{"slug": "a-proposition-decomposition-and-contradiction-driven-reasoning-system-looking", "title": "A Proposition Decomposition and Contradiction-Driven Reasoning System — Looking for Discussion and Collaborators", "summary": "An independent researcher is proposing a reasoning system that decomposes propositions into sub-components and searches for contradictions through implication relations, rather than asking a large language model to directly judge whether two propositions conflict. The design separates probabilistic candidate generation from a formal logical verifier, with the stated hypothesis that actively pursuing collisions with existing propositions discovers contradictions more efficiently under limited generation and computational budgets. The proposal remains a research design rather than a completed system, and the author is seeking discussion, criticism, alternative approaches, and collaborators to implement or test it.", "body_md": "**Abstract**\n\nI am exploring a reasoning system that uses proposition decomposition and contradiction search rather than asking an LLM to directly judge whether two propositions are contradictory.\n\nThe basic idea is to decompose propositions into smaller components, identify corresponding propositions through implication relations, and use these relations to search for contradictions. The system is intended to separate probabilistic exploration from a more reliable formal verification process.\n\nI am particularly interested in whether actively searching for contradictions can provide a useful direction for reasoning and exploration under a limited generation budget.\n\nThis is still a research design rather than a completed system. I am looking for discussion, criticism, alternative approaches, and collaborators who may be interested in implementing or testing the idea.\n\n`P → Q`, meaning that if `P` holds, `Q` follows within the adopted formal system.\nThe central system to be built is a **contradiction checker**.\n\nThe basic I/O is deliberately simple.\n\n```\nInput: Proposition P, Proposition Q\nOutput: Whether P and Q contradict each other\n```\n\nHowever, the two propositions are not compared directly as strings.\n\n**The propositions are decomposed, and implication relations between propositions are used to identify parts that can be regarded as expressing the same subject or content; the resulting identification is then used to search for contradictions.**\n\nTherefore, **proposition identification is not merely a function for checking whether a decomposition is valid; it is itself a foundation of contradiction checking.**\n\nThe central hypothesis is that, compared with generating a large number of candidates without direction, a search that actively pursues collisions with existing propositions may discover contradictions more efficiently under limited generation and computational budgets. This is a hypothesis to be supported or refuted experimentally.\n\n```\n       Proposition P        Proposition Q\n              ↓                  ↓\n       Proposition decomposition / normalization\n              ↓                  ↓\n       Acquisition of implication / compositional relations\n              ↓                  ↓\n       Proposition identification / correspondence\n                         ↓\n                  Collision search\n                         ↓\n                Formal logical verification\n                         ↓\n        CONTRADICTION / CONSISTENT / UNKNOWN\n```\n\nIf a learning system is connected later, a search loop is added here.\n\n```\nCollision candidate generation\n            ↓\n        Verifier\n            ↓\n         Reward\n            ↓\n  Next candidate generation\n```\n\n**The checker itself and the search process that looks for candidates are kept separate.**\n\nHere, identification does not mean exact string matching.\n\nA proposition is decomposed into constituent sub-propositions, and the implication relations between them and existing propositions are examined.\n\nConceptually, for example,\n\n```\nP → A → B → C\nQ → A → B → C\n```\n\nmeans that P and Q share at least the common logical structure A, B, and C.\n\nLikewise,\n\n```\nP → A\nQ → B\nB → A\n```\n\nmeans that, although P and Q are superficially different, they become comparable with respect to A.\n\nTherefore, identification and correspondence use:\n\n**Truth value and the content/structure of a proposition are treated separately.**\n\nFor example, `p` and `¬p` should be treated as corresponding positive and negative forms of the same underlying proposition, rather than as completely unrelated propositions.\n\nProposition identification has at least two uses.\n\nWhen an original proposition P is decomposed into\n\n```\nP1, P2, ..., Pn\n```\n\nthe system checks whether the content reconstructed from the decomposed elements has the same meaning, or the same specified implication relations, as P.