Existential Logicism · Pillar 3

Logical Elimination of Infinite Regress

An endless “because” is not a completed explanation.

Matthew Tripp Zejda

You ask why something happens, and someone points to an earlier event. You ask about that event, and they point to another. Each answer may tell you something useful. But extending the history is not automatically the same as explaining what makes the whole dependence structure obtain.

I call the relevant result the Logical Elimination of Infinite Regress, or LEIR. A complete grounding account cannot consist entirely of support borrowed from terms whose grounding is never supplied. It must reach nonborrowed structure. The argument identifies that requirement without pretending that naming a foundation tells us its nature.^[1]

Begin with the Explanation Read the Formal Argument

What are we asking the explanation to do?

The starting point is not uncertainty about whether anything obtains. An actual question already supplies an occurrence. ERP secures that minimum. LEIR asks a further question: what kind of dependence account can completely ground the occurrence or collection of occurrences we are trying to explain?

The word depends needs a declared meaning. A belief can depend on a reason. A physical process can depend on an operative capacity. A thing can depend on a ground for its obtaining. These relations are connected within the wider inquiry, but they are not interchangeable. A reason offered for a belief does not become a physical cause simply because both sentences contain “because.”

Choose the target first, then keep the same kind of dependence throughout the argument. The target might be a particular occurrence, a collection of present occurrences, or the obtaining of reality as such. What matters is whether the proposed account actually supplies the support it claims.

A partial explanation can be useful. It does not become a complete grounding explanation merely because its remaining dependency has been pushed farther away.

That is the problem with a pure “all the way down” answer. Every member passes its entire grounding burden to another dependent member. Nothing in the description supplies a nonborrowed basis for the structure. Infinite postponement is not greater explanatory depth. It is a refusal to complete the explanation, written in longer handwriting.

I am not criticizing an account merely because I cannot finish reading or calculating it. The question is not how long inquiry takes. It is whether the dependence that purportedly does the work is actually supplied.^[1]

What a foundation means here

Write Dep(x, y) for “x depends on y in the declared sense.” A purely dependent occurrence has at least one such outgoing dependence. A self-grounded occurrence obtains without depending on a further occurrence in that mode.

Self-grounded does not mean self-caused. It does not mean that something existed before itself and brought itself into being. The formal definition says that the relevant further dependence is absent, not that an arrow bends back into its own starting point.

Nor does the definition make any object into a foundation just because I give it the label. An adequate account must establish that its proposed structure really performs the grounding role. The endpoint condition does not, by itself, derive a physical mechanism, law, personal agent, or unique ultimate object.

Why a circle is not enough

Imagine three books arranged so that each appears to prop up the next. The circular arrangement does not explain why the books are not falling. The support has to be accounted for. Likewise, “A because B, B because C, C because A” does not obtain a nonborrowed foundation merely by returning to A.^[2]

The point is not that real feedback processes are forbidden. A cycle can be embedded in a grounded system. If a proposed whole has a separately justified capacity that supports the cycle, that capacity is doing additional explanatory work. It was not produced by drawing circular dependence arrows.

Why the endpoint cannot be nothing

A dependence chain consists of occurrences. Absolute nonbeing is the absence of every occurrence, not one special item available to finish the chain. ION excludes that alleged state or source under its realization principle. “And then nothing” cannot supply the support missing from “turtles all the way down.”

What makes the account complete?

For this argument, a complete dependence account has a nonempty set of foundations. Every target reaches some foundation through a finite dependence path, and each foundation obtains without a further dependence in the same mode. The relationship represented by the path must genuinely be the dependence being claimed.

The requirement applies to every target, but it does not require the same foundation for all of them. It also does not claim that every route in a larger network must be free of feedback. What it requires is foundation-reaching coverage: none of the targets is left with nothing but unsupplied borrowing.

A target that is already a foundation needs zero dependence steps to reach itself. There is no contradiction here. A zero-step path adds no dependence edge and does not assert that the foundation causes or supports itself in the primitive relation.

The four conditions of LEIR

The main theorem combines four conditions, each with a different role. Reality is nonempty, with ION’s necessity kept within its stated scope. An account of what obtains now must supply instantiated explanatory structure. Absolute nonbeing and purely circular borrowing do not supply foundations. Finally, the account undertakes a complete dependence explanation rather than treating indefinite postponement as the final answer.

