Abstract
It has been an open question whether or not we can define a belief revision operation that is distinct from simple belief expansion using paraconsistent logic. In this paper, we investigate the possibility of meeting the challenge of defining a belief revision operation using the resources made available by the study of dynamic epistemic logic in the presence of paraconsistent logic. We will show that it is possible to define dynamic operations of belief revision in a paraconsistent setting.
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Notes
Cf. Brown and Priest (2004) for a discussion of contradictions in the history of science and a paraconsistent logic dealing with some of those contradictions.
Cf. Priest (2006) chaps. 7 and 8.
Mares took expansion and contraction operations as primitive and defined them using the relevant logic R. He then defined revision in terms of expansion and contraction via Levi Identity. As Tanaka (2005) shows, however, some of the AGM revision postulates fail if we define revision in terms of the Levi Identity in paraconsistent logics. It is thus not clear to what extent Mares has succeeded in defining revision paraconsistently.
For a summary of paraconsistent approaches to AGM operations, see Wassermann (2011).
LP was introduced by Priest (1979).
Complete so as to guarantee the existence of greatest and least bounds which we require for the semantics of modalities, as we will see below.
Cf. Fagin et al. (1995).
For special cases of many-valued epistemic logic, for instance LP, defining knowledge in this way allows \(K\varphi \) to be evaluated as both true and false at some world. We acknowledge that there are other plausible definitions of knowledge in many-valued settings. For example, one can impose a classical definition of knowledge such that \(K\varphi \) is 1 (the top value) if it is 1 in all epistemically equivalent worlds. But this would make the logic non-paraconsistent: \(Kp\) would entail everything when \(\nu _w(p) = b\) for all \(w \in W\).
The logic we develop in this section may best be described as an ‘epistemic doxastic logic’ since \(B\varphi \) supplements rather than replaces knowledge operator. For simplicity, however, we refer to it as a ‘doxastic logic’ rather than ‘epistemic doxastic logic’.
For the concerned reader, well-foundedness guarantees the limit assumption which simplifies technical and conceptual details that are not critical in the context of this paper.
This approach was further explored in a multi-agent system in Baltag and Smets (2008).
That the end-result is a logically equivalent formula follows from the fact that every step in the transformation produces an equivalent formula, pushing the dynamic modality inside until it is attached to a propositional variable and then eliminated. Take the following simple example to illustrate the point: \([!p](q\vee r) \equiv [!p]q \vee [!q]\equiv p \rightarrow q \vee [!p]r \equiv p \rightarrow q \vee p \rightarrow r\).
Our choice of conditional is fully dictated by the semantics of \([!p]q\). For instance, the \(\mathrm{RM3}\) conditional would not suffice. The \(\mathrm{RM3}\) table for \(\rightarrow \) is given by:
Consider \([!p]q\) and take \(p = 1\) and \(q = b\). Then \([!p]q = b\), but \(p\rightarrow q = 0\). Nevertheless, one could define a belief revision \([!\varphi ]\psi \) that reduces to the RM3 arrow in the special case \([!p]q\). We leave this as an open problem.
That is, extending \(\mathrm{LP}^\rightarrow \) with any axiom or rule that produce new theorems yields a non-paraconsistent logic.
That is, adding any non-provable classical tautologies as axioms yields a non-paraconsistent logic.
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Girard, P., Tanaka, K. Paraconsistent dynamics. Synthese 193, 1–14 (2016). https://doi.org/10.1007/s11229-015-0740-2
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DOI: https://doi.org/10.1007/s11229-015-0740-2