Abstract
We propose a dynamic logic of lying, wherein a ‘lie that \(\varphi \)’ (where \(\varphi \) is a formula in the logic) is an action in the sense of dynamic modal logic, that is interpreted as a state transformer relative to the formula \(\varphi \). The states that are being transformed are pointed Kripke models encoding the uncertainty of agents about their beliefs. Lies can be about factual propositions but also about modal formulas, such as the beliefs of other agents or the belief consequences of the lies of other agents. We distinguish two speaker perspectives: (Obs) an outside observer who is lying to an agent that is modelled in the system, and (Ag) an agent who is lying to another agent, and where both are modelled in the system. We distinguish three addressee perspectives: (Cred) the credulous agent who believes everything that it is told (even at the price of inconsistency), (Skep) the skeptical agent who only believes what it is told if that is consistent with its current beliefs, and (Rev) the belief revising agent who believes everything that it is told by consistently revising its current, possibly conflicting, beliefs. The logics have complete axiomatizations, which can most elegantly be shown by way of their embedding in what is known as action model logic or in the extension of that logic to belief revision.
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25 July 2018
The original publication of the article is missing the funding information.
25 July 2018
The original publication of the article is missing the funding information.
25 July 2018
The original publication of the article is missing the funding information.
25 July 2018
The original publication of the article is missing the funding information.
Notes
Strictly binding formal belief preconditions to these terms is intended as a strong barrier to avoid pitfalls and digressions into philosophy and epistemology. This is necessary because their everyday usage is ambiguous. We have tried to stay close to dictionary meanings and reported usage. This footnote reports our findings. ‘Truthful’ is synonymous with ‘honest’. Dictionaries do not make a difference between an agent telling the truth and an agent believing that it is telling the truth. A modal logician has to make a choice. We mean the latter, exclusively. A truthful announcement may therefore not be a true announcement. It is tempting, when looking for a single term, to call a truthful agent a truthteller (a somewhat archaic usage) but that would imply that we would not require a truthteller to tell the truth, maybe stretching usage too far. So we did not, and stick to ‘truthful agent’. The dictionary meaning for the verb bluff is ‘to cause to believe what is untrue’ or ‘to deceive or to feign’. Feigning belief in \(p\) means suggesting belief in \(p\), for example by saying that you believe it, even if this is not the case. That corresponds to \(\lnot B_ap\) as precondition. However, this would make lying a form of bluffing, as \(B_a\lnot p\) implies \(\lnot B_ap\). It is common and according to Gricean conversational norms that saying something that you believe to be false is worse than (or, at least, different from) saying something that you do not believe to be true. This brings us to \(\lnot B_ap\wedge \lnot B_a\lnot p\) (\(\wedge \) is conjunction), equivalent to \(\lnot (B_ap\vee B_a\lnot p)\).
Suppose that you believe that \(\lnot p\) and that you lie that \(p\). Later, you find out that your belief was mistaken because \(p\) was really true. You can then with some justification say “Ah, so I was not really lying.”
Alternatively, we can take \([_a\varphi ]\) as a primitive operator of the logic and define truthtelling, lying and bluffing by abbreviation as, respectively, \([!_a\varphi ]\psi \leftrightarrow (B_a\varphi \rightarrow [_a\varphi ]\psi )\),
, and 
. We then need two axioms only, namely \([_a\varphi ] B_a\psi \leftrightarrow B_a[_a\varphi ] \psi \) and \([_a\varphi ] B_b\psi \leftrightarrow B_b(B_a\varphi \rightarrow [_a\varphi ] \psi )\). This suggestion is from Emiliano Lorini.From \(\lnot (B_ap\vee B_a\lnot p)\) follows \(\lnot B_ap\), and with negative introspection we get \(B_a\lnot B_ap\).
Reported in Hales (2011), an \(adf\) is a multi-agent generalization of the (single-agent) disjunctive form as in Meyer and van der Hoek (1995), p. 35 (where it is called \(\mathcal{S 5}\) normal form), and a special case of the disjunctive form as in Agostino and Lenzi (2008). An \(adf\) contains no stacks of \(B_a\) operators without an intermediary \(B_b\) operator for another agent, i.e., if \(B_a\chi \) is a subformula of an \(adf\) and \(B_a\chi ^{\prime \prime }\) is a subformula of \(\chi \) then there is an agent \(b\ne a\) and a \(\chi ^{\prime }\) such that \(B_b\chi ^{\prime }\) is a subformula of \(\chi \) and \(B_a\chi ^{\prime \prime }\) is a subformula of \(\chi ^{\prime }\).
A well-preorder is a well-founded preorder. A preorder is a reflexive and transitive relation. A relation is well-founded if every non-empty subset of the domain has a minimal element.
The former represents weak belief and the latter true strong belief. There are yet other epistemic operators in this setting: safe belief, conditional belief,\(\ldots \) We restrict our presentation to (weak) belief.
Given a plausibility epistemic model \(M\) and a state \(s\) in its domain, define \(R_a^\varphi \) as: \((s, t) \in R_a^\varphi \) iff [\(s\sim _at\), \(t\le _at^{\prime }\) for all \(t^{\prime }\) such that \(s\sim _at^{\prime }\), and \(M,t\models \varphi \)], and define the semantics of \(B_a^\varphi \) as: \(M,s\models B_a^\varphi \psi \) iff \(M,t\models \psi \) for all \(t\) such that \((s,t)\in R_a^\varphi \). We now have that
. The axioms for plausible agent lying are more complex (they also involve conditional knowledge modalities).
, and 
. We then need two axioms only, namely 
. The axioms for plausible agent lying are more complex (they also involve conditional knowledge modalities).References
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Acknowledgments
Hans van Ditmarsch is also affiliated to IMSc, Chennai, as associated researcher. I thank the anonymous reviewers of the journal Synthese for their comments, and for their persistence. I gratefully acknowledge comments from Alexandru Baltag, Jan van Eijck, Patrick Girard, Barteld Kooi, Fenrong Liu, Emiliano Lorini, Yoram Moses, Eric Pacuit, Rohit Parikh, Ramanujam, Hans Rott, Sonja Smets, Rineke Verbrugge, and Yanjing Wang. As my work on lying has a long history (from 2008 onward), I am concerned I may have forgotten to credit yet others, for which my apologies. As the editor of the special issue in which this contribution appears, Rineke had many valuable comments in addition to those by the reviewers, and was as always extremely encouraging. Emiliano spared me the embarrassment of including unsound axioms (and an incorrect action model) in the axiomatization of agent announcement logic, for which infinite thanks.
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van Ditmarsch, H. Dynamics of lying. Synthese 191, 745–777 (2014). https://doi.org/10.1007/s11229-013-0275-3
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DOI: https://doi.org/10.1007/s11229-013-0275-3