Disputatio XXIX: De Paradoxo Löwenheim–Skolemiano

On the Löwenheim–Skolem Paradox

Quaeritur

Quaeritur utrum systema formale possit determinare extensionem suam propriam, an vero iuxta theoremata Löwenheim et Skolem omnis ordo formalis habeat multitudinem modelorum diversae magnitudinis, unde sequitur relativitas veritatis ad modelum, et necessitas fontis veritatis externi.

It is asked whether a formal system can determine its own proper extension, or whether, according to the theorems of Löwenheim and Skolem, every formal order admits a plurality of models of different sizes, from which follows the relativity of truth to a model and the necessity of a source of truth external to the system.

Thesis

The Löwenheim–Skolem theorems reveal that no consistent formal theory can uniquely fix the structure of its universe, for every such theory possesses models of varying magnitude and scope. Hence, truth within a model (internal truth) differs from truth about the model (external truth). This formal distinction corresponds analogically to the theological distinction between felicity and truth: the first internal to theology’s discourse, the second dependent on divine reality beyond it.

Locus Classicus

“For now we see through a glass, darkly; but then face to face.” — 1 Corinthians 13:12

Augustine comments (De Trinitate XV.8): “Nondum est species, sed per speculum et in aenigmate.” Thomas Aquinas (ST I.12.11) interprets this as the difference between cognitio viatoris and cognitio comprehensoris, the knowledge of the pilgrim and the knowledge of the blessed. The former is mediate and partial, and the latter direct and complete. So too in logic: truth within a system (per speculum) and truth from beyond the system (facie ad faciem) are distinct orders of knowing. The formal result mirrors the metaphysical condition of creaturely understanding.

Explicatio

I. The Discovery

Between 1915 and 1920, Leopold Löwenheim and Thoralf Skolem, working independently, established two theorems foundational for modern model theory.

1. The Downward Löwenheim–Skolem Theorem: If a first-order theory has an infinite model, then it has a countable model.

2. The Upward Löwenheim–Skolem Theorem: If a theory has a model of some infinite size, then for every larger cardinal number, it also has a model of that larger size.

Together, these results imply that no first-order theory with an infinite model can control the cardinality of its universe. A theory formulated in a finite language cannot uniquely determine the size or structure of the reality it describes.

II. The Skolem Paradox

The most striking consequence arises when these theorems are applied to set theory itself, the very discipline designed to describe infinite sets. Zermelo–Fraenkel set theory (ZF) proves the existence of uncountable sets such as the set of real numbers. Yet, by the Downward Löwenheim–Skolem Theorem, ZF has a countable model, a model in which, from an external perspective, all its “uncountable” sets are actually countable!

This tension is called the Skolem Paradox. It reveals that statements true within a model (“there exists an uncountable set”) need not correspond to what is true about that model from outside it. Accordingly, the model cannot see its own countability, and its internal truth diverges from external truth.

III. Philosophical Meaning

The Skolem Paradox formally demonstrates the relativity of truth to the level of discourse. What is “true in a model” depends on the interpretation supplied from outside the model. A system cannot guarantee that its own truths are absolute; they are true within a given structure, not simpliciter.

Philosophically, this means that finitude entails perspectivality. No finite framework can encompass all possible interpretations of its own symbols. Every internal horizon is bounded by a greater horizon of meaning. The finite world’s intelligibility, therefore, is not exhausted by its own immanent relations but opens onto a transcendent ground that “models” it from beyond.

IV. Theological Analogy

Here we reach the theological analogue: The relation between internal and external truth in logic mirrors the relation between theological felicity and theological truth.

• Felicity (in Austin’s and later theological sense) denotes statements that function properly within the authorized discourse of theology, e.g., confessional utterances, liturgical speech, or inspired proclamation.

• Truth refers to correspondence between theological discourse and divine reality itself.

As the formal system cannot secure external truth by internal means, theology cannot verify divine truth by linguistic coherence alone. It must depend on the Spirit, the “external source” who bridges internal felicity and external reality. In this way, the Löwenheim–Skolem results offer a formal reflection of pneumatological mediation.

Obiectiones

Ob I. Logical positivism supposes that the Löwenheim–Skolem theorems concern formal semantics only. They tell us nothing about metaphysics or theology. To interpret them as analogies of divine truth is poetic, not logical.

Ob II. Nominalism decries that the distinction between internal and external truth is artificial. All truth is internal to a framework; there is no standpoint outside language or model.

Ob III. Postmodern relativism argues that since every theory has multiple models, there is no absolute truth. The theorems confirm that meaning is plural, not that there is an external ground.

Ob IV. Rationalistic metaphysicians hold that if external truth is required, then finite knowledge becomes impossible. We can only know within a system; appealing to an Infinite ground destroys epistemic closure.

Ob V. Finally, cautious theology itself declares that to identify divine reality with a “meta-model” risks subordinating revelation to logic. However, God is not a semantic extension but a personal will.

Responsiones

Ad I. The interpretation is analogical, not literal. Logic reveals structural truths about expression and interpretation that parallel ontological relations between being and its ground. Analogy discloses order without confusion.

