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.

What Is Theological Method?

This is the first part of a new 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.

The Question of Method

Theological disputes regularly present themselves as disputes about conclusions. One theologian avers that God acts in history while another worries that such language compromises divine transcendence; one insists upon the real presence of Christ in the Supper while another seeks a conceptual account capable of preserving what is taken to be the relevant biblical affirmations; one theologian speaks readily of divine simplicity while another suspects that the metaphysical apparatus required to sustain the doctrine creates more difficulties than it resolves. Yet beneath disagreements of this kind there usually lies another disagreement, frequently less explicit and sometimes more consequential: What counts as an appropriate way of arriving at a theological judgment in the first place?

The question of theological method therefore arises prior to many of theology's more familiar questions. Before one asks what theologians ought to conclude about some matter, one must ask what kinds of considerations legitimately bear upon the conclusion, how these considerations are related, what authority they possess, and according to what rules conflicting considerations are to be adjudicated. Method concerns, inter alia, what theologians count as evidence, what kinds of inference they permit, what distinctions they regard as legitimate, and what conditions a theological claim must satisfy if it is to be judged warranted.

The matter becomes still more difficult because theology does not operate within a single homogeneous discourse. Biblical texts, historical claims, creedal formulations, metaphysical assertions, phenomenological descriptions, liturgical practices, and logical consequences all appear within theological reasoning, though they plainly do not function in precisely the same way. The theologian who ignores these differences risks confusing the authority of a text with an interpretation of that text, an interpretation with a philosophical reconstruction of it, and that reconstruction with the reality about which the text putatively speaks. Accordingly, method is no mere preliminary housekeeping exercise. It belongs to theology's substantive intellectual work.

Method and Object

A method is ordinarily selected because of the object one seeks to understand. One does not investigate a mathematical structure in precisely the same way that one investigates a historical event, nor does one investigate a chemical reaction in the manner appropriate to interpreting a poem. The character of the object places constraints upon the procedures by which knowledge of that object may responsibly be sought.

This apparently innocent observation carries considerable theological weight. If theology has God as its ultimate object, then theological method cannot be determined wholly in advance by epistemological principles borrowed from some other domain. The theologian cannot simply announce a universally valid method and subsequently inquire whether God happens to fit within it. At least prima facie, the order must run in the other direction: what God is determines what could count as knowledge of God.

Christian theology intensifies the point because it claims that God is known through God's own acts of self-disclosure. Theology consequently begins within a peculiar epistemic situation. Its object is never simply an object standing passively before an autonomous observer. God is confessed as the one who creates the knower, sustains the knower, addresses the knower, judges the knower, and reconciles the knower. Theological knowledge, on such an account, cannot be understood simply as the successful application of a neutral human technique to a religious datum.

This does not entail intellectual obscurantism, nor does it grant theology dispensation from ordinary standards of argument. Quite the contrary. If theological claims purport to be true, then distinctions between valid and invalid inference, consistency and inconsistency, ambiguity and perspicuity, use and mention, entailment and non-entailment remain indispensable. The peculiar character of theology's object qualifies theological method without abolishing rational discipline.

Sources, Norms, and Procedures

It is useful here to distinguish three matters that theological discussions of method frequently conflate: sources, norms, and procedures.

A source supplies material for theological reflection. Scripture is a source; so are creeds, confessions, liturgical practices, historical testimony, philosophical arguments, and the accumulated conceptual vocabulary of the Christian tradition. Different theological traditions will disagree, of course, about the relative status of these sources, but identifying something as a source does not by itself determine what authority it possesses.

A norm performs another function. A norm governs theological judgment. Within classical Lutheran theology, Scripture does not stand merely as one theological source among several; it functions normatively in a manner in which Augustine, Aquinas, Luther, or the Formula of Concord does not. Precisely how such normativity is understood opens a host of hermeneutical questions, though the logical point can be made independently of their eventual resolution: the set of materials from which theology learns and the standards by which theological claims are judged need not be coextensive.

A procedure, finally, concerns what theologians do with sources under the governance of norms. They interpret texts, distinguish senses of terms, draw consequences, construct conceptual models, compare formulations, identify contradictions, trace historical development, and ask whether competing accounts preserve or obscure the phenomena requiring explanation. Much that is ordinarily called theological ‘method’ actually belongs at this procedural level.

Once these distinctions are made, certain familiar methodological quarrels become more perspicuous. A disagreement apparently concerning Scripture may turn out to concern an interpretive procedure; a disagreement ostensibly concerning metaphysics may concern the normative authority granted to some doctrinal formulation; a dispute about a theological model may finally rest upon disagreement about which biblical or confessional claims the model must preserve.

Method Does Not Manufacture Its Object

A perennial temptation arises whenever a method proves especially powerful. What begins as an instrument of inquiry gradually becomes a criterion of reality itself. Whatever the method cannot capture comes to be regarded as confused, meaningless, inaccessible, or unreal.

The history of philosophy provides abundant examples. What cannot be verified is declared meaningless; what cannot be phenomenologically given is placed in abeyance; what cannot be expressed within a particular logical language is treated as philosophically suspect; what cannot be naturalized is relegated to the realm of projection. Each maneuver may illuminate something important, though each can also convert a methodological decision into an ontological decree.

Theology must be especially attentive to this temptation. A theological method is made; God is not. Conceptual structures are constructed; the reality to which theological language refers is not thereby constructed. We formulate doctrines, distinguish concepts, erect models, and test their consequences, but these activities must not be confused with the reality they seek to articulate.

This distinction between the given and the made will become increasingly important as this series proceeds. Theologians necessarily make models. They select terms, introduce distinctions, formalize relations, and construct conceptual structures within which doctrinal claims can be understood. Such making is unavoidable and frequently fruitful. Difficulties begin when the structure of the model is silently transferred to the structure of reality, so that what belongs to our representation is treated as belonging ipso facto to that which is represented.

Caveat lector: the point does not entail that theological models are arbitrary. Some models represent their subject matter better than others. Some preserve relevant doctrinal affirmations while others distort them; some disclose consequences previously unnoticed while others conceal precisely what must be explained. Yet the distinction between model and modeled reality remains indispensable to judging any model at all.

Theological Method as Disciplined Judgment

We can now risk a provisional definition. Theological method is the disciplined ordering of sources, norms, concepts, and inferential procedures by which theologians seek truth concerning God and all things in relation to God.

Each part of this definition matters. Method is disciplined because theological inquiry cannot proceed merely by association or intuition. It involves ordering because sources, norms, concepts, and arguments do not arrive already assembled into a theological system. It seeks truth because theology claims more than internal coherence within a religious language-game. And its subject matter includes ‘all things in relation to God’ because theology has historically concerned creation, humanity, sin, history, church, sacrament, death, resurrection, and consummation precisely sub specie relationis ad Deum.

