A Formal Grammar for Reading the Gospel Miracle Narratives
Preface: What This Is, and What It Is Not
This is a formal grammar, not a discovery. It borrows one piece of vocabulary from control theory — the reachable set — and uses it to say something precise about a recurring pattern in the Gospels. It does not prove that miracles happen. It does not add empirical content to theology. What it does is take an intuition many readers already have, half-formed, and give it a shape clean enough to examine, criticize, and extend.
The intuition is this: when the Gospels narrate a healing, a resurrection, a stilled storm, they are not claiming that Jesus is simply stronger than sickness, death, or weather — a bigger force beating a smaller one in a contest fought on the same terms. They are claiming something structurally different: that the boundary the sickness or death or storm seemed to impose was never the true boundary of what was possible. The constraint looked final. It wasn’t.
That is a claim about the architecture of possibility, not about horsepower. Getting the grammar right for that claim is the whole project of what follows.
One methodological note before we begin: I will resist, throughout, the temptation to call this a law. It is a principle — a lens, a piece of grammar — and the difference matters. A law makes predictions and can be falsified. A grammar organizes how a claim is said, so that the claim can be examined honestly. This document aims only at the second thing.
1. The Central Claim, Stated First
Before any apparatus is built, the single most important move in this entire document should be visible on its own:
Y ∉ RC(X) ⇏ Y ∉ R(X)
(“Y is not reachable from X under constraint C” does NOT imply “Y is not reachable from X at all.”)
Unreachable under a given order of constraints is not the same thing as impossible. Everything else in this document is commentary on that one non-entailment. It only says: unreachable under that order. Whether the order is exhaustive of reality is a separate question — and separating those two questions, rather than letting them collapse into each other, is the entire contribution of the formalism. Ordinary speech collapses them constantly: we say a thing “cannot” happen and quietly mean “cannot happen given what I currently know of how things go.” The statement above simply refuses to let that slide pass unmarked.
1.1 Formal Definitions
Let X denote a state of some system, and let C denote the ordinary constraints that govern how X can evolve — physical law, biological process, the accumulated weight of “how things go.” Define the reachable set:
RC(X) = { Y : Y can be reached from X under constraint C }
An authority A is defined, at this stage, in the most neutral terms possible: as an agency capable of altering the conditions of reachability. No claim about the metaphysical status of A — external, internal, divine, natural — is built into the definition. Formally, authority augments the constraint regime, and the augmented reachable set contains the original as a subset:
RC(X) ⊆ RC,A(X)
so that a state excluded under the narrower regime can belong to the wider one:
Y ∉ RC(X), but Y ∈ RC,A(X)
(If one wants to retain A as an explicit operation rather than just a relation between sets, it is more accurate to say A acts on the constraint rather than on the reachable set directly — A : C ↦ CA, producing RC(X) → RCA(X). But the subset relation above is the leaner and clearer statement, and it is the one this document will use.)
It is worth being explicit about what this neutral definition buys, because it prevents the formalism from smuggling its conclusion into its premises. The argument proceeds in three separable stages, and keeping them separate is the discipline the whole project depends on:
Formal level: A → expanded reachability
Narrative level: Jesus → A
Theological level: A = divine authority
The formal level is available to anyone, regardless of what they believe about the second and third. The narrative level is an interpretive claim about what the Gospels portray Jesus as doing. The theological level is a properly theological claim about the identity of the authority in question, and it is not derivable from the first two — it has to be argued on its own terms, elsewhere. Conflating these three levels is precisely the move that turns a formal grammar into a disguised apologetic argument, and this document tries, throughout, not to make that move.
The full hierarchy, stated in its most compressed and defensible form:
Constraint → Reachability
Authority → Expanded Reachability
Faith → Participation
Transformation → Realized State
And the single sentence, beneath the boxed equation, that carries the most weight in the whole document:
Constraint determines reachability; authority determines whether the constraint is final.
2. Why This Is an Improvement Over Force-Language
Ordinary religious speech, and a great deal of popular apologetics, describes miracles in the vocabulary of contest: God’s power versus nature’s power, faith versus doubt, the greater force overcoming the lesser. This vocabulary is intuitive because it borrows from the most primitive human experience of causation — pushing, overpowering, winning. But it smuggles in a claim that the Gospel texts themselves do not obviously make: that God and the sickness, or God and the storm, are commensurable quantities on the same scale, locked in a tug-of-war that God happens to win because he is stronger.
This is worse than merely inelegant. It is theologically distorting, because it implies a genuine contest — implies that the sickness or the storm has real, competing agency, and that the outcome, in principle, could have gone the other way if the contest had been closer. It turns God into the biggest thing in a world of things, rather than something categorically different from the things in it.
