Consistent Reasoning Paradox (CRP)

• The Consistent Reasoning Paradox is a phenomenon where classical reasoning in systems with constrained memory and inference inevitably leads to contradictions.
• Research shows that CRP arises when agents’ memory updates, epistemic logic, and measurement operations conflict, especially in frameworks like quantum theory and generalized probabilistic theories.
• Investigations into CRP reveal challenges in statistical learning and logical inference, prompting structural solutions such as paraconsistent logics and dynamic type systems.

The Consistent Reasoning Paradox (CRP) encompasses a family of phenomena arising when a framework of agents, logical systems, or physical theories yields an unavoidable contradiction or impossibility via classical patterns of reasoning applied to situations where memory, inference, or expressive power is physically or formally constrained. Manifestations occur across quantum theory, generalized probabilistic theories (GPTs), classical logic, deontic systems, neural reasoning architectures, and constructive mathematics, with both theoretical and practical implications for consistency, universality, and trustworthiness of reasoning processes.

1. Formal Definition and General Frameworks

The CRP is characterized by a contradiction between reasoning principles—deduction, prediction, and memory modelling—and the underlying structure of observers, agents, or systems. In its most general operational formulation (Vilasini et al., 2019), the CRP arises whenever a physical theory:

• Models agents with physical memories updated by measurements.
• Equips agents with epistemic logic and trust relations (if agent A knows B knows φ, then A accepts φ).
• Allows agents to predict outcomes via its own probability rule (e.g., Born rule, GPT rule).
• Treats measurements as explicit physical processes that couple a system S with a memory M, requiring information-preserving memory updates; formally, a map $u: \{P_S^j\} \to \{P_{SM}^{l}\}$ must exist such that the statistical structure is retained after update.

In logical systems, paradoxes equivalent to the CRP arise in frameworks that simultaneously enforce relevance, admit broad classes of theorems, and apply strong admissibility of structural rules (Vidal-Rosset, 2019). In neural and AI settings, the paradox manifests when data-driven learners that optimize for statistical features fail to internalize logic-based reasoning, creating inconsistent generalization (Zhang et al., 2022, Bastounis et al., 5 Aug 2024).

2. CRP in Quantum Theory and Generalized Probabilistic Theories

The original Frauchiger–Renner paradox, generalized in (Vilasini et al., 2019) and formalized in (Epstein, 2 Mar 2025), exemplifies CRP in quantum mechanics and GPTs. Agents’ memories are treated as physical systems; memory update operations must preserve joint statistics so that external observers can retrodict the correlations.

In box world (a GPT with PR-box correlations), agents performing allowed operations are forced, via logical chains combining their measurements and trusted deductions, into a deterministic contradiction. E.g., deducing $a=0 \implies a=1$, enabled by strong contextuality of the PR box, illustrates conflict between classical reasoning and nonclassical correlation structure. Notably, in box world the contradiction is fully deterministic (arises for all runs) rather than conditional (post-selected) as in quantum Frauchiger–Renner experiments.

Memory evolution is modeled via classical wirings: measurement outcomes are “copied” into fresh memory boxes using irreversible but information-preserving maps; these meet Barrett’s criteria for allowed GPT operations (validity, normalization, state structure). This formalism highlights how CRP generalizes beyond quantum theory to all multi-agent setups satisfying the four core reasoning and memory principles.

3. Logical Paradoxes and Consistency

CRP is prominently realized in logical systems designed for relevance but admitting unrestricted theorem importation. The Core Logic Paradox (Vidal-Rosset, 2019) demonstrates that when the strongly admissible $L\top$ rule is applied to Tennant’s “Core logic” (intuitionistic relevant logic with the sequent $\lnot A,\,A\nvdash B$ valid), the system can simultaneously prove and disprove the same sequent with identical premises:

• Adding the theorem $\lnot A \to (A \to B)$ via $L\top$ leads to both
  • $\lnot A \to (A \to B),\, \lnot A,\, A \nvdash B$
  • $\lnot A \to (A \to B),\, \lnot A,\, A \vdash B$

This yields outright inconsistency—an internally explosive logical system—when consistency is supposed to be foundational. Here, the paradox illustrates the necessity of balancing structural rule admissibility with relevance and consistency; failure leads to duplication of consequence and paradox.