I state that last condition explicitly as A4. I require explanatory adequacy; I do not disguise that requirement as a theorem derived from bare occurrence. Given it, an account made entirely of ungrounded dependence cannot satisfy the task it has undertaken.

The result identifies what an adequate complete account must contain. Supplying the actual account remains real work, not something accomplished by wanting an explanation.

This also explains the difference between a foundation and an arbitrary stop. Saying “it just is” may end a conversation, but it does not show that the proposed endpoint provides the grounding claimed for it. I reject the substitution of a stopping label for an explanation. The complete definition and theorem below make the requirement inspectable rather than leaving it as a slogan.

Grounding is not the same as a first moment

A first event would still have to obtain and possess whatever capacities let it have consequences. Calling it first identifies its position in a sequence; it does not supply those capacities. Likewise, extending a sequence without a first member does not itself account for what sustains the sequence.

LEIR therefore does not require a first tick of a cosmic clock or a finite number of changes. Its finite paths concern the grounding coverage of the target, not a countdown through every moment in a completed physical history. The present-realization condition asks what actually does the explanatory work, not whether we can finish drawing the past.

The historical question

Aristotle distinguished several explanatory uses of “cause,” including material, form, the source of change, and end. Aquinas later distinguished dependence on a creator from a demonstration of the world’s temporal beginning. I do not inherit his creator from that distinction. I retain the need to say precisely what an explanation is explaining.^[3]

Leibniz’s example of a book of geometry copied from an earlier book sharpens the same problem. Extend the copying without a first copy. Each predecessor may explain its successor, but the extended sequence does not by itself explain why there is that series or why it has that content. Leibniz takes the argument toward God. I retain the question exposed by the example without treating his further answer as established on his authority.^[4]

Extending a dependence relation does not change the kind of explanation it supplies. The useful question is what completes the grounding, not how many times we can repeat the dependency.

What the result asks us to investigate

The nature of nonborrowed grounding is the positive question. Its internal constitution, relations, and consequences need to be made intelligible. A name such as God, law, or necessity does not do that work on its own.

This is why LEIR prepares the next inquiry without replacing it. Persistent Present Determinism develops the account of present states and their successors using its own finite-content and source-sufficiency conditions. The requirement for grounding does not, by itself, calculate a physical law or select a particular mechanism.

The formal argument

The definitions, statements, and proofs below follow the Logic-Based Engine, §4.11. Definition 4.5 specifies complete dependence explanation; Theorem 4.4 states the result and its assumptions. Open a supporting result to read its full proof.

This formal section states the Logical Elimination of Infinite Regress (LEIR) in the same formal idiom used for ERP, Tripp’s Prison, and ION. The goal is to show that, once we accept the earlier pillars and the complete-explanation requirement stated here, pure deferred dependence cannot supply a complete grounding account. Any account satisfying that requirement must include nonborrowed grounding structure.

A. Background: Occurrences, Reality, and Prior Pillars

We continue to use the basic vocabulary from the ERP and ION formal sections.

Definition 4.1. Reality and occurrences

Let ℛ be a non-empty domain of occurrences. We write Occ(x) for “x is an occurrence in reality”. Thus:

∀ x (Occ(x) → x ∈ ℛ).

Intuitively, occurrences are whatever it is that “happens” or “obtains” in reality: events, states, facts, configurations, etc.

We presuppose the following results from earlier pillars (here treated as background theorems):

ERP / Tripp’s Prison (Minimal Realism). For any epistemic agent A who tokens any substantive state S (including radical skeptical denials), there exists at least one occurrence. Formally, for any such A and S,

Tok_A(S) ⇒ ∃ x Occ(x).

In particular, a global denial cannot coherently be true of the context containing its token.
ION (No absolute nothingness). There is no possible world w in which reality is empty. In modal notation:

□∃ x Occ(x),

i.e. “there is at least one occurrence” is necessary. “absolute nothingness” is not a genuine metaphysical option under ION's realization principle.
PP (Present Principle, informally). Explanatory and causal structure must be realized in the present; at any given present, what actually grounds what is not an unrealized limit but an actually instantiated structure. An “explanation” that would only complete via a never-arrived-at infinite tail is not a true explanation of anything now.