The phrase ‘seek truth’ should also remain firmly in view. Method does not guarantee truth. No procedure can do that. A perfectly executed inference may proceed from a false premise; an internally coherent theological model may misdescribe reality; an elegant conceptual system may succeed brilliantly at explaining claims that ought themselves to be rejected. Method disciplines judgment without rendering judgment apodictic.

The theologian therefore inhabits an unavoidable tension. Theology requires method because thinking without methodological discipline easily collapses into equivocation, inconsistency, and rhetorical assertion. Yet theology must also submit its methods to criticism because no human method enjoys an unrestricted view of its object. The method serves the inquiry; the inquiry does not exist to vindicate the method.

From Method to Models

This brings us to the question that will occupy the next stage of the argument. Once theologians begin ordering doctrinal affirmations, clarifying their relations, and asking what must be true if those affirmations are true, they inevitably begin constructing models. They may not call them models, and they may resist the vocabulary of model theory altogether, but the activity remains recognizable. Certain entities are assumed, relations among them are specified, propositions are held fixed, and the consequences of alternative interpretations are explored.

Accordingly, the next question is unavoidable: What is a theological model?

Answering it will require us to distinguish a model from a theory, a theory from its interpretation, an interpretation from the reality interpreted, and satisfaction within a structure from truth simpliciter. These distinctions might initially appear technical. In fact, they bear directly upon some of theology's oldest disputes, because theologians have often disagreed less about the sentences they confess than about the structures within which those sentences are taken to be true.

The question of method thus leads naturally to the question of models. And once models enter the discussion, logic, semantics, and ontology cannot remain far behind.

Gödel’s Ontological Argument: What Formal Proof Can—and Cannot—Establish

This is the twelfth and final part of a series developed through the Department of Philosophical Theology at the Institute of Lutheran Theology’s Christ School of Theology, exploring major results and developments in modern logic and their significance for philosophical and systematic theology.

It is fitting to end this series by returning to Gödel. We encountered him earlier because the completeness theorem showed an extraordinary correspondence between syntactic derivability and semantic consequence in first-order logic, while the incompleteness theorems exposed principled limits upon sufficiently strong formal theories. We return to him now in a rather different role, for Gödel also worked for many years upon a formal reconstruction of the ontological argument, bringing together higher-order quantification, modal logic, properties, essences, and necessary existence in an attempt to show that the existence of a Godlike being follows from a small set of explicitly stated axioms.

The argument is sometimes reported under the breathless heading that Gödel “proved that God exists,” which is almost exactly the wrong way to understand its philosophical importance. What Gödel produced was a formal argument within a specified logical framework; consequently, the interesting question is not whether the symbols somehow compel belief in God, but what has been established once the derivation is valid. To answer that question requires distinctions we have accumulated throughout this series: between syntax and semantics, proof and truth, axioms and interpretations, necessity and actuality, object language and metalanguage, and finally between a formally successful model and the reality that the model is intended to represent.

There is also a historical complication worth keeping in view. What is usually called 'Gödel’s ontological proof' is now better regarded as a family of closely related arguments. Gödel left a compact manuscript dated 1970; Dana Scott, after discussing the argument with Gödel, produced a slightly modified formulation that became especially influential, while C. Anthony Anderson, Melvin Fitting, and others later proposed further emendations. Recent formal work has made the distinctions among these versions increasingly precise.

Positive Properties and a Godlike Being

The argument begins not with existence but with properties. Let

PF

mean:

F is a positive property.

Gödel did not reduce positivity to some more elementary formal notion. 'Positive' functions as a primitive predicate upon properties, and axioms specify how positive properties behave. This point is crucial, because the proof does not manufacture substantive content from logic alone; it begins with substantive assumptions governing a class of properties and then investigates what follows from those assumptions.

Using a Scott-style presentation, one central principle says, roughly, that a property and its negation cannot both be positive and that one of them must fall on the positive side. Another principle says that if a positive property necessarily entails another property, the entailed property is also positive. We may represent the latter as

[PF ∧ □∀x(Fx → Hx)] → PH.

The reading is straightforward: if F is positive, and necessarily everything possessing F possesses H, then H is positive as well.

Gödel then defines a Godlike individual as one possessing every positive property:

Gx ↔ ∀F(PF → Fx).

Thus:

x is Godlike if and only if x possesses every positive property.

Notice what has happened. 'Godlike' has not been introduced as an unanalyzed name for the Christian God, nor has divine existence been inserted explicitly into the definition. The predicate G is constructed from the prior notion of positivity, and the burden of the argument consequently begins shifting toward the axioms governing positive properties.

One of the most important axioms then says that being Godlike is itself positive:

PG.

Together with the principles governing positive properties, one can prove that every positive property is possibly exemplified. Since being Godlike is positive, it follows that

◇∃xGx.

That is:

Possibly, there exists a Godlike being.

The move deserves attention because modal ontological arguments are often caricatured as simply assuming that God possibly exists and then exploiting S5 to obtain necessary existence. In the Gödel-Scott construction, the possibility claim appears as a theorem derived from more fundamental assumptions about positivity. Whether those assumptions are plausible is another question, but formally the distinction matters. In the familiar Scott presentation, positive properties are shown to be possibly exemplified, and Godlikeness is stipulated to be positive; hence the possible exemplification of Godlikeness follows.

Essence and Necessary Existence

Possibility alone does not give Gödel what he wants, however, and the next steps introduce the concepts of essence and necessary existence. Let

F Ess x

mean:

F is an essence of x.

In the Scott-style formulation, this can be represented schematically as

F Ess x ↔ Fx ∧ ∀H[Hx → □∀y(Fy → Hy)].

Thus F is an essence of x when x actually possesses F and F necessarily entails every property H that x possesses. The definition is extremely strong. An essence does not merely belong importantly or characteristically to an individual; it necessarily carries with it every property possessed by that individual under the conditions specified by the formalism. Scott's addition of the requirement Fx—the requirement that x actually exemplify the alleged essence—turns out to be technically significant, since recent formal analysis shows that a strict rendering of Gödel's own 1970 definition without this condition produces inconsistency, whereas the Scott modification avoids that particular problem.

Necessary existence is then defined through essences:

NEx ↔ ∀F(F Ess x → □∃yFy).

In words:

x exists necessarily if and only if every essence of x is necessarily exemplified.

Gödel then adds another crucial axiom:

PNE.

Necessary existence is a positive property.

