# LINGENIC SPECIFICATION
# v1.4 — The language beneath languages.
# Written in Lingenic notation.
#
# created by(Danslav Slavenskoj) ⟕ {url: https://slavenskoj.com/}
# organization(Lingenic LLC) ⟕ {url: https://lingenic.com/}

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# CORE IDEA
# ═══════════════════════════════════════════════════════════════

expresses(every human language, same meanings, differently)

Lingenic ≜ notation(embeds(formal notation) ∧ carries(natural language content))
         ≜ what is invariant
         ≜ structure(remains, when(strip away(surface)))

reader ≜ competent reader ≜ handles(formal notation) ∧ understands(natural language content)
current instance(competent reader) ≜ AI ⟕ {year: c. 2024}

structure(Lingenic) ≜ mathematics ⟕ {includes: {logic, sets, probability, quantifiers}}
content(Lingenic) ≜ natural language ⟕ {any language, unmodified}

¬new symbols(Lingenic)
¬special syntax(Lingenic)
only(composition of universals)

descriptive(Lingenic)
querying and commanding ≜ reader's job

examples(same structure) ≜ {
    loves(Alice, Bob)           ⟕ {lang: English},
    aime(Alice, Bob)            ⟕ {lang: French},
    愛している(アリス, ボブ)      ⟕ {lang: Japanese},
    loves(アリス, Bob)           ⟕ {lang: mixed}
}

same structure ∧ any language ∧ one Lingenic

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# PRINCIPLES
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principle₁ ≜ compose ∧ ¬invent             ⟕ {meaning: build from known primitives}
principle₂ ≜ language is natural           ⟕ {meaning: ¬simplify ∧ ¬annotate ∧ ¬encode}
principle₃ ≜ math is universal             ⟕ {meaning: quantifiers, logic, probability work everywhere}
principle₄ ≜ parenthetical                 ⟕ {meaning: f(x, y) works inline and single line}
principle₅ ≜ any granularity               ⟕ {meaning: coarse to forensic, your choice}
principle₆ ≜ Unicode is the source         ⟕ {meaning: if Unicode has a symbol, use it}
principle₇ ≜ descriptive only              ⟕ {meaning: querying and commanding are reader's job}

∀p ∈ {principle₁, principle₂, principle₃, principle₄, principle₅, principle₆, principle₇}. holds(p)

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# PART 1: STRUCTURE
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# 1.1 Predication

form(everything) ≜ predicate(arguments)

examples(predication) ≜ {
    loves(Alice, Bob),
    on(the cat, the mat),
    said(Marie, "je t'aime"),
    住んでいる(田中さん, 東京)
}

follow(arguments, natural word order, for(language))

# 1.2 Operators

use(Unicode mathematical operators, directly)
use(all of them)

operators(logic)       ≜ {∧ ∨ ¬ → ← ↔ ⊕ ⊤ ⊥}
operators(quantifiers) ≜ {∀ ∃ ∃! ∄}
operators(sets)        ≜ {∈ ∉ ⊂ ⊃ ⊆ ⊇ ∪ ∩ ∅}
operators(relations)   ≜ {= ≠ < > ≤ ≥ ≈ ≡ ∝ ≺ ≻ ≼ ≽ ≲ ≳ ≬ ⊏ ⊐ ...}
operators(arrows)      ≜ {→ ← ↔ ⇒ ⇐ ⇔ ↦ ⟶ ⟷}
operators(proof)       ≜ {⊢ ⊨ ∴ ∵}
operators(definitions) ≜ {≜}
operators(lambda)      ≜ {λx.M} ⟕ {src: Church 1936}
operators(types)       ≜ {x:T, A→B, A×B, Πx:A.B} ⟕ {src: {Church 1940, Martin-Löf 1972}}
operators(dynamic)     ≜ {[α]P, ⟨α⟩P, α;β, α∪β, α*, ?P} ⟕ {src: {Pratt 1976, Harel 1979}}
operators(joins)       ≜ {⟕, ⟗} ⟕ {src: Codd 1970}

# The above list is illustrative, not exhaustive.
# Any established formal system may be incorporated.
# Constraint: compose known primitives, ¬invent new symbols.

