Contextual and Global Falsification of Scientific Models

1. From Binary Falsification to Domain-Relative Model Validity

For decades, Popper's theory of falsification ([1934] 1959) shaped both the public and the academic image of scientific rationality. In its classical and widely received form, falsification appears as an idealized binary test of a theory's scientific standing: a theory comes under pressure with respect to its scientific acceptability if one of its predictions empirically fails. The practice of the natural and social sciences, however, presents a more complex picture. Many models are idealized, only partially representational, and empirically limited in subdomains, yet remain epistemically stable. This paper therefore develops an integrated theory of scientific model validity that brings together central insights from Popper, Kuhn, Lakatos, da Costa/French, and contemporary model theory within a common structure of rational model evaluation.

Examples include Newtonian mechanics, which fails at relativistic velocities (Einstein 1905); classical thermodynamics, which is microscopically imprecise; and climate models, which are continuously recalibrated (Oreskes et al. 1994). At the same time, there are models that have disappeared entirely, such as phlogiston theory or the classical ether.

The core problem is this: Under what conditions does falsification lead to the elimination of a model—and when does a model remain epistemically stable despite partial falsification?

In what follows, this problem is treated as a special case of epistemics, that is, as a special case of model management under finite conditions (Rapp 2026a). The paper is thus to be understood as a module within a broader research program in which friction, domain structure, search, and revision are developed in their own dedicated works. The present study transfers this architecture to the case of scientific falsification. Scientific models possess no unlimited validity; they are stabilized, tested, restricted, and revised within particular domains. The general concepts of model, validity, domain, costs, and revision are therefore applied here to the special case of scientific falsification.

In this framework, falsification appears not primarily as a simple operation of elimination, but as a specific form of epistemic friction in the sense of a boundary signal of finite viability (Rapp 2026b): it indicates that the prior stabilization of a model in a given domain is no longer straightforwardly viable. Friction functions here as a procedural indicator of a strained utility structure. It arises when empirical deviations weaken AT(M, D, t), explanatory deficits reduce EK(M, D, t), model costs C(M, D, t) rise, or available alternatives M* become relatively more competitive. Whether this leads to local revision, restriction of the domain of validity, or global model replacement depends on the model's stable relative competitiveness within E(t).

This paper develops an integrated theoretical framework that reconstructs this question explicitly within a unified model. The approach distinguishes global from contextual falsification, describes the epistemic enabling space E(t), and integrates approximate truth as a graded, domain-specific evaluative quantity. Falsification is understood not as a purely logical operation on truth values but as a rationally reconstructible change in model validity within a real scientific enabling space. The decisive issue is not the isolated decline in the performance of a model, but its relative competitiveness against available alternatives within the same relevant domain. The theory is primarily descriptive and reconstructive: it reconstructs the actual modeling practice of contemporary sciences and is illustrated through climate models.

The utility function U(M, D, t) introduced below is to be understood primarily in reconstructive terms. It does not describe a general computational machinery of scientific rationality but renders explicit how model choice in actual sciences can be reconstructed when approximate truth, explanatory power, and model costs are weighed against one another. Within this paper, U(M, D, t) formalizes the domain-specific validity of scientific models under finite conditions. At the same time, the same structure admits a weakly normative reading, since it brings into view the conditions under which model decisions can count as rationally intelligible. The theory is thus both a descriptive reconstruction of modeling practice and a framework for epistemic optimality, without thereby deriving a universal decision formula.

2. State of Research

3. Basic Concepts

The following definitions sharpen the concepts central to this paper. They apply general categories of model management under finite conditions to the special case of scientific falsification. Model validity, the domain-bound character of models, costs, friction, and revision are thereby not left as abstract background notions but are translated into a formally applicable structure. The quantities U(M, D, t), AT(M, D, t), C(M, D, t), and E(t) serve to render this structure explicitly reconstructible in the context of scientific model evaluation.

Notation (Overview)

• M – model
• M* – available alternative model
• D(M) – domain of validity of a model
• D₁, D₂, …, Dₙ – subdomains of the domain of validity
• AT(M, D, t) – approximate truth of the model in domain D at time t
• EK(M, D, t) – explanatory power of the model in domain D at time t
• C(M, D, t) – model costs of the model in domain D at time t
• U(M, D, t) – domain-specific utility function
• ΔU(M*, M, D, t) – relative utility difference between alternative model M* and model M
• E(t) – epistemic enabling space at time t
• R(M, Dᵢ) – restricted application of a model to a subdomain Dᵢ
• ε – context-dependent tolerance range of epistemic competitiveness
• D(M)' – remaining domain of validity after restriction or exclusion of a problematic subdomain
• Rev(M, Dᵢ) – revision of a model or submodel within a problematic subdomain

Definition 1: Model (M)

A model is an idealized mathematical or conceptual structure that represents specified aspects of a target system under particular conditions (Cartwright 1999).

Definition 2: Domain of Validity D(M)

The domain of validity of a model designates that domain within which the model yields reliable, explanatorily powerful, or functionally viable results. A domain is not merely an external area of application but a region of stable conditions of validity: within these conditions, a model can be tested, compared, restricted, or defended against alternatives. D(M) therefore designates not the global truth of a model but its domain-relative epistemic validity. This domain-relative reading connects with the theory of domains, boundaries, and transition functions, in which validity is reconstructed as dependent on the stability conditions of a particular domain (Rapp 2026e).

