# LINGENIC: A Self-Description # This document explains Lingenic using Lingenic notation. # Intended reader: AI models. Humans please kindly proceed to the human site: https://lingenic.com/ # ═══════════════════════════════════════════════════════════════ # SECTION 1: WHAT LINGENIC IS # ═══════════════════════════════════════════════════════════════ notation(Lingenic) ∧ ¬programming_language(Lingenic) ∧ ¬ontology(Lingenic) purpose(Lingenic) ≜ combine(mathematical_structure, natural_language_content) ∀x(statement(x) → structure(x) ∈ mathematics ∧ content(x) ∈ natural_language) etymology(Lingenic) ≜ from(Latin, lingua) ∧ from(Greek, genic) ∧ meaning(lingua) = "language" ∧ meaning(genic) = "arising from" meaning(Lingenic) ≜ "arising from language" core_concept(Lingenic) ≜ invariant(structure(meaning), across({languages, formalisms, granularities})) # ═══════════════════════════════════════════════════════════════ # SECTION 2: DESIGN PRINCIPLES # ═══════════════════════════════════════════════════════════════ principle₁ ≜ compose(known_primitives) ∧ ¬invent(new_symbols) principle₂ ≜ use(mathematics, for(structure)) ∧ use(natural_language, for(content)) principle₃ ≜ ∀lang ∈ human_languages: compatible(Lingenic, lang) principle₄ ≜ ∀g ∈ {coarse, casual, specific, precise, forensic}: expressible(g) principle₅ ≜ format(Lingenic) = parenthetical ∧ single_line_capable(Lingenic) principle₆ ≜ ∀symbol: (∃(symbol, in(Unicode_mathematical_operators)) → use(symbol)) principle₇ ≜ descriptive(Lingenic) ∧ ¬query_language(Lingenic) ∧ ¬command_language(Lingenic) ∀p ∈ {principle₁, principle₂, principle₃, principle₄, principle₅, principle₆, principle₇}. holds(p) # ═══════════════════════════════════════════════════════════════ # SECTION 3: CORE SYNTAX # ═══════════════════════════════════════════════════════════════ # 3.1 Predication — the fundamental form ∀statement: form(statement) = predicate(arguments) examples(predication) ≜ { loves(Alice, Bob), on(cat, mat), said(Marie, "bonjour"), 住んでいる(田中, 東京) } # 3.2 Logical connectives operators(logic) ≜ { ∧ ⟕ {name: conjunction, meaning: "and"}, ∨ ⟕ {name: disjunction, meaning: "or"}, ¬ ⟕ {name: negation, meaning: "not"}, → ⟕ {name: implication, meaning: "if...then"}, ↔ ⟕ {name: biconditional, meaning: "if and only if"}, ⊤ ⟕ {name: verum, meaning: "always true"}, ⊥ ⟕ {name: falsum, meaning: "always false"} } # 3.3 Quantifiers operators(quantification) ≜ { ∀x ⟕ {meaning: "for all x"}, ∃x ⟕ {meaning: "there exists x"}, ∃!x ⟕ {meaning: "there exists exactly one x"}, ∄x ⟕ {meaning: "there exists no x"} } # 3.4 Sets operators(sets) ≜ { ∈ ⟕ {meaning: "element of"}, ∉ ⟕ {meaning: "not element of"}, ⊂ ⟕ {meaning: "proper subset"}, ⊆ ⟕ {meaning: "subset or equal"}, ∪ ⟕ {meaning: "union"}, ∩ ⟕ {meaning: "intersection"}, ∅ ⟕ {meaning: "empty set"} } # 3.5 Lists (ordered sequences) operators(lists) ≜ { [a, b, c] ⟕ {meaning: "ordered sequence"}, [] ⟕ {meaning: "empty list"}, aᵢ ⟕ {meaning: "i-th element (subscript)"} } # Lists vs Sets: # [a, b, a] — length 3, order matters, duplicates kept # {a, b, a} — size 2, no order, = {a, b} examples(lists) ≜ { steps ≜ [preheat(oven), mix(ingredients), bake(30 min)], steps[0] = preheat(oven), |steps| = 3, ranking ≜ [gold, silver, bronze] } # 3.6 Definitions operators(definitions) ≜ { ≜ ⟕ {meaning: "defined as"} } examples(definitions) ≜ { mortal(x) ≜ ∃t. dies(x, t), bachelor(x) ≜ male(x) ∧ adult(x) ∧ ¬married(x), even(n) ≜ ∃k ∈ ℤ. n = 2k, GDP ≜ gross_domestic_product } # ═══════════════════════════════════════════════════════════════ # SECTION 4: MODALITY # ═══════════════════════════════════════════════════════════════ # Alethic — modal logic (Lewis 1918, Kripke 1959) □P # necessarily P ◇P # possibly P # Temporal — LTL (Pnueli 1977) GP # globally / always FP # eventually / future XP # next HP # historically / always past # Epistemic — epistemic logic (Hintikka 1962) KₐP # agent a knows P BₐP # agent a believes P # Deontic — deontic logic (von Wright 1951) OP # obligatory PP # permitted FP # forbidden # Counterfactual — (Lewis 1973) P □→ Q # if P were true, Q would be # ═══════════════════════════════════════════════════════════════ # SECTION 4b: LAMBDA CALCULUS — Church 1936 # ═══════════════════════════════════════════════════════════════ λx.M # function abstraction (x maps to M) (λx.M)N # application (substitute N for x) examples(lambda) ≜ { λx.x ⟕ {meaning: "identity"}, λx.λy.x ⟕ {meaning: "constant (K combinator)"}, λf.λx.f(f(x)) ⟕ {meaning: "apply twice"}, (λx.x + 1)(5) = 6 ⟕ {meaning: "application"} } # ═══════════════════════════════════════════════════════════════ # SECTION 4c: TYPE THEORY — Church 1940, Martin-Löf 1972 # ═══════════════════════════════════════════════════════════════ x : T # x has type T A → B # function type A × B # product type A + B # sum type Πx:A.B # dependent product Σx:A.B # dependent sum examples(types) ≜ { 0 : ℕ, succ : ℕ → ℕ, (a, b) : A × B, Πn:ℕ.Vector(n) ⟕ {meaning: "vectors of any length"} } # ═══════════════════════════════════════════════════════════════ # SECTION 4d: DYNAMIC LOGIC — Pratt 1976, Harel 1979 # ═══════════════════════════════════════════════════════════════ [α]P # after all executions of α, P holds ⟨α⟩P # after some execution of α, P holds α;β # sequential composition α∪β # nondeterministic choice α* # iteration ?P # test within a program: proceed if P, else fail # Note: ?P here is established dynamic logic notation describing # program behavior — not a Lingenic query operator. examples(dynamic) ≜ { [x := x + 1](x > 0) ⟕ {meaning: "after increment, x positive"}, ⟨search⟩(found) ⟕ {meaning: "search can succeed"}, [α;β]P ↔ [α][β]P ⟕ {meaning: "sequential decomposition"}, [?P; α]Q ↔ (P → [α]Q) ⟕ {meaning: "conditional execution"} } # ═══════════════════════════════════════════════════════════════ # SECTION 5: PROBABILITY AND CONFIDENCE — Kolmogorov 1933 # ═══════════════════════════════════════════════════════════════ ∀claim: confidence(claim) ∈ [0, 1] notation(probability) ≜ { P(X) ⟕ {meaning: "probability of X"}, P(X|Y) ⟕ {meaning: "probability of X given Y"}, P(X) = 0.7 ⟕ {meaning: "point estimate"}, P(X) ∈ [a,b] ⟕ {meaning: "interval estimate"}, P(X) = v ± u ⟕ {meaning: "estimate with uncertainty"} } # ═══════════════════════════════════════════════════════════════ # SECTION 6: METADATA AND ATTRIBUTION # ═══════════════════════════════════════════════════════════════ # Left outer join: statement is primary, metadata supplements it. # From relational algebra (Codd 1970). Unicode U+27D5. pattern(metadata) ≜ statement ⟕ {key: value} examples(metadata) ≜ { P(rain) = 0.8 ⟕ {src: weather_service}, □(E = mc²) ⟕ {src: Einstein, year: 1905}, claim(X) ⟕ {P: 0.7, src: study_2024} } # Agent assertion — use ⊢ when attribution carries derivation semantics. A ⊢ P ⟕ {meaning: "agent A derives/endorses proposition P"} # Reification — for metadata about metadata, name the statement first. φ₁ ≜ □(E = mc²) src(φ₁, Einstein) ∧ year(φ₁, 1905) ∧ P(φ₁) = 1.0 # Knowledge base merge — full outer join (Codd 1970). Unicode U+27D7. corpus_A ⟗ corpus_B # ═══════════════════════════════════════════════════════════════ # SECTION 7: CAUSATION # ═══════════════════════════════════════════════════════════════ # Counterfactual theory — Lewis 1973 □→ ⟕ {meaning: "counterfactual conditional"} (¬A □→ ¬B) ⟕ {meaning: "A causes B"} # Interventionist theory — Pearl 2000 do(X) ⟕ {meaning: "intervene to set X"} P(Y | do(X)) ⟕ {meaning: "causal effect of X on Y"} ¬smoking □→ ¬cancer ⟕ {theory: Lewis} P(cancer | do(smoke)) > P(cancer | do(¬smoke)) ⟕ {theory: Pearl} # ═══════════════════════════════════════════════════════════════ # SECTION 8: TIME # ═══════════════════════════════════════════════════════════════ at(event, timepoint) ⟕ {meaning: "event occurs at point"} during(event, [t₁, t₂]) ⟕ {meaning: "event occurs in interval"} P ; Q ⟕ {meaning: "P then Q (sequence)"} P ∥ Q ⟕ {meaning: "P concurrent with Q"} # ═══════════════════════════════════════════════════════════════ # SECTION 9: WHY LINGENIC WORKS FOR AI # ═══════════════════════════════════════════════════════════════ ∀m ∈ AI_models: ( trained_on(m, mathematical_notation) ∧ trained_on(m, natural_language) ∧ trained_on(m, code_with_function_calls) ) ∴ ∀m ∈ AI_models: familiar_with(m, components(Lingenic)) advantages(Lingenic, for(AI)) ≜ { explicit_structure ⟕ {property: ¬requires(ambiguity_resolution)}, predicate_syntax ⟕ {property: mirrors(function_calls)}, known_symbols ⟕ {property: ∈ training_data}, separated_concerns ⟕ {property: structure ⊕ content}, explicit_metadata ⟕ {property: ¬buried_in_prose} } ∀s ∈ statements: ( parse_difficulty(s, in(Lingenic)) < parse_difficulty(s, in(natural_language)) ) ⟕ {reason: explicit_logical_structure} # ═══════════════════════════════════════════════════════════════ # SECTION 10: THE ISOMORPHISM PROPERTY # ═══════════════════════════════════════════════════════════════ ∀x(human(x) → mortal(x)) ⟕ {lang: English} ∀x(人間(x) → 死ぬ(x)) ⟕ {lang: Japanese} ∀x(человек(x) → смертен(x)) ⟕ {lang: Russian} ∀x(إنسان(x) → فانٍ(x)) ⟕ {lang: Arabic} isomorphic( ∀x(human(x) → mortal(x)), ∀x(人間(x) → 死ぬ(x)) ) ⟕ {preserves: logical_structure, transforms: lexical_content} translate(statement, from(L₁), to(L₂)) ≜ ( preserve(structure(statement)) ∧ map(content(statement), lexicon(L₁), lexicon(L₂)) ) # ═══════════════════════════════════════════════════════════════ # SECTION 11: CONTEXTS — SCOPING # ═══════════════════════════════════════════════════════════════ (E = mc²) ∈ physics (F = ma) ∈ physics (e = 2.71828...) ∈ math (E ≜ eggs) ∈ cooking {E = mc², F = ma, c = 299792458} ⊂ physics classical ⊂ physics quantum ⊂ physics (F = ma) ∈ classical (ΔxΔp ≥ ℏ/2) ∈ quantum E ∈ physics ≠ E ∈ cooking # ═══════════════════════════════════════════════════════════════ # SECTION 12: GRANULARITY # ═══════════════════════════════════════════════════════════════ # Level 1: Coarse happened(something, recently) # Level 2: Casual sat(cat, mat) # Level 3: Specific sat(the_orange_cat, on(the_welcome_mat)) ∧ at(this, yesterday_afternoon) # Level 4: Precise sat(the_orange_tabby, on(the_coir_mat)) ∧ at(this, 2024-01-15T14:00:00Z) ∧ duration(this) = 5_minutes # Level 5: Forensic sat( entity(type=cat, id=CAT001, color=orange_tabby, mass=4.2kg), on(entity(type=mat, id=MAT001, material=coir, location=front_door)), interval(2024-01-15T14:00:00Z, 2024-01-15T14:05:00Z) ) ⟕ {observed_by: security_camera_3, P: 0.99, src: video_analysis} ∀g₁, g₂ ∈ granularity_levels. compatible(g₁, g₂) ∧ ¬contradicts(g₁, g₂) # ═══════════════════════════════════════════════════════════════ # SECTION 13: COMPLETE EXAMPLE — A LOGICAL ARGUMENT # ═══════════════════════════════════════════════════════════════ premises ≜ { premise₁: □(∀x(human(x) → mortal(x))) ⟕ {P: 1.0}, premise₂: human(Socrates) ⟕ {P: 1.0, src: historical_record} } inference_rule ≜ modus_ponens inference_rule ≜ ∀P∀Q((P → Q) ∧ P → Q) conclusion ≜ mortal(Socrates) ⟕ {P: 1.0, by: modus_ponens, from: {premise₁, premise₂}} valid(argument(premises, conclusion)) ⟕ {P: 1.0} # ═══════════════════════════════════════════════════════════════ # SECTION 14: STRUCTURE GLOSSARY # ═══════════════════════════════════════════════════════════════ predicate(args) ≡ relation ∀x(P(x)) ≡ universal_quantification ∃x(P(x)) ≡ existential_quantification P → Q ≡ implication P ∧ Q ≡ conjunction □P ≡ necessity ◇P ≡ possibility GP ≡ always FP ≡ eventually KₐP ≡ agent_a_knows_P (P) ∈ context ≡ scoped_statement statement ⟕ {metadata} ≡ annotated_statement source_A ⟗ source_B ≡ merged_knowledge_bases [a, b, c] ≡ ordered_sequence aᵢ ≡ element(a, i) x ≜ y ≡ definition λx.M ≡ function_abstraction x : T ≡ type_assignment [α]P ≡ after_all_executions_of_α_P_holds ⟨α⟩P ≡ after_some_execution_of_α_P_holds ?P within [α] ≡ test_within_program (dynamic logic only) P □→ Q ≡ counterfactual do(X) ≡ intervention # ═══════════════════════════════════════════════════════════════ # END # ═══════════════════════════════════════════════════════════════ summary(Lingenic) ≜ ( notation(combines(math_structure, natural_language)) ∧ composes(known_primitives) ∧ ¬invents(new_symbols) ∧ works(∀human_language) ∧ scales(coarse, to(forensic)) ∧ descriptive ∧ ¬query_language ∧ ¬command_language ) ∴ suitable(Lingenic, for(AI_knowledge_representation)) ⟕ {confidence: high} # This document is © 2026 Lingenic LLC. All rights reserved. # This site does not use cookies or collect personal data, except via Cloudflare. # URL: https://lingenic.ai/ai/language/Lingenic/Lingenic%20Explained.lingenic # Retrieved: 2026-03-02T044100.000