Model Theory in Theology: From Descartes to Luther

This short talk given on August 12, 2026, at the 15th International Luther Congress in Aarhus, Denmark, introduces model theory to historians. Like all my papers, it arises from work done at the Department of Philosophical Theology, ILT Christ School of Theology. 

Two Problems We Have Not Escaped

I want to begin some distance from model theory, with two problems associated above all with Descartes. The first is the problem of the external world. How do I know what the world is like apart from my representations of it? The second is the problem of other minds. How do I know what another person thinks, means, intends, believes, or experiences when I never occupy that person’s consciousness?

Descartes formulated these problems with unusual clarity. Four centuries later, we have not made them disappear. We have mostly learned how to live intelligently with them. We do not solve either problem by somehow getting outside ourselves. Rather, we encounter something over against us, something that resists us, and we try to render it intelligible. The world does not always behave as our theories predict. Other persons say and do things our accounts of them do not anticipate. Our interpretations can fail because something confronts us that is not simply at our disposal.

We cannot solve either problem by looking down at reality from nowhere, as though we could step outside ourselves and check our account against the thing itself. But we are remarkably good at a more modest maneuver: looking left and right, comparing one account with another and both with what resists them. That is the maneuver this essay is about.

Historians Live with the Problem of Other Minds

Historians confront the second Cartesian problem every day. At a Luther Congress we ask what Luther meant, what Melanchthon believed, what a late-medieval logician was doing with suppositio, or what theological possibilities were available to an author in 1517. But we cannot climb into Luther’s mind. We have texts, vocabulary, grammatical practices, controversies, other authors, institutional settings, manuscripts, and subsequent writings.

What historians actually do is reconstruct. We construct an account under which this evidence becomes intelligible together. Different historians may agree about the words on the page and nevertheless disagree about the best account of what those words are doing. The text and its historical setting constrain us: they possess what, borrowing loosely from Charles Peirce’s language of Secondness, I shall call an over-and-againstness or “bump-up-againstness.” They do not allow us responsibly to make them say whatever we wish.

This means that the mens auctoris need not function as an independently accessible semantic court of appeal. Authorial intention remains a legitimate historical hypothesis, but what we can reconstruct is the intentionality of a textual act from publicly available historical evidence. Our reconstruction may be excellent without becoming incorrigible.

The Special Sciences Face the External-World Version

The special sciences confront an analogous problem from the other Cartesian direction. The physicist does not have uninterpreted physical reality sitting on one side of the laboratory and a theory on the other. The biologist does not inspect life in itself. Scientists encounter phenomena, measurements, regularities, anomalies, and recalcitrant observations, and they construct theoretical structures that make these intelligible.

The historian’s problem and the scientist’s problem are not identical. A manuscript, another mind, and a physical process are not the same kind of object. But structurally the inquiries share something important: neither historian nor scientist possesses a God’s-eye standpoint. Both construct accounts of something that confronts them and is capable of proving those accounts inadequate. Theories are answerable to what they seek to understand without our having to suppose that finite knowers can compare their theories with an entirely uninterpreted world.

Theology Has Both Problems at Once

Theology inherits both problems. Theologians are historians. We ask what Paul meant, what Augustine meant, what Luther meant, what Nicaea meant, and what inherited doctrinal language was doing in its original settings. But theologians also make assertions about reality: God raised Jesus from the dead; God justifies the ungodly; Christ is present in the Supper.

Theology therefore interprets witnesses to other minds while also speaking about a world in which God is soteriologically ingredient. It asks both what inherited theological assertions meant and what account of God, world, Christ, sinner, promise, sacrament, and salvation renders the subject matter intelligible. The two Cartesian problems meet inside theological inquiry.

This is why theology cannot simply borrow its method from the historian or the scientist alone. It needs a single account of inquiry general enough to cover both tasks at once—one that tells us what we are doing when we build a model of a mind we cannot enter and a model of a God we cannot view from above. That is the account model theory can supply, and it is why I turn to it now.

A Small Luther and Trutvetter Example

Let me give one small example from another paper I presented here. Luther entered the University of Erfurt in 1501, the year Jodocus Trutvetter’s Summule totius logice appeared. We possess Trutvetter’s texts and Luther’s texts. The historical question is not merely whether Luther remembered particular sentences from Trutvetter. It is what semantic and inferential possibilities belonged to the intellectual world within which Luther’s later disputation became intelligible.

Consider a Trinitarian inference discussed by Trutvetter:

This divine essence is the Father.
This divine essence is the Son.
∴ The Son is the Father.

If I assign the copula est strict numerical identity throughout, the inference is valid. If this divine essence is numerically identical with the Father and numerically identical with the Son, then Father and Son are numerically identical with one another. There is no problem with the logic, but the theology is disastrous.

