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:
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:
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):
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:
and
while
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.