Recent work in metaphysics, philosophy of science, and the theory of explanation has emphasized the structural parallels between causal, logical, and metaphysical explanation. In each domain, explanation appears to involve a tripartite structure: an explanans (that which explains), an explanandum (that which must be explained), and a principled relation that connects them. Causes explain effects by standing in law-governed relations; axioms explain theorems by inferential rules; fundamental facts explain derivative facts by relations of metaphysical dependence.
This structural alignment is not accidental, but reflects a broader aspiration toward explanatory closure: the ideal that, once the relevant principles are specified, what follows is fixed. Explanation, on this picture, consists in situating a phenomenon within a framework whose internal relations determine its place. The better the framework, the less residue remains.
There is much to recommend this ideal. It captures the power of formalization, the success of scientific modeling, and the clarity afforded by explicit inferential structures. It also motivates the widespread hope that explanation can, in principle, be rendered algorithmic: given sufficient information about initial conditions and governing principles, outcomes should be derivable.
And yet, explanatory practice itself resists this aspiration in subtle but persistent ways. Even in domains where formal rigor is maximal, explanation does not terminate merely in derivation. Judgments of relevance, adequacy, scope, and success continue to operate, often tacitly, at precisely those points where explanation appears most complete.
The question to be pursued in what follows is therefore not whether explanation works—it manifestly does—but whether explanatory success exhausts the conditions under which explanation is recognized as success. What remains operative, even where explanation appears closed?
II. Dependence Relations and the Temptation of Functionalism
The appeal of tripartite explanatory models lies in their promise of determinacy. Once the intermediary relation is fixed—causal law, inference rule, metaphysical dependence—the explanandum appears as a function of the explanans. To explain is to map inputs to outputs under stable rules.
This functional picture has been especially influential in recent metaphysics. If derivative facts depend on more fundamental facts in accordance with metaphysical principles, then explanation seems to consist in exhibiting a function from the fundamental to the derivative. Once the base facts and principles are in place, the result follows.
However compelling this picture may be, it quietly imports a further assumption: that the adequacy of the explanatory mapping is itself secured by the same principles that generate it. In other words, it assumes that once the function is specified, there is nothing left to assess.
But this assumption is false to explanatory practice.
Even in logic, where inferential rules are explicit, the correctness of a derivation does not by itself settle whether the axioms are appropriate, whether the system captures the intended domain, or whether the conclusion answers the question posed. Similarly, in metaphysics, identifying a dependence relation does not determine whether it is explanatory rather than merely formal, illuminating rather than trivial, or relevant rather than artificial.
The functional picture thus explains too much too quickly. It conflates derivability with explanatory satisfaction. The former can be fixed by rule; the latter cannot.
This gap is not accidental. It reflects a structural feature of explanation itself.
III. Explanatory Adequacy and the Irreducibility of Judgment
Consider the role of judgment in explanatory contexts that are otherwise maximally formal. In logic, the selection of axioms, the interpretation of symbols, and the identification of an intended model are not dictated by the formal system itself. In science, empirical adequacy underdetermines theory choice; multiple frameworks may fit the data equally well while differing in unification, simplicity, or fruitfulness. In metaphysics, competing accounts of grounding may be extensionally equivalent while differing profoundly in explanatory character.
In each case, explanation requires decisions that are not compelled by the formal machinery. These decisions are not arbitrary, nor are they merely psychological. They are normative: they concern what counts as explaining rather than merely deriving.
Crucially, these judgments are not external add-ons to explanation. They are conditions under which explanatory relations can function as explanations at all. A mapping from explanans to explanandum becomes explanatory only insofar as it is situated within a space of assessment in which relevance, adequacy, and success can be meaningfully evaluated.
Attempts to eliminate this space by further formalization merely reproduce it at a higher level. Meta-rules governing relevance or adequacy would themselves require criteria for correct application. The regress does not terminate in a final algorithm. What persists is the necessity of judgment.
This necessity should not be misunderstood. It does not signal a failure of rationality, nor an intrusion of subjectivity. Rather, it reveals that rational explanation presupposes a non-algorithmic space within which determinate relations can be taken as intelligible, appropriate, or successful.
Explanation, in short, presupposes intelligibility. And intelligibility is not itself a function of the explanatory relations it makes possible.