The reachable-set reformulation removes this distortion at the level of grammar, not just tone. Authority does not need to be stronger than the constraint, because authority is not on the same axis as the constraint at all. It is worth being precise here, correcting an overreach in an earlier draft of this idea: authority does not “redraw the space” of possible states, as though it were rewriting the underlying metaphysics of what can exist. That claims more than the formalism supports. What authority does, more modestly and more accurately, is change the conditions under which a state becomes reachable — it alters RC(X) without any claim about the deeper state space itself being rewritten. This is a smaller claim, and a truer one, and it is also, I think, philosophically just as forceful: it says that what looked like a wall was in fact a locked door, and a key existed that the system itself did not contain.
That is the shift worth dwelling on. Not “authority is bigger than constraint” but “authority operates on a different level than constraint” — one governs what happens within a given order of possibility, the other concerns whether that order is the last word.
One further precision matters here, because it forecloses an easy objection. The claim is not that the constraint was somehow misread or only appeared final while secretly harboring a loophole. Death really is a constraint under ordinary biology — a boundary that biology, on its own terms, does not cross. The constraint was final within the ordinary order being considered. What the narrative claims is not that this order was misunderstood, but that it is not exhaustive: that there exists a further order, or a further agency, not contained within it. Put plainly — this is not the discovery of a loophole in biology. It is the claim that biology, however correctly understood, is not the whole of the causal story. If someone objects, “but death really is irreversible under biology,” the correct response is not to dispute the biology. It is to agree with it entirely, and say: exactly — that is why the claim being made is extraordinary.
3. Relationship to Control Theory: Where the Analogy Holds, and Where It Must Stop
The term “reachable set” is not decorative borrowing. In control theory, given a system with dynamics and a set of admissible control inputs, the reachable set from an initial state is the collection of states attainable within those inputs. It is a rigorously defined, well-studied object, and one of its basic properties is monotonicity under the admissible-input set: give the controller access to a wider range of inputs — more actuators, relaxed bounds, a previously unavailable channel — and the reachable set grows, without any violation of the underlying dynamics. A target state that was unreachable under a narrow control regime can become straightforwardly reachable under a broader one.
This is the honest core of the analogy, and it is a real one: the logical move of factoring “impossible” into “impossible under these admissible transitions” is precisely the move control theory makes as a matter of course.
But here the analogy must be stopped, deliberately and by name, before it overreaches. In control theory, the controller is not external to the system in any deep sense. It obeys the same physics as the plant it acts on. Widening the admissible input set does not violate the constraint set C — it reveals that C was incompletely specified in the first place, and the “expanded” reachable set was, in a sense, there all along, waiting on a fuller accounting. The controller is a further application of law, not a suspension of it.
The theological claim this principle is built to serve wants something different and stronger: an authority genuinely external to the causal order that generates C — not an undiscovered variable within the same system, but an agency operating from outside whatever generates the system’s laws in the first place. That is not a difference of degree from ordinary control theory. It is a difference of kind, and no amount of notation can bridge it. The mathematics can express this claim with real precision. It cannot justify it. The justification, if there is one, has to come from theology proper — from claims about the identity and character of the authority in question — not from the properties of reachable sets, which are silent on whether their controller is transcendent or merely undiscovered.
Naming this limit is not a weakness of the project. It is what keeps the project honest.
4. Fit to the Gospel Narratives, Without Forcing a Single Mechanism
The real payoff of this grammar is that it lets a wide range of Gospel miracle stories be read as instances of one structural claim without requiring them to share one physical mechanism — which is where most attempts at a unified account of miracle go wrong, either by flattening every story into the same crude template or by treating each story as so sui generis that no pattern can be named at all.
Consider the range the Gospels actually offer:
Illness. The leper (Mark 1:40–42), the woman with the hemorrhage (Mark 5:25–34). Under ordinary physiological process, Whole ∉ RC(Sick) — wholeness is not on the menu of outcomes the disease process, left to itself, makes available. The narrative claims Whole ∈ RC,A(Sick).
Death. Jairus’s daughter (Mark 5:35–43), Lazarus (John 11). This is the sharpest case the formalism has to offer, because RC(Dead) is empty of “Living” not as a matter of low probability but as a matter of definition, under ordinary biological constraint. The claim being made here is not “recovery faster than expected.” It is a flat denial that the constraint set is exhaustive of what is possible for this state.
Nature. The storm (Mark 4:35–41), water into wine (John 2:1–11). Here the “system” being reconstrained is not a body at all but an environment or a material process — a different flavor of C entirely, meteorological or chemical rather than biological. The formalism absorbs this without strain, because nothing in the definition of RC(X) ties it to any one domain.
