Delta Kernel Syntax Is Qf

The quantifier-free test

In intuitionistic first-order arithmetic, formulas are built from terms using equality, falsity, conjunction, disjunction, implication, and the two quantifiers "for all" and "there exists". A formula is quantifier-free when it contains no quantifiers at all. The declaration isQF is a Boolean function that checks exactly this property: it returns true for atomic formulas like equality, and for combinations built only from the propositional connectives. It returns false the moment it encounters a universal or existential quantifier.

The test matters because the framework's proof system uses a rule called Markov's principle, which allows a certain kind of reasoning about existence. That rule is only applied to quantifier-free formulas, where truth is decidable by simple inspection. The isQF function is the gatekeeper that keeps the rule honest: it verifies that the formula in question has the right shape before the rule is allowed to fire. This is a definitional choice, not a theorem about arithmetic.

The function is defined by structural recursion on the formula. It checks each subformula in turn, and only if all parts are quantifier-free does the whole formula pass. This is a purely syntactic operation: it looks at the shape of the formula, not at its meaning. The definition is part of the δ-Kernel syntax module, which deliberately avoids a universe hierarchy, dependent types, and propositions-as-types. Formulas are plain data, and this test is plain data manipulation.

In Recognition Science, this syntax forms the forced base of the δ framework: the free distinction structure ℕδ with its recursion-licensed operations. The isQF test does not prove any arithmetic statement, does not decide whether a formula is true, and does not establish that Markov's principle is sound. It only classifies formulas by their syntactic shape. The soundness of the proof rule that uses this test is a separate matter, handled elsewhere in the framework's library.

/-- Quantifier-free test (used to keep the Markov posit rule honest:
MP is posited only for quantifier-free, hence decidable, matrices). -/
def isQF : DFormula → Bool
  | eq _ _   => true
  | fls      => true
  | conj a b => a.isQF && b.isQF
  | disj a b => a.isQF && b.isQF
  | impl a b => a.isQF && b.isQF
  | all _    => false
  | ex _     => false

/-- Quantifier-free test (used to keep the Markov posit rule honest:
MP is posited only for quantifier-free, hence decidable, matrices). -/
def isQF : DFormula → Bool
  | eq _ _   => true
  | fls      => true
  | conj a b => a.isQF && b.isQF
  | disj a b => a.isQF && b.isQF
  | impl a b => a.isQF && b.isQF
  | all _    => false
  | ex _     => false

What this page does not claim

The isQF test does not decide whether a formula is true or false. The isQF test does not prove that Markov's principle is sound. The isQF test does not establish any arithmetic fact about the natural numbers.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/DeltaKernel/Syntax.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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