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.

When Classical Logic is not Enough: Nonclassical Logics and Theological Reasoning

This is the eleventh 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.

Classical logic is extraordinarily powerful, so powerful in fact that one can easily begin to speak simply of 'logic' as though the classical system exhausted the possibilities of valid inference. The propositional calculus gives us familiar principles governing negation, conjunction, disjunction, and implication; first-order logic adds quantification and identity; model theory then permits us to specify satisfaction, consequence, validity, and interpretation with great precision. Much of the preceding series has depended upon precisely this framework, and nothing in what follows should be taken as withdrawing the confidence we have repeatedly placed in it.

Yet the development of logic in the twentieth century made clear that one need not abandon rigor in order to ask whether every feature of classical consequence is appropriate to every domain of reasoning. One can instead ask which principles are being used, what assumptions support them, and what changes when one of those assumptions is altered. This is the setting within which the various nonclassical logics emerged, not as one unified rebellion against classical logic, but as a family of formally disciplined attempts to revise particular features of consequence for particular purposes.

The plurality is important. Intuitionistic logic alters what may count as sufficient warrant for assertion and consequently declines to validate some classical principles. Relevant logics require a stronger connection between antecedent and consequent than material implication ordinarily supplies. Paraconsistent logics deny that contradiction must entail everything whatsoever, while many-valued logics permit semantic values other than the classical pair of truth and falsity. These systems do not all solve the same problem, and theology gains nothing by treating them as though they were variations upon one general theme called 'nonclassical logic'.

The more fruitful question is why theology might care about any of them.

Intuitionistic Logic: What Warrants the Assertion?

Classical logic validates the law of excluded middle:

P ∨ ¬P.

Either P or not-P.

It also validates double-negation elimination:

¬¬P → P.

If it is not the case that P is false, then P.

For classical reasoning these principles are familiar enough that one may hardly notice when they are being used. Intuitionistic logic, arising from Brouwer's philosophy of mathematics and subsequently formalized especially by Heyting, does not accept them unrestrictedly, not because the intuitionist is somehow more tolerant of contradiction, but because the standards governing assertion have changed. Under the constructive interpretation, asserting P requires an appropriate construction or proof of P, while asserting P ∨ Q requires having grounds for one disjunct or the other; consequently, the impossibility of ¬P need not itself amount to a constructive establishment of P.

The theological temptation here is obvious, and it should be resisted. It would be careless to claim that theological propositions are intuitionistic simply because faith is not mathematical proof, or to suppose that intuitionistic logic somehow captures religious trust better than classical logic. The connection is much more modest, but also more interesting, because intuitionistic logic forces us to ask a question theological argument often leaves implicit: What exactly warrants the assertion being made?

Consider the difference between

¬¬P

and

P.

Classical logic permits the passage from the former to the latter, whereas intuitionistic logic does not generally permit that inference. The distinction is useful because it forces us to ask whether showing that the denial of a proposition is untenable amounts to positively establishing the proposition itself. A theologian may successfully argue that a particular denial of divine action produces contradiction, but it remains a further question whether this alone establishes the particular positive account of divine action that the theologian wishes to defend.

One need not become an intuitionist in order to profit from the distinction. The formal system is valuable here because it makes visible an inferential step that ordinary theological prose can conceal, namely, the transition from the failure of a denial to the warrant for an affirmation.

Relevant Logic: What Has the Premise to Do with the Conclusion?

Classical material implication produces results that can initially seem peculiar. Since

P → Q

is classically equivalent to

¬P ∨ Q,

the conditional is true whenever P is false or Q is true, and consequently classical logic validates forms such as

P → (Q → P)

and

¬P → (P → Q).

These are often called paradoxes of material implication, though they are not contradictions within classical logic; they follow directly from the truth-functional definition of the conditional.

Relevant logicians ask whether a genuine relation of implication should require more than this. If we say that one proposition follows from another, should there not be some appropriate connection between the content of premise and conclusion? Relevant logics attempt to build such a requirement into the consequence relation itself, so that implication is not secured merely by the falsity of an antecedent or the independent truth of a consequent.

For theology the question is hardly peripheral, because theological discourse is saturated with conditionals. We say:

If Christ is risen, then …

If God creates ex nihilo, then …

If justification is by faith, then …

If God is immutable, then …

In such cases the theological force of the conditional ordinarily depends upon some intelligible relation between what is asserted in the antecedent and what is claimed in the consequent. We do not usually mean merely that the conditional happens to receive the value true under the truth table for material implication.