LEIR is the claim that, with the complete-explanation requirement, pure borrowing cannot be an exhaustive grounding account. An account satisfying that requirement must supply nonborrowed structure.

B. Dependence and Regress

We now introduce a general dependence relation. This is deliberately abstract so it can be read as causal, existential, or justificatory dependence, depending on context.

Definition 4.2. Dependence relation

Let Dep(x,y) be a binary relation on ℛ with the intended reading “x depends (in the relevant sense) on y”. We assume:

(D1)
Typedness: Dep(x,y) → (Occ(x) ∧ Occ(y)).
(D2)
Irreflexivity: ∀ x ¬Dep(x,x) (nothing depends on itself in the primitive sense).

Think of Dep as covering any of the following: “x is caused by y”, “x is grounded in y”, “x is justified by y”, etc. LEIR applies to any such dependence when its application satisfies the stated premises. I fix one mode throughout a derivation. A temporal predecessor is not automatically an existential ground, and an epistemic justification is not automatically a physical cause.

Definition 4.3. Dependence chains and regress

A dependence chain for an occurrence x₀ is a (finite or countably infinite) sequence ⟨ x_i⟩_{i∈ I} of occurrences with x₀ as its first element and

∀ i ∈ I∖{0} Dep(x_i−1, x_i).

We say that ⟨ x_i⟩_i∈ℕ is an infinite regress chain if I = ℕ and Dep(x_i−1,x_i) holds for all i>0.

An endless chain may revisit a member. When its members are pairwise distinct it is a strict descending chain; otherwise it contains a cycle. Neither case by itself supplies a non-dependent member. The notation describes dependence without asserting a completed physical history.

Definition 4.4. Self-grounded and purely dependent occurrences

Given a dependence relation Dep:

An occurrence f is self-grounded (w.r.t. Dep) if

Occ(f) ∧ ¬∃ y Dep(f,y).

That is, f does not depend on any further occurrence in that mode.
An occurrence x is purely dependent (w.r.t. Dep) if

Occ(x) ∧ ∃ y Dep(x,y).

Note that “self-grounded” here is structural: it says nothing about content besides non-dependence in the relevant sense. LEIR identifies the structural role required of a complete grounding account; it does not identify a particular mechanism merely from the endpoint property.

C. Why brute “nothing” and pure loops cannot ground reality

First, we exclude two obvious (but inadequate) candidates for “foundations” of a regress: termination in nothingness, and explanatory loops.

Lemma 4.1. No termination in nothingness

Let ⟨ x_i⟩_{i∈ I} be any dependence chain as in Definition 4.3. Then no “terminal point” of the chain can be absolute nothingness. That is, there is no i∈ I such that x_i is (or corresponds to) “nothing” in the sense ruled out by ION.

Proof.

By the chain definition, every x_i is an occurrence, including a singleton path with no dependence edge. Thus Occ(x_i) holds. But ION tells us that absolute nothingness is precisely the absence of all occurrences, i.e. a state in which ∀ x ¬Occ(x) holds. Hence there is no element of ℛ which both is an occurrence and “is” absolute nothingness. So the chain cannot bottom out in a member that is literally “nothing”. Any purported explanation that appeals to a terminal “nothing” is therefore incoherent in the Existential Logicism sense.

□

Lemma 4.2. Loops do not provide foundations

Suppose a finite cycle consists exclusively of borrowed dependence, with

Dep(x_i,x_i+1) (0≤ i<n), Dep(x_n,x₀).

If the cycle is offered by itself as its entire grounding explanation, with no member or declared total structure possessing nonborrowed grounding, its circularity does not supply a foundation.

Proof.

Every displayed member has an outgoing dependence and so fails the non-dependence condition of Definition 4.4. Traversing those same edges again changes neither that condition nor the explanatory status of the collection. A proposed whole with a separately justified nonborrowed capacity would add the missing ground; it would not derive that capacity from the circular edges alone. Thus this result excludes pure circular borrowing as a complete foundation, not every feedback process or every cycle embedded in a grounded system.

□

Thus neither “it ends in nothing” nor “it goes in a circle” can serve as a genuine foundation for dependence.

