Abstract
Among the non-monotonic reasoning processes, abduction is one of the most important. Usually described as the process of looking for explanations, it has been recognized as one of the most commonly used in our daily activities. Still, the traditional definitions of an abductive problem and an abductive solution mention only theories and formulas, leaving agency out of the picture. Our work proposes a study of abductive reasoning from an epistemic and dynamic perspective. In the first part we explore syntactic definitions of both an abductive problem in terms of an agent’s information and an abductive solution in terms of the actions that modify the agent’s information. We look at diverse kinds of agents, including not only omniscient ones but also those whose information is not closed under logical consequence and those whose reasoning abilities are not complete. In the second part, we look at an existing logical framework whose semantic model allows us to interpret the previously stated formulas, and we define two actions that represent forms of abductive reasoning.
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Notes
Note how the operations of adding and removing information cannot be fully specified until we fix a specific notion and a semantic model. For example, in the possible worlds semantics in which knowledge is understood as what is true in all epistemically possible situations, change in knowledge amounts to expand or shrink this set of possible worlds. But if in the same setting we work with the notion of beliefs, usually understood as what is true in only the most plausible worlds, then change in beliefs amounts only to change in the plausibility ordering.
A systematic revision of the different cases that arise can be found in Soler-Toscano and Velázquez-Quesada (2010).
More cases can be eliminated with further assumptions about the agent’s information, like truth or consistency; see Subsect. 3.4.
Physical laws can be properly considered logical formulas, but the action that we define below for abductive reasoning (Definition 17) uses a formula and a rule as different kinds of information. In this setting, it is more appropriate to consider that physical laws are rules.
In fact, non-Euclidean geometries originated when going in depth into this question.
Observe how this makes every solution consistent.
Observe how this form of abductive reasoning can be seen as a particular form of belief revision driven by the agent’s inferential abilities: she has observed and therefore knows \(\psi \), but she also knows that from \(\varphi \) she can derive \(\psi \), so she will revise her beliefs in order to incorporate \(\varphi \) into them. The relation between abductive reasoning and belief revision has been already studied, e.g., Boutilier and Becher (1995), Aliseda (2006).
The indistinguishability relation should not be confused with the equal plausibility relation, given by the intersection \(\le \cap \ge \).
This does not hold in the general case because the operations do change the model (the access set function and the plausibility relation), therefore affecting the truth-value of formulas that can see such change (formulas including \(\mathrm{A }{}\), \(\langle \sim \rangle \,{}\) or \(\langle \le \rangle \,{}\)); cf. Holliday and Icard (2010).
Strictly speaking, \(\lnot \chi \) should become \(\lnot \lnot l\), but using \(l\) makes the example clearer. Most importantly, as we explain further on, it is easy to provide Mary with the reasoning ability to get \(l\) from \(\lnot \lnot l\).
Such action is more naturally represented as the announcement of \(e\), but in our model Mary has explicitly \(\lnot \lnot e\). There are two options here. The most elegant one is to introduce an action representing Mary’s inference from \(\lnot \lnot e\) to \(e\) [see, again, Velázquez-Quesada (2010)], and then work with the announcement in its most natural form (\(e\)). Due to space reasons, here we have chosen to work with this special form of observing that the electric line works.
If \(\psi \) is not propositional, the effect of observing it is not interchangeable with an abductive step; see Holliday and Icard (2010).
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Acknowledgments
We thank Johan van Benthem and Ángel Nepomuceno-Fernández for their valuable comments and suggestions on early versions of this paper. We acknowledge support from the project FFI2011-15945-E (Ministerio de Economía y Competitividad, Spain) and the Excellence Research Project of the Junta de Andalucía P10-HUM-5844. We would also like to express our gratitude to the anonymous referees.
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Soler-Toscano, F., Velázquez-Quesada, F.R. Generation and Selection of Abductive Explanations for Non-Omniscient Agents. J of Log Lang and Inf 23, 141–168 (2014). https://doi.org/10.1007/s10849-014-9192-1
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DOI: https://doi.org/10.1007/s10849-014-9192-1