4. CRP in Neural and Symbolic Reasoning

In data-driven learning frameworks, neural models such as BERT trained on logical reasoning datasets may achieve near-perfect accuracy on familiar distributions but generalize poorly to others (Zhang et al., 2022). CRP is manifested when statistical features inherent in data sets (e.g., rule count, branching factor) dominate learning instead of logical entailment. Removing statistical features jointly is computationally infeasible; algorithmic attempts to “balance out” these features require exponentially more data. Thus, the learning process leads to models that succeed “for the wrong reasons,” generating inconsistency across sampling regimes.

AI-centric formulations of CRP (Bastounis et al., 5 Aug 2024) connect fallibility of consistent reasoning directly to human-like intelligence. Any AI forced to answer all possible equivalently phrased questions (reason consistently across all equivalence classes of descriptions) is proved to necessarily hallucinate—i.e., provide plausible but incorrect answers—infinite often. Detecting these hallucinations is strictly harder than solving the original problem; trustworthy AI must be able to say “I don’t know,” realized via the “I don’t know” function, computable only in the limit (Σ₁ class), not by explicit functional computation (Δ₁). For AGI, the implications are that certainty and explainability are fundamentally limited—an AGI cannot always explain correct answers nor be “almost sure.”

5. Paraconsistent and Subsignature Approaches to Resolving CRP

Frameworks for managing CRP via maximal consistent subsignatures (Thimm et al., 30 Aug 2024) use the forgetting operator ($\boxminus$) to project a knowledge base onto maximal consistent parts, so removing inconsistent propositions restores consistency. Minimal inconsistent and maximal consistent subsignatures are dual; the hitting set duality applies: minimal hitting sets of minimal inconsistent subsignatures are complements to maximal consistent subsignatures. Inference relations (inevitable, weak) are defined over these projections and satisfy rationality postulates from non-monotonic reasoning.

Quantitative inconsistency measures (e.g., counting minimal inconsistent subsignatures) and connections to paraconsistent three-valued logics (Priest’s logic) illustrate that reasoning over maximal consistent subsignatures recovers classical inference on consistent fragments, paralleling reasoning in minimal three-valued models. Complexity results (DP-complete, Σᵖ₂-complete, Πᵖ₂-complete) underscore that managing CRP is computationally nontrivial but theoretically robust.

6. Extensions: Justification Logic, Grounded Deduction, and Constructive Paradoxes

CRP informs the design of justification logics (Faroldi et al., 2023)—operating with explicit justifications for obligations (t : A)—so that impossible and conflicting obligations are technically distinguished rather than collapsed into a single inconsistent object. The explicit version of axiom D (jd: $\neg(t : L)$) and the no-conflicts axiom (noc: $\neg(t : A \wedge t : \neg A)$) are shown to be interderivable under appropriate constant specifications, yielding completeness for arbitrary specifications and providing the logical “firewall” to prevent CRP-type collapses in obligation reasoning.

Grounded deduction (GD) (Ford, 12 Sep 2024) uses dynamic typing and “grounding” of terms to prohibit deduction of paradoxes from ungrounded recursive definitions (e.g., “L := not L”). GD’s operation is neither classical nor intuitionistic but enforces well-typedness before a formula receives a truth value, preventing paradoxes from infecting the reasoning system.

In constructive mathematics, the use of Countable or Dependent Choice (Nezhad, 10 Dec 2024) can generate paradoxes analogous to CRP: analytic functions constructed via CAC may simultaneously have infinitely many zeros and non-zeros in a compact set, violating classical factorization theorems and highlighting that choice principles do not guarantee consistency, even when stripped down to “weaker” forms.

7. Implications and Future Directions

The CRP reveals fundamental boundaries of consistent reasoning in both theory and practice:

• Physical theories universally modeling agents’ memory lead to deterministic contradictions (box world, PR box) under classical rules, challenging universality and completeness claims (quantum mechanics).
• Logical systems must calibrate structural rule admissibility; relevance alone cannot guarantee consistency.
• Learning-based AI must reconcile statistical feature exploitation with robust, logically invariant generalization. Building trustworthy systems requires explicit mechanisms for uncertainty or “giving up.”
• Paraconsistent, justification, and dynamic type-based logics offer structural solutions for managing CRP, often at a cost of computational complexity or expressive restriction.
• The paradox persists in constructive mathematics, challenging foundational assumptions about choice and excluded middle.

In sum, the Consistent Reasoning Paradox delineates the domain of reasoning strategies constrained by the interplay of logic, memory, probability, and physical theory. It imposes non-negotiable limits on universality, consistency, and certainty, with implications for the future of AI, logic, physics, and mathematics.