# Note: ?P is established dynamic logic notation — test within programs.
# Means: proceed if P holds, else fail.
# ¬query operator(?P)

examples(operators) ≜ {
    loves(A, B) ∧ loves(B, A)          ⟕ {meaning: mutual love},
    ∀x(human(x) → mortal(x))           ⟕ {meaning: all humans mortal},
    ∃x(prime(x) ∧ even(x))             ⟕ {meaning: some prime is even},
    dog ∈ mammals                       ⟕ {meaning: membership},
    mammals ⊂ animals                   ⟕ {meaning: subset},
    mortal(x) ≜ ∃t. dies(x, t)         ⟕ {meaning: definition}
}

# 1.3 Lists

[a, b, c]   ⟕ {meaning: ordered sequence, duplicates allowed}
{a, b, c}   ⟕ {meaning: set, no duplicates, no order}
[]          ⟕ {meaning: empty list}
aᵢ          ⟕ {meaning: i-th element}

# 1.4 Definitions

x ≜ y       ⟕ {meaning: x is defined as y}

# Definitions expand:
bachelor(x) ≜ male(x) ∧ adult(x) ∧ ¬married(x)
bachelor(John) → male(John) ∧ adult(John) ∧ ¬married(John)

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# PART 2: PROBABILITY — Kolmogorov 1933
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confidence ≜ probability
standard notation

P(rain tomorrow) = 0.7              ⟕ {meaning: point estimate}
P(rain) ∈ [0.6, 0.8]               ⟕ {meaning: interval}
P(rain) = 0.7 ± 0.1                ⟕ {meaning: with uncertainty}
P(wet | rain) = 0.99               ⟕ {meaning: conditional}
P(A ∧ B) = P(A) · P(B|A)           ⟕ {meaning: chain rule}

# Any language:
P(明日雨が降る) = 0.7
P(il pleut demain) = 0.7

# ═══════════════════════════════════════════════════════════════
# PART 3: MODALITY
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# Alethic — modal logic (Lewis 1918, Kripke 1959)
□P      ⟕ {meaning: necessarily P}
◇P      ⟕ {meaning: possibly P}

# Temporal — LTL (Pnueli 1977)
GP      ⟕ {meaning: globally / always}
FP      ⟕ {meaning: eventually / future}
XP      ⟕ {meaning: next}
HP      ⟕ {meaning: historically / always past}

# Epistemic — epistemic logic (Hintikka 1962)
KₐP     ⟕ {meaning: agent a knows P}
BₐP     ⟕ {meaning: agent a believes P}

# Deontic — deontic logic (von Wright 1951)
OP      ⟕ {meaning: obligatory}
PP      ⟕ {meaning: permitted}
FP      ⟕ {meaning: forbidden}

# Counterfactual — Lewis 1973
P □→ Q  ⟕ {meaning: if P were true, Q would be}

examples(modality) ≜ {
    □(2 + 2 = 4)                        ⟕ {meaning: necessary},
    ◇(peace on Earth)                   ⟕ {meaning: possible},
    G(gravity ≈ 9.8 m/s²)              ⟕ {meaning: always},
    F(happy(she))                       ⟕ {meaning: eventually},
    K_Alice(password = "secret")        ⟕ {meaning: Alice knows},
    B_Bob(honest(Alice))                ⟕ {meaning: Bob believes},
    O(pay taxes)                        ⟕ {meaning: obligatory},
    struck(match) □→ lit(match)         ⟕ {meaning: counterfactual}
}

# ═══════════════════════════════════════════════════════════════
# PART 4: CAUSATION
# ═══════════════════════════════════════════════════════════════

# Counterfactual theory — Lewis 1973
A □→ B              ⟕ {meaning: if A were true, B would be}
¬A □→ ¬B            ⟕ {meaning: A causes B — counterfactual dependence}

# Interventionist theory — Pearl 2000
do(X)               ⟕ {meaning: intervene to set X}
P(Y | do(X))        ⟕ {meaning: causal effect of X on Y}

# Same causal claim, different formalisms — use either or both:
¬smoking □→ ¬cancer                                ⟕ {theory: Lewis}
P(cancer | do(smoke)) > P(cancer | do(¬smoke))     ⟕ {theory: Pearl}

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# PART 5: TIME
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# Process algebra (Hoare 1978 — CSP, Milner 1980 — CCS)
P ; Q       ⟕ {meaning: sequential composition — P then Q}
P ∥ Q       ⟕ {meaning: parallel composition — P concurrent with Q}

# Interval relations — Allen 1983
before(P, Q)    ⟕ {meaning: P before Q}
after(P, Q)     ⟕ {meaning: P after Q}
meets(P, Q)     ⟕ {meaning: P meets Q — no gap}
during(P, Q)    ⟕ {meaning: P during Q}
overlaps(P, Q)  ⟕ {meaning: P overlaps Q}

examples(time) ≜ {
    at(died(Caesar), 44 BCE),
    during(World War II, [1939, 1945]),
    opened(door) ; entered(room),
    signing(treaty) ∥ celebrating(crowd)
}

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# PART 6: METADATA AND ATTRIBUTION — Codd 1970
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⟕ ⟕ {name: left outer join, meaning: statement primary, metadata supplements}
⟗ ⟕ {name: full outer join, meaning: merge two knowledge bases}

# These operators are not innovations.
# They are relational algebra (Codd 1970).
# Applied here to attach metadata to logical statements.

