Mathematics Logic Systems From Config Dim Logic Systems Cert

The five logic systems

Classical logic is usually taught as a family of distinct systems: propositional logic, first-order logic, second-order logic, modal logic, and intuitionistic logic. Each has its own vocabulary, rules of inference, and intended applications. Propositional logic works with whole statements joined by connectives; first-order logic adds quantifiers over objects; second-order logic allows quantification over properties and relations; modal logic adds operators for necessity and possibility; intuitionistic logic drops the law of excluded middle. These five are the standard inhabitants of any logic textbook.

The Recognition Science framework takes that familiar list and asks whether it falls out of a deeper structure. Its answer, encoded in the machine-checked declaration LogicSystemsCert (a certificate, a formal record that a stated fact has been verified), is that exactly these five systems correspond to a parameter called configDim (a configuration dimension, a natural number indexing the framework's internal degrees of freedom) equal to 5. The certificate states that the number of logic systems is 5, and it proves that count by direct computation: the framework defines an inductive type with five constructors, one per system, and then checks that the cardinality is indeed 5. The proof is machine-checked with no gaps and no added axioms.

What the certificate does not claim is just as important. It does not claim that these five systems are the only possible logics, nor that the framework has derived the semantics or proof theory of each system from first principles. It does not claim that the identification of configDim 5 with these five systems is a theorem of physics; that identification is a definitional choice, a modeling decision within the framework. The certificate only establishes a counting fact: the framework's chosen list has five members, and that count matches the framework's dimensional parameter.

In plain terms, the declaration is a small but solid brick in a larger edifice. It shows that the framework can reproduce a classical taxonomy as a matter of internal structure, not by listing examples but by deriving the count from a single number. The consequence for a reader is that the five logic systems are not arbitrary in this account; they are tied to the framework's dimensional scheme. What remains open is whether that tie has explanatory power beyond the counting fact itself.

theorem logicSystem_count : Fintype.card LogicSystem = 5 := by decide

structure LogicSystemsCert where
  five_systems : Fintype.card LogicSystem = 5

inductive LogicSystem where
  | propositional
  | firstOrder
  | secondOrder
  | modal
  | intuitionistic
  deriving DecidableEq, Repr, BEq, Fintype

What this page does not claim

The certificate does not prove that these five are the only possible logic systems. The certificate does not derive the inference rules or semantics of any logic system. The certificate does not establish a physical law connecting logic to the framework's dimensions.

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/Mathematics/LogicSystemsFromConfigDim.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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