Fixed-Point Theorems and the Ethics of Radical Transparency: A Logic-First Treatment
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908524272943104 |
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| author | Alpay, Faruk Alakkad, Hamdi |
| author_facet | Alpay, Faruk Alakkad, Hamdi |
| contents | This paper establishes a formal framework, grounded in mathematical logic and order theory, to analyze the inherent limitations of radical transparency. We demonstrate that self-referential disclosure policies inevitably encounter fixed-point phenomena and diagonalization barriers, imposing fundamental trade-offs between openness and stability.
Key results include: (i) an impossibility theorem showing no sufficiently expressive system can define a total, consistent transparency predicate for its own statements; (ii) a categorical fixed-point argument (Lawvere) for the inevitability of self-referential equilibria; (iii) order-theoretic design theorems (Knaster-Tarski) proving extremal fixed points exist and that the least fixed point minimizes a formal ethical risk functional; (iv) a construction for consistent partial transparency using Kripkean truth; (v) an analysis of self-endorsement hazards via Löb's Theorem; (vi) a recursion-theoretic exploitation theorem (Kleene) formalizing Goodhart's Law under full disclosure; (vii) an exploration of non-classical logics for circumventing classical paradoxes; and (viii) a modal $μ$-calculus formulation for safety invariants under iterative disclosure.
Our analysis provides a mathematical foundation for transparency design, proving that optimal policies are necessarily partial and must balance accountability against strategic gaming and paradox. We conclude with equilibrium analysis and lattice-theoretic optimality conditions, offering a principled calculus for ethical disclosure in complex systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_06055 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fixed-Point Theorems and the Ethics of Radical Transparency: A Logic-First Treatment Alpay, Faruk Alakkad, Hamdi Logic Computer Science and Game Theory 03B70, 03F40, 91A80 F.4.1; I.2.0; F.3.1; K.4.1 This paper establishes a formal framework, grounded in mathematical logic and order theory, to analyze the inherent limitations of radical transparency. We demonstrate that self-referential disclosure policies inevitably encounter fixed-point phenomena and diagonalization barriers, imposing fundamental trade-offs between openness and stability. Key results include: (i) an impossibility theorem showing no sufficiently expressive system can define a total, consistent transparency predicate for its own statements; (ii) a categorical fixed-point argument (Lawvere) for the inevitability of self-referential equilibria; (iii) order-theoretic design theorems (Knaster-Tarski) proving extremal fixed points exist and that the least fixed point minimizes a formal ethical risk functional; (iv) a construction for consistent partial transparency using Kripkean truth; (v) an analysis of self-endorsement hazards via Löb's Theorem; (vi) a recursion-theoretic exploitation theorem (Kleene) formalizing Goodhart's Law under full disclosure; (vii) an exploration of non-classical logics for circumventing classical paradoxes; and (viii) a modal $μ$-calculus formulation for safety invariants under iterative disclosure. Our analysis provides a mathematical foundation for transparency design, proving that optimal policies are necessarily partial and must balance accountability against strategic gaming and paradox. We conclude with equilibrium analysis and lattice-theoretic optimality conditions, offering a principled calculus for ethical disclosure in complex systems. |
| title | Fixed-Point Theorems and the Ethics of Radical Transparency: A Logic-First Treatment |
| topic | Logic Computer Science and Game Theory 03B70, 03F40, 91A80 F.4.1; I.2.0; F.3.1; K.4.1 |
| url | https://arxiv.org/abs/2509.06055 |