Ad II. To deny any standpoint beyond a system is self-refuting, for the assertion itself pretends to transcend the system it describes.The very recognition of frameworks implies an external perspective.

Ad III. The multiplicity of models does not entail relativism but dependence. That truth is manifold within systems implies that there must exist a unifying act among them. Otherwise, plurality becomes unintelligible.

Ad IV. Appeal to the Infinite does not abolish finite knowledge but secures it. Just as the meta-theory is the condition for model-theory’s truth, so too the external ground is the condition for internal intelligibility. 

Ad V. God is not a meta-model but the living ground of truth itself.
The analogy is formal, not an ontological identity. It shows that even logic intimates the same structure that theology names personally as Word and Spirit.

Nota

The Löwenheim–Skolem paradox is more than a logical curiosity. It is a parable of theology’s condition. The paradox teaches that a system’s internal truth may remain intact even when interpreted within a domain smaller than its intention. Thus, the same structure that claims to describe an uncountable reality can find a countable model in which all its sentences are true. This means that while within the model, truth is preserved, from the standpoint of the greater reality, that very truth is inadequate.

So too with theology: the veritas interna of faith—truth within the system of grace—is coherent and authorized by the Spirit, yet it cannot exhaust the veritas externa of the infinite God it confesses. While the Church’s language can constitute a faithful model, it cannot articulate the totality of divine reality. This is not error but mercy. Accordingly, incomprehensibility is the form of participation. To speak truly of God is to speak within a model sustained by God Himself, while knowing that the truth spoken exceeds the model that bears it.

Hence the paradox reveals the theological structure of all knowledge of God: truth in the model depends on truth about the model, just as immanence depends on transcendence. The Spirit guarantees the coherence of the inner discourse while the Logos grounds its correspondence to reality. Their unity is the event of revelation: the infinite entering the finite without ceasing to be infinite.

In this way, Löwenheim and Skolem unwittingly formalized a metaphysical law: The finite can truly mirror the infinite only because the infinite freely indwells the finite.

Determinatio

From the foregoing it is determined that:

1. The Löwenheim–Skolem Theorems reveal that every formal system admits models of many sizes. Internal coherence does not yield external uniqueness.

2. Truth within a model (veritas interna) and truth about a model (veritas externa) are formally distinct. The former depends on the latter for interpretation.

3. The finite order, therefore, mirrors the condition that it is self-consistent yet semantically open. Thus, its meaning cannot be secured from within but requires reference to a transcendent source.

4. Theologically, this formal relation corresponds to the distinction between felicity and truth. Theology’s internal felicity (Spirit-authorized speech) depends upon external truth (the reality of God) which the Spirit mediates.

5. Hence, the Löwenheim–Skolem Paradox becomes a logical parable of participation. The finite model cannot perceive its own countability. In the same manner, creation cannot grasp its own dependence. Both require an Infinite perspective in which their truths are integrated and completed.

Therefore, the multiplicity of models is not chaos but a sign, an index of the Infinite intelligibility that sustains every finite order of meaning.
What logic calls the meta-model, theology calls the Logos; what logic calls interpretation from without, theology names the act of the Spiritus Veritatis.

Transitus ad Disputationem XXX

The Skolem paradox discloses the difference between truth within and truth about a system—between felicity and correspondence.Theology must interpret this difference in light of the Spirit’s authorization and the Logos’s adequation. 

Hence the next inquiry, Disputatio XXX: De Veritate Interna et Externa Theologiae, examines how finite theological language can be both internally faithful and externally true of God.

Disputatio XXVIII De Systemate Incompleto et Veritatis Factore Infinito

On the Incomplete System and the Infinite Truthmaker

Quaeritur

Quaeritur utrum systema finitum, si sit consistent, possit continere veritatem suam propriam, an vero, iuxta theoremata incompleti Gödeliana, omnis ordo finitus necessario referat ad veritatis fontem extra se—ad infinitum veritatis factorem.

It is asked whether a finite system, if consistent, can contain its own truth, or whether—according to Gödel’s incompleteness theorems—every finite order must necessarily refer to a source of truth beyond itself, to an infinite truthmaker.

Thesis

Gödel’s incompleteness results demonstrate formally what metaphysics has long intuited: The finite cannot ground the totality of its own truth. Every consistent formal system sufficient for arithmetic contains true statements it cannot prove. Hence, truth exceeds derivation, and the complete explanation of truth demands participation in something transcending the finite system.

Locus Classicus

“Great is our Lord, and abundant in power; his understanding is infinite.”
— Psalm 147:5

Aquinas comments: “Intellectus divinus est infinitus, quia adaequat veritatem ipsius Dei, quae est infinitum esse.” (STI.14.6.) The divine intellect alone comprehends all truth as being identical to being. Human or finite systems of reason, by contrast, express truth participatively, that is, as reflections of the infinite intellect. Thus, the logic of finitude corresponds to the metaphysics of participation.

Explicatio

I. The Context of Gödel’s Discovery

In 1931, 25 year-old Kurt Gödel, an Austrian logician, published “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme” (Monatshefte für Mathematik und Physik 38, 1931). His goal was to investigate the limits of formal systems such as the Principia Mathematica by Whitehead and Russell, which sought to derive all mathematical truths from a finite set of axioms through mechanical rules of deduction.

