On Meta-Theoretical Discourse and Formal Intelligibility
Quaeritur
Utrum intelligibilitas formalis systematum logicorum et mathematicorum praesupponat discursum metalinguisticum irreducibilem ad linguam obiectivam; et utrum hic excessus non solum epistemicus sed ontologicus sit, ita ut ipsa possibilitas significationis in rebus fundetur; et utrum hic fundus intelligibilitatis recte intelligatur ut spatium teleologicum, quod systemata formalia non efficiunt sed quod ipsa attrahit et constituit.
Whether the formal intelligibility of logical and mathematical systems presupposes a metalinguistic discourse irreducible to object language; and whether this excess is not merely epistemic but ontological, such that the very possibility of signification is grounded in things themselves; and whether this ground of intelligibility is rightly understood as a teleological space which formal systems do not produce but which draws them forth and constitutes them.
Thesis
Formal systems do not generate intelligibility. They presuppose it. Every object language capable of truth conditions relies upon a meta-discourse that cannot be fully internalized without loss of the very properties that render the system intelligible. This excess is not merely epistemic but ontological. The possibility of meaning precedes formalization and belongs to the structure of reality itself.
This irreducible space of intelligibility may be described as teleological: not as an imposed purpose or subjective projection, but as the permanent possibility of meaningful determination that draws formal systems into being and coordinates their interpretation. Metalanguage thus testifies to an order of meaning that no formal system can exhaust, yet without which no formal system can be what it is.
Locus classicus
Gödel, Über formal unentscheidbare Sätze (1931)
“Es gibt innerhalb eines jeden hinreichend mächtigen formalen Systems wahre Sätze, die innerhalb dieses Systems nicht beweisbar sind.”
“There are, within every sufficiently powerful formal system, true propositions that cannot be proven within that system.”
Gödel’s result is not merely technical. It reveals that truth outruns formal derivability and that the conditions for recognizing truth are not fully capturable by the system whose truths are in question.
Peirce, Collected Papers 5.121
“Thirdness is the mode of being of that which is such as it is, in bringing a second and a first into relation.”
Peirce’s category of Thirdness names mediation, lawfulness, and intelligible continuity. It points beyond dyadic relations to the conditions under which relations can be meaningful at all.
Aristotle, Metaphysics Γ.4 (1006a)
τὸ αὐτὸ ἅμα ὑπάρχειν τε καὶ μὴ ὑπάρχειν ἀδύνατον
“It is impossible for the same thing to belong and not belong to the same thing at the same time.”
The principle of non-contradiction is not derived from a system; it governs the possibility of systemhood itself.
Explicatio
The inquiry into metalanguage arises not from philosophical curiosity but from the internal limits of formalization itself. Whenever a formal system is sufficiently expressive to represent arithmetic, syntax, or inference, it becomes possible to ask questions about the system as a system: about its consistency, its completeness, its interpretability, and its truth conditions. These questions are not posed within the object language alone but from a vantage that speaks about the system. This vantage is meta-discourse.
Gödel’s incompleteness theorems make this structural distinction unavoidable. The encoding of syntactic relations by Gödel numbering allows statements about provability to be represented within arithmetic. Yet the recognition of undecidable truths still requires a standpoint that distinguishes truth from provability. That distinction is not eliminable. Even when meta-statements are formalized, the act of recognizing the adequacy of that formalization occurs at a higher level still. The meta recedes as it is formalized. What is gained in rigor is offset by a renewed excess.
This phenomenon is not accidental. It reveals something essential about intelligibility itself. Formal systems can model relations, generate derivations, and define extensions. What they cannot do is generate the conditions under which their own operations are meaningful. The possibility of interpretation is not a theorem of the system; it is the horizon within which the system can appear as intelligible at all.
This horizon is not merely epistemic. It is not simply a limitation of human cognition or a defect in symbolic manipulation. It belongs to the nature of formal structures themselves. A system that could exhaustively account for its own intelligibility would collapse the distinction between object language and metalanguage, thereby eliminating the very conditions that make interpretation possible. Meaning would be flattened into mechanism, and truth into derivability.
To say this is not to disparage formal rigor. On the contrary, it is formal rigor that reveals the necessity of this distinction. Logic itself teaches that intelligibility cannot be fully objectified without remainder. The meta is not an embarrassment to formalism; it is its condition.
This irreducible excess may be clarified by reconstructing Peirce’s notion of Thirdness. Thirdness is not merely a category of mediation within thought. It names the lawful continuity that makes relations intelligible. It is that by virtue of which signs signify, laws govern, and inference is possible. In this sense Thirdness is not added to dyadic relations; it is what allows relations to be relations rather than brute collisions.
What Peirce names phenomenologically, we may here name ontologically. The intelligibility that coordinates formal systems is not imposed from outside but belongs to the structure of reality. Formal systems are not self-originating. They are drawn into being by the possibility of meaning that precedes them. This possibility is not itself formal, yet it is not indeterminate. It orders, constrains, and directs formalization without being reducible to it.