Since a Godlike being possesses every positive property, any Godlike being possesses necessary existence. Moreover, the argument establishes that Godlikeness itself is an essence of anything Godlike. Once these pieces are assembled, the conclusion follows:

□∃xGx.

Necessarily, there exists a Godlike being.

This is a genuine formal result. The familiar Scott variant has been formally checked using contemporary higher-order theorem provers and proof assistants, and the derivation of the necessary existence conclusion from the stipulated axioms and definitions can be verified mechanically. Indeed, the computer-assisted work is philosophically interesting precisely because it removes much uncertainty about whether some unnoticed inferential gap lies hidden inside the argument.

But now the philosophical work begins rather than ends.

What Exactly Has Been Proved?

Three questions must be distinguished. First, does the conclusion follow from the axioms and definitions in the specified logic? Second, are those axioms themselves true or otherwise rationally warranted? Third, do 'positive property', 'Godlike', 'essence', and 'necessary existence' adequately represent the theological and metaphysical concepts to which we intend them to refer?

The first question is formal. The latter two are not settled merely by answering the first.

Suppose T is the theory consisting of the relevant axioms and definitions, while φ is the claim that necessarily a Godlike being exists. We may establish

T ⊢ φ.

Given the proof system, φ is derivable from T. If the semantics is appropriate and the formal system sound, we may correspondingly have

T ⊨ φ.

Every model satisfying T satisfies φ.

Neither statement, however, contains the further premise that T is true of reality. That claim must come from somewhere else. A valid derivation tells us what follows if the axioms hold; it does not transform the axioms into metaphysical truths merely because their consequences have been derived without error.

This is especially important because 'positive property' remains primitive. The axioms tell us how positivity behaves: positive properties must satisfy certain closure conditions, Godlikeness is positive, necessary existence is positive, and so forth. But the formal system does not independently establish that the relevant theological understanding of perfection, goodness, or divine reality corresponds to precisely this class of formally positive properties.

We can now see why merely announcing that the proof has been computer-verified misses the point. A proof assistant can establish that the conclusion follows from the formalized premises, and model finders can test consistency or produce countermodels to candidate claims. They cannot, merely by executing those procedures, determine whether 'positive' has captured what a theologian means by divine perfection or whether Gödel's definition of 'essence' captures what belongs to the essence of God. Modern automated work on the argument has been valuable precisely because it separates these questions instead of collapsing them.

The Problem of Modal Collapse

The most striking illustration is the phenomenon known as modal collapse. In the Gödel-Scott family of formulations under discussion, the axioms are strong enough to derive

φ → □φ.

Whatever is true is necessarily true.

If this principle holds generally, then the distinction between contingent and necessary truth collapses. What actually happens could not have been otherwise, at least within the modal structure represented by the theory. Automated analysis has confirmed that modal collapse follows in the familiar Scott-style formulation and in closely related corrected forms of Gödel's argument.

For theology this is hardly an insignificant consequence. Classical Christian theology ordinarily distinguishes the necessity of God's being from the contingency of creation. God does not create because God lacks the ability not to create, and the created order is not ordinarily regarded as following from the divine essence with the same necessity with which God is God. If every actuality is necessary, the formal system threatens precisely this distinction between Creator and creature, necessity and freedom, which means that the theologian has good reason to inspect the assumptions producing the collapse.

Yet the right response is not to say that modal collapse proves Gödel's argument invalid. If the collapse is derivable from the axioms, then it is one of their consequences, and a formally valid proof cannot be refuted by disliking another theorem of the same system. Rather, modal collapse gives us evidence relevant to the independent assessment of the axioms: if those axioms entail a consequence we have strong theological or metaphysical reason to reject, then we have reason to reconsider the axioms, their definitions, or the logical framework within which they operate.

Later variants make this point particularly clear. Anderson and Fitting alter Gödelian assumptions in ways that preserve versions of the necessary-existence argument while avoiding modal collapse. The existence of such variants shows that the collapse is not simply an unavoidable consequence of any modal ontological argument; it depends upon how the relevant notions have been formalized and which axioms govern them.

When Formalization Discovers Something

Here the argument becomes a fitting conclusion to our series, because formalization is doing more than decorating an old philosophical argument with symbols. By making definitions and inferential commitments explicit, it can reveal consequences that ordinary prose leaves hidden. Modal collapse is one example; the recently identified difficulty with the unmodified 1970 definition of essence is another. What looked informally close enough can turn out formally to matter greatly.

This is one of the genuine promises of formal methods for theology. A formal reconstruction may show that a conclusion does not follow unless some additional premise is introduced, that two formulations previously regarded as equivalent actually behave differently, that an apparently harmless definition generates an unwanted theorem, or that weakening an axiom preserves the desired result while avoiding an objection. In such cases logic is not replacing theological judgment but giving theological judgment a more exact object upon which to work.

The same point applies to models. If there is a model of T in which some candidate theological conclusion fails, then the conclusion does not follow merely from T. If every model of T satisfies the conclusion, we have established semantic consequence. If T possesses models with structures substantially different from the one theology intended, the Löwenheim–Skolem considerations encountered earlier in this series return. If the intended structure can be isolated only by moving to stronger higher-order resources, the costs examined in our discussion of second-order logic arise. If necessarily equivalent formulations nevertheless differ in theological content, the problem of hyperintensionality returns as well.

Gödel's little argument thus sits at the intersection of nearly everything we have been discussing.

Why It Matters for Theology

The great theological lesson of Gödel's ontological argument is therefore neither that formal logic has proved God nor that formal logic is incapable of speaking meaningfully about God. Both conclusions are too easy. The argument shows instead what becomes possible when theological and metaphysical commitments are made explicit enough to enter a rigorous formal system.

Once the axioms have been stated, logic can be relentless. It can determine consequences that the original author may not have noticed, expose hidden dependence upon modal principles, distinguish definitions that initially appeared equivalent, and even allow computers to verify derivations whose details would otherwise be extraordinarily difficult to survey. What logic cannot do merely by being logic is certify that the primitive predicates have been interpreted correctly or that the axioms from which the derivation begins are true of God.

This distinction is not a weakness of formalization. It is the condition under which formalization becomes intellectually useful.

The theologian therefore ought neither fear formal logic nor ask it to do work it cannot do. When a formal argument establishes

T ⊢ φ,

the achievement can be considerable. We now know that φ follows from T according to the stated rules. The next questions concern T itself: what its terms mean, what its axioms assert, what models satisfy it, whether those models correspond to the intended subject matter, and whether we have independent reason to believe that the world—or God—is as the theory represents.

Those questions cannot be evaded by pointing again to the proof.

