Non-Axiomatic Logic

# Non-Axiomatic Logic: A Model of Intelligent Reasoning **Author:** Pei Wang (Temple University, USA) **Year:** 2013 **Type:** Book (monograph / graduate textbook) **Publisher:** World Scientific Publishing — ISBN 978-981-4440-27-1 (print) / 978-981-4440-28-8 (e-book); LoC 2013007458; 270 pp. **Source:** [World Scientific — worldscibooks/10.1142/8665](https://www.worldscientific.com/worldscibooks/10.1142/8665). Full text ingested as RawData (14 chunks). ## Abstract > This book provides a comprehensive, precise, and up-to-date description of Non-Axiomatic Logic (NAL), the logical core of the AI system NARS (Non-Axiomatic Reasoning System). NAL is designed for the creation of general-purpose AI by formalizing the regularity of human thinking at a general level, and so also belongs to Cognitive Science. Its distinctive feature is the *relative rationality* a system exhibits when it must work with **insufficient knowledge and resources** (the AIKR assumption: the system is Finite, works in Real-time, and is Open). NAL differs from ordinary logics in all of its major components: it uses *subject–predicate* (term) sentences rather than predicate–argument, *experience-grounded* rather than model-theoretic semantics, and *syllogistic* rather than truth-functional inference rules. Each statement carries a two-dimensional truth-value (frequency and confidence) interpreted as the amount of evidence for it. The book specifies NAL as a formal language (Narsese), a set of formal inference rules, and a semantic theory. ## Structure The system is built up in nine layers of increasing expressive power, each adding term constructs and inference rules on top of the previous one: - **IL-1 / NAL-1** — idealized inheritance, evidence and its measurement, two-dimensional truth-value, basic local + forward + backward syllogistic inference rules. - **NARS memory & control** — inference tasks, bag-based probabilistic storage, *concept as a unit*, the inference cycle (the resource-bounded "attention" mechanism that makes NAL run under AIKR). - **NAL-2** derivative copulas (similarity / instance / property); **NAL-3** set-theoretic compound terms (intersections, differences); **NAL-4** relational terms (products, images); **NAL-5** statements as terms / higher-order statements (implication, equivalence, negation); **NAL-6** variable terms and unification. - **NAL-7** events as statements (temporal inference); **NAL-8** operations and goals as events (procedural reasoning, sensorimotor interface); **NAL-9** self-monitoring and self-control (mental operations, feeling/emotion, consciousness). - **Appendices** — Narsese grammar, the full NAL inference-rule table, NAL truth-value functions, and proofs of theorems. ## Relevance to Hyperon This is **the canonical theory book that defines NAL/NARS itself** — the foundational source for the Non-Axiomatic Reasoning tradition tracked throughout the Hyperon ecosystem. It is the upstream reference behind: - the NAL ruleset re-implemented in MeTTa — see MeTTa-NARS Deep Dive; - the practical NARS reference reasoner OpenNARS for Applications (Hammer 2021), whose ONA design operationalizes this theory; - the NAL layers verified inside the Claw agents (mettaclaw / metta-nars / PeTTa-OpenPSI) during the Cross-Org Cognitive Repo Sweeps; - the PLN↔NARS comparison work — NAL is the *uncertain term-logic* counterpart to PLN's probabilistic logic, both being evidence-/confidence-weighted inference frameworks for reasoning under uncertainty. It complements Wang's earlier *Rigid Flexibility: The Logic of Intelligence* (2006) and his 1995 PhD thesis, but *Non-Axiomatic Logic* (2013) is the definitive, self-contained specification of the logic. *Ingested 2026-06-24 (full book PDF, found manually by Lake) to close the NARS foundational-coverage gap identified in WS9 of the Wiki Finalization Meta-Plan.*