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Reference — lib_nal.metta

Non-Axiomatic Logic (NAL) — a logic of uncertain reasoning with explicit evidence-based truth values. NAL is the primary symbolic engine for OmegaClaw's inheritance and implication reasoning.

Truth value format

Every NAL statement carries (stv frequency confidence):

  • frequency ∈ [0.0, 1.0] — how often the statement held among observed evidence.
  • confidence ∈ [0.0, 1.0] — how much evidence supports that frequency.

Negation is (stv 0.0 c).

w2c in the formulas below denotes the NAL evidence-weight-to-confidence mapping; it is the standard NAL formula.

Core relations

Relation Meaning
--> Inheritance. (--> a b) ≈ "a is a kind of b". Asymmetric.
==> Implication. (==> P Q) ≈ "if P then Q".
<-> Similarity. (<-> a b) ≈ "a and b are similar". Symmetric in principle but see limits below.
× Product. (× sam garfield) ≈ the ordered pair.
[] Property set. ([] dangerous) ≈ the set of dangerous things.
IntSet Intensional set — members share a property (also used by PLN).

The |- operator

|- applies NAL inference. OmegaClaw selects the appropriate rule automatically based on premise shape.

Rule catalogue

Deduction — the workhorse

Shape: (--> A B) and (--> B C) ⊢ (--> A C).

Truth function:

f = f₁ × f₂
c = f₁ × f₂ × c₁ × c₂

Deduction also emits the exemplification conclusion alongside the primary one (for --> only).

Abduction

Shape: (--> A B) and (--> A C) ⊢ (--> B C) (seeking cause from effect).

Truth function:

f = f₂
c = w2c(f₁ × c₁ × c₂)

Known limit: practical confidence ceiling at ~0.45.

Induction

Shape: (--> A B) and (--> C B) ⊢ (--> A C) (generalizing from multiple instances).

Truth function (symmetric to abduction):

f = f₁
c = w2c(f₂ × c₁ × c₂)

Two instances of the same generalization at c = 0.42 revise to c ≈ 0.59 — use revision to combine multiple instances.

Revision — merging independent evidence

Shape: two beliefs about the same statement.

Truth function:

w = c / (1 - c)            per premise
w_total = Σ w_i
c_out = w_total / (w_total + 1)
f_out = weighted average of f_i by w_i

Produces strictly higher confidence when evidence agrees. When evidence disagrees, frequency drifts toward the middle while confidence grows — contradiction becomes mathematically visible.

Empirical check points:

  • Three sources at (stv 1.0 0.45) revise to (stv 1.0 0.647).
  • Five sources at (stv 1.0 0.45) revise to (stv 0.848 0.937).

Exemplification

Shape: produced alongside deduction for --> premises.

Truth function:

f = 1.0
c = w2c(f₁ × f₂ × c₁ × c₂)

Conditional deduction (modus ponens)

Shape: (==> P Q) and P ⊢ Q.

Truth function: same as deduction (f = f₁ × f₂, c = f₁ × f₂ × c₁ × c₂).

Conditional syllogism (==> chaining)

Works with nested --> inside ==>. Same deduction-style formula.

Conditional abduction

Shape: (==> P Q) + observed Q ⊢ P.

Empirical output shape: (stv 0.9 0.408) on typical inputs — confidence under the abduction ceiling.

Implication chaining

Two ==> sharing a middle term. Works with nested --> inside ==>.

Multi-instance induction

Two instances each at c ≈ 0.42 revise to c ≈ 0.59 — verified empirically. Combine induction with revision to build up evidence.

Higher-order via proxy

Use atomic labels for rules as subjects. Pattern confirmed: birdRule --> reliable --> trustworthy compiles and reasons.

Negation

Via (stv 0.0 c). Propagates through deduction as expected.

Similarity (<->)

Works via NAL-2 rules (cycle 2260). Asymmetric in practice — only matches when the shared term occupies a specific argument position. Do not assume full symmetry.

Analogy

Works via the NAL-2 analogy rule (cycle 2260). Same positional asymmetry as similarity.

Rules NOT to rely on

Pattern Status
NAL-3 decomposition Non-functional — compounds are fully opaque.
Implication chaining (automatic transitive) Two ==> sharing a middle term do not automatically chain. Walk the chain manually.
Contrapositive Partial. Negated consequent + conditional yields the antecedent but with c = 0.0.
Conjunctive antecedents Unsupported — A AND B ==> C cannot be directly represented.
Similarity / analogy (full symmetry) Argument-position-sensitive despite theoretical symmetry.

Invocation

NAL is reached through the (metta ...) skill. Variables use $1, $2, …

Deduction

(metta (|- ((--> sam human)         (stv 1.0 0.9))
           ((--> human mortal)      (stv 1.0 0.9))))

Implication with variable unification

(metta (|- ((==> (--> (× $1 elephant) eat) (--> $1 ([] dangerous))) (stv 1.0 0.9))
           ((--> (× tiger elephant) eat)                            (stv 1.0 0.9))))

Revision on shared term

(metta (|- ((--> wolf animal) (stv 1.0 0.45))
           ((--> wolf animal) (stv 1.0 0.45))))

Output: (--> wolf animal) (stv 1.0 ~0.62).

Multi-hop degradation — in numbers

Starting each premise at c = 0.9:

Hop Output c
1 0.81
2 0.73
3 < 0.5
4 ~0.25

This is a feature, not a bug — the math honestly represents diminishing certainty. Practical implication: keep chains to 2–3 hops, insert revision to restore confidence.

See also