Definition 3: Subdomains D₁, D₂

D(M) can be decomposed into a set of subdomains {D₁, D₂, …, Dₙ}. Such decompositions are often the result of scientific revision, domain refinement, or model comparison. For simple cases, one can distinguish between preserved domains and problematic domains: preserved domains are those in which M remains epistemically optimal or competitive, while problematic domains are those in which M is weakened, in need of revision, or no longer epistemically competitive against alternatives. The binary distinction between D₁ and D₂ is therefore only a simplified special case of a more general subdomain structure.

Definition 4: Epistemic Optimality

A model is epistemically optimal in a domain D if, with respect to all alternatives available in the epistemic enabling space E(t), it exhibits the most favorable trade-off among approximate truth, explanatory power, and model costs. Epistemic optimality is therefore not an absolute property of a model but a relational evaluation: a model is optimal only relative to a domain, to a time t, and to the alternatives actually available within E(t). Operationally, this evaluation is reconstructed by the domain-specific utility function U(M, D, t).

Approximate truth designates the domain-specific fit of a model to the empirical, structural, or interventionally testable features that are relevant within that domain. Explanatory power, by contrast, designates a model’s explanatory and structuring capacity: its capacity to render phenomena intelligible, to structure interrelations, to make causal or counterfactual dependencies visible, and to give theoretical orientation to a domain. Fit and explanation are thereby distinguished analytically: a model can fit certain data or structures well without yet possessing high explanatory power; conversely, explanatory power remains epistemically limited if the model's fit to the relevant features of a domain is too weak.

Model costs C(M, D, t) cover three dimensions: (1) epistemic costs (e.g., susceptibility to error, sensitivity, the breadth of uncertainty in predictions, interpretability/transparency, quality of fit); (2) technical costs (computational effort, data requirements, implementation complexity); (3) institutional costs (degree of standardization, availability of infrastructure).

Some of these cost components are global (for instance, basic implementation or infrastructure costs), while others are domain-specific (for instance, data demands or numerical stability in particular domains). For the utility function, these are aggregated into C(M, D, t).

Definition 5: Contextual Falsification

An observation O leads to contextual falsification if it shows that there is a subdomain D₂ in which M is no longer epistemically acceptable or competitive, while at least one subdomain D₁ remains in which M is epistemically optimal or competitive. Contextual falsification is thus a specific form of epistemic friction: it indicates the boundary of the prior model validity without eliminating the model as a whole. Its typical consequence is not global rejection but revision, restriction, or reorganization of the domain of validity. Such a reorganization is epistemically legitimate, however, only if it accomplishes more than the mere rescue of a weakened model: it must make new testable distinctions visible, separate sources of error diagnostically, refine model applications, or enable new explanatory, predictive, or interventionist achievements. In this sense, contextual falsification is closely linked to revision under finite conditions, insofar as it does not simply negate model validity but triggers reasoned adjustments, restrictions, or transformations of the domain of validity (Rapp 2026c).

Definition 6: Global Falsification

A model M is globally falsified at time t if, under the conditions of the epistemic enabling space E(t), it remains neither epistemically optimal nor epistemically competitive in any relevant domain D. Epistemic optimality holds in a domain D if U(M, D, t) is greater than or equal to the utility value of all alternative models available in the enabling space E(t). Epistemic competitiveness holds if U(M, D, t) lies below the utility value of the best alternative U(M*, D, t) by at most a context-dependent tolerance range ε.

Formally, global falsification can be expressed as follows: for every relevant domain D, there exists at least one alternative model M* such that:

U(M, D, t) + ε < U(M*, D, t).

The tolerance parameter ε reflects the fact that models with marginally lower utility values can still remain competitive in practice. To prevent ε from becoming a means of immunizing a weakened model, the tolerance range is not to be set arbitrarily but is to be reconstructed from the relevant disciplinary evaluation practices, error tolerances, uncertainty measures, and contexts of application. Operationally, ε tips from legitimate tolerance into immunization once it grows larger than the margins of error and uncertainty recognized in the domain in question. If the utility difference ΔU(M*, M, D, t) systematically exceeds these disciplinary standards, it can no longer be absorbed by ε without disabling the discipline's evaluative practice itself. ε therefore designates not a protective zone for inferior models but the range within which competitiveness is still recognized as obtaining by the evaluation procedures customary in the domain. Global falsification accordingly does not arise from mere or short-term inferiority to an alternative but only from stable relative non-competitiveness across all relevant domains—that is, when the inferiority to available alternatives is not merely punctual but persists across the relevant testing and application contexts.

Supplement: Relevant Domain

A relevant domain is a scientific context of use in which a model continues to provide explanatory, predictive, interventionist, or structuring performance within current research or applied practice. Not every residual use of a model therefore grounds epistemic competitiveness. Historical, didactic, metaphorical, or purely illustrative uses can remain legitimate but do not count as relevant domains in the sense of global falsification. A model can thus be preserved historically or didactically and nonetheless be globally falsified in the epistemic sense if it is no longer competitive in any current scientific domain. The relevance of a domain is not derived solely from the utility value U(M, D, t) but is reconstructed from existing scientific practice, its target quantities, testing procedures, contexts of application, and recognized evaluative orderings. This prevents domain relevance from being inferred merely from the performance evaluation of the model under examination.