IV. Theory Choice, Model Adequacy, and the Limits of Formal Closure
The persistence of judgment becomes especially visible in contexts of theory choice and model adequacy, where formal success does not settle explanatory priority. In such cases, multiple frameworks may satisfy all explicitly stated constraints while nevertheless differing in their capacity to illuminate, unify, or orient inquiry. The choice among them is not determined by additional derivations, but by evaluative considerations that are internal to rational practice yet irreducible to rule.
This phenomenon is familiar across domains. In logic, distinct formal systems may validate the same set of theorems while differing in expressive resources or inferential economy. In the philosophy of science, empirically equivalent theories may diverge in their explanatory virtues—simplicity, coherence, depth, or integration with neighboring domains. In metaphysics, competing accounts of dependence or fundamentality may agree extensionally while offering incompatible explanatory narratives.
What is striking in these cases is not disagreement as such, but the form disagreement takes. The dispute is not over whether a rule has been followed correctly, nor over whether a derivation is valid. It concerns whether a framework makes sense of the phenomena in the right way—whether it captures what is explanatorily salient rather than merely formally sufficient.
No finite list of criteria resolves such disputes without remainder. Attempts to formalize explanatory virtues inevitably encounter the same problem they seek to solve: the application of the criteria themselves requires judgment. To ask whether a model is sufficiently unified, sufficiently simple, or sufficiently illuminating is already to presuppose a background sense of what counts as unity, simplicity, or illumination here rather than there.
This does not imply that theory choice is subjective, conventional, or arbitrary. On the contrary, the judgments involved are responsive to real features of the domain under investigation. But responsiveness is not compulsion. The domain constrains judgment without dictating it. Explanatory rationality thus occupies a space between determination and indifference—a space in which reasons can be given, criticized, refined, and sometimes revised, without being reduced to algorithmic selection.
The significance of this point is often underestimated because it emerges most clearly at moments of philosophical maturity rather than at the level of elementary practice. When a framework is first introduced, its power lies in what it enables. Only later, once its success is established, does the question arise of how that success is to be assessed, limited, or compared with alternatives. At that stage, explanation turns reflexive: it must account not only for its objects, but for its own adequacy as explanation.
What becomes apparent in such moments is that explanatory closure is never purely internal to a system. Even the most formally complete framework remains dependent on a space of evaluation in which its claims can be judged relevant, sufficient, or illuminating. This space is not itself a further theory competing with others. It is the condition under which theories can compete meaningfully at all.
The persistence of this evaluative dimension should not be regarded as a temporary limitation awaiting technical resolution. It is a structural feature of rational inquiry. Explanation advances not by eliminating judgment, but by presupposing it—quietly, continuously, and indispensably.
V. Articulation, Revision, and a Limit Case for Algorithmic Explanation
The limits identified above become especially clear when we consider not the objects of explanation, but the activity of explanation itself: the practices of articulation, revision, and defense through which theoretical frameworks are proposed and sustained. These practices are not peripheral to rational inquiry. They are constitutive of it. Yet they sit uneasily within accounts that aspire to explanatory closure through algorithmic or law-governed relations alone.
Consider a familiar kind of case from the history of twentieth-century psychology and philosophy of science: a theorist committed to a thoroughly naturalistic and algorithmic account of human behavior undertakes the task of writing a systematic defense of that very account. The activity involves drafting, revising, responding to objections, anticipating misunderstandings, and adjusting formulations in light of perceived inadequacies. The goal is not merely to produce text, but to get the account right—to articulate it in a way that clarifies its scope, resolves tensions, and persuades a critical audience.
From the standpoint of the theory being defended, the behavior involved in this activity may be describable in causal or functional terms. One may cite conditioning histories, environmental stimuli, neural processes, or computational mechanisms. Such descriptions may be true as far as they go. But they do not yet explain what is explanatorily central in the context at hand: namely, why this articulation rather than another is judged preferable, why a given revision counts as an improvement rather than a mere change, or why the theorist takes certain objections to matter while setting others aside.
These judgments are not epiphenomenal to the enterprise. They are what make the activity intelligible as theorizing rather than as mere behavior. To revise a manuscript because a formulation is inadequate is to operate with a norm of adequacy that is not supplied by the causal description of the revision itself. To aim at persuasion is to treat reasons as bearing on belief, not merely as inputs producing outputs.
