Abstract
We review and examine in this paper the validity of the different axioms (and inference rules) of knowledge and belief and relating knowledge to belief which have been proposed in the epistemic !epistemic logic literature. In doing so, we encounter many of the problems that epistemic !epistemic logic has had to face in its relatively short (modern) history and provide relevant pointers for more details. We also contribute to this area by providing conditions under which the notion of belief can be formally defined in terms of knowledge , and vice versa. We also prove that certain convoluted axioms dealing only with the notion of knowledge can be derived from understandable interaction axioms relating knowledge and conditional belief !conditional .
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Notes
- 1.
One of these two examples is the following. Suppose that Smith has strong evidence that ‘Jones owns a Ford’ (1) (for instance, Jones has owned a Ford ever since Smith has known him). Then, because of (1) and by propositional logic !propositional , Smith is also justified in believing that ‘Jones owns a Ford or his friend Brown is in Barcelona’ (2), even if Smith has no clue where Brown is at the moment. However it turns out that Jones does not own a Ford and that by pure coincidence Brown is actually in Barcelona. Then, (a) (2) is true, (b) Smith believes (2), and (c) Smith is justified in believing (2). So Smith has a true and justified belief in (2). Intuitively, however, one could not say that Smith knows (2).
- 2.
Fagin et al. (1995) and Meyer and van der Hoek (1995) are the standard textbooks in computer science dealing with epistemic !epistemic logic. Also see the survey Gochet and Gribomont (2006) for a more interdisciplinary approach and Halpern (2003) for a broader account of the different formalisms dealing with the representation of uncertainty.
- 3.
The notion of justification is dealt with in the field of justification logic (Artemov and Fitting 2011). Logical models !formal , logical of (un)awareness have been proposed in economics (Heifetz et al. 2006) and artificial intelligence (Fagin and Halpern 1987) with a recent proposal in Halpern and Rêgo (2009). Some models for the notion of epistemic surprise can be found in Aucher (2007) and Lorini and Castelfranchi (2007).
- 4.
There are a number of logical frameworks that deal with rational agency: Cohen and Levesque’s theory of intention (1990), Rao and Georgeff’s BDI architecture (Georgeff and Rao 1991; Rao and Georgeff 1991), Meyer et al.’s KARO architecture (van Linder et al. 1998; Meyer et al. 2001), Wooldridge’s BDI logic LORA (2000) and Broersen et al.’s BOID architecture (2001).
- 5.
See Aucher (2010) for more details on the perfect external approach and its connection with the other modeling approaches, namely the internal and the imperfect external approaches.
- 6.
Gochet (2007) reviews the various attempts to formalize the notion of knowing how in artificial intelligence and logic.
- 7.
See Deschene and Wang (2010) for a survey of approaches to computer security issues which use epistemic !epistemic logic.
- 8.
- 9.
Wheeler’s argument against axiom B is based on two theorems derivable in the logic KTB. One of them is the following: \(K(\varphi \rightarrow K\psi ) \rightarrow (\neg K\neg \varphi \rightarrow \psi )\). If \(\varphi\) stands for ‘the agent sees some smoke’ and ψ stands for ‘there is fire’, then the consequent of this theorem states that if the agent considers it possible that he sees some smoke (without necessarily being sure of it), then there is fire. This conclusion is obviously counterintuitive.
- 10.
- 11.
- 12.
The modal operators of weak !weak and strong !strong , full belief are denoted “\(B\varphi\)” and “\(C\varphi\)” respectively in Lenzen (1978).
- 13.
These two operators are respectively denoted “\({C}^{\psi }\varphi\)” and “\({B}^{\psi }\varphi\)” in (Lamarre and Shoham 1994).
- 14.
- 15.
In both philosophy and computer science , there is formalization of the internal point of view. Perhaps one of the dominant formalisms for this is auto-epistemic !epistemic logic (Moore 1984, 1995). In philosophy, there are models of full !strong , full belief like the one offered by Levi (1997) which is also related to ideas in auto-epistemic !epistemic logic. See Aucher (2010) for more details on the internal approach and its connection to the other modeling approaches, namely the imperfect and the perfect external approaches.
