LogicSystem
The inductive type enumerates the five canonical logic systems tied to configDim equal to 5. Researchers proving completeness of the Reality Recognition Framework cite it to index the logic component of universal embeddings. The definition lists the five constructors and derives Fintype, Repr, and related instances in a single inductive block.
claimLet $L$ be the inductive type whose constructors are propositional, first-order, second-order, modal, and intuitionistic logic.
background
The module Mathematics.LogicSystemsFromConfigDim identifies five canonical logic systems with configDim D = 5. The upstream structure LogicSystem from UltimateIsomorphism supplies the simplified interface consisting of a proposition type Prop' and a provability relation proves : Prop' → Prop' → Prop. This enumeration supplies the concrete finite domain over which the framework quantifies logic embeddings.
proof idea
The declaration is the inductive definition itself. Five constructors are listed explicitly and the deriving clause installs DecidableEq, Repr, BEq, and Fintype with no additional tactics or lemmas.
why it matters in Recognition Science
The type is the domain for logic_embeds and FrameworkComplete in UltimateIsomorphism, which together establish that every logic system embeds into universalStructure. It thereby contributes the logic clause to reality_recognition_framework_complete. Within Recognition Science the five systems realize the configDim = 5 requirement that closes the logic part of the completeness argument.
scope and limits
- Does not axiomatize the inference rules internal to any listed system.
- Does not construct or verify embeddings into universalStructure.
- Does not connect the systems to spatial dimension D = 3 or to constants such as phi.
- Does not prove any meta-theoretic properties beyond the derived decidability instances.
formal statement (Lean)
15inductive LogicSystem where
16 | propositional
17 | firstOrder
18 | secondOrder
19 | modal
20 | intuitionistic
21 deriving DecidableEq, Repr, BEq, Fintype
22