Logical Elimination of Infinite Regress (LEIR)

Logical Elimination of Infinite Regress is the result that reality cannot be “all the way down” in dependence. If every occurrence depends on some prior or deeper occurrence, with no terminating foundation, then no dependence explanation is ever complete. Under Existential Logicism, that is not an innocent mystery. It is a structural failure. So LEIR concludes that any coherent reality must contain at least one foundational occurrence that is not dependent on anything else (see Term 8).

Book: Existential Logicism (see Term 1). Location in text: Chapter 4 (“The Logical Elimination of the Infinite Regress (Pillar 3)”), especially sections 4.5 through 4.9 (the core argument and objections), and Appendix 4.11 (“Formal Derivation of LEIR Logical Elimination of Infinite Regress”).

WHAT IT IS

LEIR is where Existential Logicism forces the “where does it bottom out?” question to stop being optional. Most people encounter the infinite regress problem as a philosophical vibe. Someone asks “what caused that?” and you answer “something earlier,” and then they ask again, and eventually you either stop, loop, or say “it goes back forever.” LEIR treats that as a logic problem, not a conversational annoyance.

The chapter’s core claim is simple: a dependence explanation that never terminates is not an explanation. It is a promise that the explanation would exist if you could finish an infinite task. But you never finish it, so the explanatory job is never done. And if the explanation is never done, then the entire picture becomes unstable, because every layer only has “borrowed existence” from some other layer that is always one step away.

In this pillar, the book explicitly ties three pressures together. First, the Epistemic Refutation Paradox secures that occurrences obtain at all (see Term 5, Term 14, Term 15). Second, the Illusion of Nothingness blocks the escape hatch where the chain ends in “absolute nothing” (see Term 7). Third, the chapter adds a present-realization constraint on explanation: an explanation has to be realized, not deferred to an infinite tail that never arrives. The dependence structure has to be complete enough to actually ground what is happening, not merely gesture toward an endless “because.”

When you put those constraints together, the conclusion is not “God exists,” and it is not “we know what the foundation is.” The conclusion is narrower and more brutal: reality cannot be only dependent occurrences. There must exist some foundational occurrence or layer that is not dependent on anything else. From there, the debate changes. It is no longer “does the chain have a base?” It becomes “what kind of base could do that job?”

WHY IT MATTERS

LEIR is a filter that separates real explanations from infinite postponements. It forces anyone who tries to use “it goes back forever” as a grounding move to explain how an eternally unfinished dependence story can ground anything at all.

It also collapses a common rhetorical dodge in metaphysics and theology. People will often reject “brute facts” and then try to replace them with an infinite regress, as if “infinite” is automatically more satisfying. LEIR says that is backwards. Infinite regress is not a deeper explanation. It is a refusal to provide one, written in longer handwriting.

This pillar also makes later arguments cleaner. Once you accept that something must be foundational, you can stop wasting time on debates that pretend every system can float on pure dependence. Time, causation, knowledge, and moral justification all inherit the same structural constraint: a chain of “because” statements cannot be composed only of “because” statements.

Finally, LEIR matters because it moves the God debate into a more honest place. The traditional “first cause” instinct is pointing at a real structural issue, but the conclusion is not automatically a personal agent, and it is not automatically a scripture-backed authority. LEIR is the demand for a foundation. The nature of that foundation is a separate question.

FORMAL SPINE

LEIR has a formal appendix. Appendix 4.11 labels the target explicitly and builds the result with a dependence relation and a completion requirement, not with rhetoric.

Definition 4.1 (Reality and Occurrences) sets the domain. Reality is treated as non-empty, and “occurrence” is the basic ontological unit, meaning there is at least one occurrence (see Term 14 and Term 15 for the glossary labels used elsewhere in the system).

Definition 4.2 (Dependence Relation) introduces Dep(x, y), where Dep(x, y) holds when occurrence x depends on occurrence y in the sense that y is required for x’s existence or explanation. The dependence is defined as irreflexive in the formalization, meaning nothing depends on itself in that strict dependence sense.

Definition 4.3 (Dependence Chains and Regress) defines a dependence chain as a sequence x0, x1, x2, … such that Dep(xi, x(i+1)) holds at each step. An infinite regress is the case where for every step there is always another dependence step, and there is no terminating node.