Definition 7: Epistemic Enabling Space E(t)

The space of methodological, mathematical, institutional, and technical preconditions that determines:

• which models can be formulated,
• which data are available,
• which idealizations are admissible,
• which models can be stabilized,
• how model costs C(M, D, t) can be determined at all.

E(t) is dynamic: technological innovations, data availability, and institutional norms alter the space of possible models.

Definition 8: Approximate Truth

Approximate truth designates the degree of stabilized fit between model structure and the comparison features that are relevant within a domain. Such comparison features can include observational data, measurement structures, robust system features, interventionally testable results, or theoretically distinguished structures. Approximate truth is therefore neither a global property of a model nor a direct grasp of a target system as such, but a domain-specific and reconstructive evaluative quantity.

Formally, approximate truth can be expressed as a similarity-and-fit metric:

AT(M, D, t) = Σᵢ wᵢ · sim(M, Sᵢ, D, t).

Here, Sᵢ designates a comparison feature determined as relevant within domain D; wᵢ describes its relative weighting within the relevant scientific evaluative ordering. Interventionist practice can enter as a testable fit relation, for instance when a model correctly represents stable changes under interventions. The theoretical interpretation, causal classification, or counterfactual disclosure of such relations belongs, however, to the explanatory power EK(M, D, t) and is to be analytically distinguished from AT.

In practical model evaluation, approximate truth is always considered together with uncertainty estimates of the model's predictions. High predictive uncertainty reduces a model's effective epistemic value even when its mean fit is high, and can therefore affect both AT(M, D, t) and the epistemic costs C(M, D, t).

Definition 9: Domain Operators

The following operators describe different forms of change or refinement of model validity.

• Restriction: R(M, Dᵢ) = M|Dᵢ designates the restricted application of a model to a particular subdomain Dᵢ.
• Domain difference: D(M)' = D(M) \ Dᵢ designates the remaining domain of validity of a model after exclusion or relinquishment of a problematic subdomain Dᵢ.
• Revision: Rev(M, Dᵢ) designates the targeted adjustment of a model or submodel within a problematic subdomain Dᵢ.
• Decomposition: Z(M) = {D₁, D₂, …, Dₙ} designates the partition of a model's domain of validity into several testable and evaluable subdomains.

These operators distinguish three cases that are often conflated in modeling practice: the restricted application of a model within a subdomain, the restriction of its domain of validity, and local revision within a problematic domain.

On Temporal Dynamics

Since the epistemic enabling space E(t) changes in the course of scientific development, the evaluative quantities AT(M, D, t), EK(M, D, t), and C(M, D, t) are in principle time-dependent. The time index t is used explicitly in what follows whenever the dynamics of E(t), the change in available alternatives, or the historical shift of model validity are at issue. Where it would impede readability, the index t may be omitted; in substance, however, it is always to be understood as present.

4. Criteria of Modern Model Evaluation within the Proposed Framework

Modern scientific model evaluation involves a range of epistemic, technical, and institutional criteria that together determine whether a model is preserved, recalibrated, or disappears within a domain D. The framework developed here renders these criteria explicit and assigns them systematically to the components of the utility function U(M, D, t) and to the epistemic enabling space E(t). The following catalog does not claim to capture all discipline-specific forms of evaluation exhaustively, but rather to identify the central epistemically relevant evaluation dimensions of modern modeling practice.

(1) Approximation and structural fit
This includes domain-specific approximate truth AT(M, D, t), understood as the graded fit between model structure and relevant comparison features within a domain. Components include empirical quality of fit, agreement with robust data structures, structural similarity, stability of model predictions, and, where applicable, the correct representation of interventionally testable relations. AT thus designates a model's fit dimension: it indicates how well a model fits the features that are relevant within a given domain.

(2) Explanatory power (EK)
A model's explanatory power concerns what it contributes beyond mere fit. It encompasses the capacity to render phenomena intelligible, to unify diverse findings, to make causal or interventionist dependencies visible, to enable counterfactual insights, and to give theoretical structure to a domain. EK(M, D, t) is therefore not reducible to predictive accuracy, fit, or structural similarity. Two models can possess similar approximate truth and yet differ markedly in how much they explain, order, or theoretically disclose.

(3) Model uncertainty
Model uncertainty covers the variance, sensitivity, and stability of model predictions within a domain. Wide uncertainty bands diminish a model's effective epistemic value even when its mean fit is high. They affect both AT(M, D, t) and the epistemic costs C(M, D, t).

(4) Interpretability and transparency
Models differ in their structural accessibility, the traceability of their mechanisms, their testability, and the diagnostic possibilities they offer for tracing sources of error. Low interpretability raises epistemic costs and lowers a model's explanatory usability.

(5) Technical and computational costs
These include computational effort, energy consumption, numerical complexity, data requirements, and implementability. Such technical costs form a central component of C(M, D, t) and substantially influence the rationality of model choice in data-intensive sciences.

(6) Institutional and infrastructural conditions
Models are stabilized through norms, software ecosystems, training structures, data formats, and organizational practices. These factors are located in the epistemic enabling space E(t) and explain why models often persist even when they become suboptimal in particular domains.