To understand the significance of this, we must review some key notions. A formal system may be thought of as a rigorously defined language governed by rules. While its syntactic component consists of symbols and derivations its semantic component is concerned with truth and meaning about numbers or other entities to which it refers. For such a system to be a satisfactory foundation of mathematics, it must have two crucial properties:

1. Consistency: No contradiction can be derived within the system.

2. Completeness: Every true statement expressible in the system can be derived from the system's axioms.

Gödel’s work proved that these two properties cannot coexist in any finite system capable of expressing arithmetic.

II. Gödel’s First Incompleteness Theorem

Gödel showed in this proof how to assign to each formula and proof a numerical code, a process that is now called Gödel numbering. By this ingenious device, statements about formulas could become statements about numbers. He then constructed a sentence G that effectively says of itself,

“This statement is not provable within this system.”

If the system is consistent, it cannot prove G, for to do so would render it inconsistent, that is, it would prove a falsehood. Yet if the system is consistent, G is in fact true, since its unprovability makes the assertion it contains correct. Hence, G is true but unprovable within the system. The upshot of this is this: No consistent, sufficiently expressive finite system can be complete. Simply put, there will always exist true propositions that escape its derivations.

III. Gödel’s Second Incompleteness Theorem

Gödel then proved a deeper corollary, that no consistent system can prove its own consistency. But to show that its axioms are non-contradictory, one must appeal to a meta-system, to a higher language standing outside the system itself. Hence, every finite logical order depends on another for its assurance of truth and coherence.

IV. Philosophical Significance

Gödel’s theorems thus reveal a structural transcendence of truth over formal expression. They are not merely mathematical curiosities but demonstrations of a universal condition of finitude, that truth always surpasses the framework that tries to contain it. Every closed system that seeks to explain itself without remainder either collapses into contradiction or appeals to a higher order.

Metaphysically, this mirrors the ancient insight that the finite requires the infinite as its truthmaker. The correspondence between logical form and ontological order is not accidental but structural: just as a formal system needs a meta-system to ground its truth, so the finite world needs a transcendent act of being to ground its reality.

What Gödel discovered formally, metaphysics already discerned existentially: veritas non est intra ordinem finitum nisi per participationem veritatis infiniti.

Obiectiones

Ob I. Formalists like David Hilbert hold that the incompleteness theorems apply only to mathematical systems, not to reality. They concern symbols and proofs, not the metaphysical order of being.

Ob II. Scientific empiricism argues that science does not need to be “complete” in this logical sense. Explanatory power depends on observation, not on formal derivation. Thus, Gödel’s results have no bearing on physical intelligibility.

Ob III. Reductive naturalists claim that the analogy between formal systems and the finite world is metaphorical, and thus to move from logical incompleteness to ontological dependence is an illicit category jump.

Ob IV. Skeptics of many kinds opine that Gödel’s theorem requires arithmetic within a system, and that nature is not a formal calculus. Accordingly, it is meaningless to say that the universe is “incomplete” in the Gödelian sense.

Ob V. The cautious theologian claims that appealing to Gödel to prove divine necessity risks confusing logic with revelation. God’s infinity is not a corollary of syntax but a matter of faith.

Responsiones

Ad I. Gödel’s theorems indeed concern formal systems, yet they express a universal relation between expression and truth. Wherever truth is represented within a finite structure, that structure cannot exhaust it. The logical limit mirrors an ontological condition.

Ad II. Scientific explanation presupposes coherence and consistency within its theories. Gödel shows that such coherence cannot be self-guaranteed; it must be received from a higher frame. Hence, the dependence of empirical science on deeper intelligibility is reinforced, not diminished.

Ad III. The analogy is legitimate when carefully drawn. Formal systems model the relation of expression to truth; the finite world models the relation of being to its source. In both, self-sufficiency proves impossible; participation becomes the only path to completeness.

Ad IV. The universe is not a calculus, yet our reason reflects its structure through logic.To say that the world is “Gödelian” is not to mathematize it but to recognize that finitude, even in its most abstract forms, cannot close upon itself.

Ad V. The appeal to Gödel is not a theological proof but a formal analogy. It illuminates by example what theology asserts by revelation: that all truth in the finite is truth by participation in the Infinite Word.

Nota

Gödel’s theorem exposes not merely a boundary of formal systems but a metaphysical structure, for the finite, in order to remain consistent, must remain open to what it cannot contain. Incompleteness is thus not a defect but the mark of dependence. The object system’s unprovable truths are signs of an order beyond itself, an order upon which its very coherence rests.

In theology, this structure mirrors creation’s relation to its Creator. The creature is a consistent finite system whose truth is guaranteed only by participation in the infinite. Every finite logos, to be true, must be grounded in a Logos that transcends it; every rational discourse presupposes an unspoken act that makes discourse possible.

Hence Gödel’s discovery becomes a theological axiom: truth cannot be self-enclosed. Simply put, there must exist an actus essendi veritatis, an infinite truthmaker, by whom the finite is both comprehensible and incomplete. Logical incompleteness is thus a formal echo of the metaphysical participation of the finite in the divine, and the incompleteness of the finite itself. It reveals that closure is illusion, and openness to transcendence is the very condition of truth.