- 16.
The classification is as follows. If X, Y, Z are epistemic operators, \(X\varphi \rightarrow Y Z\varphi\) are called positive introspection formulas, \(\neg X\varphi \rightarrow Y \neg Z\varphi\) are called negative introspection formulas, \(XY \varphi \rightarrow Z\varphi\) are called positive extraspection formulas, \(X\neg Y \rightarrow \neg Z\varphi\) are called negative extraspection formulas, and \(X(Y \varphi \rightarrow \varphi )\) are called trust formulas.
- 17.
Here is the proof:
$$\displaystyle{\begin{array}{l@{\quad }l@{\quad }l} 1\quad &K\varphi \rightarrow B\varphi \quad &\text{Axiom KB1} \\ 2\quad &K\neg K\varphi \rightarrow B\neg K\varphi \quad &\text{KB1}: \neg K\varphi /\varphi \\ 3\quad &B\varphi \rightarrow \neg B\neg \varphi \quad &\text{Axiom D}\\ 4\quad &B\neg \varphi \rightarrow \neg B\varphi \quad &\text{3, contraposition} \\ 5\quad &B\neg K\varphi \rightarrow \neg BK\varphi \quad &4: K\varphi /\varphi \\ 6\quad &\neg K\varphi \rightarrow K\neg K\varphi \quad &\text{Axiom 5}\\ 7\quad &\neg K\varphi \rightarrow B\neg K\varphi \quad &\text{6,2, Modus Ponens} \\ 8\quad &\neg K\varphi \rightarrow \neg BK\varphi \quad &\text{7,5, Modus Ponens} \\ 9\quad &BK\varphi \rightarrow K\varphi \quad &\text{8, contraposition}. \end{array} }$$ - 18.
Note that the axiom \(({B}^{\psi }(\varphi \rightarrow \varphi ^{\prime}) \wedge {B}^{\psi }\varphi ) \rightarrow {B}^{\psi }\varphi ^{\prime}\) and the inference rule from \(\varphi\) infer \({B}^{\psi }\varphi\) are both derivable in \((\mathsf{P})_{{B}^{\psi }}\). Therefore, \((\mathsf{P})_{{B}^{\psi }}\) is also a normal modal !modal logic.
- 19.
That is, S4 plus .4: \((\varphi \wedge \hat{K}K\psi ) \rightarrow K(\varphi \vee \psi )\); see Sect. 5.5.3.
- 20.
That is, S4 plus .3.2: \((\hat{K}\varphi \wedge \hat{ K}K\psi ) \rightarrow K(\hat{K}\varphi \vee \psi )\); see Sect. 5.5.3.
- 21.
For this definition to be consistent, we have to add another constraints that Stalnaker does not mention: in this definition, knowledge should only deal with propositional facts belonging to the propositional language \(\mathcal{L}_{0}\). Indeed, assume that the agent believes non-p (formally B¬p). Then clearly the agent knows that she believes non-p by KB2 (formally KB¬p). However, assume that p is actually true. If we apply this definition of knowledge , then, if she learnt that p (which is true), she should still believe that she believes non-p (formally BB¬p), so she should still believe non-p (formally B¬p), which is of course counterintuitive. This restriction on propositional knowledge does not produce a loss of generality because we assume that the agent knows everything about her own beliefs and disbeliefs.
- 22.
- 23.
Lenzen uses axiom KB3′ instead of KB3, but one can easily show that the replacement does not invalidate the proposition.
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Acknowledgements
I thank Manuel Rebuschi and Franck Lihoreau for helpful comments on this paper. I also thank the anonymous English native speaker referee for detailed comments.
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Aucher, G. (2014). Principles of Knowledge, Belief and Conditional Belief. In: Rebuschi, M., Batt, M., Heinzmann, G., Lihoreau, F., Musiol, M., Trognon, A. (eds) Interdisciplinary Works in Logic, Epistemology, Psychology and Linguistics. Logic, Argumentation & Reasoning, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-03044-9_5
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