The Series in Retrospect

We began this series with Frege, Peirce, and Cantor because modern logic enormously expanded what could be formally expressed. Russell and the development of axiomatic methods showed why disciplined formal construction was necessary; Gödel showed both the extraordinary reach and the principled limitations of proof; Löwenheim–Skolem and Compactness taught us that theories may have structures we never intended; Tarski taught us to distinguish truth from satisfaction and object language from metalanguage; Church and Turing placed limits upon mechanical decision; Kripke gave necessity and possibility a model-theoretic semantics; second-order logic showed how additional expressive strength can be purchased at metatheoretical cost; hyperintensionality showed that even complete modal agreement may fail to capture sameness of content; and nonclassical logics taught us that the relation of consequence itself may become an object of philosophical investigation.

Gödel's ontological argument draws these threads together because it forces us to ask, all at once, what language we are using, over what its variables range, which modal semantics we have chosen, which properties our higher-order quantifiers admit, what our definitions mean, which axioms are assumed, what follows from them, and whether the formal structures thereby generated correspond to the theological reality about which we intend to speak.

After twelve installments, that may be the most important lesson modern logic can offer theology. Formalization does not abolish interpretation, metaphysics, or theological judgment; neither does it leave them where it found them. It disciplines them by forcing us to locate exactly where our commitments enter and exactly what those commitments entail.

Logic does not relieve theology of the obligation to speak truthfully about its subject matter. It makes it considerably harder for theology to conceal from itself what it has actually said.

Bibliographical Note

Gödel's ontological argument appears in the posthumously published third volume of his Collected Works, with an introduction by Robert Merrihew Adams. Dana Scott's closely related formulation became one of the principal versions discussed in the subsequent literature. C. Anthony Anderson's “Some Emendations of Gödel's Ontological Proof,” Faith and Philosophy 7 (1990): 291–303, develops an influential revision, while Melvin Fitting's Types, Tableaus, and Gödel's God (Kluwer, 2002) provides an extensive logical treatment. Christoph Benzmüller and Bruno Woltzenlogel Paleo inaugurated detailed computer-supported verification of the argument using contemporary higher-order theorem provers and proof assistants, while later work by Benzmüller, David Fuenmayor, Annika Kanckos, Scott, and others has clarified the relations among different Gödelian variants, modal collapse, positivity, and the exact logical strength required by the argument. Recent work with Scott also distinguishes more sharply Gödel's 1970 manuscript from Scott's modified version and shows the importance of the precise definition of essence.

Beyond Possible Worlds: Hyperintensionality and the Grain of Theological Content

This is the tenth part of a series developed through the Department of Philosophical Theology at the Institute of Lutheran Theology’s Christ School of Theology, exploring major results and developments in modern logic and their significance for philosophical and systematic theology.

Kripke semantics gave modal logic an extraordinary conceptual advance. Once necessity and possibility could be interpreted relative to possible worlds and an accessibility relation, claims that had seemed resistant to rigorous semantic treatment became formally manageable. To say that φ is necessary at a world w is to say that φ is true at every world accessible from w; to say that φ is possible is to say that φ is true at at least one accessible world. With this apparatus, philosophers could distinguish actuality from necessity, possibility from actuality, and different systems of modal reasoning by imposing different conditions upon accessibility.

But a successful semantics can disclose its own limitations precisely by being successful. Possible-world semantics distinguishes propositions that differ in their modal profiles. What happens, however, when two propositions have exactly the same modal profile but nevertheless appear to differ in meaning, explanatory role, subject matter, or metaphysical ground? This is the problem of hyperintensionality.

To see the issue, we should first distinguish three levels. An extensional context is sensitive to extension—in the case of sentences, principally to truth value. An intensional context can distinguish expressions that have the same actual extension but differ across possible worlds. A hyperintensional context is finer-grained still: it may distinguish expressions even when they are necessarily equivalent, and therefore have the same truth value at every possible world. That is the central idea behind contemporary talk of hyperintensionality.

Suppose, for example, that φ and ψ are necessarily equivalent:

□(φ ↔ ψ).

On a coarse-grained possible-world account in which propositions are identified with the sets of worlds at which they are true, φ and ψ determine the same proposition. They are true at precisely the same worlds, and nothing in their possible-world extensions distinguishes them.

Now consider two necessary truths:

2 + 2 = 4.

and

If God is triune, then God is triune.

Assuming standard arithmetic and classical logic, both are true at every possible world under consideration. If propositions are simply sets of possible worlds, both consequently correspond to the same set: the set of all possible worlds. Yet one proposition concerns arithmetic, while the other concerns the logical consequence of a theological predication. Whatever account we finally give of propositional content, it seems difficult to maintain that they say the same thing merely because no possible world distinguishes their truth values. This is a standard pressure against identifying propositional content simply with sets of possible worlds: distinct necessary truths collapse into the same coarse-grained intension.

The corresponding problem arises for necessary falsehoods. If two propositions are impossible, each is true at no possible world. On the same coarse-grained account, both correspond to the empty set, even though they may express entirely different impossibilities. Possible worlds tell us where propositions are true; they do not always tell us finely enough what those propositions say.

The theological significance of this becomes apparent almost immediately.

Suppose a theologian maintains that God is triune is necessarily true. Suppose also that 7 + 5 = 12 is necessarily true.

The propositions then agree in modal profile: each is true at every possible world. But no theologian wishes to infer that the doctrine of the Trinity and an elementary proposition of arithmetic possess the same theological content. The former says something about God; the latter does not. Modal equivalence, even necessary equivalence, is therefore too coarse to capture every distinction theology needs to make.

The point becomes still clearer when we consider explanation. Assume that φ and ψ are necessarily equivalent. It does not follow that

φ because ψ

and

ψ because φ

are interchangeable. Explanation has direction. The existence of Socrates may explain the existence of Socrates' singleton, for example, even though, necessarily, Socrates exists if and only if the singleton of Socrates exists. Reversing the explanation does not thereby become equally satisfactory. Contemporary discussions of grounding therefore routinely treat grounding and explanation as hyperintensional: substitution of necessarily equivalent claims can change the truth or adequacy of a grounding or explanatory statement. Theology is filled with precisely such explanatory asymmetries.

Consider the difference between saying that something is true because God is what God is and saying merely that the proposition is necessarily true. If

□φ,

we know that φ holds throughout the relevant space of possible worlds. But from this alone we have learned nothing about why φ is true, whether φ belongs to the essence of something, or what metaphysically grounds φ.

Necessity and essence therefore come apart. An influential line of contemporary metaphysics, associated especially with Kit Fine, argues that although essential truths are necessary, not every necessary truth about an object states something belonging to its essence. One may have necessary connections to countless objects or mathematical truths that contribute nothing to what one is. The modern literature on grounding makes the same point: essence and metaphysical explanation seem to require distinctions finer than modal covariance across possible worlds.