Determinatio

From the foregoing it is determined that:

1. Gödel’s incompleteness theorems formally demonstrate the incapacity of the finite for self-completion. Every consistent system depends upon truths it cannot contain and upon a meta-system it cannot itself generate.

2. Truth transcends formal derivation. Just as no calculus can produce all truths of arithmetic, no finite ontology can account for its own intelligibility.

3. Consistency requires transcendence. The assurance that a system is non-contradictory always arises from a higher standpoint.
   Ontologically, this implies that the finite world’s coherence depends on an Infinite act of being.

4. The Infinite functions as the universal truthmaker. The meta-system for logic corresponds analogically to the Creator for creation: the necessary being in whom all contingent truths are grounded and from whom their coherence flows.

5. Therefore, Gödel’s result, though mathematical in form, reveals a metaphysical truth: the finite is intelligible only by participation in the Infinite. The world’s incompleteness is not deficiency but sign — a structural openness to the Infinite intellect whose understanding is unbounded.

Hence, the incompleteness of systems becomes a formal witness within reason to the metaphysical participation of all truth in God — in quo sunt omnes thesauri sapientiae et scientiae absconditi (Colossians 2:3).

Transitus ad Disputationem XXIX

If every finite order requires an infinite truthmaker, how can finite language and models still signify truly? The following disputation explores the paradox of internal and external truth uncovered by the Löwenheim–Skolem theorem, showing how the structure of theology mirrors the relation between truth in a model and truth about it. 

We turn, therefore, to Disputatio XXIX: De Paradoxo Löwenheim–Skolemiano, wherein we examine how truth within a model and truth about that model diverge, and how this divergence reveals the theological relation between faith’s internal coherence and the infinite reality of God.

Disputatio XXVII De Essentiis Dispositionalibus

On Dispositional Essences

Quaeritur

Quaeritur utrum necessitas legum naturae possit sufficienter explicari per essentias dispositionales ipsarum rerum, an vero talis explicatio tandem recidat in naturalem essentialismum sine fundamento ontologico, qui rursus ad participationem infiniti redigitur.

It is asked whether the necessity of natural laws can be adequately explained by the dispositional essences of things themselves, or whether such an explanation ultimately collapses into an unfounded natural essentialism that once again requires participation in the Infinite.

Thesis

Dispositional essentialism seeks to ground the laws of nature in the intrinsic powers of entities. Accordingly, each thing, by virtue of what it is, behaves as it does. Laws are thus expressions of essence, not external constraints. Yet finite essence itself requires grounding for its actuality and coordination. Therefore, the appeal to dispositional essences displaces but does not resolve the need for an infinite ground of law.

Locus Classicus

“In him we live and move and have our being.” — Acts 17:28

The early Fathers, having read Paul in light of Hellenic metaphysics, interpreted this as a declaration that all powers and movements within creation presuppose divine causality. For example, Basil of Caesarea (Hexaemeron I.5) taught that “every natural power is the gift of divine energy,” and Aquinas affirmed that “omnis operatio naturae est actus Dei in natura” (ST I.105.5). Thus, even when power is intrinsic to a creature, its being and operation participate in the act of the Creator.

Explicatio

Dispositional essentialism emerged in late twentieth-century metaphysics as a reaction against both Humean regularity and Armstrong’s relational realism. This view aims to secure necessity without appeal to external laws or transcendent governance. Philosophers such as Brian Ellis, C. B. Martin, and Stephen Mumford argued that laws do not govern things from without but that the flow from within from the very essences or natures of entities. Accordingly, an electron repels another not because a law commands it, but because repulsion belongs to its nature. The behavior is thus essential and not contingent.

In this view, every natural property is dispositional; it is defined by its powers and tendencies. To possess a charge, mass, or spin just is to manifest appropriate dispositions under suitable conditions. Laws of nature are thus derivative descriptions of the necessary behaviors of these dispositional essences. Therefore, there are no separate laws or external principles, but only powers whose exercise constitutes the order of nature.

This approach elegantly restores necessity to the finite without invoking extrinsic governance. But the question remains: Whence the unity of this system of powers? If every essence carries its own necessity, what guarantees the coherence of those necessities across the totality of the world? Why do distinct powers not conflict or dissolve into chaos? While the finite essence, to be actual, must exist and operate within a coherent totality of being, that totality cannot itself be one of the powers. Rather, it must be the condition of their coexistence and harmony.

Hence, while dispositional essentialism succeeds in moving the locus of necessity inward—from external law to internal essence—it fails to remove the need for ontological participation. While essence, in so far as it is essence, is an intelligible structure of being, powers, however intrinsic, can only be participatory modes of a deeper enabling act.

Obiectiones

Ob I. According to Ellis in 2001, the essence of each natural kind explains its behavior. Thus, no further metaphysical foundation is required, and to demand more is to mistake explanation for regression.

Ob II. Martin argued in 2008 that dispositions constitute causal grounds for their manifestations. Since power is primitive and self-explanatory, the world’s order is the network of powers acting according to their natures.

Ob III. Mumford in 2004 argued that laws are supervenient on dispositional essences, and hence add nothing ontologically to them. Thus, the finite order is self-sufficient so long as it consists of stable powers and their mutual tendencies.