Possession and uncleanness. The Gerasene demoniac (Mark 5:1–20). This is the hardest case for the grammar, and honesty requires saying so plainly. The relevant “state” and “constraint” here are not straightforwardly physical; they involve categories — purity, social exclusion, cosmological agency — that resist reduction to a state space. The formalism can still be applied, loosely, but it is at its most metaphorical exactly here, and pretending otherwise would be a failure of the intellectual honesty this project claims for itself.
What the grammar buys, across this range, is the ability to say these are the same kind of claim without saying these are the same mechanism. The unifying structure is not physical but narrative-theological: an ordinary constraint defines a boundary; the story asserts the boundary is not final; an authority external to the constraint-generating order is invoked as the reason why. That commonality is real and worth naming, independent of whether the underlying constraint in a given story is biological, meteorological, or social-cosmological.
A caution belongs here too. The Gospels do not use this vocabulary, and John in particular resists a purely constraint-violation reading of these events — he calls them sēmeia, signs, whose primary function is to point past themselves toward who Jesus is, not to assert a metaphysical claim about the architecture of possibility as such. The grammar captures a real structural feature of these narratives. It should not be mistaken for how the narratives understand themselves.
5. Faith as Participation, Not Power Source
This is arguably the point at which the grammar does its most important theological work, and it deserves the sharpest statement possible.
A widespread but shallow reading of the Gospels treats faith as a causal input — a spiritual force multiplier, such that sufficient faith produces the miracle and insufficient faith explains its absence. This reading draws support from real passages: Jesus “could do no deed of power” in Nazareth “because of their unbelief” (Mark 6:5–6); “your faith has made you well” (Matthew 9:22). But generalized into the mechanism of the whole system, this reading becomes something close to cruel. It makes a sick person’s recovery a referendum on the adequacy of their own spiritual performance, and by strict logical converse, it makes unhealed suffering the sufferer’s fault.
The reachable-set grammar structurally forecloses this reading, because it places faith in a different logical position from the start. Faith does not act on C. It does not expand RC(X). All of that work belongs to A alone:
A → possibility
Faith enters only after the possibility exists:
F → participation
The refinement from receptivity to participation matters here, and it is worth being explicit about why. Receptivity suggests something passive — an open door, a willingness not to obstruct. But the Gospel portraits of faith are rarely passive. Bartimaeus shouts over a crowd trying to silence him (Mark 10:46–52). The hemorrhaging woman pushes through a crowd to touch a garment (Mark 5:27). The paralytic’s friends dig through a roof (Mark 2:1–5). Peter climbs out of a boat onto open water (Matthew 14:28–29). This is not openness; it is movement toward a possibility that has already been made available. Participation captures that movement without smuggling back in the claim that the movement itself generated the possibility. Grace, in this architecture, is the gift of the newly opened state; faith is the act of stepping into it.
This gives three distinct causal or logical roles — textually more accurate, and theologically cleaner than the “faith as force” reading:
A → possibility F → participation G → gift
Authority opens. Faith participates. Grace gives. And critically: F ≠ A. Faith does not create the possibility — this is the point the whole architecture has been built to preserve.
And it explains, without strain, why several of the most dramatic miracle stories — the storm, Lazarus, the wedding at Cana — involve no invocation of anyone’s faith as precondition at all. Under the “faith as power source” reading, this is an awkward exception. Under “faith as participation,” it is exactly what you would expect: participation can be one thread among the causes of a story’s shape without being required for every instance of A’s operation.
6. The Broader Philosophical Implication
Set the theology aside for a moment and a genuinely interesting general question remains: what changes if a system’s endogenous constraints are not treated as identical with the boundary of reality?
This is not a new question, and it would be dishonest to present it as one. It runs through Hume’s argument that a miracle is, by definition, a violation of what “firm and unalterable experience” has established, and his consequent skepticism toward testimony to such violations. It runs through Aquinas’s distinction between what is contrary to nature simpliciter and what is merely contrary to nature’s ordinary course. The core point does not need a survey of the philosophy-of-science literature to make; it needs only one sentence, already implicit in the control-theory discussion above: a model’s reachable set is not necessarily identical to reality’s total possibility space.
What the reachable-set language contributes is not a new answer to the old question but a cleaner way of locating exactly where a disagreement about it lives. Two people can agree completely about RC(X) — about everything that is possible under known constraint — and disagree only about whether C exhausts the actual constraint-generating order of reality, or is instead a locally reliable, well-confirmed, but non-exhaustive description of it. That is a far narrower and more honest place to locate a disagreement about miracles than most popular debate manages, where the two sides usually end up arguing past each other — one side defending the reliability of C as a model (which is not in dispute), the other asserting the existence of A (which is the actual point of contention).