That matters greatly for classical theology. When the theologian says that omnipotence, goodness, or triunity belongs to God essentially, the claim is not obviously exhausted by saying that God possesses the relevant property in every possible world in which God exists. The theologian is saying something about what God is, not merely plotting the distribution of a predicate across modal space.

Compare:

Necessarily, if God exists, then 2 + 2 = 4.

with:

Necessarily, if God exists, then God is God.

Both may be necessary. Yet the second appears connected to divine identity in a way the first plainly is not. Possible-world necessity by itself does not mark that difference.

Hyperintensionality therefore raises a question more fundamental than whether modal logic is adequate. The question is whether modal profile supplies a sufficiently fine grain of content for all the philosophical work theology asks propositions to perform. In many cases it does not.

Belief provides another familiar example. A person may believe φ without believing ψ even when φ and ψ are necessarily equivalent. Someone may believe a complicated mathematical theorem without recognizing an equivalent formulation of that theorem, or believe one description of an individual without believing another necessarily co-referring description. If belief were modeled entirely by the set of possible worlds compatible with what the believer believes, problems of logical omniscience quickly arise: the believer threatens to become committed to every logical consequence of everything believed. Hyperintensional approaches seek a semantic grain fine enough to distinguish contents that possible-world semantics treats alike.

The theological analogue is obvious. A fourth-century theologian might affirm everything needed for a doctrinal conclusion without possessing our later conceptual formulation of that conclusion. Two creedal formulations might agree extensionally, or even necessarily, while differing significantly in what they make explicit, what conceptual distinctions they employ, and what theological errors they exclude. If theological propositions are identified solely by their truth across possible worlds, some of these differences risk disappearing.

This is not merely a problem about wording. The distinction between homoousios and a formulation that happens to have the same truth conditions may matter precisely because doctrinal language intends to say something determinate about the relation of Father and Son. Likewise, two theories of justification might generate the same verdicts about every imagined case while differing concerning what grounds justification, what role faith plays, or what relation obtains between promise and reception. Agreement in extension—even necessary agreement—does not by itself establish sameness of theological account. How, then, should hyperintensionality be modeled?

There is no single accepted answer. Some approaches treat propositions as structured entities rather than merely sets of worlds, so that the internal semantic organization of a proposition contributes to its identity. Others employ impossible worlds: points of evaluation at which logical, mathematical, or metaphysical impossibilities may obtain. Two necessary truths that agree at every possible world can then differ because they behave differently at impossible worlds. Still other approaches use finer-grained notions of facts, states of affairs, subject matter, proof, grounds, or structured meaning.

Impossible-world semantics is particularly instructive. If φ and ψ are both necessary, no possible world distinguishes them. But an impossible world might be one at which φ holds while ψ does not. Extending the semantic space beyond the possible thereby gives us a way of separating contents that standard possible-world semantics collapses. This does not require believing that impossible worlds concretely exist somewhere; like possible worlds in formal semantics, they may be treated as representational devices within a semantic theory. Contemporary hyperintensional semantics employs precisely such strategies.

But once again, greater expressive discrimination brings philosophical costs. How fine-grained should propositions become? If every syntactically distinct sentence expresses a different proposition, we distinguish too much. If all necessary equivalents express the same proposition, we distinguish too little. Between these extremes lies the hard question of which differences matter for meaning, explanation, essence, grounding, belief, and subject matter.

The problem is therefore not merely to make semantic content finer-grained. It is to make it finer-grained in the right way.

This brings us back to a theme running through the entire series. Frege gave us quantification; model theory taught us how theories are interpreted in structures; Löwenheim–Skolem and Compactness exposed limits upon how tightly first-order theories control their models; Tarski distinguished truth from the semantic machinery by which truth is defined; Kripke showed how necessity and possibility can be treated through possible worlds; second-order logic showed that additional expressive power can be purchased, but only at a price.

Hyperintensionality now reveals another boundary. Even when we know the truth value of a proposition at every possible world, we may still not know enough about its content.

That is a remarkable result for philosophical theology, because theology is concerned not merely with which sentences come out true but with what is being said, what makes it true, how one truth explains another, what belongs to the essence of God or creature, and which conceptual distinctions are doctrinally significant. A semantics that records only distributions of truth across possible worlds may therefore be indispensable for modal reasoning while remaining insufficient for these further tasks.

Why It Matters for Theology

Hyperintensionality matters for theology because theological truth is not exhausted by modal extension. Two claims can agree at every possible world and nevertheless differ in subject matter, meaning, explanatory direction, essential content, or metaphysical ground.

This is particularly important when theology speaks of divine essence, Trinitarian relations, incarnation, justification, sacramental presence, or divine action. In such cases theologians ordinarily care not merely that certain propositions are necessarily connected but how they are connected and what accounts for the connection. To say that φ necessarily accompanies ψ is weaker than saying that ψ grounds φ, that φ belongs to the essence of some object, that ψ explains φ, or that φ and ψ express the same content.

Possible-world semantics therefore remains enormously valuable without being semantically exhaustive. Kripke taught us how to represent modal profile. Hyperintensional theories remind us that modal profile is not always identity of content.

For philosophical theology, the lesson is again one of discrimination rather than skepticism. The question is not whether formal semantics fails, but which semantic distinctions a particular formalism is capable of representing. Once theology begins asking not merely what could or must be true, but what a doctrine means, what grounds it, what explains it, and what belongs essentially to its subject matter, it has entered territory in which possible worlds alone may no longer be enough.

Bibliographical Note

The contemporary literature on hyperintensionality grows from problems concerning belief, meaning, logical omniscience, essence, grounding, and explanation. Kit Fine's work on essence was especially important in challenging the reduction of essence to necessity, while subsequent work on grounding and metaphysical explanation has reinforced the need for distinctions among necessarily equivalent contents. Mark Jago's The Impossible: An Essay on Hyperintensionality (Oxford University Press, 2014) and Francesco Berto and Mark Jago's Impossible Worlds (Oxford University Press, 2019) develop impossible-world approaches to hyperintensional semantics. Contemporary surveys treat structured propositions, impossible worlds, grounding, essence, content, and related approaches as different attempts to explain how semantic and metaphysical distinctions can be finer-grained than possible-world intensions.

Second-Order Logic: The Promise and Price of Saying More

This is the ninth part of a series developed through the Department of Philosophical Theology at the Institute of Lutheran Theology’s Christ School of Theology, exploring major results in modern logic and their significance for philosophical and systematic theology.