D. Infinite regress and incomplete explanation

The remaining regress option is a genuinely infinite descent with no minimal self-grounded member and no loop. LEIR’s central claim is that, given the earlier pillars, this cannot yield a complete explanation of reality, and thus cannot be the whole story.

We make explicit what “complete explanation” means in the Existential Logicism sense.

Definition 4.5. Complete dependence explanation

For a nonempty target X⊆ℛ and one declared dependence mode, a complete dependence explanation has a foundation set F satisfying:

(E1)
F⊆ℛ and F≠∅.
(E2)
For every x∈ X there are f∈ F, a finite n≥0, and a path x₀,…,x_n∈ℛ such that

x₀=x, x_n=f, Dep(x_i,x_i+1) (0≤ i<n).
(E3)
Each f∈ F obtains and has no outgoing dependence in that mode.

If x is itself a foundation, choose n=0 and x₀=x=f. This zero-step path asserts no edge Dep(f,f).

The condition is foundation-reaching coverage of every target. It does not require one common foundation for all targets, and it is weaker than requiring every path in the entire dependence graph to be well-founded. Other paths may contain feedback or cycles. The displayed relation must genuinely express the dependence claimed; drawing an arbitrary path is not an explanation of its physical or epistemic mechanism.

Infinite regress, on its own, is the situation where (E2) is attempted with F=∅, and every member is purely dependent with no f as in (E3).

Lemma 4.3. Infinite regress never reaches a foundation

Suppose that for all x∈ℛ there exists y with Dep(x,y), and that there are no self-grounded occurrences (i.e. no f with Occ(f)∧¬∃ y Dep(f,y)). Then no nonempty set X⊆ℛ admits a complete dependence explanation in the sense of Definition 4.5.

Proof.

Assume for contradiction that some non-empty X has a complete explanation via ⟨ F,Dep⟩. By (E1), F is non-empty, and by (E3) each f∈ F is self-grounded. But this contradicts the global assumption that there are no self-grounded occurrences. So no such F can exist. Hence there is no complete dependence explanation for any non-empty X under the hypothesis that every occurrence is purely dependent and no self-grounded element exists.

□

Thus, if we demand complete explanation rather than a forever-unfinished chain, infinite regress is structurally inadequate: it never reaches anything that could count as a foundation in the Existential Logicism sense.

E. LEIR: Reality cannot be “all the way down”

We can now state LEIR explicitly.

Theorem 4.4. LEIR – Logical Elimination of Infinite Regress

Fix a nonempty target X and a dependence mode. Assume:

(A1)
Reality is nonempty, with ION's realization-based necessity understood in its stated scope.
(A2)
A complete account of what obtains now must supply actually instantiated explanatory structure, not an unrealized tail that would have to finish first.
(A3)
Absolute nonbeing and purely circular borrowing do not supply foundations (Lemmas 4.1 and 4.2).
(A4)
The account undertakes a complete dependence explanation of X in the sense of Definition 4.5. I require this explanatory adequacy rather than accepting indefinitely deferred borrowing as the final answer.

Then every account satisfying these conditions must contain a nonempty foundation set F such that each x∈ X has a finite path to some f∈ F. No account containing only pure ungrounded dependence satisfies that demand.

If such a complete account is realized by actuality, its foundations are real. A demand that there be an explanation is not, without that realization claim, a proof that a particular explanation has already been supplied.

Proof.

An account satisfying (A4) satisfies (E1)–(E3). By (E1) it contains at least one foundation, and by (E2) every target reaches some foundation in finitely many links, possibly zero. A universe of exclusively dependent members with no foundation would violate (E3), as Lemma 4.3 establishes. The excluded termini in (A3) cannot substitute for the missing condition.

The universal quantifier ranges over targets and is followed by the existential choice of a foundation. It does not assert a unique f shared by all targets. Thus complete adequacy forces grounded coverage, while the actual internal constitution of each ground remains something the account must establish. I adopt (A4) as an explanatory requirement; the conditional proof does not manufacture actuality from the desire for an explanation.

□

F. Remarks and Corollaries

Corollary 4.6. Self-grounded reality is not a brute stop

The grounding role required by LEIR is not an arbitrarily selected last member of a chain. Absolute nonbeing, pure circular borrowing, and indefinitely deferred dependence do not meet the stipulated complete-explanation requirement. Some nonborrowed grounding must be supplied by an account that does meet it.