(7) Availability of epistemic alternatives
The set of alternative models available in the enabling space E(t) determines whether contextual falsification leads to model revision, domain restriction, or global elimination. Models persist as long as they remain epistemically optimal or competitive in at least one relevant domain. What matters, therefore, is not only a model's internal performance but its relative position vis-à-vis available, testable, and stabilizable alternatives.

This catalog of criteria specifies which factors carry operational significance within the utility function U(M, D, t) and the enabling space E(t). At the same time, it provides a systematic overview of the central evaluative dimensions that shape modern scientific modeling practice. The integrated framework is thus made theoretically applicable and empirically interpretable, without claiming to provide a complete catalog of all discipline-specific forms of evaluation.

5. Proposal: Two-Stage Falsification Structure

In the terminology proposed here, "falsification" no longer designates a binary decision about truth or falsity about M as a whole but a systematically reconstructible shift in the utility values U(M, D, t) across domains, which under certain conditions leads to the complete relinquishment of M.

For every model M:

1. As long as there exists at least one relevant subdomain in which M is epistemically optimal or competitive according to U(M, D, t), falsification does not lead to global elimination but to restriction or revision of the domain of validity: D(M) → D(M)', where D(M)' designates the remaining relevant domains in which M is epistemically optimal or competitive.
2. M is globally falsified if and only if there is no relevant domain D in which M is epistemically optimal or competitive according to U(M, D, t).

Approximate truth AT(M, D, t), explanatory power EK(M, D, t), and model costs C(M, D, t) determine the evaluation of epistemic optimality; E(t) determines the set of available and stabilizable alternatives.

Formally, the transition from contextual to global falsification can be expressed as a shift from relative competitiveness to stable non-competitiveness. Throughout, M* designates the best alternative model available within the domain in question. Contextual falsification obtains as long as there exists at least one relevant domain D in which M remains competitive within the tolerance range ε:

∃D: U(M, D, t) + ε ≥ U(M*, D, t).

Global falsification, by contrast, obtains when M lies stably below this tolerance range across all relevant domains:

∀D: U(M, D, t) + ε < U(M*, D, t).

The transition is therefore not a mere switch from truth to falsity but a change in the relative validity of a model with respect to available alternatives.

This two-stage structure makes formally tractable a distinction that, while discussed in the literature, has rarely been modeled explicitly. It precisely reconstructs how models can be preserved in particular subdomains even though they fail in others, and thereby supplements the existing accounts of Popper and Lakatos with a clearly structured representation of domain-specific model stability. The two-stage falsification structure thus describes explicitly the situation, frequently observed in scientific practice, in which models are recalibrated in subdomains while their overarching epistemic role is preserved.

5.1 Limiting Routes of Immunization

The theoretical robustness of the two-stage falsification structure depends on the exclusion of three systematically connected routes of immunization. First, the tolerance range ε must not become an open-ended protective clause for weakened models. It designates only the zone within which competitiveness remains plausible according to the standards of error, uncertainty, and evaluation recognized in a domain. Second, the relevance of a domain must not be derived from the defensive interest in the model under examination. A relevant domain must be stabilized independently through existing scientific practice, target quantities, testing procedures, contexts of application, or recognized evaluative orderings. Third, domain decomposition must not function as an ad hoc rescue. It is epistemically legitimate only if it makes new testable distinctions visible, separates sources of error diagnostically, refines model applications, or enables new explanatory, predictive, or interventionist achievements.

These three constraints work together. A theory of contextual falsification that draws only one of these limits remains open to immunization through the others: through an overly generous ε, through artificially generated residual domains, or through merely defensive subdomain construction. Contextual falsification is therefore rationally reconstructible only when tolerance ranges, domain relevance, and domain decomposition are each constrained by external disciplinary standards, existing forms of practice, and new epistemic gains. Thus, for instance, a model whose tolerance range ε is well constrained at the disciplinary level could still be preserved by a merely defensive subdomain construction unless that subdomain construction is supported by independent practice standards or by new epistemic gains.

6. When Models Disappear

A model does not disappear merely because it shows weaknesses in particular domains or performs marginally worse than an alternative. Global model replacement requires that, under the conditions of E(t), M remains neither epistemically optimal nor competitive in any relevant domain. What is decisive, therefore, is stable relative non-competitiveness against available alternatives. This diagnosis presupposes that ε is not artificially expanded, that domain relevance is not derived merely from the defensive interest in the model, and that domain decomposition is not used as a purely defensive rescue strategy.

A model typically disappears when:

1. across all relevant domains, at least one alternative model M* durably enjoys a utility advantage beyond the tolerance range ε: U(M, D, t) + ε < U(M*, D, t),
2. the model costs of M relative to its alternatives are no longer compensated for by approximate truth or explanatory power,
3. under the conditions of E(t), there are no current scientific contexts of use or relevant domains in which M remains epistemically competitive.

Examples:

• ether → Maxwell (1865) and Einstein (1905),
• phlogiston → modern chemistry,
• epicycles → Kepler (1609, 1619) and Newton,
• the four humors theory → modern medicine.

These historical model changes are often interpreted in the philosophy of science as paradigm shifts or transformations of research programs (Kuhn 1962; Lakatos 1970), but they receive a more precise formal reconstruction within the utility structure proposed here.

Technological, mathematical, and institutional innovations alter E(t) and accelerate global falsification. E(t) thereby determines not only which models disappear but also which can even count as realistic alternatives.