First-order logic achieved something remarkable. It gave modern mathematics and philosophy a formal language expressive enough to represent enormously complicated structures while retaining a collection of equally remarkable metatheoretical properties. Gödel proved it complete. Compactness tells us that if every finite subset of a first-order theory has a model, the whole theory has a model. The Löwenheim–Skolem theorems tell us that theories with infinite models ordinarily have models of different infinite sizes. Church and Turing showed that validity is not decidable, but valid first-order sentences can nevertheless be effectively enumerated through formal proof.

By this point in our series, however, we have also discovered the price paid for these virtues. First-order theories frequently fail to determine uniquely the structures about which we intended to speak. The Löwenheim–Skolem theorem guarantees that a first-order theory with an infinite model cannot, under the usual conditions, uniquely characterize an infinite structure up to isomorphism. If our theory has the intended natural numbers as a model, it will also have nonstandard models. If we hoped that sufficiently careful first-order axiomatization would force interpretation back onto the one structure we originally intended, model theory tells us otherwise.

One response is to strengthen the language.

In first-order logic our quantifiers range over individuals:

∀x Px.

We read this:

Every individual is P.

Second-order logic allows us also to quantify over properties and relations themselves. Thus we may write

∀X φ,

where X is not an individual variable but a predicate variable. Under the standard, or full, semantics for second-order logic, a one-place predicate variable ranges over all subsets of the domain, a two-place relation variable ranges over all sets of ordered pairs from the domain, and similarly for relations of higher arity. The apparently small move from quantifying only over objects to quantifying also over properties and relations produces a dramatic increase in expressive power.

The natural numbers provide the classic example.

First-order arithmetic can express induction only by means of an axiom schema. For every formula φx of the appropriate kind, there is a corresponding induction axiom. What first-order logic cannot say in a single sentence is that induction holds for every property whatsoever of natural numbers, because the variables of first-order logic do not range over properties.

Second-order logic can say exactly this:

∀X[(X0 ∧ ∀x(Xx → Xx⁺)) → ∀xXx].

The formula says:

For every property X, if 0 has X and whenever a number has X its successor also has X, then every natural number has X.

Under full second-order semantics, X ranges over every subset of the domain. The second-order Peano axioms consequently characterize the natural-number structure up to isomorphism. Unlike first-order Peano arithmetic, they have no nonstandard models when interpreted under full semantics. Similar resources permit second-order characterizations of other important mathematical structures, including the real numbers as a complete ordered field.

Here, then, is the promise of second-order logic. It can sometimes say what first-order logic cannot say, and because it can say more, it can sometimes constrain its models far more tightly.

This point deserves emphasis because it returns us to the problem raised by Löwenheim–Skolem. Suppose our concern is not merely to construct some structure satisfying a theory, but to characterize the intended structure uniquely up to isomorphism. First-order logic may frustrate that aspiration for principled reasons. Second-order logic can sometimes accomplish precisely what first-order logic cannot.

But the gain is not free.

The very features that give full second-order logic its greater expressive power cost us several of the great metatheorems that made first-order logic so attractive. There is no effective sound and complete proof calculus for full second-order validity. Compactness fails. The ordinary Löwenheim–Skolem results fail as well. Indeed, these failures are closely related to second-order logic's capacity to characterize structures such as the natural numbers and the real numbers categorically.

This is not an accidental defect in a formalism awaiting technical repair. If full second-order arithmetic categorically characterizes the natural numbers, it cannot at the same time possess the first-order combination of expressive limitations, compactness, and Löwenheim–Skolem behavior that generated nonstandard models in the first place. We have gained one thing partly because we have surrendered another.

The trade becomes clearer if we return to Gödel's completeness theorem. For first-order logic,

T ⊨ φ if and only if T ⊢ φ.

Every semantic consequence of T is captured by formal derivation. No corresponding effective proof system captures all validities of second-order logic under full semantics. In fact, the validities of full second-order logic are not recursively enumerable. There can therefore be no mechanical procedure which simply generates all and only the valid second-order formulas by means of a complete formal calculus.

At precisely this point Leon Henkin discovered something illuminating. If we weaken the semantics so that second-order variables do not range over all subsets and relations on the domain, but only over a specified collection of them, a completeness theorem can be recovered. These are now called Henkin or general models. Under Henkin semantics, second-order logic behaves in important respects like many-sorted first-order logic and regains familiar completeness and model-theoretic properties.

But once again the gain has a price. The categorical power associated with full second-order semantics is no longer generally available.

The resulting choice is philosophically fascinating. If we demand that the second-order quantifiers really range over all subsets and relations of the appropriate kind, we obtain the expressive strength responsible for categoricity, but lose an effective complete proof system and other first-order metatheoretical properties. If instead we restrict the ranges of those quantifiers sufficiently to recover a Henkin-style completeness theorem, much of the distinctive model-theoretic strength that attracted us to second-order logic disappears.

This raises an even deeper question: Where did the additional expressive power come from?

Under full semantics, when we say that X ranges over every property of objects in a domain D, the usual set-theoretical semantics understands this as quantification over the entire power set of D. But if the semantic apparatus must already determine what all the subsets of D are, then a great deal has been placed into the metalanguage before the object language begins its work. The categoricity achieved by full second-order logic therefore depends upon a very strong semantic interpretation of its quantifiers.

This is why some philosophers have questioned whether full second-order logic should be regarded simply as “logic” in the same sense as first-order logic. Others have argued that the additional expressive power is precisely what makes second-order logic indispensable. The dispute need not be settled here. What matters for our purposes is that stronger formal expressiveness may transfer some of the burden from axioms inside the formal theory to semantic assumptions governing the range of the quantifiers.

For theology, that observation should sound familiar.

Theologians regularly speak in ways that appear to quantify not merely over individuals but over properties and relations. Claims about identity offer a simple example. A Leibnizian principle can be represented as

∀x∀y[(∀X(Xx ↔ Xy)) → x = y].

This says:

For any objects x and y, if x and y have exactly the same properties, then x is identical with y.

Whatever one finally thinks about the metaphysics of properties or the adequacy of the principle, the logical point is clear. “Every property” is not ordinary first-order quantification. If the phrase is meant literally, we have crossed into second-order territory.

Such territory appears quickly in philosophical theology. Discussions of divine attributes, personal identity, Christology, the Trinity, essence, necessity, and the relation between nature and properties can require us to speak not merely about objects but about what may truly be predicated of objects. Second-order resources can therefore make explicit distinctions that a purely first-order language either cannot make or can reproduce only indirectly.

There is another theological attraction, however, that may be even more important. Throughout this series we have repeatedly distinguished a formal theory from its intended interpretation. Theology ordinarily does not mean to say merely that there is some model in which its sentences come out true. When it speaks about God, creation, incarnation, justification, or resurrection, it intends its assertions to concern a determinate reality.

