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On the factivity of implicit intersubjective knowledge

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Abstract

The concept of knowledge can be modelled in epistemic modal logic and, if modelled by using a standard modal operator, it is subject to the problem of logical omniscience. The classical solution to this problem is to distinguish between implicit and explicit knowledge and to construe the knowledge operator as capturing the concept of implicit knowledge. In addition, since a proposition is said to be implicitly known just in case it is derivable from the set of propositions that are explicitly known by using a certain set of logical rules, the concept of implicit knowledge is definable on the basis of the concept of explicit knowledge. In any case, both implicit and explicit knowledge are typically characterized as factive, i.e. such that it is always the case that what is known is also true. The aim of the present paper is twofold: first, we will develop a dynamic system of explicit intersubjective knowledge that allows us to introduce the operator of implicit knowledge by definition; secondly, we will show that it is not possible to hold together the following two theses: (1) the concept of implicit knowledge is definable along the lines indicated above and (2) the concept of implicit knowledge is factive.

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Notes

  1. See Fagin et al. (1995) and Hoek and Meyer (1995) for standard introductions to this topic.

  2. See Fagin and Halpern (1988), Fagin et al. (1995) and Levesque (1984) for an introduction. In (Fagin and Halpern (1988), pp. 317–318), we find: “To represent the knowledge of agent \(i\), we allow two modal operators \(K_{i}\) and \(X_{i}\), standing for implicit knowledge and explicit knowledge of agent \(i\), respectively. Implicit knowledge is the notion we have been considering up to now: truth in all worlds that the agent considers possible. On the other hand, an agent explicitly knows a formula \(\upvarphi \) if he is aware of \(\upvarphi \) and implicitly knows \(\upvarphi \). Intuitively, an agent’s implicit knowledge includes all the logical consequences of his explicit knowledge”. A similar characterization is given in (Cresswell (1972), p. 11), and (Stalnaker (1999), pp. 241–242) .

  3. This kind of approach is proposed in Konolige (1986). See Ågotnes and Alechina (2007) and Jago (2009) for further developments.

  4. The system introduced in the following section is a modification of the system DES4 of dynamic epistemic logic described and discussed in Duc (1997). DES4 was introduced without an appropriate semantics, but a Kripke style semantics was then presented in Ågotnes and Alechina (2007). This system has some shortcomings, due to the fact that not every formula of the language in which the system is formulated is implicitly knowable: in particular, not every axiom is implicitly knowable. This limitation is counterintuitive and hinders a direct interpretation of implicit knowledge as knowledge deriving from a possible chain of inferential steps, since every axiom is surely accessible in a unique inferential step. However, it is worth noticing that the system we are going to introduce is more powerful than DES4, so that the limitative conclusions we will prove are valid with respect to DES4 as well. The present modification originates in the works about explicit logic of knowledge presented in Artemov (2008). Actually, the operator of implicit knowledge can be viewed as the existential generalization of the explicit operators adopted in this paper.

  5. A consequence of axiom FK2 is that, if a proposition \(\upvarphi \) is knowable, \(\langle \mathbf{F}\rangle \mathbf{k}(\upvarphi )\), then it is not possible to explicitly know that \(\upvarphi \) is not explicitly known, \(\mathbf{k}\lnot \mathbf{k}(\upvarphi )\). This consequence is acceptable insofar as the concept of knowledge captured by k is the concept of intersubjective and stable knowledge. Indeed, knowledge of \(\lnot \mathbf{k}(\upvarphi )\) is not stable, provided that \(\upvarphi \) is assumed to be knowable.

  6. DE* is a weakened version of the system DES4 proposed in Duc (1997).

  7. The idea of assigning set of formulas to worlds in order to model what is explicitly known is proposed in Fagin and Halpern (1988) and developed in Fagin et al. (1995), Fitting (2005) and Artemov (2008) in different directions.

  8. This is in accordance with the way in which the outcomes of scientific experiments are reported.

  9. The standard modal parts of the proof of completeness are omitted. See (Blackburn et al. (2001), Chap. 4), for details.

  10. FKI provides us with a second strategy to cope with the problem of logical omniscience. As highlighted in (Stalnaker (1999), p. 242), there are two general strategies for facing the problem. According to a first one, we can extend the ordinary concept of belief by introducing the distinction between explicit and implicit belief. This is the strategy commonly proposed. According to a second one, we can extend the ordinary concept of agent, by idealizing her epistemic power along the lines proposed here.

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  1. Department of Philosophy, Catholic University of Milan, L. A. Gemelli 1, 20123 , Milan, Italy

    Alessandro Giordani

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  1. Alessandro Giordani
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Giordani, A. On the factivity of implicit intersubjective knowledge. Synthese 191, 1909–1923 (2014). https://doi.org/10.1007/s11229-013-0381-2

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