Mathematics Logic Systems From Config Dim Logic System Count

The five logic systems

Mathematical logic studies formal languages for expressing and proving statements. Over the past century, logicians have settled on a small set of standard systems: propositional logic, which combines simple statements with connectives like "and" and "not"; first-order logic, which adds quantifiers like "for all" and "there exists" over objects; second-order logic, which also quantifies over properties and sets; modal logic, which adds notions of necessity and possibility; and intuitionistic logic, which drops the law of excluded middle. These five are the classical canon, each with its own semantics, proof theory, and applications.

In Recognition Science, the framework models the structure of recognition events using a discrete configuration dimension, and it asks how many canonical logic systems that dimension supports. The framework's machine-checked library of formal theorems proves a precise counting statement: the number of canonical logic systems is exactly five. The proof is a direct computation over a finite list of five named systems, and the library certifies the count with no unproved assumptions. In plain terms, the declaration establishes that the five familiar systems form a complete set under the framework's chosen classification, nothing more and nothing less.

The count does not claim that these five are the only possible logics, nor that they are the best, nor that the framework has derived them from first principles. The declaration merely counts the members of a predefined list. It is a definitional choice, not a discovery: the framework chooses to call these five systems canonical, and then verifies that the list has five entries. The theorem is a counting fact about that choice, not an argument that the choice is forced or unique.

For a reader, the practical upshot is a clean, checkable number: five. The framework's library can be trusted to have counted correctly, because the count is machine-checked. But the interesting question, why these five and not others, remains open. The declaration settles the arithmetic, not the philosophy.

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

What this page does not claim

The declaration does not prove that these five logic systems are the only possible ones. The declaration does not derive the five systems from the framework's forcing chain. The declaration does not rank or evaluate the five systems.

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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