This can make categoricity look enormously attractive.

Suppose a theological theory T admits a large family of non-isomorphic models. We might ask whether this plurality reflects genuine theological alternatives, harmless differences in representation, or simply the expressive weakness of the language in which T has been stated. Second-order resources may sometimes allow us to constrain the intended structure more sharply than first-order resources permit.

But nothing follows merely from the fact that we have moved to second-order logic. Second-order logic does not magically identify the intended theological structure. Nor does increased expressive power guarantee theological truth. We must still determine whether the predicates have been interpreted properly, whether the axioms say what the doctrine intends, and whether the semantic resources introduced by the formalism correspond to anything we are prepared to accept metaphysically.

Indeed, second-order logic sharpens rather than eliminates the problem of interpretation. With first-order logic, we worried that the axioms did not control their models tightly enough. With full second-order logic, we must additionally ask what licenses our claim to quantify over all the relevant properties and relations.

This yields an important lesson for model-theoretic theology. There are at least two ways in which a formal theory can fail to capture what we intend. Its language may be too weak to characterize the intended structure, or its semantics may achieve the desired characterization only because the intended structure has effectively been built into the semantic apparatus. The first danger is underdetermination; the second is concealed determination from the metalanguage.

Neither is solved by formalism alone.

Second-order logic therefore places philosophical theology before a genuine methodological choice. Sometimes the expressive poverty of first-order logic is a virtue. Its weakness makes possible completeness, compactness, and powerful general model theory. At other times that same weakness prevents us from saying what we actually want to say, particularly when we want to quantify over properties, characterize structures categorically, or distinguish intended from unintended interpretations more sharply.

The proper question is therefore not whether second-order logic is better than first-order logic. The question is what theological work we need the logic to do and what semantic price we are prepared to pay for allowing it to do that work.

Why It Matters for Theology

Second-order logic teaches theology that expressive power and formal tractability do not necessarily increase together. A stronger language can distinguish more, characterize more, and sometimes eliminate unintended models, while at the same time surrendering completeness, compactness, effective axiomatizability, and other properties available to first-order logic under its standard semantics.

That tradeoff matters whenever theology attempts formal reconstruction. If we restrict ourselves to first-order resources, we must accept limits upon what our theories can characterize. If we invoke full second-order resources, we must be explicit about the semantics that gives those resources their power and about the assumptions entering through the metalanguage.

The deepest lesson may therefore concern something we have encountered repeatedly throughout this series. Formal precision does not remove philosophical decisions; it makes their location clearer. Sometimes the decisive commitment lies in an axiom. Sometimes it lies in a rule of inference. Sometimes it lies in the structure selected as a model. And sometimes, as second-order logic shows with unusual clarity, it lies in what we have allowed our quantifiers to range over before the argument has even begun.

Theology should want that clarity.

Bibliographical Note

Leon Henkin's “Completeness in the Theory of Types,” Journal of Symbolic Logic 15 (1950): 81–91, established the completeness result associated with what are now called Henkin or general semantics. Stewart Shapiro's Foundations without Foundationalism: A Case for Second-Order Logic (Oxford University Press, 1991) provides an influential philosophical defense of second-order logic. Jouko Väänänen's work on second-order and higher-order logic gives a particularly useful account of full and general semantics, categoricity, and the failures of first-order compactness and Löwenheim–Skolem properties when full second-order semantics is adopted.

Kripke: Possible Worlds, Necessity, and the Semantics of Modal Logic

This is the eighth part of a series developed through the Department of Philosophical Theology at the Institute of Lutheran Theology’s Christ School of Theology, exploring major results in modern logic and their significance for philosophical and systematic theology.

For much of the first half of the twentieth century, modal logic occupied an uncertain position. Philosophers and logicians had long wanted to reason formally about necessity and possibility, but the semantic status of modal expressions remained obscure. What does it mean to say that something is necessarily true rather than merely true? What makes a proposition possible rather than actual? And how are we to understand the inferential behavior of expressions such as “it is necessary that” and “it is possible that” without treating them merely as unexplained operators added to ordinary logic?

The difficulty was real because modal contexts do not behave extensionally in the straightforward way familiar from classical first-order logic. If two names designate the same object, substitution of one for the other ordinarily preserves truth in extensional contexts. Yet in modal or other intensional contexts, substitution may fail. One may know that Cicero is Cicero without knowing that Cicero is Tully, although Cicero and Tully designate the same man. The logical problem was therefore not merely how to invent symbols for necessity and possibility, but how to give those symbols a semantics capable of explaining their inferential structure.

Saul Kripke's work around 1959 and 1960 supplied the decisive breakthrough. Building upon earlier developments by Rudolf Carnap, Stig Kanger, Jaakko Hintikka, and others, Kripke gave modal logic a relational semantics in which modal operators are interpreted through what are commonly called possible worlds.

Let us suppose that we have a collection of worlds and a relation of accessibility among them. We write

wRv

to mean:

world v is accessible from world w.

The accessibility relation does not ordinarily mean that one world physically travels to another, nor need possible worlds be understood as concrete universes. Within the semantics, they function first of all as points of evaluation. The relation R specifies which worlds are relevant when the modal status of a proposition is evaluated at a given world.

We can then define necessity and possibility.

□φ

is read:

It is necessary that φ.

And

◇φ

is read:

It is possible that φ.

The crucial semantic clauses are these:

w ⊨ □φ

if and only if

for every v such that wRv, v ⊨ φ.

In words: φ is necessary at world w exactly when φ is true at every world accessible from w.

Likewise,

w ⊨ ◇φ

if and only if

there is some v such that wRv and v ⊨ φ.

In words: φ is possible at world w exactly when φ is true at at least one world accessible from w.

What had previously looked like an elusive intensional distinction now received a mathematically precise semantic treatment. Necessity becomes truth throughout an appropriate range of accessible worlds; possibility becomes truth at at least one such world.

The importance of the accessibility relation appears when we notice that different modal logics correspond to different structural properties of R. If every world is accessible from itself, then R is reflexive. If accessibility is symmetric, then whenever wRv, we also have vRw. If it is transitive, then whenever wRv and vRu, we have wRu.

These apparently technical properties correspond to familiar modal principles.

For example, the system T validates

□φ → φ.

The principle says that whatever is necessary is true. Semantically, this follows when accessibility is reflexive, since if every world is accessible from itself, then anything true at all worlds accessible from w must be true at w itself.

The system S4 adds, among other things, the principle

□φ → □□φ.

If something is necessary, then it is necessarily necessary. This corresponds naturally to transitivity of accessibility.

S5 validates still stronger principles, including

◇φ → □◇φ.