Ob IV. Naturalistic Metaphysics claims that to appeal to an Infinite act is unnecessary duplication. If dispositions suffice for explanation, positing divine participation is a metaphysical surplus.

Ob V. Sometimes the theological tradition assumed that grounding the powers of things directly in the infinite may risk erasing natural causality. But the integrity of secondary causes requires that creatures possess genuine powers of their own.

Responsiones

Ad I. While essence may explain behavior, it does not explain existence. To say “the electron repels because it is its nature to repel” still leaves unasked why such a nature exists at all. Essence is formal cause and being is act. The latter cannot be derived from the former without reference to a self-sufficient act of existence.

Ad II. Power cannot be self-explanatory, for power is always power to act.The actuality of its exercise depends on a larger order within which it operates. Without a unifying act of being, powers remain mere potentialities without coherence.

Ad III. Supervenience explains correspondence but not causation. That laws supervene on essences tells us that essence and law covary, not why such correlation obtains. The dependence relation itself requires grounding.

Ad IV. Appealing to the infinite is not an additional move but a natural completion in the order of explanation.The Infinite is not another entity among the powers but the act in which all finite essences receive their actuality and unity. Without such an act, the multiplicity of powers lacks ontological coherence.

Ad V. Participation does not abolish finite agency but founds it. Creatures possess true powers because the infinite communicates actuality to them. Their independence as secondary causes is secured by the divine act that continuously sustains them in being.

Nota

Dispositional essentialism rightly perceives that the necessity of nature arises from within things themselves. It holds that each being acts according to what it is, and thus, its tendencies are not imposed from without but flow from its essence. The very intelligibility of this insight, however, betrays its limit, for the intrinsic power of a thing explains its manner of acting, not its capacity to act at all. Accordingly, the essence that disposes toward activity still requires an act that gives it existence and coherence.

Hence, the metaphysical question beneath dispositional essentialism is not why things act as they do, but why there are things capable of acting at all. To say that the stone falls because it has mass, or that the charge repels because it is charged, presupposes the ontological act by which stone and charge subsist. Clearly, while the essence disposes, only the act sustains.

The theological transformation of this view is participation. On this view, every finite power is a communicated potency; it receives from the Infinite Act not only its existence but its coordination with all others. The unity of law in the world, that is, the harmony among dispositions, is thus the reflection of the divine unity that gathers all powers into a single order of being. Nature’s lawfulness is the shadow of grace: finite essences cooperate because they share in one act of creation.

Dispositional essentialism, therefore, contains a veiled confession:
to affirm inner necessity is already to acknowledge the immanence of the divine act within creation. The Spirit is the bond that makes powers conspire toward intelligibility, and the Logos is the act through which each essence becomes dispositional at all. Necessity, properly understood, is participation in the divine constancy by which all things are held in being.

Determinatio

From the foregoing it is determined that:

1. Dispositional essentialism internalizes necessity but does not abolish dependence. Thus, finite essences are intelligible structures whose actuality presupposes a unifying act of being.

2. The unity of natural order cannot arise from a plurality of isolated powers. Coordination among dispositions requires an ontological ground transcending them.

3. Essence without act is impotent. The existence and operation of every power presuppose an act that is not itself one power among others. They thus presuppose an infinite act of being.

4. The participation of finite essences in the Infinite corresponds to the metaphysical structure of creation. As Augustine said, “Omne bonum quod habet creatura, habet participando” (De Diversis Quaestionibus 83.46). Powers are real and finite, and their actuality is participatory.

5. Hence, dispositional essentialism, though the most promising finite account, nonetheless points beyond itself. Its truth lies not in rejecting participation but in clarifying the mode of it: each finite power is a share in the creative act that sustains and orders all powers.

Therefore, the necessity of natural law is neither imposed from without nor self-generated from within. It arises from the participation of dispositional essences in the infinite act of being, in the Word through whom all powers subsist and in the Spirit who continuously actualizes their operation.

Transitus ad Disputationem XXVIII

The unity of nature cannot be secured from within the multiplicity of powers. The next disputation therefore asks whether finite systems, even when internally coherent, can ever be complete in themselves. 

We proceed to Disputatio XXVIII: De Systemate Incompleto et Veritatis Factore Infinito, in which the Gödelian structure of dependence reveals that every finite necessity presupposes an infinite act of truth.

Disputatio XXVI: De Universalibus Immanentibus et Necessitatione

On Immanent Universals and Necessitation

Quaeritur

Quaeritur utrum necessitas legum naturae sufficienter explicari possit per relationes reales inter universalia immanentia, an vero talis explicatio aut in regressum, aut in factum brutum, aut in participationem infiniti deveniat.

It is asked whether the necessity of natural law can be adequately explained by real relations among immanent universals, or whether such an explanation must ultimately collapse into regress, brute fact, or participation in the Infinite.

Thesis

Immanent realism explains regularity by positing universals instantiated within things and connected by real relations of necessitation. Yet the nexus that binds one universal to another either becomes an infinite regress, a brute primitive, or an implicit participation in an infinite unity. Thus, while immanent universals preserve realism, they cannot close the circle of explanation within the finite.