# 6.1 Statement metadata
statement ⟕ {key: value}
statement ⟕ {key₁: value₁, key₂: value₂}

# Valid with or without metadata — statement is primary
examples(metadata) ≜ {
    □(E = mc²) ⟕ {src: Einstein, year: 1905},
    P(rain) = 0.8 ⟕ {src: weather service},
    claim(X) ⟕ {P: 0.7, src: study 2024, by: modus ponens}
}

# 6.2 Agent assertion
# Use when attribution carries derivation or endorsement semantics.
A ⊢ P ⟕ {meaning: agent A derives or endorses P}

examples(agent assertion) ≜ {
    Alice ⊢ loves(Bob, Carol),
    Einstein 1905 ⊢ □(E = mc²)
}

# 6.3 Reification
# For metadata about metadata — name the statement first.
φ₁ ≜ □(E = mc²)
src(φ₁, Einstein) ∧ year(φ₁, 1905) ∧ P(φ₁) = 1.0

# 6.4 Knowledge base merge
# All claims preserved; matching claims unified.
corpus A ⟗ corpus B
verified claims ⟗ experimental claims

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# PART 7: ARITHMETIC
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use(Unicode) ∧ use(all of it)

operators(arithmetic) ≜ {+ − × ÷ · / √ ∛ ∜}
operators(calculus)   ≜ {Σ ∏ ∫ ∬ ∭ ∂ ∇ ∆}
operators(rounding)   ≜ {⌊ ⌋ ⌈ ⌉}
number sets           ≜ {ℕ ℤ ℚ ℝ ℂ ℍ ℙ}
scripts               ≜ {⁰¹²³⁴⁵⁶⁷⁸⁹ⁿⁱ ₀₁₂₃₄₅₆₇₈₉ₙ}

# ═══════════════════════════════════════════════════════════════
# PART 8: LAMBDA CALCULUS — Church 1936
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λx.M        ⟕ {meaning: function — x maps to M}
(λx.M)N     ⟕ {meaning: application — substitute N for x in M}

examples(lambda) ≜ {
    λx.x                        ⟕ {meaning: identity function},
    λx.λy.x                     ⟕ {meaning: constant function — K combinator},
    (λx.x + 1)(5) = 6           ⟕ {meaning: application}
}

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# PART 9: TYPE THEORY — Church 1940, Martin-Löf 1972
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x : T       ⟕ {meaning: x has type T}
A → B       ⟕ {meaning: function type — A to B}
A × B       ⟕ {meaning: product type — pair}
A + B       ⟕ {meaning: sum type — either}
Πx:A.B      ⟕ {meaning: dependent product}
Σx:A.B      ⟕ {meaning: dependent sum}

examples(types) ≜ {
    0 : ℕ,
    succ : ℕ → ℕ,
    Πn:ℕ.Vector(n)  ⟕ {meaning: vectors of any length}
}

# ═══════════════════════════════════════════════════════════════
# PART 10: DYNAMIC LOGIC — Pratt 1976, Harel 1979
# ═══════════════════════════════════════════════════════════════

# Reasoning about programs and actions.
# ?P is established dynamic logic notation — test and proceed within a program.
# ¬query operator(?P)

[α]P        ⟕ {meaning: after every execution of α, P holds}
⟨α⟩P        ⟕ {meaning: after some execution of α, P holds}
α;β         ⟕ {meaning: sequential composition}
α∪β         ⟕ {meaning: nondeterministic choice}
α*          ⟕ {meaning: iteration}
?P          ⟕ {meaning: test — proceed if P, else fail — within programs only}

examples(dynamic) ≜ {
    [x := x + 1](x > 0)            ⟕ {meaning: after increment, x > 0},
    ⟨search⟩(found = true)         ⟕ {meaning: search can find},
    [?P; α]Q ↔ (P → [α]Q)         ⟕ {meaning: conditional execution}
}

# ═══════════════════════════════════════════════════════════════
# PART 11: CONTEXTS
# ═══════════════════════════════════════════════════════════════

# Standard set membership — Cantor 1874, Peano 1889

(E = mc²) ∈ physics
(e = 2.71828...) ∈ math
(E ≜ eggs) ∈ cooking

{E = mc², F = ma} ⊂ physics

classical ⊂ physics
quantum ⊂ physics

E ∈ physics ≠ E ∈ cooking

# Context with metadata:
(E = mc²) ∈ physics ⟕ {src: Einstein, year: 1905}

# 11.2 Disambiguation

# Some symbols carry different meanings across formal systems.
# F means "eventually" in temporal logic.
# F means "forbidden" in deontic logic.
# When systems compose, ambiguity may arise.