So what must est, essentia, Father, and Son be doing semantically for Trutvetter’s discussion to become intelligible? And when Luther later reasons with inherited scholastic logical vocabulary, what larger structure of interpretation makes both Trutvetter and Luther intelligible?

Here I need to point out that I am using the word interpretation in a way different from its hermeneutical use familiar to many of us. In formal semantics an interpretation assigns semantic values to the non-logical vocabulary of a language—in the simplest cases, the things to which its terms apply. Two readers may agree completely on the visible words and disagree about the meanings being assigned to them.

In historical work we are not pretending that Trutvetter’s or Luther’s prose is itself a formal language. We construct structured interpretations of the texts, and where useful we can regiment selected claims formally. One historian may model the relevant predication one way, another differently. We can then ask which model makes more of Trutvetter’s text, Luther’s text, and the historical relation between them intelligible. We can even ask whether a larger space of intelligibility can accommodate both without erasing their differences.

Theological Language Presents the Same Problem

The same issue appears when we turn from the historical reconstruction of theologians to the theological tradition and the Word of God that confront us. People of good will can agree on the language of theology while interpreting its terms very differently.

Lutherans may say, “We are justified by grace through faith.” But what is the semantic work of justification? What is grace? What sort of relation between God and the sinner is being asserted? Or Christians may agree that “Christ is present in the Eucharist” while operating with very different understandings of Christ, presence, body, sign, promise, and sacrament.

Theological disagreement therefore frequently occurs because the same sentences are satisfied under different models. A model may preserve familiar vocabulary while changing the larger structure in which the vocabulary functions. The question then becomes whether one model preserves more of the theological subject matter, whether another changes the subject, and whether several admissible models remain possible.

Model Theory Makes the Structure Explicit

Only now do I want to introduce model theory in its technical sense—and I will keep the symbols to the minimum that actually does work for us.

A theory, syntactically understood, is simply a set of sentences. Call it T:

T={φ1​,φ2​,φ3​,…}

Read this as: T is just the set of claims the theory makes.

An interpretation assigns semantic values to the non-logical terms of a language. A model is a structure under such an interpretation in which the relevant sentence or theory is true. When a structure M makes a sentence φ true under the interpretation supplied by M, we say that M satisfies φ, and we write:

$$M\models \phi$$

Read ⊨ simply as “satisfies” or, less technically, “makes true.”

A model of a theory satisfies every sentence in it. The collection of all such models is what logicians call Mod(T):

$$Mod(T)=\{M\mid M\models T\}$$

Read this as: Mod(T) is the class of models that make all of T’s sentences true.

And now we can state formally something historians and theologians encounter constantly. One theory can admit more than one model:

M1​⊨T

and

M2​⊨T,

while

~(M1​ = M2​).

The final expression says only that the two models are not numerically identical. More importantly for our purposes, they may differ substantially in the structures under which the same sentences are true. Both can satisfy the theory without representing its subject matter in the same way.

Satisfaction alone therefore does not tell us which model should be preferred. Some candidate models can be excluded because they violate constraints constitutive of the subject matter. An interpretation of a text that cannot accommodate its actual wording may cease to be an interpretation of that text. A theological model that preserves the word resurrectiononly by abandoning the identity of the one said to have been raised may have changed the subject rather than merely offered a less attractive account.

Among the models that remain admissible, we compare. We ask about consistency, coherence, explanatory scope, fecundity, parsimony, applicability, historical adequacy, capacity to accommodate recalcitrant evidence, and sometimes elegance or beauty. These virtues are not mechanically commensurable. There is no algorithm that adds them together and announces the correct model.

This is where I find Kant’s reflektierende Urteilskraft—reflecting judgment—useful. We compare, discriminate, and judge without pretending that an antecedently possessed universal rule mechanically determines the result. I call the structured field within which such comparative judgments occur teleo-space.

The position is fallibilist. Further texts, evidence, experiences, arguments, or distinctions may force us to reorder the models. But fallibility does not imply that every model has equal epistemic standing. We can give reasons why one is better than another.

Nor do we need a God’s-eye standpoint from which model and uninterpreted reality can be placed side by side. We cannot look down from nowhere. But we are remarkably good at looking left and right. We can compare models. We can see that one explains something another leaves obscure, preserves a distinction another destroys, or accommodates an obstinate text or experience another must suppress.

Model theory therefore does not solve Descartes’ problems by giving finite creatures access to reality without mediation. It does something more useful. It clarifies the structure of responsible inquiry when such access is unavailable.

The historian constructs models of other minds from texts that resist interpretation. The scientist constructs models of a world that resists theory. The theologian does both: we interpret witnesses, and we make claims about a reality in which God is soteriologically ingredient.