This conclusion identifies a necessary explanatory role. The absence of an outgoing dependence edge does not, by itself, derive the foundation's internal mechanism, its laws, or a unique physical identity. Calling an unexplained candidate “self-grounded” is not a substitute for supplying that account.

Corollary 4.7. No regress into a void

Given ION and LEIR, no explanatory chain can terminate in a metaphysical void. Every successful explanation must either:

terminate in self-grounded structure within reality, or
be recognized as incomplete (and thus not a genuine “all the way down” theory).

The popular picture “turtles all the way down, and then nothing” is not a coherent description of reality on Existential Logicism grounds.

Remark 4.8. Mode-neutrality and application

The same formal test may be applied in different dependence modes only after its premises are checked in that mode. ERP supplies a performative occurrence anchor for epistemic denial. A present causal explanation must identify the actual capacities through which a process continues. An existential account must not borrow actuality from a supposedly real exterior omitted from its own totality.

These are related tasks, not interchangeable relations. LEIR states what a complete dependence account must contain; it does not derive a physical law from an epistemic premise or a first event from a grounding relation.

Remark 4.9. Finite moments, unbounded history

LEIR does not claim that there must be a first temporal moment or a finite number of changes. It only claims that:

at any given present, the dependence structure that actually explains what is happening has foundation-reaching coverage in the sense of Definition 4.5, and
the existence of reality as such is self-grounded rather than deferred to anything beyond reality.

In this way, Existential Logicism allows unbounded variation and change while denying that reality’s existence or present obtaining could be structured as a pure infinite regress.

Questions and replies

Why can an infinite chain not be the explanation?

The target is a complete grounding explanation made only of purely dependent members. Under the stated requirement, each target must reach nonborrowed grounding. Repeating or extending dependencies without providing that grounding does not satisfy the condition. Individual links may still provide useful partial explanations. Lemma 4.3 states the exact failure, and the main theorem makes A4 explicit.

Mathematics uses infinite structures. Does that refute LEIR?

A mathematically specified infinite sequence is not the same claim as a complete explanation of actuality. Its definition, rule, or proof belongs to the mathematical account; it does not automatically supply a physical carrier or a ground for reality. LEIR does not refute the sequence by counting its members. It asks whether the proposed grounding account meets its declared explanatory task.

Could the whole loop supply what its members lack?

A whole with a separately justified nonborrowed capacity would supply additional grounding, rather than deriving that capacity from circular borrowing alone. Lemma 4.2 excludes the pure circular account. It does not prohibit feedback or deny that organized wholes can have relevant capacities. The account still has to identify and justify the capacity it invokes.

Does “self-grounded” mean something creates itself?

No. Definition 4.4 says the occurrence has no outgoing dependence in the relevant mode. It does not assert Dep(f, f). When a target already is a foundation, the zero-step path in Definition 4.5 adds no edge at all. Reachability with zero steps and primitive self-dependence are different relations.

Does a finite grounding path require a beginning of time?

No. A grounding path is not automatically a history of earlier events. Remark 4.9 explicitly separates the requirement for foundation-reaching coverage from a first temporal moment or a finite number of changes. Naming an earliest state would still leave its actuality and capacities to be explained.

Must every occurrence reach one unique foundation?

The displayed result says every target reaches some foundation. It does not say there is one unique foundation shared by every target. Those quantifiers are different. Nor does it require every alternative route through the complete network to be well-founded. Definition 4.5 states exactly which coverage condition is required.

Why not call the endpoint a brute fact and stop?

The absence of an outgoing edge identifies a structural endpoint. It does not, by itself, explain why that candidate obtains or how it does the work attributed to it. Calling the endpoint “self-grounded” does not remove this obligation. Corollary 4.6 distinguishes the necessary grounding role from an arbitrarily selected stopping point.

Is the complete-explanation requirement an assumption?

Yes. I state it as A4 and explain the task it sets. The theorem derives what an account satisfying that requirement must contain. It does not claim that the demand alone discovers a particular realized foundation. Rejecting A4 changes the explanatory undertaking; it does not show that pure borrowing meets the requirement while somehow avoiding its consequences.