This is not yet an argument for replacing classical implication. It may instead be an argument for recognizing that many theological uses of 'if … then …' express more than the material conditional was ever intended to capture. The important point is therefore methodological: before formalizing a theological conditional, one must determine what sort of inferential relation the natural-language formulation is attempting to express.

Relevant logic helps precisely because it refuses to allow us to ignore that question.

Paraconsistent Logic: What Follows from Contradiction?

Perhaps no family of nonclassical logics is more immediately attractive to theologians, and perhaps none is more easily abused, than paraconsistent logic. Classical logic validates the principle commonly called explosion:

P, ¬P ⊢ Q.

From a contradiction, anything follows.

The principle does not mean that Q bears some hidden relation to P. Rather, once both P and ¬P have been admitted into a classical theory, every sentence becomes derivable, and the theory consequently loses its ability to discriminate among conclusions. In that technical sense, contradiction produces triviality.

Paraconsistent logics reject explosion. In a paraconsistent consequence relation, it is not generally the case that

P, ¬P ⊨ Q

for arbitrary Q, and therefore inconsistent information can be reasoned from without permitting every proposition to follow. What must be emphasized, however, is that paraconsistency does not by itself entail that contradictions are true; it entails only that contradiction need not produce inferential collapse.

This distinction is especially important in theology, where doctrines are often said loosely to be 'paradoxical' or even 'contradictory'. Christ is divine and human; God is one and three; the believer is righteous and sinful; God acts while creatures genuinely act. Yet none of these formulations has the form

P ∧ ¬P

unless one has first identified 'human' with 'not divine', 'three' with 'not one', or otherwise made the predicates contradictory in the same respect and under the same description.

Indeed, much of the history of Christian doctrine can be read as sustained resistance to exactly such conflations. Chalcedonian Christology does not say that Christ is finite and not finite in the same respect; Trinitarian doctrine distinguishes essence from person; the Lutheran formula simul iustus et peccator does not require that righteousness and sin be predicated univocally in the same respect. The logical discipline here lies not in invoking paraconsistency too quickly, but in determining first whether a genuine contradiction exists.

Paraconsistent logic becomes genuinely interesting when we confront a theological corpus, a historical tradition, or a developing theory that actually contains inconsistent commitments. Must everything then follow? A paraconsistent framework says no, and that can be useful when analyzing historically layered materials, competing doctrinal formulations, or theories under revision, because one can study the consequences of inconsistency without first pretending that the inconsistency is absent and without allowing the system to become trivial.

It is therefore essential to distinguish paraconsistency from dialetheism. The former concerns the behavior of consequence in the presence of contradiction; the latter is the metaphysical thesis that some contradictions are in fact true. One may use paraconsistent logic as a formal tool without thereby committing oneself to the reality of true contradictions, and theology should preserve that distinction with some care.

Many-Valued Logic: Must Every Proposition Be Simply True or False?

Classical propositional logic operates with two truth values, true and false. Many-valued logics generalize this architecture by permitting additional semantic values, though the significance of those additional values varies considerably from system to system. Some contain three values, others finitely many, and still others infinitely many; moreover, the extra values need not always be understood as degrees of truth, since they may instead represent indeterminacy, lack of information, semantic defect, or some other feature of the evaluation.

The theological temptation must again be controlled. The existence of many-valued logics does not establish that theological truth itself comes in degrees, nor does it show that mystery or doctrinal controversy requires intermediate truth values. What these logics do show is that bivalence is a semantic choice that can be examined rather than silently presupposed in every domain.

Suppose, for example, that we consider a predicate such as

x is mature in faith.

At what precise point does this predicate become true? If there is no sharp boundary, the issue may concern vagueness rather than either contradiction or theological confusion. Similar difficulties arise with predicates such as 'orthodox', 'responsible', 'culpable', 'spiritually mature', and even, in some contexts, 'alive' and 'dead', where biological or conceptual boundaries may be difficult to draw sharply.

A many-valued semantics offers one family of ways of representing such cases. It is not the only family, since supervaluationism, epistemicism, contextualism, and other theories compete with it, but the formal possibility is philosophically useful because it prevents us from assuming without argument that every semantically difficult case must still admit a sharp classical assignment of exactly one of two values.

The lesson for theology is therefore not that truth is fuzzy. It is that the semantics appropriate to a theological predicate must be investigated rather than assumed.

Which Logic for Theology?