The general move here — treating a model’s edge as provisional rather than as reality’s edge — is not, on its own, a religious move. It is one of the more sober habits available to a careful thinker in any field. What theology adds, and adds distinctively, is not the general move but a specific and much stronger claim riding on it: that the edge, in at least some cases, is crossed not by better modeling within the same order but by an agency ontologically outside the order that generates the model’s own terms. That addition is not something the formalism can supply. It has to be argued on other grounds entirely — grounds belonging to theology and metaphysics, not to the mathematics of reachable sets, which remain, honestly, silent on the question of what kind of thing A ultimately is.
The payoff sentence, stated as plainly as I can manage it:
The miracle is not necessarily a violation of reality; it is a claim that the reality ordinarily accessible to the system is not the whole of reality.
This distinction also has the virtue of blocking the weakest and most common form of miracle apologetics — the move from “science currently says X is impossible” directly to “therefore God did Y,” which conflates the limits of a model with the limits of the world and thereby earns, rightly, the contempt of anyone paying close attention. The claim this grammar actually supports is subtler and more defensible: science tells us what is reachable under a specified model of the world’s constraints; the theological question is whether that model exhausts the causal order of reality — a legitimate philosophy-of-religion question, distinct from any claim about what science has or has not shown.
7. Honest Limitations
Four should be named without softening, because a project that claims intellectual honesty as a value has to demonstrate it here, where it costs something.
(a) Risk of vacuity. As it stands, the schema is close to unfalsifiable by construction. A is defined as whatever expands RC(X) to include Y. Left there, “Y happened because something capable of making Y happen occurred” is true but empty — a restatement of the outcome, not an explanation of it. The schema only gains content once A is independently characterized, as Christian theology does characterize it, by appeal to the specific identity, character, and prior claims made about Jesus in the broader narrative and tradition. That independent characterization is where the actual theological argument lives. The reachable-set notation organizes the claim; it does not make the case for it.
(b) C is never fully specified. In control theory, the admissible-input set is precisely defined; that precision is the whole reason the mathematics works. In the theological application, C is gestured at — “ordinary constraints” — but never rigorously delimited. This is acceptable for a metaphor. It would be a serious problem for a claim to genuine formal rigor. The notation borrows the appearance of precision from a field where the terms really are precise, while deploying terms that, here, are not. Naming this gap is better than letting the notation quietly imply a rigor the project doesn’t actually have.
(c) The externality of A is stipulated, not derived. Nothing in reachable-set mathematics distinguishes a controller genuinely outside a causal order from a controller that is simply an undiscovered variable within a larger, more completely specified version of the same order (§3). All of the theological weight rests on the former reading; the formalism is neutral between the two and cannot adjudicate.
(d) The heterogeneity that makes the schema flexible also makes it hard to test. The framework’s ability to cover illness, death, nature, and possession without forcing a single mechanism is a real virtue (§4). It is purchased, however, by leaving “state” and “constraint” loose enough to be retrofitted to whatever a given narrative reports, which risks the schema functioning as a label applied after the fact rather than a claim that could, even in principle, come out otherwise. A more rigorous next stage of this project would need to fix C independently of each narrative’s outcome — specifying, in advance and for each category of case, what the relevant constraint actually consists in — so that “Y ∉ RC(X)” is a substantive claim rather than a redescription of the story’s ending.
To develop this further with integrity would require, at minimum: an independent (textual, historical, or systematic) account of what individuates A from an unusually powerful but still endogenous cause; explicit, case-by-case specification of C prior to knowing the narrative’s outcome; and constant vigilance against letting the elegance of the notation stand in for an argument the notation cannot itself supply.
8. Directions Worth Pursuing, Without Hype
Pedagogically, the distinction at the heart of this document — Y ∉ RC(X) does not entail Y ∉ R(X) — is a genuinely useful teaching device independent of any theological application. It gives students a precise way to distinguish the limits of a model from the limits of the world, which is a distinction worth having clearly in hand well before one ever gets to the question of miracles. It would sit comfortably in an introductory philosophy-of-science or philosophy-of-religion course as a tool for making “miracle-talk” precise enough to argue about well, whatever conclusion a given student ultimately reaches.