\n\nFor a simple AND decomposition, for example,\n\n```\nP ↔ (P1 ∧ P2 ∧ ... ∧ Pn)\n```\n\ncan be verified.\n\nWhen comparing P and Q, the propositions derived from both are first put into correspondence.\n\nThen, if a relation such as\n\n```\nP → A\nQ → ¬A\n```\n\nis found for the same subject/content, this becomes a candidate for testing whether P and Q can hold simultaneously.\n\nLonger paths are also allowed.\n\n```\nP → A → B → C\nQ → D → ¬C\nD → C\n```\n\nIn this case, P and Q respectively imply C and ¬C, so they form a collision candidate.\n\n**Therefore, proposition identification/correspondence is not merely a preprocessing step for contradiction checking; it constructs the comparison space in which contradictions can be discovered.**\n\nThe initial version primarily targets propositional logic.\n\nInput:\n\n```\nP\nQ\n```\n\nOutput:\n\n```\nCONTRADICTION\nCONSISTENT\nUNKNOWN\n```\n\nDefinitions:\n\n“Failure to find a contradiction” is not treated as “consistent.”\n\nWithin domains such as propositional logic where finite search is possible, the system should be deterministic whenever possible.\n\nP and Q are not compared in a single step.\n\nBoth are decomposed, and propositions that can be derived from them are expanded step by step.\n\nConceptually, paths such as\n\n```\nP → A → B → C\nQ → D → E → ¬C\n```\n\nare constructed, and the system checks whether positive and negative forms eventually appear for content that can be identified as the same.\n\nThe ideal form is to explore combinations of propositions derived from P and Q as broadly as possible and discover collisions such as\n\n```\nA and ¬A\nB and ¬B\nC and ¬C\n...\n```\n\nHowever, enumerating all consequences leads to combinatorial explosion, so the search process selects promising directions.\n\nCandidate generation is not treated as undirected random search.\n\nFor example, given an existing proposition P, the search objective is:\n\nGenerate a proposition that may collide with P.\n\nHowever, the system must consider the possibility that a search may degenerate into simply generating `¬P` repeatedly. The appropriate reward structure is itself an experimental variable.\n\nInitial reward candidates:\n\n```\nCollision found              → positive reward\nNo collision found           → low / 0 reward\nCould not be formalized      → low / negative reward\n```\n\nAs needed, additional terms such as the number of inference steps to the collision, decomposition validity, and redundancy may be added.\n\n**A “non-trivial collision” is not required as a formal definition.**\n\nThe central comparison is:\n\n```\nUndirected search\nvs.\nCollision-seeking search\n```\n\nwith the collision discovery efficiency compared under the same generation and computational budgets.\n\nDecomposition is not simply “the more fine-grained, the better.”\n\nAt minimum, the following properties are required.\n\nLogical content should be preserved between the original and decomposed forms.\n\n```\nOriginal proposition\n       ↓ decomposition\nP1, P2, ..., Pn\n       ↓ reconstruction\nEquivalent to the original proposition / satisfies the specified implication relations\n```\n\nThe system checks whether each component is actually necessary.\n\nIf removing an element Pi still allows the original proposition to be fully preserved from the remaining elements, Pi may be treated as a redundant component.\n\nConceptually, one can verify conditions such as:\n\n```\nP1 ∧ ... ∧ Pn entails P\nP1 ∧ ... ∧ ¬Pi ∧ ... ∧ Pn does not entail P\n```\n\nA component is not rewarded merely for making the representation finer.\n\nThe current specification does not claim that this problem is necessarily NP-hard. The intended claim is only that searching for minimal sufficient decompositions or minimal evidence sets may cause combinatorial explosion.\n\nPropositions are not treated as mere strings, but as nodes connected by relations.\n\nExample:\n\n```\nP → A\nA → B\nB → C\nQ → D\nD → C\n```\n\nThis shows that P and Q both imply C.\n\nWith negation included,\n\n```\nP → A → B → C\nQ → D → ¬C\n```\n\ncan produce a collision candidate.\n\nThe proposition graph is not the final truth checker.\n\nIts main roles are to:\n\nFinal logical checking is performed by a separate formal verifier.\n\nNot every proposition needs to be compared with every other proposition.\n\nFirst, implication relations, structure, and correspondence of subjects are used to narrow the set to propositions that may be describing the same thing.\n\nAfter candidate correspondences are produced, an LLM may be used for semantic comparison of atomic propositions that cannot be formally matched.\n\nDo not insist on complete mechanical determination here. The LLM is used as an auxiliary mechanism for constructing comparison targets, **not as the final contradiction checker**.\n\nUnknown propositions are not forcibly assigned a truth value.