At this point one might ask which logic theology should use, but the question is too coarse if it is understood as demanding one system for every theological task. There is no reason to suppose that theology needs a single nonclassical logic to replace classical logic across the board, and there is every reason to retain classical first-order logic for the enormous range of theological reasoning for which its proof theory, semantics, and inferential behavior are entirely adequate.

The existence of nonclassical logics does not overthrow classical logic. What it does overthrow is the assumption that every feature of classical consequence lies beyond philosophical examination. Intuitionistic logic asks what licenses assertion; relevant logic asks what connection implication should require between premise and conclusion; paraconsistent logic asks whether inconsistency must entail triviality; many-valued logic asks whether every semantic domain is adequately represented by exactly two truth values.

What has happened, accordingly, is not an abandonment of logic but a deepening of the philosophy of logic, because logical consequence itself has become an object of investigation. Earlier in this series we asked what follows from a theory, what structures satisfy it, whether the intended structure can be characterized, whether truth can be defined within the relevant language, whether consequences can be mechanically decided, and whether possible-world semantics supplies a sufficiently fine-grained account of content. Nonclassical logic now asks a question prior to many of those questions: Which relation of consequence are we employing when we say that one proposition follows from another?

The answer cannot simply be read off from the theological subject matter. The doctrine of the Trinity does not announce that its proper formal reconstruction must be classical, relevant, paraconsistent, or intuitionistic, nor does the Incarnation tell us in advance what sort of logical system best represents the relations among its propositions. One must first determine what the doctrine actually asserts, whether its apparent tensions are genuine contradictions or merely differences of respect, what kinds of conditionals occur within the argument, and what semantic distinctions the doctrine itself requires.

Only after that work has been done does the choice of formal machinery become philosophically responsible.

There is a danger in both directions. One can force every theological claim into classical form and conclude that whatever does not fit must be confused, or one can invoke a nonclassical logic whenever a doctrine appears difficult and thereby protect a defective formulation from criticism by simply changing the consequence relation. Neither procedure is satisfactory, because in both cases logic is being selected before the theological and semantic work has been done.

The choice of logic should instead follow from an analysis of the inferential phenomena one is attempting to represent. If the problem is vagueness, paraconsistency may be beside the point; if the problem is inconsistent information, many-valuedness may not address it; if the issue concerns the relation between antecedent and consequent, intuitionistic logic does not automatically solve it. Different logics revise different structures, and therefore there is no generic escape hatch labeled 'nonclassical'.

Why It Matters for Theology

Nonclassical logic matters for theology because it reveals that our conception of consequence already contains philosophical commitments. Classical logic gives powerful and often entirely appropriate accounts of theological reasoning, but its principles are better understood when we know what alternatives would look like and which features of inference those alternatives modify.

The discipline imposed by nonclassical logic is therefore double. We must specify exactly which classical principle appears inadequate for the task before us, and we must also identify what is gained and what is lost when that principle is revised. Merely changing the logic does not settle the theological question, since every alteration in consequence brings with it new semantic and proof-theoretic commitments of its own.

For theology this discipline can be salutary, because it forces distinctions that theological rhetoric too easily obscures. Apparent contradiction must be distinguished from genuine contradiction; material implication from explanatory or relevant connection; lack of proof from falsity; vagueness from inconsistency; mystery from contradiction; and formal tolerance of inconsistency from the metaphysical claim that some contradiction is actually true.

These are not logical niceties added after the theological work is complete. They belong to the conditions under which theology can say clearly what it means to say.

The most important lesson, therefore, is not that theology needs another logic, but that theology should know what its logic is doing, what it permits, what it forbids, and why. Once that much has been learned, the existence of alternative logics becomes less threatening and more useful, because each can be treated not as a rival worldview but as an instrument for testing the assumptions built into a particular account of consequence.

This brings us naturally to the final installment of the series. We began with Frege, Peirce, and Cantor, where modern logic dramatically enlarged the resources available for formal expression; we moved through model theory, completeness, incompleteness, compactness, truth, computability, modality, second-order logic, and hyperintensionality; and we have now reached the point at which consequence itself can be formally varied.

It is therefore fitting to return at the end to Gödel. His ontological argument brings together much of the machinery accumulated along the way: quantified modal logic, higher-order resources, necessity, possibility, formal derivation, and the distinction between the validity of an argument and the truth or adequacy of the axioms from which it proceeds. The final question will not be whether logic can 'prove God', but something more precise and, I think, more interesting: what exactly has been established when a theological argument has been successfully formalized and proved valid?

