Abstract
There is a strong context dependency in meaning of modalities in natural languages. Kratzer [9] demonstrates how to deal with this problem within possible world semantics. In this paper, we propose to interpret epistemic modalities in background of an epistemic state. Our analysis is a meta-linguistic one and we extensively use the proof-theoretic consequence relation. We define, then, a belief structure and introduce a belief structure revision operator. We call this framework Logic of Belief Structures (LBS). Then, we apply LBS to formalization of belief revision and interpretation of conditionals and investigate the relationship between belief revision and conditionals. Furthermore, we propose two types of conditionals, epistemic and causal conditionals.
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Notes
- 1.
Because \((p \rightarrow p)\) is a FOL-theorem, it holds: \(MIGHT_{T_2} (p \rightarrow p)\) iff \(T_2\) is consistent.
- 2.
Here, function tr is the translation function from English sentences to FO-sentences.
- 3.
Note that it holds: \(KNOWN_{T_{9b}}\) \(p_{keys} \Rightarrow MUST_{T_{9b}}\) \(p_{keys}\). See (7e).
- 4.
- 5.
According to definition of \(v^s_W\), \(v^s_W(top(BS,k)) = \{w \in W\): all formulas in top(BS, k) are true in \(w\}\).
- 6.
These orders are a modification of comparative possibility in Lewis [11, p. 52].
- 7.
In domain cons-max(BS), \(\preccurlyeq _{BS}\) is also reflexive and connected.
- 8.
In domain cons-max(BS), \(\approx _{BS}\) is also reflexive. Thus, in cons-max(BS), \(\approx _{BS}\) is an equivalence relation.
- 9.
- 10.
However, AGM-theory has a nice correspondence with the probability theory [4, Chap. 5]. Our approach is difficult to relate with a probability theory.
- 11.
- 12.
Here, we represent counterfactual conditional with \(\mapsto \).
- 13.
Gärdenfors [4, Sect. 4.5] gives an insightful description of Grove’s system.
- 14.
Some parts of Lakatos’ discussion on scientific research programs in [10] can be described within LBS. In belief structures of scientists, basic theories are more trusted than their auxiliary hypotheses (\(BT > AH\)). Suppose that the set nO of observation data is consistent with BT but inconsistent with \(BT \cup AH\). In such a case, scientists would try to find the set nAH of new auxiliary hypotheses such that \(nO \cup BT \cup nAH\) is consistent. In this way, a basic theory can be protected against new anomalies.
- 15.
This research was supported by Grant-in-for Scientific Research, Scientific Research C (24520014): The Construction of Philosophy of Science based on the Theory of Multiple Languages. Finally, I would like to thank two reviewers for useful comments.
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Nakayama, Y. (2015). Formal Analysis of Epistemic Modalities and Conditionals Based on Logic of Belief Structures. In: Murata, T., Mineshima, K., Bekki, D. (eds) New Frontiers in Artificial Intelligence. JSAI-isAI 2014. Lecture Notes in Computer Science(), vol 9067. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48119-6_4
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