Locus Classicus

“He is before all things, and in him all things hold together.”
— Colossians 1:17

Patristic and scholastic theology interpreted this not as poetic hyperbole but as a metaphysical statement. Athanasius (Contra Gentes 41) held that creatures “stand fast by participation in the Word.” Aquinas, commenting on the same verse, wrote: “In ipso omnia constant, quia ipse est ratio essendi et ordinis in rebus.”(Super Colossenses I.17.) The order and interrelation of created forms thus depend upon the Logos as their unifying act.The attempt to ground such order solely in finite relations among universals severs form from source and leaves unity unexplained.

Explicatio

D. M. Armstrong, seeking a realist alternative to both Humean descriptivism and Platonic transcendence, developed a theory of immanent universals in What Is a Law of Nature? (Cambridge University Press, 1983) and A World of States of Affairs (1997). For Armstrong, universals are not abstract entities existing apart from things but real features instantiated in rebus. A natural law is then a relation of necessitation between such universals:

N(F, G) means that every instance of F is necessarily also an instance of G.

For example, the law “All electrons repel each other” corresponds to a relation N (being an electron, repelling other electrons). This N-relation is itself a real universal connecting others, not a mere linguistic rule.

Armstrong’s system preserves a realist ontology, for lawfulness exists in the world, not in our descriptions. It also avoids Platonism by keeping universals immanent.Yet the decisive problem lies in the status of the necessitation relation itself.

If N is simply another universal, it must stand in further relations explaining how it binds F and G—relations such as N′(N, F, G)—and so on ad infinitum. If N is primitive, we are left with unexplained necessity. If N is grounded in the overall structure of being, that structure functions as a transcendent unity, in effect, a metaphysical participation in the Infinite.

Thus Armstrong’s account, while internally rigorous, cannot ultimately provide a self-sufficient finite explanation. It gives us the mechanics of law but not its metaphysical coherence. The problem is not empirical but ontological: what makes the system of immanent relations one and necessary rather than a contingent web of co-instantiated properties?

Obiectiones

Ob I. In 1983 David Armstrong argued that immanent universals provide the ontological structure science presupposes. The relation of necessitation N is real and sufficient. No further grounding is needed.

Ob II. Moderate realism claims that by positing universals in rebus rather than ante res, we respect the finitude of creation and avoid both Humean nominalism and Platonic abstraction.

Ob III. Scientific pragmatism holds that the theory of immanent universals aligns well with scientific practice, which operates by discovering relations among properties, not by appealing to transcendent causes.

Ob IV. Empiricists argue that an infinite ground multiplies entities beyond necessity. The unity of laws is a consequence of the shared structure of matter and fields, not of any higher participation.

Ob V. The theologicus cautus ("cautious theologian") opines that to require an infinite explanation of finite order threatens to erase the integrity of secondary causes and the natural autonomy of creation.

Responsiones

Ad I. To say that N(F, G) is real explains that the relation exists, not why it obtains. Unless N itself is grounded, the account halts in primitive necessity. A brute tie between universals is no advance over the brute law it replaces.

Ad II. Immanent universals are indeed within things, yet their coordination across all things remains unexplained.The in rebus does not by itself yield the per se unity of the real. Participation in a higher act of being is required for coherence among universals.

Ad III. Scientific adequacy differs from metaphysical sufficiency.
Empirical inquiry describes how properties are correlated; metaphysics asks why such correlation is necessary. Armstrong’s ontology presupposes the unity it should explain.

Ad IV. The claim that matter and fields explain law simply restates the problem at a lower level. For the structure of matter and fields is itself law-like and requires grounding. Invoking the material order as ultimate converts contingent structure into absolute necessity without reason.

Ad V. Participation in the Infinite does not annul finite autonomy but establishes it. Only what is grounded in the Infinite can act coherently according to its own nature. The Spirit’s causal presence secures the creature’s integrity by making its lawfulness possible.

Nota 

The idea of universal immanence captures a profound half-truth.
On the one hand, it rightly discerns that the divine is not absent from the world but intimately present within it, sustaining the being of all that is. On the other hand, when detached from the transcendence that grounds it, immanence collapses into necessity without freedom, into an all-encompassing process in which God and world dissolve into one another.

Theology must therefore distinguish immanentia participationis from immanentia identitatis. In the first, the divine act is interior to all things as their sustaining cause; in the second, the divine and the creature are confused as modes of one process. But true immanence is participatory, not monistic. God is within all things precisely because all things are within God.

Necessity, when viewed in this light, is not mechanical but the mark of divine fidelity. The stability of natural law expresses the constancy of the creative Word, whose will does not waver. The same act by which God gives being gives order, and thus the regularity of the world is grace made habitual. Accordingly, immanence and necessity are not rival to transcendence and freedom; they are its temporal manifestation.

The absolute dependence of creatures upon the divine act entails that while God is more interior to them than they are to themselves, yet God remains infinitely beyond them. The world’s necessity is therefore double. It is necessary in itself because the divine act holds it in being, and yet contingent before God, who freely gives it.

Universal immanence, properly understood, is the metaphysical form of providence:the Creator’s continuous presence as the reason for the world’s coherence. This providence is the “necessity of grace,” the steady rhythm of divine constancy through which all that is remains possible.