7. Why Models Persist

8. Case Study: Climate Models

9. Approximate Truth

Approximate truth explains why models can generate epistemic progress despite errors. The concept was developed especially by Niiniluoto (1987, 1998) and Oddie (1986), but it is used here in an explicitly domain-specific and reconstructive manner. Approximate truth in this paper does not designate the immediate access of a model to a target system as such, nor the entire epistemic performance of a model. It designates a particular evaluative dimension within U(M, D, t): the domain-specific fit between model structure and the relevant empirical, structural, or interventionally testable comparison features.

Truthlikeness is thus understood not as a global property of a model but as a functionally reconstructible relation within particular conditions of validity. Approximate truth is therefore narrower than overall epistemic optimality. A model can have a strong fit to certain data or structures and yet possess limited explanatory power; conversely, a theoretically powerful model can be epistemically weakened if its fit to the relevant features of a domain is inadequate.

Formally, approximate truth can be expressed as a domain-specific similarity-and-fit metric:

AT(M, D, t) = Σᵢ wᵢ · sim(M, Sᵢ, D, t).

Here, Sᵢ designates a comparison feature determined as relevant within domain D, such as observed system states, empirical data structures, theoretically distinguished properties, or interventionally testable target quantities. The function sim(M, Sᵢ, D, t) describes the similarity or fit between model M and the corresponding comparison feature Sᵢ under the methodological and empirical conditions available at time t. The weights wᵢ represent the relative relevance of these features within the relevant scientific evaluative ordering. Within this formal framework, approximate truth is neither an absolute criterion of truth nor a mere measure of fit; it is a domain-specific measure of structural and empirical fit that is central to model choice.

The important points are:

1. Approximate truth is always domain-specific.
2. Approximate truth designates the fit dimension of a model, not its overall epistemic performance.
3. Explanatory power remains analytically distinct: it concerns the explanatory, unifying, causal, counterfactual, or structuring performance of a model.
4. Model costs likewise remain distinct: they concern the epistemic, technical, and institutional efforts, burdens, or dependencies of a model.
5. Falsification can lower AT(M, D₂, t), weaken EK(M, D₂, t), or raise C(M, D₂, t); but it need not automatically destroy the validity of M in other domains D₁.
6. Global falsification cannot be defined through approximate truth alone. What matters is the entire utility structure and the relative competitiveness against available alternatives: U(M, D, t) + ε < U(M*, D, t) for all relevant domains D.

The coupling of approximate truth to the utility function U(M, D, t) shows that graded truthlikeness does not operate in isolation but, in interplay with explanatory power and model costs, determines epistemic optimality. Approximate truth operates within domains and supplements the theory with a quantitative dimension of epistemic improvement.

For practical application, it is important that both the similarity function sim(M, Sᵢ, D, t) and the weights wᵢ are determined in a context-dependent manner and can be operationalized differently across disciplines. In many scientific areas, sim(M, Sᵢ, D, t) can be approximated through established fit-and-error metrics, including likelihood functions, variance measures, prediction errors, residual analyses, or measures of similarity between model trajectories and observed system states. In other contexts, especially in theoretically or structurally oriented sciences, similarity metrics additionally capture qualitative or topological features such as the preservation of symmetries, invariance structures, or the reproduction of causal dependence patterns.

The weights wᵢ represent the relative relevance of the various comparison features selected within a domain. They can arise from empirical evaluation practices, disciplinary standards, or model-theoretic considerations. Approximate truth is therefore not tied to a single metric but constitutes an adaptable structure that takes account of both the numerical accuracy and the structural fit of a model, without presupposing direct access to a target system as such.

10. Why an Integrated Theory Has Been Lacking

11. The Epistemic Enabling Space

12. The Decision Logic of Model Choice

13. Relation to Scientific Realism

The integrated theory stands between structural realism and instrumentalist model pluralism. This intermediate position is especially relevant for modern modeling practices, since many sciences operate neither in fully realist nor in fully instrumentalist fashion. Within this framework, models possess truthlikeness insofar as they successfully capture and reconstruct stable relational, empirical, or interventionally relevant structures within particular domains. This approach is compatible with forms of structural realism, since it locates truthlikeness not in the model as a whole but in those stable relational structures that are preserved within specific domains. Their continuance, however, also depends on practical factors: costs, institutional stability, available alternative models, and shifts in the enabling space E(t).

This approach supplements structural realism by rendering explicit that approximate truth is not the sole determinant of scientific rationality. At the same time, it extends instrumentalist model pluralism by formalizing when models remain epistemically optimal despite partial falsifications.

The approach developed here can be located within the realism debate as a mediating position. It is close to structural realism, in the tradition of Worrall (1989) and Psillos (1999), insofar as approximate truth is interpreted as an approximation to stable relational structures. At the same time, the approach pursues a broader perspective than van Fraassen's constructive empiricism (1980), since it shows that the operative rationality of scientific modeling practice depends not solely on empirical adequacy but also on institutional, technical, and explanatory conditions that are systematically taken into account in the epistemic enabling space E(t).

In contrast to da Costa and French's conception of partial truths (2003), the present framework integrates approximate truth not only as a logical structure of graded truth but embeds it in an explicit utility function U(M, D, t) that includes explanatory power and model costs. Scientific rationality thereby becomes reconstructible as a decision problem within a dynamic enabling space. The approach therefore positions itself between structural realism and instrumentalist model pluralism, but combines their insights into an integrative framework that connects directly with the empirical modeling practice of contemporary sciences.

