Project Overview
Conjectural Reasoning Theory begins from a simple fracture in the usual story of knowledge. Hume shows that finite observations cannot justify universal claims. Popper replaces justification with conjecture and refutation: knowledge grows by proposing ideas and eliminating those that fail. But that leaves a deeper persistence problem. If a conjecture is destroyed every time its representation, context, or encoding changes, then knowledge does not accumulate. It merely flashes, survives locally, and disappears.
CRT asks what kind of system could let knowledge persist across those changes.
Abstract
Conjectural Reasoning Theory is a research program for grounding machine intelligence in a Popperian account of objective knowledge. Rather than treating knowledge as statistical regularity extracted from observations, CRT treats knowledge as persistent executable structure: a network of conjectural relations that survives by satisfying internal constraints and by remaining reusable across changes in representation.
Contemporary transformer-based systems provide the motivating contrast. They introduce candidate structure through training, but the survival of that structure is governed by loss over encoded data. Their internal organization is therefore selected through World-2 representation rather than through autonomous World-3 constraint. This helps explain why such systems remain vulnerable to hallucination, distributional dependence, and catastrophic forgetting.
CRT proposes an alternative architecture. The untyped lambda calculus supplies a representation-independent substrate for executable structural conjectures. Claims are validated by reduction, extended by stochastic mutation, stabilized by coherence, and pruned through decay. A separate World-2 interface then conjectures encodings and decodings between physical states and autonomous World-3 structures. The result is a roadmap toward systems capable of cumulative objective knowledge growth rather than mere inductive approximation.
Project Contents
2 Popperian Requirements for Knowledge-Producing Systems
2.1 World-3 Requirements: Structural Conjecture 2.2 World-2 Requirements: Encoding Conjecture 2.3 World-2 First: Transformer-Based Language Models 2.4 In Search of World 33 The Bounds of Executability
3.1 The Structure of Transformations 3.2 The Turing Bridge 3.3 Structural Autonomy 3.4 CoherenceProblem of Induction
Hume’s problem of induction shows that no finite set of observations can justify a universal claim or guarantee a future regularity. Any attempted defense of induction either assumes induction in advance or becomes circular. This breaks the idea that knowledge is built upward from observation by justification.
Popper’s response is to abandon justification. Knowledge begins with conjectures, which are then exposed to criticism and empirical testing. Observations do not confirm theories. They eliminate theories that conflict with the world. This gives knowledge an evolutionary structure: variation supplies conjectures, criticism supplies selection.
CRT begins where that story becomes incomplete. Selection can remove errors, but selection alone does not explain cumulative growth. For complex knowledge to accumulate, successful conjectural structure must persist long enough to be reused, extended, recombined, and criticized in new contexts. The real problem is therefore not merely refutation. It is persistence under change.
Objective Knowledge and Survival
Popper’s World 3 names the objective content of knowledge: arguments, theories, problems, proofs, and relations that are not reducible to any individual mind that happens to grasp them. A person may misunderstand a proof, forget a theory, or encode an idea badly, while the structure of the proof or theory remains what it is.
CRT turns that philosophical distinction into an architectural one. Structural conjectures belong to World 3. They concern executable relations among claims. Encoding conjectures belong to World 2. They concern how physical states, symbols, observations, or actions are mapped onto those structural relations.
This separation is essential. If structural survival depends directly on the current encoding, then every representational failure threatens the knowledge itself. A genuine World 3 must instead remain autonomous: physical interaction may challenge how we access a structure, but it should not immediately overwrite the structure being accessed.
World-3 Requirements
A World-3 system is not merely a database of claims. It must be a domain where conjectural structures can be introduced, tested by internal constraint, composed into larger structures, and retained when they continue to function.
- Conjectural introduction. Structural conjectures must be introducible without justification. The system must allow new executable possibilities to appear before they have been proven useful.
- Logical determinacy. There must exist internal constraints capable of eliminating structural conjectures. In CRT, those constraints arise from execution: a claimed reduction either reaches the required term under the relevant conditions or it does not.
- Cumulative structure. Surviving structural conjectures must be persistent and composable. They must be capable of becoming parts of larger claims instead of being dissolved into a single opaque parameter state.
World-2 Requirements
World 2 is the interface layer. It does not invent objective structure directly. It conjectures how physical situations should be encoded into World-3 terms and how World-3 mediators should be decoded into physical actions.
- Conjectural introduction. Encoding conjectures must be introduced without justification. A system must be able to guess how a physical situation maps onto structural terms before it knows whether that mapping is correct.
- Physical determinacy. Physical constraints must be capable of refuting encoding conjectures. If two internally convergent routes produce physically divergent outcomes, the interface has failed.
- Relational structure. World-3 structural constraints must determine which combinations of encodings are admissible. A single encoding may look plausible in isolation but fail when required to preserve a larger structural configuration.
- Non-authoritarian access. World 2 may exercise World-3 structures, but it must not directly decide which structures survive. This protects objective structure from being erased by local representational failure.
Transformer-Based Language Models
Transformer-based language models provide a useful negative case. They do introduce candidate structure through initialization and training, and their encodings are tested against physical training data. But the survival of internal structure is governed by loss over encoded examples. The data-facing interface therefore becomes authoritarian: it directly determines what internal organization remains.
This violates the CRT requirement for structural autonomy. The model’s internal relations do not persist as independent, composable conjectures governed by stable logical constraints. They persist as a parameter configuration inseparable from the training distribution that produced it. When the distribution changes, previous structure can be overwritten. Catastrophic forgetting is not an accident here. It is a symptom of architecture.