Bibliographical Note

The modern study of intuitionistic logic grows from L. E. J. Brouwer's philosophy of mathematics and Arend Heyting's subsequent formalization of intuitionistic reasoning. Alan Ross Anderson and Nuel Belnap's work on entailment became foundational for relevance logic. Stanisław Jaśkowski and Newton da Costa were among the major pioneers of paraconsistent logic, while subsequent work by Graham Priest and others developed its philosophical implications. Jan Łukasiewicz's work on three-valued logic, initially associated especially with future contingents, helped initiate the systematic study of many-valued logics. These traditions should not be treated as a single alternative to classical logic, since each revises different features of classical consequence for different formal and philosophical purposes.

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.

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.

Tarski: Truth, Satisfaction, and the Limits of a Language Speaking About Itself

This is the sixth 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.

By the time modern logic had developed the resources of quantification, set theory, axiomatic systems, and model theory, a question that had always accompanied logic could no longer be postponed: What does it mean to say that a sentence is true? Although philosophers had, of course, asked about truth from antiquity onward, Alfred Tarski showed that once the question is raised within a sufficiently precise formal setting, one must distinguish matters that ordinary language easily allows us to run together. In particular, one must distinguish a sentence from the language in which we speak about that sentence, derivability from truth, and truth simpliciter from truth relative to an interpretation or structure.

This distinction became decisive because the formal languages developed by Frege, Russell, Hilbert, Gödel, and others were powerful enough to express increasingly complicated claims, while at the same time making possible new forms of semantic self-reference. Once a language becomes sufficiently expressive, allowing it unrestrictedly to contain its own truth predicate invites familiar paradoxes, most famously the liar: a sentence saying of itself that it is not true. Tarski's achievement was not merely to warn against such paradoxes, but to show how the notion of truth could nevertheless be defined rigorously for formal languages, provided that we carefully distinguish the language under investigation from the metalanguage in which its semantic properties are described.

The resulting conception of truth is often called the semantic conception of truth. Its most famous intuitive requirement is represented by what Tarski called Convention T. A satisfactory definition of truth should entail instances of the following form:

“Snow is white” is true if and only if snow is white.

The point is not the example, which is deliberately trivial, but the logical form. On the left we mention a sentence; on the right we use language to state the condition under which that sentence is true. A theory of truth must connect sentence and world without simply identifying the two, and it must do so from a standpoint in which the sentence itself can be referred to as an object of semantic investigation.

For first-order languages, however, truth is reached through the more basic notion of satisfaction. Suppose that M is a structure for a language L, with domain D, and that s is an assignment of objects in D to the variables of L. We then write

M ⊨ φ[s]

which is read:

The formula φ is satisfied in structure M under assignment s.

This notation matters because an open formula such as Px is not, strictly speaking, true or false independently of an assignment to x. It is satisfied in M under s when the object assigned by s to x belongs to the extension of P in M. Thus, if

s(x) = a,

then

M ⊨ Px[s]

just in case a belongs to the extension of P in M.

The recursive clauses then proceed through the logical structure of formulas. Negation, conjunction, disjunction, and the other connectives receive satisfaction conditions in terms of their component formulas, while quantifiers are handled by varying assignments. Thus,

M ⊨ ∃x φ[s]

if and only if there is some a in D such that

M ⊨ φ[s[x ↦ a]].

Likewise,

M ⊨ ∀x φ[s]

if and only if, for every a in D,

M ⊨ φ[s[x ↦ a]].

What initially looks like a technical device turns out to be philosophically important, because the semantic relation between language and structure is built up compositionally. We do not begin with an unexplained global notion of truth and then apply it indiscriminately. We define what it is for atomic formulas to be satisfied, specify how satisfaction behaves under the logical operations, and arrive finally at truth for sentences, which, because they contain no free variables, are satisfied or not satisfied independently of the particular assignment.

Accordingly, for a sentence σ we may write

M ⊨ σ

and read this:

σ is true in M,

or, equivalently,

M satisfies σ.

At this point an important distinction becomes unavoidable. To say that σ is true in M is not yet to say that σ is true simpliciter, unless M is being taken as the intended interpretation. Model theory deliberately allows many structures to interpret the same formal language, and therefore the same sentence may be true in one structure and false in another. The semantics tells us what follows once an interpretation has been fixed; it does not, merely by giving us the formal semantics, determine which interpretation is the one about which we intended to speak.