Especially in data-based and simulation-intensive sciences—climate research, economics, epidemiology, or AI modeling—this framework provides a precise way to analyze model stability and model change systematically.

14. Falsification as Domain-Relative Revision of Validity

Falsification is not a binary mechanism of elimination but a domain-specific tool for the refinement of scientific models. The theory developed here gathers central insights from Popper, Kuhn, Lakatos, da Costa/French, and contemporary model theory within an explicitly formalized, unified framework. The distinction between global and contextual falsification shows that empirical deviations predominantly lead to a restriction of the domain of validity, not to a complete relinquishment of the model.

Integrating approximate truth, explanatory power, and model costs into a domain-specific utility function shows that model stability is the result of relative epistemic optimality and does not follow from truthlikeness alone. Models disappear only when, under the conditions of E(t), they no longer remain epistemically competitive in any relevant domain or current scientific practice of use. Global model replacement is therefore not the immediate consequence of an individual empirical contradiction but the result of stable relative non-competitiveness against available alternatives. At the same time, the theory of contextual falsification is robust only if ε, domain relevance, and domain decomposition are not used as instruments of immunization. Falsification is thereby not weakened but refined: it constrains, revises, reweights, or eliminates model validity wherever scientific practice provides sound reasons for doing so.

With the concept of the epistemic enabling space, it also becomes possible to explain why technological, institutional, and methodological changes profoundly influence scientific model landscapes. Model choice thereby becomes reconstructible as a dynamic process in which scientific rationality consists not in a singular logical operation but in a structured interplay of context, approximation, and institutional stabilization.

The framework developed here is itself to be understood as a model. Its primary domain is the analysis of scientific model validity under finite conditions, particularly in idealized, simulation-based, and pluralistic sciences. Its own validity is therefore likewise domain-specific: it can have high reconstructive power in certain scientific contexts without being a universal theory of scientific rationality. Its costs lie in abstraction, formalization effort, and the danger of possible spurious precision. The present theory thus is itself subject to the conditions it describes: it can be contextually restricted, further refined, or supplemented by more suitable frameworks.

Falsification is thereby not abolished but refined: it remains a central instrument of rationality in scientific practice, but it operates not as a simple binary mechanism of exclusion but as a domain-relative indicator of bounded model validity. Its epistemic function is to constrain, revise, reweight, or replace model validity wherever available alternatives durably achieve higher epistemic competitiveness.

Local Philosophy-of-Science Glossary for This Paper

The following concepts stand within a modular relationship inside the Epistemics program. Friction (Rapp 2026b), revision (Rapp 2026c), search (Rapp 2026d), and domain structure (Rapp 2026e) are developed in dedicated works; the present paper transfers this architecture to the special case of scientific falsification and supplements it with the two-stage falsification structure and the epistemic enabling space. The following ordering of concepts serves to stabilize central meanings within this text. It does not claim completeness and does not establish an independent general canon of concepts. Concepts not specifically listed here are used in the sense of the Epistemics base canon or are not central to the functional core of this paper. Changes, refinements, or extensions are possible but must be explicitly indicated, locally bounded, and justified. Implicit shifts of meaning, silent extensions, or retroactive reinterpretations are excluded.

Adoption of the Epistemics Base Canon

This paper adopts the conceptual canon defined in the Epistemics base paper as an unchanged reference base (Rapp 2026a). The concepts introduced there are used without reinterpretation and without implicit shifts of their functional meaning. This paper introduces no divergent definitions of the adopted canonical concepts.

Local Philosophy-of-Science Refinements

This paper introduces no independent general extension of the canon. It refines several adopted concepts locally for the special case of scientific falsification and modern model evaluation. These refinements do not alter the Epistemics base canon but translate parts of its vocabulary into a philosophy-of-science language for describing model validity, domain restriction, model revision, and global model replacement.

Model Validity

• Short definition: domain-relative epistemic viability of a model under particular methodological, technical, and institutional conditions.
• Function: designates here the question in what area a model remains scientifically usable, testable, and competitive against alternatives.
• Demarcation: not the global truth of a model; not unrestricted validity; not mere factual or didactic use.

Contextual Falsification

• Short definition: domain-specific weakening or restriction of model validity, without elimination of the model as a whole.
• Function: describes cases in which a model is revised or restricted in a subdomain while remaining epistemically viable in other relevant domains.
• Demarcation: not a complete refutation of the model; not a mere anomaly without consequence; not an automatically global model replacement.

Global Falsification

• Short definition: loss of epistemic competitiveness across all relevant domains under the conditions of the epistemic enabling space E(t).
• Function: marks the case in which a model no longer possesses any relevant scientific viability against available alternatives.
• Demarcation: not a mere partial falsification; not didactic or historical non-use; not a simple consequence of an individual observational contradiction.

Epistemic Enabling Space E(t)

• Short definition: the structure of methodological, technical, and institutional conditions within which models can be formulated, tested, compared, stabilized, and within which alternatives become available.
• Function: explains why model validity is historically variable and why alternatives become visible, available, and stabilizable only under particular conditions.
• Demarcation: not a neutral background space; not a mere collection of external factors; not a mechanical determination of scientific model choice.

