What is a Theological Model?

This is the second part of a series developed through the Department of Philosophical Theology at Christ School of Theology, examining theological method, models, interpretation, and the logical and semantic conditions of theological inquiry.

From Method to Model

The first essay in this series argued that theological method concerns the disciplined ordering of sources, norms, concepts, and inferential procedures by which theologians seek truth concerning God and all things in relation to God. That formulation already implied something more. Once theologians begin arranging claims, distinguishing terms, identifying relations, and asking what follows from what, they are constructing representations of theological reality. They may speak instead of doctrinal systems, conceptual schemes, metaphysical accounts, explanatory frameworks, or simply interpretations, though in each case the activity exhibits a recognizable structure: certain claims are held fixed, certain entities are posited or presupposed, certain relations are specified, and the resulting structure is examined for coherence, adequacy, and explanatory power.

This is the point at which talk of theological models becomes useful, though it can also become treacherous. The word ‘model’ has several senses, and theologians often move among them without noticing the transition. A model can be an exemplar to be imitated; it can be a simplified representation of some object; it can be a theoretical construction employed to explain phenomena; and, in logic, it can be a structure in which the sentences of a theory are satisfied. These senses overlap, though they are hardly identical. If we wish to speak perspicuously about theological models, some semantic housekeeping is therefore required.

The purpose of such housekeeping is not to reduce theology to formal logic. That would merely substitute one methodological imperialism for another. Yet model theory supplies distinctions that theologians badly need, because it forces us to distinguish a language from an interpretation of that language, a theory from a structure satisfying that theory, and satisfaction within a structure from truth simpliciter. Once those distinctions are seen, several perennial theological confusions become considerably easier to diagnose.

Theory and Model

Suppose we begin with a language L containing certain predicates, relations, constants, and perhaps other logical or nonlogical expressions. Within that language we formulate a set of sentences T. This set of sentences constitutes a theory in the broadly logical sense. The theory says certain things, or at least provides sentences capable of saying them once its expressions are interpreted.

A model of T is then a structure M in which every sentence belonging to T is satisfied.

In abbreviated form:

M ⊨ T

This is read: ‘M is a model of T,’ or more explicitly, ‘every sentence in T is true in M under the relevant interpretation.’

Already the theological significance should be apparent. The theory and the model are not identical. T consists of sentences. M is a structure relative to which those sentences receive an interpretation and are satisfied. If one forgets this distinction, one can easily slide from talking about a doctrinal formulation to talking as though the formulation itself were the theological reality.

Consider, for example, a highly simplified Trinitarian theory containing sentences expressing that the Father is divine, the Son is divine, the Spirit is divine, the Father is not the Son, the Son is not the Spirit, and the Father is not the Spirit. These sentences constrain acceptable interpretations. They rule some structures out. Yet they do not, merely by being stated, determine a unique metaphysical account of divine triunity. More than one structure may satisfy the same set of sentences, depending upon how ‘divine,’ ‘person,’ identity, relation, and the domain itself are interpreted.

That is precisely why theological models arise. They attempt to exhibit ways in which a set of theological assertions might jointly be true.

A Model Is an Interpretation

In elementary model theory, one often represents a structure in something like the following way:

M = ⟨D, I⟩

Here D is a nonempty domain of objects, while I is an interpretation assigning semantic values to the nonlogical vocabulary of the language. If P is a one-place predicate, I assigns to P some subset of D; if R is a two-place relation, I assigns to R some set of ordered pairs from D; if a is a constant, I assigns to a some member of D.

Thus, if P means ‘is divine,’ then Pᴹ is the extension of P in M: the objects in the domain that satisfy the predicate. If R means ‘begets,’ then Rᴹ consists of those ordered pairs ⟨x,y⟩ such that x stands in the begetting relation to y.

The notation can appear forbidding only until its philosophical point is grasped. A model interprets a language by assigning structure to its vocabulary. It tells us what the terms range over, which objects instantiate which predicates, which objects stand in which relations, and under what conditions the sentences of the theory come out true.

