Löb-Safe Logics for Reflective Agents (2408.09590v2)

Abstract: Epistemic and doxastic logics are modal logics for knowledge and belief, and serve as foundational models for rational agents in game theory, philosophy, and computer science. We examine the consequences of modeling agents capable of a certain sort of reflection. Such agents face a formal difficulty due to L\"ob's Theorem, called L\"ob's Obstacle in the literature. We show how the most popular axiom schemes of epistemic and doxastic logics suffer from L\"ob's Obstacle, and present two axiom schemes that that avoid L\"ob's Obstacle, which we call Reasonable L\"ob-Safe Epistemic Doxastic logic (${LSED}^R$) and Supported L\"ob-Safe Epistemic Doxastic logic (${LSED}^S$).

• The paper demonstrates that standard epistemic and doxastic logics crash under Löb's Obstacle due to their handling of reflective reasoning.
• The authors develop LSED and LSED* by weakening introspection axioms to ensure sound and complete formal systems for reflective agents.
• The study highlights implications for AI and game theory by providing robust models that better capture human-like, incomplete reasoning.

L\"ob-Safe Logics for Reflective Agents

The paper titled L\"ob-Safe Logics for Reflective Agents, authored by Seth Ahrenbach and colleagues, presents an examination of the formal difficulties encountered by agents capable of reflective reasoning, particularly in the context of epistemic and doxastic logics. The core issue addressed in the paper is the so-called "L\"ob's Obstacle", deriving from L\"ob's Theorem in provability logic, and its implications for standard modal logics used to model knowledge and belief in rational agents.

Background and Problem Statement

Reflective agents are those capable of reasoning about self-referential sentences or propositions. Traditional epistemic logic such as $\mathit{S5}$ and doxastic logic like $\mathit{KD}45$ are foundational in modeling knowledge and belief, respectively. However, these logics face a significant challenge when accounting for reflective reasoning. The problem originates from L\"ob's Theorem, which states that if a system can prove that "if $\Box\varphi$ then $\varphi$" where $\Box$ denotes a modality like provability or knowledge, then it can also prove $\Box\varphi$.

This theorem implies a contradiction in systems where reflective reasoning is modeled, as it would entail proving any proposition $\varphi$. For epistemic and doxastic logics with certain properties, known as L\"ob Conditions—namely, the expressibility of L\"ob sentences, standard modal operator axioms (such as K, 4), and the rule of necessitation—this leads to the derivation of L\"ob's Theorem and thus crashes the logic.

Analysis of Crashing Logics

The paper systematically analyzes standard epistemic and doxastic logics to show how they crash into L\"ob's Obstacle:

1. S5 Epistemic Logic: Due to the presence of Axiom 4 (positive introspection) and the truth axiom, this logic meets all L\"ob Conditions and necessarily crashes, leading to a situation where $K_i \varphi$ for all $\varphi$.
2. S4 Epistemic Logic of Hintikka: Despite rejecting negative introspection, Hintikka's inclusion of positive introspection and the truth axiom also makes this logic crash due to L\"ob's Theorem.
3. Kraus and Lehmann’s System: A combined system for knowledge and belief aiming to model agents with incomplete information also crashes due to its inclusion of positive belief introspection and consistent belief axioms.
4. KD45 Doxastic Logic: This standard model of belief, due to its adherence to positive and negative belief introspection and belief consistency, similarly crashes into L\"ob's Obstacle.

These logics fail to provide a consistent foundation for reflective reasoning agents, highlighting a need for alternative formal systems that can avoid L\"ob's Theorem while remaining suitable for representing agents’ knowledge and beliefs.

Proposed Solutions: L\"ob-Safe Logics

To address these challenges, the authors propose two alternative logics: LSED (Reasonable L\"ob-Safe Epistemic Doxastic logic) and LSED* (Supported L\"ob-Safe Epistemic Doxastic logic). The key innovation in these logics is to adjust the axioms governing belief to avoid the problematic L\"ob Conditions.

1. LSED: This logic replaces the positive introspection axiom for belief with a weaker axiom termed "Reasonable Belief" (RB), expressed as $B_i \varphi \rightarrow B_i K_i \varphi$. This ensures that agents’ beliefs are based on reasonable or evidential support without necessitating positive introspection, thereby avoiding L\"ob's Theorem.
2. LSED*: An even weaker formulation, this logic adopts the "Supported Belief" (SB) axiom, $B_i \varphi \rightarrow \Diamond_i K_i \varphi$, indicating that beliefs should be evidentially possible without stringent support requirements.

Soundness, Completeness, and Implications

The new L\"ob-Safe logics maintain soundness and completeness through the application of the Sahlqvist-van Benthem algorithm, ensuring robust formal properties. These logics are particularly suitable for representing human-like reasoning agents who interact with reality and possess incomplete knowledge.

Conclusion and Future Work

The paper effectively illustrates the limitations of current epistemic and doxastic logics in handling reflective agents by highlighting their susceptibility to L\"ob's Obstacle. By proposing LSED and LSED*, the authors provide viable alternatives that maintain consistency while supporting reflective reasoning. This work opens new avenues for developing formal systems in AI capable of capturing the nuanced reasoning processes of advanced agents. Future research may explore practical applications in game theory and artificial intelligence, particularly focusing on how these L\"ob-Safe logics can be leveraged to build more robust models of rational cooperation and decision-making in dynamic environments.