Our models are fallible. More than one may satisfy the assertions with which we begin. Some can nevertheless be excluded; others can be comparatively ordered. We can offer what I would call a reasoned account: a publicly criticizable, corrigible judgment about how the subject matter may be independently of our present act of modeling it.

We cannot look down from nowhere. But we can look left and right. And perhaps model theory can help theology become clearer about what it is doing when it does.

Prolegomena to Disputationes Theologicae III: Reference Before Proclamation

“Theological language cannot proclaim what it has first failed to name.”

This essay forms part of Prolegomena to Disputationes Theologicae, a series in philosophical theology produced through the Department of Philosophical Theology at Christ School of Theology. Together these essays articulate the methodological foundations of the larger Disputationes Theologicae project by recovering the proper order of theological inquiry. The series proceeds from the conviction that theology exists because these questions exist and that theology's first responsibility is to render Christian doctrine intelligible without diminishing, translating away, or replacing the reality to which it refers. Having argued that theology must first render its judgments intelligible, the present essay asks the next necessary question: How does theological language genuinely refer to God? Only language that truly refers can be truthfully proclaimed.

This essay is the third of six Prolegomena to Disputationes Theologicae, a series in philosophical theology produced through the Department of Philosophical Theology at Christ School of Theology. Together, these essays articulate the methodological foundations of the larger Disputationes Theologicae project by recovering the proper order of theological inquiry. They proceed from the conviction that theology exists because these questions exist and that theology’s first responsibility is to render Christian doctrine intelligible without diminishing, translating away, or replacing the reality to which it refers.

The preceding essay argued that intelligibility is not the source of theological truth but a condition under which theological claims may be responsibly affirmed or denied. Theology therefore seeks conceptual clarity before it renders judgment. Yet intelligibility alone cannot complete theology’s task. One may understand perfectly well what a sentence means while remaining uncertain whether it is about anything at all.

The next question therefore arises necessarily:

How does theological language become genuinely about God?

This question is prior to proclamation. The priority at issue is not necessarily temporal. The preacher need not first complete a philosophical theory of reference before proclaiming the gospel. The priority is logical and theological. Proclamation cannot create its own referent. It cannot make itself speech about God merely through rhetorical power, ecclesial authorization, existential effect, or the sincerity of the one who speaks.

One cannot proclaim what one’s language has failed to identify.

A sermon may be rhetorically compelling, existentially arresting, ecclesially sanctioned, and even morally transformative while remaining uncertain in its reference. Before theology asks whether proclamation is faithful, effective, or life-giving, it must ask whether the language of proclamation continues to name the reality of which prophetic and apostolic testimony speaks.

Theological language therefore requires more than intelligibility.

It requires reference.

Reference is among the most neglected questions in modern theology. Enormous attention has been given to meaning, interpretation, narrative, language games, performative utterance, communal practice, existential appropriation, and rhetorical effect. These inquiries have often been illuminating. Language does form communities, shape perception, order practices, and open possibilities of existence. The question, however, is whether the reality about which theology speaks is constituted by these linguistic and communal activities or whether those activities remain answerable to a reality they did not create.

The decisive question is simple:

What makes theological discourse about God rather than merely about religion?

Theology does not merely analyze religious consciousness. It does not merely describe ecclesial practices, preserve inherited vocabularies, narrate communal identities, or interpret human experiences of ultimacy. It claims to speak about God: the God of Abraham, Isaac, and Jacob; the Father of Jesus Christ; the God who creates, judges, reconciles, raises the dead, and promises the consummation of creation.

Unless these claims genuinely refer beyond the linguistic practices in which they are expressed, theology has exchanged its subject matter for its own discourse. It may continue to use the word ‘God,’ but the word may now designate only a moral ideal, a communal self-understanding, an existential possibility, a cultural memory, or the symbolic horizon of human meaning. The vocabulary remains, while the subject has quietly changed.

Reference must therefore be distinguished from several closely related notions.

Reference is not meaning. A sentence may be intelligible even when its principal terms fail to identify anything real.

Reference is not truth. A statement may successfully identify its subject while predicating something false of it. Reference makes truth and falsity possible; it does not by itself determine which obtains.

Reference is not warrant. A person may possess reasons for believing a claim even though the terms employed in that claim do not refer as the speaker assumes.

Reference is not exhaustive understanding. Speakers frequently refer successfully while possessing incomplete, confused, or partially mistaken conceptions of that to which they refer. Referential success does not require conceptual mastery.

Nor is reference identical with existential appropriation, ecclesial participation, or performative effect. These may accompany successful reference, and proclamation may indeed become a means through which God addresses the hearer. Yet neither personal transformation nor communal use can by itself guarantee that the language employed remains about the God whom Christian witness claims to name.

Theology therefore requires a distinct account of reference.