Theological argument does this constantly, usually without the notation. When a theologian says that ‘person’ in Trinitarian discourse denotes a subsistent relation rather than an individual substance, an interpretation has been supplied. When another theologian says that ‘nature’ in Christology is neither a concrete individual nor merely a set of properties, another interpretation has been proposed. When a theologian distinguishes between divine action as temporal causal intervention and divine action as the sustaining condition of creaturely actuality, the semantic structure assigned to ‘acts’ has changed. The dispute may accordingly appear grammatical while being fundamentally ontological.

Models and Reality

At this point, however, a caveat becomes indispensable. In formal semantics, a model is often treated simply as a mathematical structure. In theology, matters cannot end there, because theological assertions purport to concern reality. A model may satisfy a theory without thereby establishing that the theory is true of God. This distinction is absolutely fundamental.

Suppose M ⊨ T. We have shown that there is at least one interpretation under which every sentence in T is satisfied. We have not thereby shown that T is true simpliciter. Still less have we shown that M reproduces the ontological structure of divine reality.

The distinction can be put bluntly. Consistency is not truth. Satisfiability is not actuality. Model-theoretic success is not metaphysical vindication.

Yet the converse error must also be resisted. Because a model does not establish truth, it does not follow that models are theologically unimportant. A model can reveal that some set of claims is jointly satisfiable; it can expose hidden assumptions; it can show that an apparent contradiction depends upon an equivocation; it can reveal commitments that were previously unnoticed; and it can make competing interpretations sufficiently explicit that genuine disagreement becomes visible.

Models therefore perform an indispensable mediating function. They stand between language and reality, not because reality itself is constructed, but because our intellectual access to complex claims requires structured representations.

The Given and the Made

This is one place where the distinction between the given and the made becomes especially useful. The theological model is made. Its vocabulary is selected, its distinctions introduced, its relations specified, and its inferential consequences explored. God, if Christian theology is correct, is not made by this activity. Neither is Christ's incarnation, the resurrection, justification, or the presence of Christ in the Eucharist constituted by the conceptual structures through which theologians seek to understand them.

There is therefore an asymmetry between reality and representation. The theologian can revise the model without revising God. One can replace one account of divine simplicity with another, refine a Christological ontology, or abandon an inadequate construal of sacramental presence without supposing that the divine reality altered while the theologian was sharpening the concepts.

This asymmetry might seem obvious, though theological controversies repeatedly obscure it. A favored conceptual scheme acquires such authority that criticism of the scheme is experienced as criticism of the doctrine itself. Conversely, rejection of a particular metaphysical model is sometimes mistaken for rejection of the reality that model sought to articulate. The theologian who distinguishes these levels gains considerable freedom. One can defend the doctrine while criticizing a model of the doctrine; one can preserve the confession while revising an ontology thought necessary to sustain it.

This distinction is especially important in philosophical theology, where metaphysical language carries enormous explanatory power and corresponding danger. Categories such as ‘substance,’ ‘relation,’ ‘property,’ ‘essence,’ ‘person,’ ‘nature,’ ‘event,’ and ‘state of affairs’ can illuminate theological claims, though none arrives innocent of philosophical history. They bring inferential consequences with them. Model construction allows those consequences to become visible.

The Underdetermination of Models

A further complication now appears. A theory may have many models.

In logical notation:

M₁ ⊨ T
M₂ ⊨ T
M₃ ⊨ T

Each of these structures satisfies T, though the structures themselves may differ substantially.

The theological importance of this possibility can scarcely be exaggerated. A doctrinal formulation may constrain interpretation without determining a unique metaphysical account. Indeed, much theological disagreement may arise precisely because multiple models satisfy what disputants regard as nonnegotiable doctrinal claims.

The Nicene confession, for example, rules out certain accounts of the Son. Chalcedon rules out certain Christologies. The Lutheran Confessions exclude certain interpretations of justification and sacramental presence. Yet exclusion is not the same thing as complete determination. A confession can establish boundaries without specifying a single exhaustive ontology within those boundaries.

This is not an argument for doctrinal relativism. Quite the opposite. If several models satisfy some doctrinal theory T, one must ask what further criteria might discriminate among them. Biblical adequacy, confessional fidelity, historical continuity, explanatory scope, semantic perspicuity, ontological economy, inferential stability, and philosophical coherence may all become relevant. Yet these criteria themselves must be ordered, and their authority is not self-interpreting. We have thereby returned to method.