On this view, hallucination is also not merely a failure of factual lookup. It is the behavior of a quasi-inductive system extending statistical regularities beyond the domain in which those regularities were supported. The model approximates patterns in the observable distribution, but it does not recover the autonomous World-3 structure that would explain those patterns.
In Search of World 3
If current learning systems fail because their structural survival is governed by representation, then the next step is to find a domain whose rules are internal. We need a substrate where structures can be introduced conjecturally, composed with one another, executed, refuted, and retained without depending on any particular physical encoding.
This is why the project turns to Church and the λ-calculus. Lambda calculus supplies a minimal formal environment in which transformations can be expressed without specifying the machine that realizes them. It does not begin with hardware, memory registers, or training data. It begins with terms and reductions.
In CRT, this makes the λ-calculus a candidate World-3 substrate. Its terms can function as structural conjectures. Its reductions can validate or refute executable relations. Its compositional closure allows surviving structures to become parts of larger structures. The remainder of the framework asks what additional dynamics are needed for such structures to persist, decay, and accumulate.
The Structure of Transformations
Church’s λ-calculus provides a minimal symbolic system for executable transformation. Variables, application, abstraction, and β-reduction are enough to express computation without appealing to a particular machine. This is the formal skeleton CRT uses to model structural conjectures.
Variables: x
Application: (M N)
Abstraction: λx.M
Reduction: (λx.M)N → M[x := N]A λ-term is not treated merely as a symbolic expression. It is treated as a candidate executable structure. A claim such as A -(B)-> C asserts that applying B to A reduces to C. The order in which reductions are attempted may vary, but confluence ensures that if a terminating normal form exists, it is unique.
This gives CRT a precise way to talk about structural success and failure. A claim is not true because it is asserted. It survives only if execution reaches the required structure under the relevant constraints.
The Turing Bridge
Turing clarified mechanical procedure by describing an abstract machine with explicit states, symbols, and transition rules. This gave computation an operational foundation and showed that Church’s λ-definable functions coincide with Turing-computable functions.
For computer science, Turing’s model became the dominant image of computation: a machine evolving step by step through encoded states. For CRT, however, that strength is also a limitation. A machine-centered account emphasizes realization. A World-3 account must emphasize structure that remains invariant across realizations.
The bridge matters because it legitimizes Church’s abstraction without surrendering to machine-specific description. Turing shows that λ-calculus captures effective procedure. CRT then returns to Church because representation-independence is exactly what an autonomous World 3 requires.
Structural Autonomy
Structural autonomy is the principle that World-3 structures must be governed by relations internal to the structural domain itself. They may be accessed through physical encodings and exercised by physical systems, but their survival must not be directly authored by those encodings.
The λ-calculus supports this because composition and reduction are intrinsic to the domain. Given terms M and N, their application is itself a term. If reduction terminates, confluence fixes the normal form independently of reduction path. Compatibility and incompatibility are therefore determined by executional structure rather than by external representation.
Knowledge is not represented by successful hypotheses alone. It is represented by persistent executable structures whose survival is determined internally, while physical representations compete for access to those structures.
The rest of CRT develops the dynamics required for such a domain to grow: firing initiates bounded traversal, mutation introduces candidate mediators, coherence stabilizes recursively reusable claims, and decay removes unsupported structure. World 2 then connects this autonomous World 3 to the physical world through conjectural encodings and decodings.
Coherence
A single successful transformation is not enough to build a persistent computational structure. A chain can only continue if each transformation prepares the conditions required for the next one. Constructor theory makes this idea explicit by separating a substrate from the constructor acting upon it.
A -(b)-> C
T = {A → C}Here the substrate begins in state A, while b is the constructor implementing the task. Executing the task produces a new substrate state C.
Now suppose this is only the first step of a larger computation.
A -(b)-> C -(d)-> EFor the chain to continue, producing the next substrate state is not sufficient. The resulting state must also prepare the constructor required for the following transformation. In other words, constructor and substrate exchange roles.
A -(b)-> C
b -(C)-> d
C -(d)-> EThe first task tells us how the constructor transforms the substrate. The shifted task tells us how the resulting substrate transforms the constructor. Together they recursively prepare one another, allowing the computation to continue without introducing an external controller.
Biology provides a familiar example. A gene-expression system transforms the state of a cell by producing proteins. Those proteins then alter the regulatory state of the cell, changing which genes can be expressed next. The constructor changes the substrate, and the substrate changes the constructor. The process recursively prepares its own continuation.
A simple catalyst behaves differently. Catalase converts hydrogen peroxide into water and oxygen, but the products do not construct the next catalyst required for a larger explanatory chain. The reaction completes, yet the network does not recursively prepare itself.
CRT generalizes this idea to executable structures. A claim is coherent only when execution remains valid after repeatedly shifting the constructor-substrate boundary. Coherence therefore becomes the mechanism that allows local transformations to compose into persistent World-3 structures rather than isolated computations.
Research Notes
Execution-Validated Claims Network
Develop the EVC as a forgetful network of lambda terms where firing, bounded traversal, excitation, mutation, and decay generate selection pressure over executable structure.
World-2 Interface
Formalize encoding and decoding conjectures as hypotheses that map physical states and actions onto autonomous World-3 structures without controlling their survival.
Constructor-Theoretic Framework
Reformulate CRT in constructor-theoretic terms, making coherence, substrate-constructor role exchange, and World-3 persistence explicit as constraints on possible transformations.
Full Draft
The current manuscript develops the philosophical motivation, formal substrate, execution-validated network dynamics, and World-2 interface in detail.
PDF Constructing World 3: Execution-Validated Claims Network