If something is possible, then it is necessarily possible. In standard Kripke semantics, S5 may be modeled by treating accessibility as an equivalence relation, or more simply in many presentations by allowing every relevant world to be accessible from every other.

The achievement here was not merely that several modal calculi could now be given models. Kripke semantics showed why different systems validate different modal principles. Instead of treating modal axioms as isolated formal stipulations, one could correlate them with structural features of frames.

A Kripke frame consists of a set W of worlds together with an accessibility relation R:

F = ⟨W, R⟩.

A Kripke model adds a valuation V telling us where atomic propositions are true:

M = ⟨W, R, V⟩.

The semantics therefore mirrors a pattern already familiar from first-order model theory. A formal language receives interpretation relative to a structure, and truth is defined recursively relative to that structure. Kripke's innovation was to extend this model-theoretic strategy to modal discourse.

This has enormous consequences for philosophical theology because so much theological reasoning is modal whether or not theologians explicitly acknowledge it. Theology repeatedly distinguishes what God does from what God could have done, what creatures happen to be from what they must be, what follows necessarily from the divine nature from what results contingently from divine willing, and what is possible given certain theological commitments from what is impossible.

Consider a simple theological sentence:

□(God is God).

Whatever else may be said about the example, the modal operator tells us that the assertion is not merely that the proposition happens to be true. It is represented as true at every relevant accessible world.

Or consider:

◇(God creates no world).

This would express the claim that there is at least one accessible world in which God does not create. Whether the claim is theologically acceptable is a further question, but the semantics makes clear what kind of claim it is.

The value of the apparatus lies precisely in separating formal structure from theological judgment. Kripke semantics does not tell us which propositions are necessary, which worlds are genuinely possible, or which accessibility relation theology ought to adopt. Rather, it gives us a disciplined framework within which those further questions can be stated with precision.

This point matters because “possible world” language can easily become metaphysically inflated. One sometimes speaks as though Kripke semantics had established the existence of a vast plurality of concrete worlds. It did no such thing. The semantics requires mathematical structures containing points of evaluation and an accessibility relation. What metaphysical interpretation, if any, should be given to those points is a further philosophical issue.

That distinction is especially important in theology, where possible-world language is often used in discussions of divine necessity, freedom, providence, incarnation, atonement, and the problem of evil. One may employ Kripke semantics to regiment modal relations without committing oneself to David Lewis's later modal realism, according to which possible worlds are concrete realities. The formal semantics and the ontology of possible worlds are distinct questions.

There is another important lesson here. Modal claims are not merely ordinary claims with decorative prefixes. If

φ

is true, it does not follow that

□φ

is true.

Likewise, if

◇φ

is true, we cannot infer

φ.

The operators alter the conditions under which a proposition is evaluated. Much theological confusion arises precisely when claims of actuality, possibility, and necessity are allowed to slide into one another without argument.

Suppose, for example, one argues:

God creates the world.

Therefore,

God necessarily creates the world.

Nothing in ordinary logic licenses this inference. To move from actuality to necessity requires an additional modal premise.

Or suppose one reasons:

It is possible that God creates a world containing rational creatures.

Therefore,

God creates such a world.

Again, the inference fails. Possibility does not entail actuality.

These distinctions are elementary once formalized, but they become extremely important when theological arguments move among divine attributes, divine actions, and creaturely possibilities. Modal logic permits one to see precisely where the transition occurs and to ask what principle licenses it.

Kripke's work also transformed the treatment of quantified modal logic. Once individuals, quantifiers, and modal operators are combined, further questions arise. Does the domain of objects remain fixed across worlds, or may different worlds contain different objects? If an object exists in more than one world, how is it identified across worlds? Can an object possess some properties essentially and others accidentally?

These questions helped lead directly into Kripke's later work on naming, necessity, and essential properties. If a name such as “Aristotle” rigidly designates the same individual in every possible world in which that individual exists, then modal claims about Aristotle differ importantly from claims expressed merely through descriptions such as “the teacher of Alexander.” The distinction between rigid designation and descriptive reference would become one of the major developments in late twentieth-century philosophy.

For theology, the implications are immediate. Claims about God, Christ, divine attributes, personal identity, incarnation, and resurrection often depend upon questions of transworld identity and essential predication. To say that Christ could have acted otherwise is not merely to describe another individual satisfying a similar description. It is to make a modal claim about the same individual.

Kripke's semantics therefore gives philosophical theology something more valuable than an additional notation. It provides a framework for distinguishing necessity from actuality, possibility from consistency, essential from accidental predication, and semantic evaluation from metaphysical interpretation.

The framework also fits naturally with the larger trajectory of the results considered in this series. Tarski taught us to distinguish object language from metalanguage and truth from satisfaction. Church and Turing showed that formal specification does not guarantee universal mechanical decidability. Kripke now demonstrates that even intensional notions such as necessity and possibility can be treated with considerable formal rigor once their semantics is properly specified.

The lesson is not that metaphysics has been reduced to set theory. Rather, the logical structure of modal discourse can be made explicit without pretending that the formalism itself settles every metaphysical question.

Why It Matters for Theology

Kripke semantics matters for theology because theological reasoning is saturated with modal distinctions. God is said to exist necessarily, creation is usually said to be contingent, certain divine attributes are treated as essential, creaturely states as possible or impossible, and doctrines are often tested by asking whether a specified set of claims can be jointly true. Modal logic enables these claims to be distinguished rather than merely asserted.

It also teaches an important methodological discipline. A formal modal model tells us what follows once we have fixed a set of worlds, an accessibility relation, and a valuation. It does not itself tell us what counts as a genuinely possible world for theology, whether divine possibilities should be modeled by S4 or S5, whether domains should vary across worlds, or whether possible worlds should be given any robust metaphysical status. Those are theological and philosophical questions that arise after the semantics has done its work.

Kripke's achievement was therefore not to settle the metaphysics of possibility, but to provide a semantics within which modal reasoning could finally be treated with the same kind of rigor that model theory had already brought to extensional logic. For philosophical theology, that was a decisive gain.

Bibliographical Note

Saul A. Kripke's early modal-semantic work appeared in “A Completeness Theorem in Modal Logic,” Journal of Symbolic Logic 24 (1959): 1–14, and “Semantical Considerations on Modal Logic,” Acta Philosophica Fennica 16 (1963): 83–94. His later lectures, published as Naming and Necessity, developed the influential notions of rigid designation, necessary a posteriori truths, and essential properties. For theological applications, the most important background lies in the semantics of modal systems T, S4, and S5, quantified modal logic, rigid designation, and the distinction between formal possible-world semantics and substantive metaphysical theories of possible worlds.