Comparatively, the same grammar could be applied, honestly and without pre-committing to any single tradition’s outcome, to wonder-narratives in other religious literatures — rabbinic accounts, hadith, Buddhist sutras. This comparative test would be valuable precisely because it would quickly expose where the schema’s flexibility becomes a liability (§7d) and where, if anywhere, it tracks a structural feature that recurs across traditions rather than one built to fit a single set of texts. That is the honest way to find out whether this grammar has identified something real about how “boundary-transcending authority” gets narrated, or whether it is an artifact of having been designed with one set of stories already in mind.
Closing
The claim this document has tried to make precise is not that miracles occur, nor that the theological identification of Jesus’s authority has thereby been established. It is narrower and, I think, more defensible: that the Gospel narratives, read carefully, are not making a claim about superior force within a shared contest. They are making a claim about the non-finality of an ordinary boundary — and that this specific claim, once separated from the force-competition language that usually surrounds it, can be stated with a clarity that neither inflates it into science nor deflates it into mere metaphor.
Whether that claim is true is a question this grammar was never built to answer. What it can do — and I think does do, honestly — is make sure the question being asked is the right one.
Addendum: Formal Justifications
The claims made throughout this document rest on a small number of set-theoretic facts. None of them are deep; their value lies entirely in being stated precisely enough to check, which is the whole point of using this vocabulary in the first place. What follows are not proofs in any sense that requires specialized mathematics — they are the kind of short, checkable justification that ordinary set theory supplies for free, laid out explicitly so that no step in the argument depends on an unstated assumption.
A.1 Non-entailment: Y ∉ RC(X) ⇏ Y ∉ R(X)
Claim. Membership of Y outside a constrained reachable set does not, by itself, imply membership of Y outside the unconstrained reachable set.
Justification. Let R(X) denote the total set of states reachable from X under no restriction beyond whatever governs reality as such, and let C pick out a sub-collection of transitions available under some more limited regime. By construction, the constrained reachable set is a subset of the total one:
RC(X) ⊆ R(X)
This holds simply because every transition permitted under a restricted rule set is a fortiori a transition permitted under no restriction — restricting the rules can only remove reachable states, never add ones the unrestricted case lacked. From RC(X) ⊆ R(X), elementary set logic gives: Y ∉ R(X) ⇒ Y ∉ RC(X) (if Y is outside the larger set, it is outside the smaller one too, since the smaller is contained in the larger). But the converse does not hold. Y ∉ RC(X) places Y outside a subset; it says nothing about whether Y lies inside or outside the larger superset, because a subset can fail to contain an element that the full set still contains. The only way the converse would hold is if RC(X) = R(X) — that is, if the constraint were already exhaustive of everything reachable, which is precisely the substantive claim in dispute, not something available for free.
This is the entire formal content of the central claim. Its force is not in its difficulty — there is none — but in the fact that ordinary speech routinely elides exactly this step, treating “ruled out under our best current model” as interchangeable with “ruled out.”
A.2 Monotonicity under augmented authority: RC(X) ⊆ RC,A(X)
Claim. Introducing an authority A that adds permissible transitions to a regime cannot shrink what was already reachable; it can only hold the reachable set fixed or enlarge it.
Justification. Let C denote the original set of permissible transitions and let CA ⊇ C denote the augmented set once A’s additional transitions are admitted — this containment is simply what it means for A to add transitions rather than remove them. Any sequence of transitions available under C remains available under CA, since every transition permitted by C is, by the containment just stated, also permitted by CA. Therefore any state reachable via a C-permitted path is also reachable via a CA-permitted path (namely, the same path). Hence:
RC(X) ⊆ RC,A(X)
This is the formal basis for saying that authority expands reachability rather than redirects or competes with it: nothing available under the ordinary regime is lost when A is introduced, and the interesting content of any given case lies entirely in whether the containment is strict — whether RC,A(X) contains at least one state, such as Y, that RC(X) lacked.
A.3 What is not established by A.1–A.2
It should be stated plainly that neither result above does any of the following, and no amount of restating them more elaborately would change that:
- Neither establishes that any particular A exists.
- Neither establishes that Y ∈ RC,A(X) for any particular Y of theological interest (resurrection, healing, calmed storm) — that is a substantive claim about a specific case, not a consequence of the general framework.
- Neither distinguishes an A genuinely external to the order generating C from an A that is merely an undiscovered variable within a more completely specified version of C (the limitation already named in §3 and §7c above).
What A.1 and A.2 establish is only the logical space in which the theological claim can be stated without contradiction or category error: that a constrained impossibility does not entail an absolute one, and that an authority which adds permissible transitions cannot, merely by doing so, remove any that were already available. Everything beyond that — whether such an authority exists, and whether it acted in any given narrated case — is theology, history, and testimony, not mathematics. The addendum’s purpose is to make sure the mathematics is not asked to carry more than this.
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