Models Can Reveal More Than We Put Into Them

An additional feature of model construction deserves notice. Models can disclose consequences that were not obvious when the model was first built. Once one assigns meanings, specifies relations, and formalizes commitments, one may discover that the resulting structure entails something unexpected.

This is among the genuine intellectual benefits of formal and semi-formal theological modeling. Human beings are notoriously poor at keeping track of large networks of conceptual commitments. We affirm A because it appears independently plausible, B because it accords with a doctrinal source, and C because it solves a philosophical difficulty, only later to discover that A, B, and C jointly imply D, which nobody wanted.

A model can force this discovery.

The point is analogous to proof in logic. One often knows the premises before knowing their consequences. Formalization renders latent structure explicit. Theology has sometimes resisted such procedures out of fear that formal methods somehow diminish mystery, though the objection is misplaced. Logic does not render divine reality transparent; it renders our own commitments more transparent. Those are very different achievements.

If there is mystery, it belongs to the object. Confusion belongs to us. The theologian ought not baptize the latter with the name of the former.

The Model Is Not the Doctrine

We can now state another distinction that deserves almost aphoristic emphasis: a theological model is not identical with the doctrine it models. Doctrine ordinarily consists of normative or authoritative theological claims embedded within an ecclesial, biblical, and historical context. A model is a structured interpretation intended to show how such claims might be understood as jointly true.

Consequently, criticism of a model does not ipso facto constitute rejection of a doctrine. Nor does successful defense of a model necessarily establish the doctrine itself. The levels must be held apart.

This distinction is particularly useful when theology encounters modern metaphysics. One theologian may model divine omniscience through possible worlds, another through propositions, another through divine self-knowledge, and still another through a metaphysics of truthmaking. They may disagree profoundly about ontology while confessing substantially the same doctrine. Alternately, two theologians may employ nearly identical metaphysical machinery while differing at the doctrinal level because they assign different normative status to biblical or confessional claims. Without a distinction between doctrine and model, these disagreements become almost impossible to classify.

What Makes a Model Theological?

We can now risk a provisional answer. A theological model is a structured interpretation constructed to represent how a set of theological claims may be jointly true and how the entities, properties, and relations presupposed by those claims might be understood.

The definition is intentionally modest. A model does not claim exhaustive correspondence with divine reality. It does not manufacture its object. It does not replace Scripture, confession, proclamation, or worship. Rather, it makes conceptual commitments explicit so that they can be examined.

The adjective ‘theological’ matters because theological models are not free-standing constructions. They arise within an inquiry governed by theological sources and norms. Their adequacy cannot therefore be measured solely by elegance, simplicity, or formal consistency. A model of the Trinity that is internally impeccable but incompatible with the claims that generated Trinitarian doctrine is a poor theological model, however admirable it may be as an exercise in metaphysics.

Conversely, appeals to theological authority do not absolve a model from philosophical scrutiny. If a proposed interpretation is inconsistent, equivocal, semantically unstable, or ontologically extravagant, such defects matter. Theology qua theology may have distinctive sources and norms, but it does not thereby enter a logical sanctuary in which contradiction becomes virtue.

Toward Satisfaction and Truth

We are now in a position to see the next problem. If a model supplies an interpretation under which theological sentences are satisfied, what exactly is the relation between satisfaction and truth?

When we write

M ⊨ φ

we say that φ is satisfied in M. But theology wants eventually to say more than that some sentence comes out true under some interpretation. It wants to ask whether what φ says is actually the case.

That transition is neither automatic nor trivial. It carries us from formal semantics toward questions of reference, truth, realism, and theological knowledge. It also forces us to confront one of the most seductive confusions in philosophical theology: the tendency to mistake truth-in-a-model for truth about reality.

Accordingly, the next essay in this series will ask: What does it mean for a theological claim to be satisfied, and what does satisfaction have to do with truth?

That question will take us further into model theory, though it will also bring us closer to theology's oldest intellectual ambition: not merely to construct coherent representations of God, but, insofar as creaturely judgment permits, to say truly what is the case.