The Christian answer does not begin with the human capacity to reach God through description, inference, religious experience, or conceptual construction. It begins with God’s capacity to identify himself. God does not first become the referent of theological discourse when human beings devise a sufficiently adequate name. God gives himself to be named.

Israel does not invent the God of Abraham, Isaac, and Jacob. The Church does not construct the Father of Jesus Christ by adopting a distinctive religious vocabulary. God publicly identifies himself through acts and words: in the calling of Israel, the prophetic witness, the incarnation of the Word, the crucifixion and resurrection of Jesus Christ, the apostolic testimony, and the scriptural form in which this testimony is normatively received.

Human language refers because human beings have first been addressed.

Reference is therefore receptive before it is expressive.

This ordering distinguishes Christian theology from theories that construe theological language primarily as the projection of religious consciousness, the grammar of ecclesial life, or the symbolic articulation of human existence. Theology speaks because it has first been spoken to. It names because God has first made himself identifiable.

Yet revelation does not eliminate philosophical questions concerning reference. It creates them. Once God has acted and spoken, theology must ask how names, predicates, narratives, metaphors, and doctrines continue to refer to the God who has revealed himself. It must ask how reference remains stable through historical distance, linguistic change, doctrinal development, cultural translation, and the inevitable partiality of human understanding.

Divine self-disclosure is therefore the ground of theological reference, but it is not a substitute for theological discipline.

The problem is not merely whether the Church has retained the same words. The same expression may be preserved while its referent is altered. Nor does referential continuity require that every generation possess precisely the same descriptions or conceptual schemes. Different descriptions may identify the same reality, while identical descriptions may be employed within fundamentally different ontologies.

Theology must therefore distinguish continuity of vocabulary from continuity of reference.

This is also why theological interpretation cannot terminate in textual analysis alone. Texts possess linguistic forms, historical settings, and authorial intentions. These are indispensable to interpretation. Yet prophetic and apostolic authors do not finally intend only their own acts of writing. They intend realities. They bear witness to what God has done, whom God has identified himself to be, and what God has promised.

Theological interpretation consequently asks not only what a text meant within its first historical context, but what reality the text identifies and whether contemporary theological speech remains answerable to that same reality.

The order is therefore theological before it is hermeneutical:

God acts and speaks.

Prophetic and apostolic witnesses identify the one who has acted.

Scripture normatively bears this witness.

The Church receives, interprets, and confesses Scripture.

Doctrine tests whether the Church’s speech preserves the identity of the one witnessed to.

Proclamation addresses the hearer in the name of this same God.

The legitimacy of proclamation depends upon preserving rather than replacing this referential order. Proclamation does not establish the identity of God by its own occurrence. It becomes genuine proclamation when the God who has identified himself in Israel and in the crucified and risen Jesus Christ remains the one about whom—and through whose agency—the proclamation speaks.

The claim that reference precedes proclamation therefore does not deny that God acts through proclamation. It identifies the condition under which such a claim is intelligible. God may address the hearer through the proclaimed Word because the proclaimed Word does not invent the one who speaks through it. Its authority is derivative. Its referent is antecedent. Its efficacy, when granted, is divine.

This also explains why philosophical theology remains indispensable. Philosophy does not discover or manufacture the referent of Christian theology independently of revelation. Revelation has already identified the one about whom theology speaks. Philosophical theology clarifies the logical, semantic, and ontological conditions under which theological language may continue to refer faithfully to this God.

It distinguishes naming from description, reference from predication, identity from attributed properties, and continuity of terminology from continuity of subject matter. It asks how speakers may successfully refer under conditions of partial understanding, how descriptions may change without changing the referent, and how apparently identical theological expressions may conceal incompatible accounts of reality.

These distinctions are not external constraints imposed upon theology. They are instruments of theological accountability. Without them, theology may preserve traditional vocabulary while replacing its subject with something conceptually more manageable.

Reference is therefore neither a merely linguistic achievement nor a merely historical inheritance. It is the continuing discipline of remaining answerable to the God whose self-disclosure first made theological language possible. It is the refusal to allow the Church’s words, practices, experiences, or conceptual systems to become substitutes for the reality to which they are ordered.

Theology may revise its descriptions.

It may refine its concepts.

It may correct its inherited models.

It may discover that some of its predicates were confused, inadequate, or false.

What it may not do is quietly change the subject while continuing to speak as though nothing decisive has happened.

Reference precedes proclamation because proclamation can proclaim as gospel only what it has first received as God’s self-identification. Where reference fails, proclamation becomes religious speech about the community’s own meanings. Where reference is preserved, proclamation may remain answerable to the God who acts, speaks, judges, reconciles, and promises.

Only once the referent has been identified does the question of predication properly arise. We may then ask not merely whether theological language is about God, but whether what it says about God is true.

The next question therefore follows necessarily:

Under what conditions can theological judgments be true?