Determinatio

From the foregoing it is determined that:

1. Armstrong’s immanent realism preserves ontology but not ultimacy. The N-relation that ties universals together is either another universal (regress), an unexplained primitive (brute fact), or a reflection of a deeper unity (participation).

2. Finite relations cannot ground universal coherence. The multiplicity of universals demands a unifying act that is not itself one among them. Without such an act, law remains accidental coordination.

3. The appeal to the Infinite is not extrinsic but intrinsic. The very notion of “necessitation” implies participation in an unconditioned ground of necessity. The Infinite is the metaphysical horizon within which immanent universals receive their order.

4. The participation of universals in the Infinite corresponds to the theological doctrine of the Logos. As the eternal form of all forms, the Logos is the ratio essendi and ratio ordinis of finite properties. Law, in this light, is the reflection of divine intelligibility within creation.

5. Hence, immanent realism, while the most sophisticated of finite explanations, points beyond itself. Its internal coherence is the sign of participation, not self-sufficiency. In the Infinite Word, the many relations of the finite find their unity; in the Spirit, they find their continuous actuality.

Therefore, the necessity of natural law cannot rest in the N-relations of universals alone but requires the participation of all finite forms in the Infinite act of being — in ipso omnia constant.

Transitus ad Disputationem XXVII

If universal immanence reveals that the divine act is interior to all things as the ground of their necessity,then theology must next inquire how that immanent act manifests within the distinct natures of creatures themselves. After all, divine constancy, to be real, must articulate itself in finite structures of power and tendency. The same Word who upholds all being also orders its operations; the regularity of nature is the expression of this inner form.

Hence we turn to Disputatio XXVII: De Essentiis Dispositionalibus,
in which we ask whether the necessity of natural law arises from the intrinsic powers of things, or whether even these dispositions, in all their apparent autonomy, depend upon participation in the infinite act that both constitutes and coordinates them.

Disputatio XXV: De Regularitatibus Humeanis

On Humean Regularities

Quaeritur

Quaeritur utrum leges naturae sint tantum descriptiones constantium eventuum in mundo, an vere exprimant necessarias rationes essendi quae exigunt causam extra ipsam seriem eventuum.

It is asked whether the laws of nature are merely descriptions of the constant conjunctions of events in the world, or whether they express necessary relations of being that require a cause beyond the series of events themselves.

Thesis

The Humean account of natural law as mere regularity confuses description with necessity and drains law from having real ontological force. If laws are nothing but patterns within phenomena, then the world lacks any genuine principle of order. Accordingly, to recover necessity, the finite must once again refer beyond itself to an infinite ground in which the possibility of law is constituted.

Locus Classicus

“He makes his sun rise on the evil and on the good, and sends rain on the just and on the unjust.”
— Matthew 5:45

The constancy of divine action in nature, understood here as fidelitas Dei, was read by Augustine (Enarrationes in Psalmos 147.18) and Aquinas (ST I.103.8) as evidence that natural regularities are not self-existent but proceed from a sustaining cause. Luther, in his lectures on Genesis, described the continuance of natural order as the “mask of God” (Larva Dei), behind which divine agency preserves creation. Thus, constancy itself is a sign of dependence, not autonomy.

Explicatio

The Humean conception of law arose in the Treatise of Human Nature (1739–40) and the Enquiry Concerning Human Understanding (1748), where David Hume denied that we ever perceive necessary connection between events. From this epistemic premise, he concluded that laws of nature are nothing more than uniform patterns of experience: a "constant conjunction" of similar events leading the mind to expect one after the other. On this view, necessity is not in things but in thought; it is a “habit of imagination.”

In the twentieth century, David Lewis sought to preserve Hume’s metaphysics while providing a systematic account of laws. In A Subjectivist’s Guide to Objective Chance (1980) and Philosophical Papers, vol. II (1986), Lewis articulated the Best-System Analysis (BSA) in which laws are the generalizations that occur given the axioms of the simplest and strongest deductive system that best summarizes all particular facts: the “Humean mosaic” of local matters of fact. A law, then, is whatever appears in the optimal balance between simplicity and strength in describing the total history of the world.

The appeal of this position is its ontological economy: no mysterious necessities, only patterns; no governing principles, only description.
Yet its cost is high: it leaves the world without internal order or modal depth. The regularity of events may be observed, but the reason for that regularity is left unspoken. On Humeanism, the universe is a sequence without syntax, a film of contiguous frames in which connection is projected by the mind. In such a world, the word law is metaphor; nothing obliges events to recur, and the distinction between possible and impossible collapses into mere fact and non-fact.

Theologically, this view is untenable. It denies both creation’s intelligibility and divine fidelity. To call law a mental convenience is to deny that the world speaks truthfully of its Maker. Reason, however, testifies otherwise, for the constancy of nature presupposes an underlying act of being that makes regularity possible.

Obiectiones

Ob I. Already in 1748 David Hume had argued that all necessity arises from habit. We never perceive any power or connection in nature. Therefore, what we call a law is only an observed uniformity in experience.