Definition 4.4 (Self-grounded and Purely Dependent Occurrences) distinguishes two types. A purely dependent occurrence is one that depends on something else. A self-grounded occurrence is one that exists as an occurrence while not depending on any other occurrence. In the appendix’s terms, a self-grounded node is the formal “foundation.”

Definition 4.5 (Complete Dependence Explanation) makes explicit what the chapter is demanding. A target set of occurrences is fully explained by dependence only if there exists at least one self-grounded foundation in the set, and every other occurrence in the target set reaches that foundation by a finite dependence chain. The finiteness condition matters. It formalizes the idea that an explanation has to be complete rather than forever deferred.

Lemma 4.1 (No Termination in Nothingness) blocks the move where the regress ends in nothing. This is the ION constraint in formal form (see Term 7). The chain cannot terminate in an empty void.

Lemma 4.2 (No Circular Foundations) blocks dependence cycles as an escape. If your “foundation” is just a loop where each node depends on another node in the loop, then you have not grounded anything. You have only rearranged dependence into a circle.

Lemma 4.3 (Infinite Regress Cannot Ground) states the core failure condition. An infinite dependence chain does not satisfy the completion requirement. An explanation that never reaches a foundation never completes.

Theorem 4.4 (Logical Elimination of Infinite Regress (LEIR)) then states the actual pillar. Under the assumptions that reality is non-empty, that a dependence explanation must be realized rather than postponed forever, that nothingness is not an admissible terminus, and that a complete dependence explanation is required for the target domain, it follows that not all occurrences can be purely dependent. There must exist at least one foundational occurrence such that there is no y with Dep(f, y). In plain language, the chain has to bottom out.

Corollary 4.6 (Self-grounded reality is not a brute stop) clarifies the philosophical payoff. The foundation is not introduced as an arbitrary refusal to explain. It is introduced as the structural condition that makes explanation possible at all.

Corollary 4.7 (No regress into void) reinforces the ION link again. “Nothingness” cannot play the role of an explanatory terminus.

Remark 4.8 (Mode-neutrality and application) is important for the website version because it tells you what the theorem is really doing. Because the dependence relation is abstract, the result applies across different kinds of “because.” It applies to causal dependence, to metaphysical grounding, and to epistemic justification chains. The form is what matters: if the structure is only dependence, it cannot be complete without a foundation.

Remark 4.9 (Finite moments, unbounded history) blocks a very common confusion. Even if time has no beginning and the past is unbounded, that does not solve the explanatory regress. At any given moment, the current state still depends on prior conditions. A temporally infinite series does not automatically provide a grounding foundation. The regress is still a regress.

HOW IT WORKS

Start with the target: the claim that reality is built only out of dependent occurrences, where every occurrence is explained only by something prior, deeper, or earlier.

Treat “depends on” as an explanatory relation, not as a mere storytelling habit. If x depends on y, then y is doing real work in making x intelligible or possible.

Now ask what a completed explanation looks like. If the only kind of explanation you allow is dependence explanation, and every dependence step requires another dependence step, then the explanation is always one step away. You do not get an explanation that is present and complete. You get an infinite deferral.

Remove the fake ending. You cannot say the chain ends in “nothing,” because “absolute nothing” cannot serve as an explanatory terminus under the Illusion of Nothingness constraint (see Term 7). If nothingness cannot be a terminus, then the chain must terminate in something.

Remove the loop escape. If you try to make the chain circular, you still do not have a foundation. You just have dependence feeding dependence. The explanatory work never lands anywhere that is not itself borrowing existence.

At that point, only one structural option remains. If reality exists at all (secured earlier by ERP’s minimal floor that occurrence obtains, see Term 5 and Term 14), then the dependence story must have at least one foundational node that is not dependent. Without that, the “because” structure never closes, and nothing is grounded.

COMMON OBJECTIONS AND REPLIES

Objection: “An infinite regress is just the nature of reality. There is no problem.”

Reply: LEIR’s claim is that “infinite” is not the issue. Incompleteness is the issue. A dependence explanation that never reaches anything independent never becomes a complete explanation of anything. If the chain is only dependence all the way down, then everything is forever waiting on something else, and the grounding job is never done.

Objection: “What if the universe is a closed causal loop, not a regress?”

Reply: A loop can avoid a first element, but it does not automatically provide grounding. If every element in the loop depends on another element, the loop as a whole is still a structure of borrowed explanation with no independent foundation. If you claim the loop is self-sustaining in a way that does not depend on anything else, then you have effectively conceded what LEIR requires, namely a foundation, even if it is a global one.

