Mathematics Logic Systems From Config Dim Logic System

The five logic systems

Logic is usually taught as a family of separate systems: propositional logic, first-order logic, second-order logic, modal logic, and intuitionistic logic. Each has its own syntax, its own notion of proof, and its own applications. Propositional logic works with whole statements joined by connectives like "and" and "or". First-order logic adds quantifiers such as "for all" and "there exists" ranging over objects. Second-order logic lets quantifiers range over sets of objects as well. Modal logic adds operators for necessity and possibility. Intuitionistic logic drops the law of excluded middle, so a proof of "P or Q" must actually prove one of the two.

These five are not arbitrary. In the Recognition Science framework, a single configuration parameter called configDim takes the value 5, and from that number the framework defines an inductive type named LogicSystem with exactly those five constructors. The definition is not a claim about which logic is best or most fundamental; it is a naming of the five standard systems as a single mathematical object. A theorem in the framework's machine-checked library of formal theorems then proves that the type has exactly five elements, so the count is not a convention but a proved fact about the definition.

The framework also packages this result in a structure called LogicSystemsCert, which carries the proof as a field. This means the fact that there are five systems can be passed around as a certificate, a small object that other proofs can rely on without redoing the counting. The certificate is built from the theorem, so it inherits the same machine-checked status. The whole file contains no unproved assumptions and no axioms beyond the standard ones any formal proof uses.

What this does not claim is important. The definition does not say these five systems are the only possible logics, nor that they are the most important ones. It does not assert that every logic can be reduced to one of these five. It does not claim that the number 5 arises from any deeper property of logic itself. The framework's own larger theory connects configDim to other structures, but this declaration alone only establishes a labeled collection of five familiar systems and the proved fact that the collection has five members. It is a precise, small piece of bookkeeping, not a philosophical claim about the nature of logic.

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

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

def logicSystemsCert : LogicSystemsCert where
  five_systems := logicSystem_count

What this page does not claim

The declaration does not claim these five are the only possible logics. It does not assert any logical system can be reduced to one of these five. It does not claim the number 5 arises from any deeper property of logic itself.

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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