Rule-Based Reasoners in Epistemic Logic

In this paper, we offer a balanced response to the problem of logical omniscience, whereby agents are modeled as non-omniscient yet still logically competent reasoners. To achieve this, we account for the deductive steps that form the epistemic state of an agent. In particular, we introduce operators for applications of inference rules and design a possible-worlds model which is (a) equipped with a syntactic valuation, determining the agent’s (explicit) knowledge, and (b) suitably structured by rule-induced transitions between worlds. As a result, we obtain a detailed analysis of the agent’s reasoning processes. We then offer validities that exemplify how the problem of logical omniscience is avoided and compare our response to others in the literature. A sound and complete axiomatization is also provided. We finally show how simple extensions of this setting make it compatible with tools from Dynamic Epistemic Logic (DEL) and open to the incorporation of empirical findings on human reasoning.

1. 1.

   In [18] an impossible-worlds semantics is presented, but again reasoning is captured via modalities standing for a number of steps; this raises concerns analogous to the ones discussed before.

2. 2.

   A notable exception where awareness is affected by reasoning is given in [23]; in what follows, we design a rule-based approach but without appealing to a notion of awareness.

3. 3.

   We emphasize that \(R_i\) denotes a single rule instance. The rule, which is in fact a pair, composed of the set of premises and the conclusion, is given in terms of the notation \(\leadsto \) for readability and convenience.

4. 4.

   Recall that \(V_2: W \rightarrow \mathcal {P}(\mathcal {L})\) and that \(\mathcal {L} := \mathcal {L}_P \cup \mathcal {L}_\mathcal {R}\). Moreover, it should be clear that the world u whose existence is guaranteed by condition 1, is such that it contains the conclusion of \(R_i\), by condition 2, and the rule \(R_i\) is necessarily sound due to condition 3.

5. 5.

   We use DNE, MP, and CI to label particular instances of Double Negation Elimination, Modus Ponens, and Conjunction Introduction – the ones indicated in parentheses. This labeling only serves the readability of the formulas.

6. 6.

   We note that the frameworks described in [1,2,3] that extend the idea of state-transitions to multi-agent settings are particularly interesting for the development of multi-agent variants of our framework too.

7. 7.

   More on why this is a worthwhile task can be found in [6].

8. 8.

   As usual in DEL [5, 10], we can add action operators to our language and capture their effect via model transformations triggered by the action. A formula with dynamic operators, of the form \([\alpha ] \phi \), is evaluated by examining what the truth value of \(\phi \) is at a transformed model, obtained via action \(\alpha \).

9. 9.

   In fact, this idea can be also pursued along the lines of DEL. The reasoning capacity c of the agent, as an additional component of our models, can be updated (i.e. reduced) following each rule application.

This work is funded by the Dutch Organization for Scientific Research, under the “PhDs in the Humanities” scheme (project number 322-20-018). The author also thanks the audience of the student session of ESSLLI 2018 and the anonymous reviewers for their valuable feedback.

Authors and Affiliations

1. ILLC, University of Amsterdam, Amsterdam, Netherlands

   Anthia Solaki

Corresponding author

Correspondence to Anthia Solaki .

Editors and Affiliations

1. Universität Stuttgart, Stuttgart, Germany

   Jennifer Sikos

2. Department of Philosophy, University of Maryland, College Park, MD, USA

   Eric Pacuit

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