\n\n```\nDecidable within the formal system\n    → formal verification\n\nNot decidable within the formal system\n    → UNKNOWN\n```\n\nSearch and experiments for acquiring external information are future extensions and are not the center of the minimal implementation.\n\nAfter the checker is completed, an LLM may be connected as a search process.\n\n```\nLLM\n ↓\nGenerate proposition decompositions / derived propositions / collision candidates\n ↓\nFormal verifier\n ↓\nReward\n ↓\nNext candidate generation\n```\n\nMultiple candidates are generated and evaluated through the verifier.\n\nAt this stage, reinforcement learning such as GRPO may be used.\n\nIn production, LoRA / QLoRA may be applied to an existing LLM, and Unsloth or similar tools may be used as the training infrastructure. The training infrastructure itself is not the research target.\n\nThe first experiment is not intended to determine whether an “intelligent AI” can be built.\n\nIt tests the following single point:\n\n**Under the same generation budget, can collision-seeking search discover more contradictions than undirected search?**\n\nThe comparison should be controlled for quantities such as:\n\n```\nNumber of generated candidates\nToken count\nComputation time\n```\n\nCandidate evaluation measures include:\n\n```\nNumber of discovered contradictions / number of generated candidates\nNumber of discovered contradictions / token count\nNumber of discovered contradictions / computation time\n```\n\nA toy dataset may be used initially, but **the production code and toy-specific code should not be designed as separate systems**. The same design should allow the model size to be changed while preserving the system being evaluated.\n\nIn principle, the LLM is **not** used to make the final judgment “Do these two propositions contradict?”\n\nThis is the most important point.\n\nDo not treat a formally proven equivalence and an LLM judgment that two statements are similar as the same kind of result.\n\nMutual information represents statistical dependence over probability distributions.\n\nImplication represents a relation within a logical system.\n\nThe present method is not proposed as a direct alternative definition of mutual information. It is positioned as a method for making comparable information explicit as logical structure.\n\nDo not claim that every natural-language proposition can be determined with certainty.\n\nFirst specify the formal system being targeted, and then seek as much rigor and completeness as possible within that scope.\n\n**Conserve data and reduce hypotheses with small experiments before spending large amounts of computation.**\n\n**Use intuition to generate hypotheses, but use explicit definitions, predictions, and falsification conditions during verification.**\n\n**The following supplementary note describes the broader motivation that led me to focus on trial-and-error search and the direction of reasoning.**\n\nThe history of academic discoveries has, in most cases, been driven by trial and error. If there were a universal algorithm capable of solving every unsolved problem, one would expect it to have been discovered during the last 2,000 years. It has not. For details, please search for the P vs NP problem.\n\nSince there is no universal algorithm for problem solving, the only way to approach unsolved problems may be to “search through them the hard way.” Therefore, is it really correct to try to solve unsolved problems with an algorithm called an LLM? Even if trial and error can be performed inside a model, I suspect it would remain very limited.\n\nThere is research suggesting that OpenAI’s o1, particularly the o3 series, can exhibit trial-and-error behavior, and I think this is directionally correct. However, many problems remain:\n\nPersonally, I think the last two are especially important. I have tried to tackle them, but they are quite difficult.\n\nWhenever I think about the problems of AI, extrapolation seems to appear as an issue everywhere. But perhaps humans also extrapolate by performing an enormous amount of search.", "url": "https://wpnews.pro/news/a-proposition-decomposition-and-contradiction-driven-reasoning-system-looking", "canonical_source": "https://discuss.huggingface.co/t/a-proposition-decomposition-and-contradiction-driven-reasoning-system-looking-for-discussion-and-collaborators/180653#post_1", "published_at": "2026-09-21 01:41:07+00:00", "updated_at": "2026-09-21 01:52:35.917824+00:00", "lang": "en", "topics": ["artificial-intelligence", "large-language-models", "ai-research", "ai-agents"], "entities": [], "alternates": {"html": "https://wpnews.pro/news/a-proposition-decomposition-and-contradiction-driven-reasoning-system-looking", "markdown": "https://wpnews.pro/news/a-proposition-decomposition-and-contradiction-driven-reasoning-system-looking.md", "text": "https://wpnews.pro/news/a-proposition-decomposition-and-contradiction-driven-reasoning-system-looking.txt", "jsonld": "https://wpnews.pro/news/a-proposition-decomposition-and-contradiction-driven-reasoning-system-looking.jsonld"}}