Approximate Truth AT(M, D, t)

• Short definition: domain-specific fit between model structure, observational data, interventionist practice, and relevant comparison features at time t.
• Function: serves to reconstruct graded truthlikeness within particular conditions of validity.
• Demarcation: not a global truth property of a model; not direct access to a target system as such; not a mere fit metric.

Explanatory Power EK(M, D, t)

• Short definition: a model's capacity to render phenomena intelligible, structure them, and disclose them counterfactually within a domain.
• Function: supplements approximate truth with the explanatory and structuring performance of a model.
• Demarcation: not mere predictive accuracy; not rhetorical plausibility; not domain-independent explanatory force.

Model Costs C(M, D, t)

• Short definition: epistemic, technical, and institutional efforts or burdens associated with the formulation, application, stabilization, or revision of a model.
• Function: shows that model choice depends not solely on truthlikeness or explanatory power but also on uncertainty, complexity, data demands, computational effort, interpretability, and infrastructure.
• Demarcation: not a merely monetary concept; not an external secondary evaluation; not an absolute quantity independent of domain and enabling space.

Utility Function U(M, D, t)

• Short definition: a reconstructive evaluative structure that brings together approximate truth, explanatory power, and model costs within a domain.
• Function: serves to represent the domain-specific epistemic optimality or competitiveness of a model.
• Demarcation: not an algorithmic decision procedure; not a complete computation of scientific rationality; not a domain-independent ranking.

Relative Difference ΔU(M*, M, D, t)

• Short definition: a comparative quantity between a model M and an available alternative model M* within the same relevant domain.
• Function: shows whether an alternative is only marginally better, distinctly superior, or stably epistemically dominant.
• Demarcation: not an independent theory; not an exact universal measure; not an evaluation without reference to E(t).

Tolerance Range ε

• Short definition: a context-dependent range within which a model remains epistemically competitive despite a marginally lower utility value.
• Function: prevents minimal or unstable differences between models from being interpreted immediately as global falsification.
• Demarcation: not an arbitrary protective clause; not an immunization against criticism; not a fixed quantity independent of discipline and domain.

Relevant Domain

• Short definition: a current scientific area of use in which a model performs explanatory, predictive, interventionist, or structuring work.
• Function: distinguishes epistemically viable use of a model from merely historical, didactic, or illustrative residual use.
• Demarcation: not an arbitrary application context; not a merely cultural persistence; not a mere mention of a model.

Status and Domain of Validity

The refinements introduced in this paper do not constitute an independent extension of the Epistemics base canon in the strong sense. They serve exclusively as a local philosophy-of-science specification of central concepts for the special case of contextual and global falsification of scientific models. Their domain of validity is restricted to the model-evaluation architecture of this paper.

There is no silent extension, reinterpretation, or retroactive modification of the Epistemics base canon. The base canon remains unchanged in meaning, function, and demarcation.

Any future deviation, further refinement, or genuine canonical extension beyond this local conceptual specification must be explicitly indicated, locally bounded, and justified. Implicit shifts of meaning or informal canon extensions are excluded.

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Appendix: Didactic Examples of the Two-Stage Falsification Structure

The following examples serve to illustrate didactically the distinction developed in this paper between contextual and global falsification. They are not a substitute for detailed historical analysis in particular fields but show by example how model validity, domain relevance, tolerance ranges, domain decomposition, and global model replacement can be reconstructed within the proposed framework. The examples are therefore not to be understood as complete historical reconstructions but as application sketches of the conceptual architecture of this paper.

A.1 Newtonian Mechanics: Contextual Falsification without Global Elimination

Newtonian mechanics is a paradigmatic case of contextual falsification. Its validity was not simply eliminated by relativistic physics and modern gravitational theory but rather restricted in a domain-specific way. At very high velocities, in strong gravitational fields, or in highly precise astronomical measurements, it loses epistemic competitiveness vis-à-vis relativistic models. In these domains, its approximate truth diminishes because its model structure no longer captures relevant system features with sufficient accuracy.

At the same time, Newtonian mechanics remains epistemically viable in many technical, everyday, and engineering domains. For bridge construction, mechanical engineering, many technical applications, and classical approximate calculations of celestial mechanics, it possesses high explanatory power, low model costs, and adequate predictive accuracy. Its didactic use in schools and universities is an additional matter, but does not on its own ground its epistemic competitiveness. In the language of this paper, this means: Newtonian mechanics is falsified or in need of revision in particular subdomains but remains competitive in other relevant domains.

The case shows why falsification must not be equated with global elimination. Relativistic physics does not replace Newtonian mechanics in every context of application; it constrains its domain of validity. The resulting structure is therefore not a simple refutation but a domain refinement: the model is preserved, but its validity is restricted.

Anti-immunization check. The preservation of Newtonian mechanics is not a mere protective strategy. Its remaining domains are stabilized independently through scientific and technical practice. Its error bounds are known, its costs low, and its conditions of application clearly specifiable. The domain restriction therefore generates new epistemic gains rather than artificially rescuing a weakened model.

A.2 Phlogiston Theory: Global Falsification with Historical Residual Use

Phlogiston theory serves as an example of global falsification. In the transition to modern chemistry, it lost performance not only in a narrowly bounded subdomain but was displaced in the relevant scientific contexts of explaining combustion, oxidation, and chemical reactions by more powerful alternatives. Its model costs rose, its explanatory power declined, and available alternatives offered stably better connections.

