therefore, N has a perfect nature.

• 8. Therefore, N is a perfect being.

Cumulative strategy

The cumulative strategy makes use of various philosophical arguments with the aim of presenting a general case in favor of theism as the most plausible and probable explanation for Stage II, also known as the identification Stage. In this way, all theoretical resources are used together to establish the theistic conclusion. Such strategies, also known as cumulative cases, combine the conclusions reached in various arguments to strengthen the theistic worldview, i.e., the existence of God as the best explanation for observed phenomena.²⁴

Cumulative cases have historically been the way in which philosophers and theistic thinkers have defended their worldview due to the integrative and systematic capacity that arguments from natural theology provide when it comes to understanding the nature of God, the ultimate foundation of reality, as well as the phenomena of reality. Furthermore, the advantage of this type of strategy is that it does not require an exhaustive analysis of the attributes or properties of the entity involved, as the conclusions established in each of the arguments reveal an aspect of its nature that, together, forms a complete image of Divinity.

For example, cosmological arguments allow us to establish, as we have seen, the existence of a fundamental or necessary entity with causal powers. To this conclusion, we can add the conclusion of teleological arguments, which postulate the existence of an intelligent mind, the cause of the order and fine-tuning of the cosmos. On the other hand, moral arguments provide us with a basis to believe that morality (ontological or epistemological) is grounded in a morally perfect entity who serves as the ultimate and objective standard for its existence. At this point, the different independent conclusions reached through these arguments help us elucidate the nature of this entity, allowing us to infer that its properties align with the traditional concept of God in philosophical theism. Thus, theism presents itself as the best metaphysical theory when addressing these phenomena, as the postulated entity (God) possesses all the necessary explanatory resources to ground them. Therefore, the arguments provided by natural theology serve as independent pieces of evidence in favor of the theistic model, which, taken together, allow us to construct a complete picture of the nature of God and His explanatory role with respect to the various domains of reality.

Conclusion

Throughout this article, we have addressed an analysis of the three main strategies that have been proposed to "bridge the gap" between Stage I (establishing the existence of a first cause, necessary being, or fundamental entity) and Stage II (the identification of this being as God) in cosmological arguments. Firstly, the abductive strategy allowed us to establish, based on various criteria or theoretical virtues, that theism presents itself as the best theory about the nature of N, the necessary being in Stage I. Secondly, the deductive strategy has provided us with numerous independent arguments to individually establish each of the divine attributes from the notion of necessary existence, aseity, or fundamentality, in conjunction with various metaphysical principles and theses. Thirdly, the cumulative strategy has provided us with an argumentative framework from which to establish the explanatory superiority of theism by using various arguments that appeal to different phenomena and aspects of God.

However, these strategies can be used to strengthen intermediate conclusions of a similar nature in other arguments favorable to theism, not limited solely to cosmological arguments. Therefore, and based on the strategies and arguments proposed throughout the article, the so-called "Gap Problem" can be satisfactorily addressed, strengthening the theistic case and providing new solutions for future research in this specific area of natural theology. Finally, we will present a summary of the results achieved in the following diagram Fig. 1:

Fig. 1

The divine nature in stage II