Whitehead’s notion of prehension may serve as an analogy. Prehensions are not actual entities but the permanent possibilities of actualization. They are not events but the conditions under which events can occur meaningfully. In an analogous way, intelligibility is not itself a formal structure but the permanent possibility of formal meaning. It is that by which formal systems can be interpreted, related, and evaluated.
This is why attempts to algorithmize theory change inevitably fail. To formalize the rules by which theories are revised presupposes a prior understanding of relevance, adequacy, and success—concepts that themselves resist algorithmic capture. The criteria of revision always exceed the system being revised. The ladder by which the system ascends cannot be retained within the system without contradiction.
Wittgenstein’s Tractatus gestures toward this limit. What cannot be said must be shown. Yet showing is not mute. It is a mode of intelligibility that precedes explicit articulation. There is no seeing that could not, in principle, be spoken—but the speaking presupposes the very space it attempts to articulate. The ladder cannot be climbed unless it already stands.
Kant’s distinction between determining and reflecting judgment clarifies this further. Determining judgment subsumes particulars under given rules. Reflecting judgment seeks the rule under which particulars may be unified. The former may be formalized. The latter resists algorithmic closure. Reflecting judgment operates within a teleological space: it seeks coherence, purposiveness, and meaning without presupposing a determinate schema. This space is not subjective whim. It is the condition under which object languages can be coordinated at all.
Thus intelligibility is teleological not because it aims at a humanly imposed end, but because it orients formal structures toward meaning. Formal systems are “pulled into being” by this space. They do not emerge ex nihilo. They are responses to a prior call of intelligibility that is written into the structure of reality itself.
Objectiones
Ob I. If intelligibility exceeds formal systems, then rigor is compromised and mathematics collapses into metaphysics.
Ob II. Metalanguage reflects only human cognitive limitation, not any ontological feature of reality.
Ob III. Teleology introduces purpose into domains governed solely by efficient causality.
Ob IV. If intelligibility cannot be formalized, then it cannot be known or discussed without contradiction.
Responsiones
Ad I. Rigor is not compromised but clarified. Formal precision reveals the limits of formalization. To acknowledge these limits is not to abandon rigor but to respect its conditions.
Ad II. The recurrence of metalanguage is not contingent upon human psychology. It arises from the structure of formal systems themselves. Any intelligence capable of truth would confront the same distinction.
Ad III. Teleology here names orientation toward meaning, not extrinsic purpose. It does not replace efficient causality but grounds the intelligibility of causal explanation.
Ad IV. Intelligibility can be discussed analogically and architectonically without being reduced to an object language. Such discourse does not eliminate the meta; it inhabits it knowingly.
Nota
This disputation marks a decisive transition within the Disputationes. Up to this point, the inquiry has examined language, truth, causality, participation, and manifestation within the horizon of theological speech. Here the investigation turns explicitly to the conditions under which any discourse—logical, scientific, or theological—can be intelligible at all.
The significance of Gödel’s results is not exhausted by their mathematical application. They reveal that truth is not coextensive with formal derivability and that intelligibility requires a standpoint irreducible to object language. This insight aligns logic with philosophy at its deepest level. Logic does not eliminate metaphysics; it summons it.
The reconstruction of Peircean Thirdness offered here is not semiotic but ontological. It prepares the way for a theology of intelligibility by showing that meaning is not a human projection but a feature of reality. Formal systems respond to intelligibility; they do not create it.
This recognition quietly undermines every reductionist account of reason. It shows that the space in which meaning arises is not manufactured by minds but discovered by them. The humanities and mathematics converge here, not in method but in vocation: both seek the conditions of intelligibility.
Theological implications are now unavoidable, though not yet asserted. If intelligibility belongs to the structure of reality, then meaning is not accidental. If meaning is not accidental, then the question of Logos presses itself forward—not as hypothesis, but as the name for the ground of intelligibility itself.
Determinatio
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Formal systems presuppose intelligibility and do not generate it.
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No sufficiently expressive system can internalize the conditions of its own truth.
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The distinction between object language and metalanguage is irreducible.
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This irreducibility is ontological, not merely epistemic.
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Intelligibility constitutes a teleological space of meaning.
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Formal systems are drawn into being by this space rather than constituting it.
Transitus
If the intelligibility of formal systems presupposes an irreducible metadiscursive horizon, and if this horizon is not merely epistemic but grounded in the being of things themselves, then intelligibility cannot be understood as an accidental feature of formalization. It belongs instead to the structure of reality as ordered toward meaning.
What has now emerged is the necessity of law—not yet as a catalog of determinate principles, nor as primitive axioms internal to any one system, but as the ontological condition under which intelligibility may take stable, communicable form. Formal systems do not generate this order; they respond to it. They are drawn into articulation by a prior normativity that renders meaning possible and truth answerable.
The question therefore presses beyond the limits of metalinguistic excess toward the nature of order itself. How is intelligibility stabilized without being exhausted? How does teleological attraction give rise to lawfulness without collapsing into mechanism or necessity? And in what sense may laws be said to participate in the very ground of intelligibility they articulate?
These questions compel the inquiry to move from the conditions of meaning to the structures of order by which meaning abides. We therefore turn to the consideration of law—not as an artifact of formal reason, but as a mode of being through which intelligibility is sustained and made communicable.