This point connects directly with the Löwenheim–Skolem and Compactness results considered in the preceding essays. Those theorems showed that first-order theories frequently possess models very different from the structures one might initially have intended. Tarski now gives us the semantic machinery for stating the matter precisely. If a theory T has many models, then

M₁ ⊨ T,

M₂ ⊨ T,

M₃ ⊨ T,

and so forth,

may all hold even though the structures M₁, M₂, and M₃ differ substantially. Satisfaction tells us whether a structure makes the sentences of the theory true; it does not by itself confer intendedness upon that structure.

The distinction between truth and provability is equally important. If T is a theory and φ a sentence, then

T ⊢ φ

says that φ is derivable from T by the formal proof rules, whereas

T ⊨ φ

says that every model of T satisfies φ.

Gödel's completeness theorem connects these two notions for first-order logic:

T ⊢ φ if and only if T ⊨ φ.

But the equivalence does not erase the conceptual distinction. The expression on the left concerns syntactic derivability; the expression on the right concerns semantic consequence. Indeed, the importance of Gödel's theorem lies precisely in the fact that two independently defined notions—proof and semantic consequence—turn out to coincide for first-order logic.

Here Tarski's work makes a contribution that theology ought to notice, although perhaps not in the way theologians sometimes suppose. The result does not establish that truth is ineffable, that human language cannot speak about God, that propositions fail before transcendence, or that theological language must finally dissolve into mystery. None of these claims follows from Tarski. What does follow is more disciplined and, for theology, more useful: whenever we speak about the truth of sentences belonging to a language, we must distinguish the sentences themselves from the semantic framework within which their truth conditions are being specified.

That distinction becomes especially important when theology moves between biblical language, doctrinal formulation, philosophical reconstruction, and formal representation. Suppose, for example, that a theological theory contains the sentence

∀x(Fx → Cx),

read:

Everything that is finite is created.

A theologian may ask whether the sentence follows from some theological theory T, whether it is satisfied in some model M of that theory, whether it expresses accurately what the theological sources intend, or whether it is in fact true of reality. Those are related questions, but they are not identical questions. Formal semantics can illuminate their relations precisely because it does not allow them simply to collapse into one another.

The distinction also bears upon theological metalanguage. Creeds, confessions, biblical propositions, and doctrinal assertions ordinarily occur within historically developed languages whose terms already bear substantial semantic weight. When the theologian begins to say what those sentences mean, under what conditions they are true, what follows from them, or what models satisfy them, the theologian has moved, whether explicitly or not, into a metalanguage. Once that movement is recognized, one can ask more carefully whether the metalanguage merely explicates the theological language, whether it transforms it, or whether it imports ontological and semantic commitments that the original language itself did not possess.

For philosophical theology, therefore, Tarski's importance lies not in providing a theological theory of truth, but in teaching us how much must already be distinguished before such a theory can responsibly be attempted. A sentence, its proof, its interpretation, the structure in which it is satisfied, and the reality about which it is intended to speak belong to different logical relations, even though theological discourse often moves rapidly among them. The semantic conception of truth disciplines that movement because it forces us to say, at each stage, what language we are using, what structure we have fixed, and what relation we are asserting between them.

The consequence is not skepticism but precision. Tarski does not tell us that truth escapes language; he shows us that language can speak rigorously about truth only when it respects the distinctions required by its own semantic functioning. For theology, which must continually speak both within its inherited language and about that language, this is no small achievement.

Why It Matters for Theology

Tarski's work gives philosophical theology at least four enduring lessons. First, truth must be distinguished from provability. Second, satisfaction in a model must be distinguished from truth under an intended interpretation. Third, theological object-language must be distinguished from the metalanguage by which theologians analyze its meaning and truth conditions. Fourth, formal adequacy does not by itself establish theological adequacy, since a formal structure may satisfy a theory without yet being the structure about which the theology intends to speak.

These distinctions become increasingly important as theology employs formal logic, model theory, possible-world semantics, or other forms of analytic reconstruction. The more powerful our formal languages become, the more necessary it becomes to know exactly which claims belong to the formal system and which claims concern the interpretation of that system. Tarski's achievement was to make that difference visible with a precision that modern theology can scarcely afford to ignore.

Bibliographical Note

Alfred Tarski's classic statement appears in “The Concept of Truth in Formalized Languages,” originally published in Polish in 1933 and later translated into English in Logic, Semantics, Metamathematics. His broader semantic conception is presented accessibly in “The Semantic Conception of Truth and the Foundations of Semantics,” Philosophy and Phenomenological Research 4 (1944): 341–376. For contemporary treatments, see standard introductions to model theory and philosophical logic under the topics of satisfaction, semantic truth, object language, metalanguage, and Tarski's undefinability theorem.