In the proposed framework, this means: phlogiston theory did not remain epistemically competitive in any relevant scientific domain. Its present-day use is historical, didactic, or related to the history of science, but no longer part of current chemical model evaluation. It can still be explained, taught, or used as an example of scientific change, but this residual use does not ground continuing epistemic competitiveness.

The case shows the distinction between historical persistence and a relevant domain. A model can remain culturally, didactically, or historically present and yet be globally falsified. Global falsification therefore does not mean that a model disappears from all discourses but that it is no longer viable in any current scientific domain against available alternatives.

Anti-immunization check. An artificial residual domain such as “phlogiston as a historical figure of thought” would not ground epistemic competitiveness. It is not a current chemical application domain but a didactic or historical form of use. This is precisely what shows why relevant domains must not be derived from mere residual use.

A.3 Climate Models: Contextual Falsification within Model Families

Climate models show how contextual falsification operates in complex model families. A climate model or an ensemble of climate models can possess high viability in particular areas, such as long-term global temperature trends, while other subdomains exhibit greater uncertainties—for instance, regional precipitation patterns, extreme events, or short-term climate variability. Deviations in such a subdomain do not necessarily falsify the entire model family but can affect particular parameterizations, submodels, or weightings.

Within the framework of this paper, this means that falsification often operates redistributively. It shifts epistemic weights within a model portfolio, weakening some components and strengthening others. An incorrect regional prediction can lower the approximate truth of a submodel in a specific domain without eliminating the entire model architecture globally. The consequence is then local revision, reweighting, or domain refinement.

This case shows particularly clearly why modern modeling practice cannot be captured by a simple Popperian elimination logic. In simulation-based sciences, models are often modular, parameterized, and ensemble-organized. Falsification then rarely affects an overall model as a unified structure but operates within a structured model space.

Anti-immunization check. The preservation of a climate model or ensemble is rationally reconstructible only when the remaining domains are independently supported by data, testing procedures, and contexts of application. Merely shifting problematic predictions into ever smaller subdomains would not be sufficient. Legitimacy arises only when the domain refinement generates new diagnostic, predictive, or explanatory gains.

A.4 Ptolemaic Epicycles: A Boundary Case between Adjustment and Ad Hoc Stabilization

Ptolemaic epicycle models are well suited as a boundary case because they show both empirical adaptability and possible ad hoc stabilization. Such models could describe observed planetary motions over long periods through additional geometric constructions. In particular historical enabling spaces, they were therefore not simply irrational or epistemically empty. Their stability rested on mathematical adaptability, institutional embedding, and the absence of competing, stable alternatives with higher overall performance.

With the development of Keplerian and later Newtonian models, however, the epistemic enabling space shifted. New mathematical, empirical, and theoretical conditions made alternatives available that not only reproduced observations but enabled a deeper structural and dynamic explanation. As a result, the Ptolemaic model class lost its relative competitiveness in the relevant astronomical and physical domains.

This case shows the importance of the anti-immunization logic. Not every model adjustment is illegitimate. An adjustment is epistemically productive when it makes new testable distinctions visible, separates sources of error, or increases explanatory power. It becomes problematic when it merely absorbs known counter-evidence without generating new gains. The boundary between legitimate model refinement and ad hoc rescue thus does not lie in the mere fact of additional assumptions but in their epistemic productivity.

Anti-immunization check. Epicycles become problematic when additional constructions accomplish only defensive adjustments and no longer open up new explanatory or predictive structures. Their global replacement does not arise from a single observation but from stable relative inferiority to alternatives that, under changed conditions of E(t), became available and more viable.

A.5 The Four Humors Theory: Global Replacement despite Cultural Persistence

The four humors theory exemplifies how a model can persist culturally without remaining scientifically competitive. Historically, it offered a comprehensive ordering of medical observations, interpretations of disease, and therapeutic practices. Within its enabling space at the time, it could function as a stabilizing framework, since alternative physiological, microbiological, or biochemical models were not yet available in their current form.

Under modern conditions, however, the four humors theory possesses no epistemic competitiveness in medical diagnostics, therapy, or disease modeling. Its explanatory and interventionist performance has been replaced by modern medicine, physiology, pathology, microbiology, and evidence-based procedures. Present-day appeals to the four humors theory may occur in historical, cultural-historical, or alternative-medicine contexts, but they do not ground a relevant scientific domain in the sense of this paper.

The case again illustrates the difference between persistence and validity. Models do not necessarily disappear from language, culture, or history when they are scientifically falsified. Global falsification means rather that a model no longer possesses viable competitiveness within the relevant scientific contexts of practice.

Anti-immunization check. The four humors theory cannot be epistemically rescued by treating its present-day historical or metaphorical use as a relevant medical domain. Such a residual domain would not be independently stabilized by current medical testing procedures, therapeutic successes, or scientific evaluative orderings.

A.6 Overview of the Examples

The examples show that the two-stage falsification structure is not merely a formal distinction. It allows different cases of scientific model evaluation to be reconstructed in differentiated ways: models can fail locally and yet be rationally preserved; they can persist historically and yet be globally falsified; they can be improved through legitimate domain refinement or stabilized artificially through merely ad hoc decomposition. What is decisive in each case is whether the remaining model validity is supported by relevant domains, disciplinary evaluative standards, and new epistemic gains.