Importantly, the difficulty here is not that the theory fails to predict or describe the behavior in question. It may do so successfully. The difficulty is that prediction and description do not exhaust explanation in this context. What remains unexplained is how the theorist’s activity can be understood as responsive to reasons—as governed by considerations of correctness, clarity, and relevance—rather than as merely following a causal trajectory.
One might attempt to extend the theory to include meta-level explanations of these practices. But such extensions merely relocate the problem. Any account that treats theoretical articulation as the output of a function—however complex—must still presuppose criteria by which one articulation is taken to be better than another. Those criteria cannot themselves be generated by the function without circularity. They must already be in place for the function to count as explanatory rather than as merely generative.
Consider a function d that specifies the dependency relations by virtue of which a metaphysical system M is explained on the basis of more fundamental objects, properties, relations, or states of affairs F. On this view, F together with d metaphysically explains M.
The question that immediately arises concerns the status of d itself. Is d something that admits of explanation, or is it not? If d is explained, then there must be some more basic function p in virtue of which d obtains. But once this path is taken, it is difficult to see how an infinite regress is avoided, since the same question must then be raised concerning p.
Suppose, alternatively, that d is not in need of explanation—that it is primitive, incorrigible, or somehow self-evident. This move, however, is problematic. Why should a metaphysical dependency function enjoy a privileged status denied to laws of nature or other explanatory principles? One might argue that certain transformation rules in logic possess a form of self-evidence or decidability, but this cannot plausibly be extended to metaphysical dependency relations. If it could, metaphysics would collapse into a formal logical system, contrary to its actual practice.
The difficulty, then, is not that metaphysical explanation fails, but that modeling it as a function obscures the normative and non-algorithmic judgments that are required to identify, assess, and deploy dependency relations in the first place.
This point does not target any particular theory as incoherent or self-refuting. The issue is structural, not polemical. Explanatory frameworks that aspire to algorithmic completeness necessarily presuppose a space in which articulation, revision, and defense are assessed as norm-governed activities. That space is not eliminated by successful explanation; it is activated by it.
The case thus serves as a limit test. Where explanation turns reflexive—where it must account for its own articulation and adequacy—the aspiration to closure gives way to dependence on evaluative judgment. The theorist’s practice reveals what the theory itself cannot supply: the conditions under which its claims can be meaningfully proposed, criticized, and improved.
VI. Explanatory Ambition and a Structural Constraint
The preceding analysis does not challenge the legitimacy of algorithmic, causal, or formally articulated explanation. Nor does it deny the success of contemporary explanatory frameworks in their respective domains. What it challenges is a specific aspiration: the hope that explanation can be rendered fully self-sufficient—that once the relevant relations are specified, nothing further is required for explanatory adequacy.
What emerges instead is a structural constraint on explanatory ambition. Explanatory relations, however rigorous, do not determine their own adequacy as explanations. They presuppose a space in which relevance, success, and improvement can be meaningfully assessed. This space is not external to rational inquiry, nor does it compete with formal explanation. It is internal to the very practice of offering, revising, and defending explanations as such.
This conclusion should not be misunderstood as reintroducing subjectivism, voluntarism, or irrationalism. The judgments involved are constrained by the domain under investigation and answerable to reasons. But they are not compelled by rules alone. Explanation constrains judgment without exhausting it. The possibility of error, disagreement, and revision is not a defect of rational inquiry but a condition of its vitality.
Nor does this conclusion invite a regress to foundational doubt. The space of judgment at issue is not a prior theory awaiting justification. It is operative wherever explanation functions successfully. To recognize its indispensability is not to abandon explanatory rigor, but to acknowledge what rigor already presupposes.
The temptation to explanatory closure is understandable. It reflects the genuine power of formal systems and the desire to secure rationality against arbitrariness. But when closure is taken to be complete, it obscures the very practices through which explanations gain their standing. What is lost is not explanation itself, but intelligibility—understood as the condition under which explanation can count as illuminating rather than merely generative.
The upshot, then, is modest but firm. Explanation does not collapse into derivation, because rational inquiry cannot dispense with judgment. This is not a contingent limitation to be overcome by future theory, but a permanent feature of explanatory practice. Any account that neglects it risks mistaking formal success for explanatory sufficiency.
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