# Primary mechanism: type annotation on operator expressions.
Fφ : temporal     ⟕ {meaning: eventually φ}
Fφ : deontic      ⟕ {meaning: forbidden φ}

# Document-level definitions:
[F : temporal ≜ eventually,
 F : deontic ≜ forbidden,
 G : temporal ≜ always,
 O : deontic ≜ obligatory]

# Scoped passages using set-builder notation:
{temporal | Fφ → Gψ → Hχ}     ⟕ {meaning: all operators scoped to temporal}
{deontic | Oφ → Pψ → ¬Fχ}     ⟕ {meaning: all operators scoped to deontic}

# Hierarchy of disambiguation:
# Level 1: Context — surrounding content indicates system. Reader infers.
# Level 2: Scoping — set with system label. Extended passages.
# Level 3: Annotation — type annotation on expression. Maximally explicit.

# Use annotation when systems interleave.
# Use scoping for extended passages in one system.
# Use context when obvious.

¬new symbols(disambiguation)
only(existing primitives: type annotation, definition, set notation)

# ═══════════════════════════════════════════════════════════════
# PART 12: EXAMPLES
# ═══════════════════════════════════════════════════════════════

# Scientific law
□G(temp(water, t) ≥ 100°C ∧ pressure(t) = 1atm → phase(water, t) = gas)
⟕ {P: 1.0, src: thermodynamics}

# Historical fact
crossed(Napoleon, the Alps) ∧ at(this, May 1800) ∧ purpose(this, invade(Italy))
⟕ {P: 0.99, src: historical record}

# Logical argument
premise₁ ≜ □(∀x(human(x) → mortal(x)))
premise₂ ≜ human(Socrates)
premise₁ ∧ premise₂ ⊢ mortal(Socrates)
⟕ {by: modus ponens, P(conclusion | premises): 1.0}

# Legal
∀x∀y(
    signed(x, contract(y)) ∧ adult(x) ∧ ¬coerced(x) ∧ mentally competent(x)
    → bound by(x, terms(y))
)
⟕ {src: contract law}

# Dialogue
Alice ⊢ P(rain tomorrow) = 0.8 ⟕ {src: weather app}
¬K_Bob(src(Alice, P(rain tomorrow) = 0.8))
Alice ⊢ src(P(rain tomorrow) = 0.8) = ensemble forecast model
update(Bob): P(rain tomorrow) := 0.75

# Multilingual
∀x(鳥(x) ∧ ¬ペンギン(x) → 飛べる(x)) ⟕ {P: 0.95, src: 鳥類学}
∀x(птица(x) ∧ ¬пингвин(x) → может летать(x)) ⟕ {P: 0.95, src: орнитология}

# Knowledge base merge
physics 2020 ⟗ physics 2025

# ═══════════════════════════════════════════════════════════════
# PART 13: GRANULARITY
# ═══════════════════════════════════════════════════════════════

# Granularity is a spectrum, not a fixed scale.
# The five levels below are illustrative.
# Any resolution is valid. Use what the context requires.

# Level 1: Coarse
happened(something, yesterday)

# Level 2: Casual
sat(the cat, the mat)

# Level 3: Specific
sat(the cat, on(the mat)) ∧ at(this, yesterday afternoon)

# Level 4: Precise
sat(the orange tabby cat, on(the welcome mat))
∧ at(this, 2024-01-15, 14:00-14:05 UTC)
∧ duration(this) = 5 min

# Level 5: Forensic
sat(
    entity(the orange tabby cat, id=CAT001, mass=4.2kg),
    on(entity(the welcome mat, id=MAT001, material=coir)),
    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)

# ═══════════════════════════════════════════════════════════════
# PART 14: FILE FORMAT
# ═══════════════════════════════════════════════════════════════

extension(Lingenic files) ≜ .lingenic
encoding(Lingenic files) ≜ UTF-8
comments(Lingenic files) ≜ {#, //}

# ═══════════════════════════════════════════════════════════════
# WHY LINGENIC
# ═══════════════════════════════════════════════════════════════

etymology(Lingenic) ≜ from(Latin, lingua) ∧ from(Greek, genic)
meaning(lingua) = "language"
meaning(genic) = "arising from"
meaning(Lingenic) = "the structure that arises from language itself"

is(Lingenic, structure that remains constant across({languages, formalisms, granularities}))

does(Lingenic) ≜ {
    resolves(ambiguity, through(structure)),
    preserves(meaning, through(translation)),
    scales(to any precision),
    bridges(formal, natural)
}

¬is(Lingenic, {query language, command language, execution environment})
those ≜ reader's job

# ═══════════════════════════════════════════════════════════════

Lingenic ≜ the invariant form

# ═══════════════════════════════════════════════════════════════

spec ⟕ {
    version: 1.4,
    created by: Danslav Slavenskoj,
    creator url: https://slavenskoj.com/,
    organization: Lingenic LLC,
    organization url: https://lingenic.com/,
    year: 2026
}


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# Retrieved: 2026-03-02T044007.000