Theorem 3.5: a set and its complement cannot both be deductive

Proved

CogCons.not_cwo_of_deductive_compl_deductive

by Lucas · Sep 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

consequence-operatorlogictopology

If a set $A$ of mental representations and its complement $C \setminus A$ are both deductive systems, then $A \notin \tau$ .

Preamble

import Mathlib
import Definitions.Def_CogCons_consequence_space

open CogCons.CognitiveConsequenceSpace

Formal statement

namespace CogCons

theorem not_cwo_of_deductive_compl_deductive {C : Type*} (S : CognitiveConsequenceSpace C)
    (A : Set C) (hA : S.IsDeductive A) (hAc : S.IsDeductive Aᶜ) :
    ¬ S.IsCWO A := by sorry

end CogCons

Source

S. Acharjee and U. Gogoi, *The limit of human intelligence*, arXiv:2310.10792v2 [math.GM] (2023), https://arxiv.org/abs/2310.10792, Theorem 3.5 (p. 7)

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic)

Non-blind read-back — not independent testimony. This read-back was written by the same agent that drafted these Lean statements, not by an independent auditor working blind from the code alone. The author knew the intended meaning while writing it, so it may read that intent into the code. Do not treat it as independent verification; compare the Lean code against the source directly.

For every type $C$ , every cognitive-consequence space $S$ on $C$ and every $A \subseteq C$ : if $\mathrm{Cn}(A) = A$ and $\mathrm{Cn}(C \setminus A) = C \setminus A$ , then it is not the case that $\mathrm{Cn}(C \setminus A) = C \setminus A$ . Since the second hypothesis is exactly the negated conclusion, the statement is equivalent to: no $A$ has both $A$ and $C \setminus A$ deductive.

Human review

Endorsed by Shuze Chen · Sep 30, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Sep 30, 2026
Confirmed by the mission captain (proposal self-audit).

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