Does God end the regress?

That names a proposed terminus. The account must still explain in what sense it supplies nonborrowed grounding. A definition or title does not establish the required capacity. LEIR does not derive a preference-bearing person, revelation, or scriptural authority from the grounding requirement. Those additional claims have their own arguments, examined in Finite Mind, Finite God.

What if a complete explanation is impossible?

Then the claim of complete explanation has been abandoned, not fulfilled by an infinite regress. Local explanations can still be assessed as local explanations. I do not treat a refusal of the complete task as proof that nothing is ever explained; I distinguish it from an account claiming to have supplied the complete grounding.

Does LEIR by itself prove all of the later physics?

No. Remark 4.8 requires the premises to be checked in each dependence mode. A physical law, a uniquely fixed successor, or a particular carrier needs the conditions and evidence appropriate to that claim. LEIR supplies a constraint on complete explanation; the later work must show how its proposed structure meets it.

Terms used here

Existential Logicism (EL): The connected framework in which occurrence, finite access, grounding, the present, consciousness, and moral agency are examined through their respective arguments.
Logical Elimination of Infinite Regress (LEIR): The result that a complete dependence account satisfying its stated conditions cannot contain only ungrounded borrowing. Every target must reach nonborrowed grounding in the declared mode.
Occurrence and reality: An occurrence is something that happens or obtains in the relevant actuality. The formal domain ℛ is nonempty; Occ(x) identifies an occurrence in that domain.
Dependence mode: The particular relation under examination, such as causal, existential, or justificatory dependence. One mode is fixed throughout a derivation.
Dep(x, y): The occurrence x depends on occurrence y in that declared sense. Typedness keeps both within the occurrence domain; irreflexivity excludes a primitive self-edge.
Purely dependent occurrence: An occurrence with at least one outgoing dependence in the selected mode.
Self-grounded occurrence: An occurrence with no outgoing dependence in the selected mode. This is a structural condition, not self-causation or a complete description of its internal mechanism.
Target set X: The nonempty collection of occurrences whose dependence explanation the account undertakes.
Foundation set F: The nonempty set of occurring, nondependent endpoints required by the complete account. Different targets may reach different members.
Foundation-reaching coverage: Every target has a finite dependence path to some foundation. A target already in the foundation set can have a path of length zero.
Complete dependence explanation: The structure satisfying Definition 4.5: a nonempty foundation set, finite coverage of each target, and nondependence of each foundation in the fixed mode.
Present realization (PP): The requirement that the operative structure claimed to explain what obtains now be actually instantiated, not left to an unrealized tail that must finish first.
ERP and ION: ERP supplies the occurrence anchor. ION excludes absolute nonbeing as an admissible actuality under its realization principle. They play different roles within LEIR and do not erase A4.
Nonborrowed grounding and an arbitrary stop: Nonborrowed grounding must perform the explanatory role assigned to it. Merely choosing where to stop asking questions is not an account of that role.

Sources and further reading

Matthew Tripp Zejda, Existential Logicism: A Logic-Based Engine for Navigating Reality. Chapter 4; §4.11, “Formal Argument: Logical Elimination of Infinite Regress.” The numbered definitions, results, and proofs on this page follow that formal section. Read about the Engine.
Matthew Tripp Zejda, Existential Logicism: A Logic-Based Tool for Navigating Reality. Chapter 4, especially the distinctions among present realization, circular borrowing, arbitrary stops, and complete grounding. Read about the Tool.
Historical passages discussed in Chapter 4. Aristotle, Physics, Book II, chapter 3, translated by Robin Waterfield, edited by David Bostock. Thomas Aquinas, Summa Theologiae, I, question 46, article 2, translated by the Fathers of the English Dominican Province. Read Aquinas’s question 46.
Gottfried Wilhelm Leibniz, “On the Ultimate Origination of Things.” In Philosophical Essays, translated and edited by Roger Ariew and Daniel Garber, Hackett, 1989. The copying example is discussed in the Logic-Based Engine, §4.8.1.
Matthew Tripp Zejda, Existential Ground. Chapters 7 and 9 develop the related distinctions among reconstructed history, operative grounding, and temporal firstness. Those applications do not replace the dependence definition or premises stated here.

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