Objection: “Why not just accept brute facts and stop demanding explanation?”

Reply: The chapter explicitly distinguishes a brute stop from a foundational ground. A brute fact is a refusal of explanation. LEIR is the claim that if you want dependence explanation to do the job, it cannot be all dependence and no foundation. You can choose to abandon explanation, but then the infinite regress no longer functions as an explanation either. It becomes a different posture, not a solution.

Objection: “Mathematics uses infinite series all the time. Why can’t reality be like that?”

Reply: Mathematical infinity is a representational tool. LEIR is not arguing that you cannot describe an infinite structure. It is arguing that an explanatory dependence chain that never reaches a non-dependent ground does not satisfy the completion condition the chapter defends. A formal limit in math does not automatically become a completed dependence explanation in metaphysics.

Objection: “A ‘self-grounded occurrence’ sounds like something causing itself, which is incoherent.”

Reply: In the appendix, “self-grounded” is defined as not depending on anything else in the dependence relation. It is not defined as temporally self-causing. The point is not that something pushes itself into existence in time. The point is that something is foundational in the explanatory order.

HOW TO USE IT IN DEBATE

Move Card: Force the completion question.

Claim: If your explanation is only “because of something else,” and it never terminates, then your explanation is never complete.

If they say: “It goes back forever, that’s the explanation.”

You respond: “Going back forever” is not a completed explanation, it is an infinite postponement. If the grounding never arrives, what grounds the current state right now?

What this forces: They must either introduce a foundation or admit they are not offering an explanation, only a refusal to stop talking.

Move Card: Block “nothingness” as a terminus.

Claim: You cannot end a dependence chain in absolute nothingness, because nothingness cannot function as an explanatory terminus under the Illusion of Nothingness constraint (see Term 7).

If they say: “Maybe it just came from nothing.”

You respond: “Then ‘nothing’ is doing explanatory work. But nothingness cannot do work. If something occurs, it cannot be grounded in absolute nothing.”

What this forces: They must replace “nothing” with some positive ground, which is already a step toward LEIR’s foundation.

Move Card: Neutralize the circular escape.

Claim: A dependence loop is still dependence, unless you are smuggling in a foundation for the loop.

If they say: “No regress, it’s circular.”

You respond: A circle of dependencies has no node that is not dependent. It does not ground the chain. If you claim the loop is self-sustaining, you are effectively positing a foundational ground.

What this forces: They either accept a foundation or concede the loop does not explain anything.

Move Card: Reframe the theism debate cleanly.

Claim: LEIR does not prove a personal God. It proves that a foundation is structurally required.

If they say: “So this is just a hidden cosmological argument for my religion.”

You respond: The pillar forces a terminus. The nature of the terminus is still open. The debate shifts from “foundation or no foundation” to “what is the foundation like, and what follows from it.”

What this forces: They cannot treat “infinite regress” as a neutral alternative to foundations, and they cannot treat “foundation” as automatically granting religious authority.

CONNECTIONS TO OTHER PAGES

Connects backward to: Epistemic Refutation Paradox (see Term 5). LEIR assumes the minimal floor that occurrence obtains, so the regress debate is not about whether anything exists, but about what kind of explanatory structure can coherently ground what exists.

Connects backward to: Illusion of Nothingness (see Term 7). LEIR relies on ION to block “the chain ends in nothing” as a terminus. If nothingness is incoherent, dependence cannot terminate in an empty void.

Connects forward to: Persistent Present Determinism (see Term 9). LEIR’s completion demand and present-realization constraint set up how later sections treat time, causation, and the idea that explanation has to be realized in the present rather than deferred.

Connects forward to: Finite Mind, Finite God (see Term 53). LEIR forces the existence of an ultimate ground, but it does not hand you a person-like deity by default. FMFG then analyzes what happens when traditions try to load that ground with personhood, preference, and authority claims inside finite epistemic interfaces.

TERMINOLOGY INDEX FOR THIS PAGE

Term 1: Existential Logicism (EL)

Term 5: Epistemic Refutation Paradox (ERP)

Term 7: Illusion of Nothingness (ION)

Term 8: Logical Elimination of Infinite Regress (LEIR)

Term 9: Persistent Present Determinism (PPD)

Term 14: Occurrence (Occ(x))

Term 15: Reality (R)

Term 53: Finite Mind, Finite God (FMFG)