Ob II. In 1980 David Lewis claimed that to treat laws as abstract necessities adds ontological baggage. The world is a mosaic of local facts. The “Best System” captures their pattern without positing mysterious connections.

Ob III. Empiricists say that science requires only prediction, not metaphysical grounding. Whether laws “exist” beyond description is irrelevant to the success of physics.

Ob IV. According to nominalism, the notion of an infinite truthmaker is incoherent. Necessity is linguistic convenience; to speak of grounding is to confuse semantics with ontology.

Ob V. Liberal theology avers that reading divine causality into natural regularity is to return to pre-critical metaphysics. Lawfulness may express God’s reliability metaphorically, but it requires no metaphysical participation.

Responsiones

Ad I. Observation alone yields correlation, not connection. If necessity were merely mental habit, then any sequence could become law through repetition, which contradicts both experience and reason. Our recognition of constant conjunction presupposes that reality itself is structured for recurrence.

Ad II. The Best-System Analysis transforms the ontological into the epistemic. It tells us what generalizations we find simplest, not why the world is ordered so as to be summarized. The balance of simplicity and strength explains convenience, not causation.

Ad III. Science as practice may ignore metaphysical foundations, but the intelligibility of science presupposes them. If the universe were pure contingency, induction itself would be unjustified. The success of empirical prediction depends upon a real order antecedent to observation.

Ad IV. To reduce necessity to language is self-refuting, for the reduction itself claims necessity. Semantic regularity cannot explain ontological stability. Laws must be more than linguistic; they must participate in a structure of being.

Ad V. The metaphysical reading of divine causality is not regression but completion. Scripture’s description of the world’s constancy as divine faithfulness grounds natural order in personal being. To deny participation is to make the creation silent about its Creator.

Nota 

This disputation confronts the decisive modern rupture in the understanding of law and causality. Whereas the preceding disputations discerned in natural order the reflection of divine reason and the imprint of legibus primitivis, modern empiricism, epitomized by Hume, denies that any necessity or intrinsic connection binds events together. According to this view, so-called “laws of nature” are but habits of observation—regular successions that the mind projects as necessity out of custom. The world thus loses its inner ratio; causality becomes expectation, and order a fiction of the perceiving subject.

In this disputation we examine the implications of this Humean reduction. Can theology accept a cosmos governed only by constant conjunction without undermining the very possibility of providence, creation, and divine intelligibility? If natural law is but descriptive regularity, how can the world be a medium of revelation or a site of divine action? The question touches not only metaphysics but also epistemology, for in the collapse of necessity the mind itself loses participation in the rational structure of being.

Therefore Disputatio XXV tests the theological coherence of the modern naturalist paradigm. It contrasts the participatory order of lex aeterna and legibus primitivis with the secularized uniformity of Humean regularity, seeking to determine whether a purely empirical account of law can sustain the intelligibility of creation or whether it inevitably dissolves the Logos into the contingencies of perception. In this way, it prepares the path toward the retrieval of a richer metaphysical realism in Disputatione XXVI: De Immanentia Universali et Necessitate.

Determinatio

From the foregoing it is determined that:

1. The Humean and Lewisian accounts evacuate natural law of real necessity. By reducing law to description, they destroy the very distinction between order and coincidence.

2. Regularity theory fails the coordination test. It cannot explain why distinct patterns harmonize across domains,why electromagnetism and gravitation, time and entropy, compose one coherent cosmos.

3. Reason’s demand for sufficient cause (PSR) re-emerges. If law is mere pattern, PSR is violated, and intelligibility perishes.The mind’s refusal to accept brute regularity is itself evidence of participation in an Infinite intelligibility.

4. The finite order requires a ground that is both necessary and self-explanatory. Such a ground cannot lie within the Humean mosaic; it must transcend it while remaining immanent as its condition.

5. Hence, the constancy of nature is participatory: the world’s regularities exist not ex se but per participationem in the Infinite act of being. What the theologian calls divine providence, the philosopher names the unconditioned truthmaker of order.

Therefore, the Humean view of law as regularity fails both scientifically and metaphysically. While it can describe, it cannot explain; while it can record, it cannot ground. Thus, the world is not a mosaic of inert facts but a living participation in the Logos, in whom all order holds together.

Transitus ad Disputationem XXVI

In the preceding disputation it was shown that the Humean conception of law as mere regularity severs the bond between mind and world, dissolving necessity into habit and rendering the cosmos unintelligible from within. The world thus appears as a theater of sequences without reasons—an ordered surface lacking interior logos. Against this view, theology must ask whether there remains a deeper mode of necessity, one not imposed from without nor abstracted by thought, but immanent in things themselves as their participation in divine reason.

This question turns our attention from the empirical pattern to the metaphysical structure that makes such pattern possible. If order is real, it must inhere in being as such; universals must be operative within the concrete, not floating above it. How, then, are these universalia immanentia to be conceived? Are they divine ideas within things, the ontological forms of their participation in the Logos, or the very conditions of their acting and being acted upon?

Therefore we advance to Disputationem XXVI: De Universalibus Immanentibus et Necessitatione, in which we shall inquire how necessity arises from within the order of being itself, how universals dwell in the particular as formative presence, and how through this immanent structure the world remains transparent to the intelligibility of God.