True lies

Abstract

A true lie is a lie that becomes true when announced. In a logic of announcements, where the announcing agent is not modelled, a true lie is a formula (that is false and) that becomes true when announced. We investigate true lies and other types of interaction between announced formulas, their preconditions and their postconditions, in the setting of Gerbrandy’s logic of believed announcements, wherein agents may have or obtain incorrect beliefs. Our results are on the satisfiability and validity of instantiations of these semantically defined categories, on iterated announcements, including arbitrarily often iterated announcements, and on syntactic characterization. We close with results for iterated announcements in the logic of knowledge (instead of belief), and for lying as private announcements (instead of public announcements) to different agents. Detailed examples illustrate our lying concepts.

Appendices

Appendix 1: Action models

Assume an epistemic model $M = ( S, R, V )$, and a formula $\varphi$. We recall the semantics of believed public announcement (arrow elimination semantics for public announcement).

$$\begin{aligned} \begin{array}{lcl} M,s\models [\varphi ] \psi&\quad \text{ iff }&M|\varphi ,s\models \psi \end{array} \end{aligned}$$

where epistemic model $M|\varphi = ( S, R^\varphi , V )$ is as M except that for all $a\in A$, $R^\varphi _a\ := \ R_a\cap \ (S\times [\![\varphi ]\!]_M)$; and where $[\![\varphi ]\!]_M := \{ s\in S\mid M,s\models \varphi \}$.

The semantics of truthful public announcement (state elimination semantics for public announcement) is as follows.

$$\begin{aligned} \begin{array}{lcl} M,s\models [\varphi ] \psi&\quad \text{ iff }&M,s\models \varphi \text { implies } M|_\varphi ,s\models \psi \end{array} \end{aligned}$$

where $M|_\varphi = ( S', R', V')$ is such that $S' = [\![\varphi ]\!]_M$, $R' = R \cap ([\![\varphi ]\!]_M \times [\![\varphi ]\!]_M)$, and $V' = V \cap [\![\varphi ]\!]_M$.

These different semantics are the same in the following important sense. If the announcement formula is true, the believed announcement semantics and truthful announcement semantics result in bisimilar models. In other words, as bisimilar models have the same logical theory: they cannot be distinguished by a formula in the logic, such as the resulting beliefs of the agents.

Action model logic is a generalization of public announcement logic, namely to (possibly) non-public actions. We present the version with factual change. In the language we only have to replace the public announcement modalities $[\varphi ]\psi$ with action model modalities $[\mathsf {M},\mathsf {s}]\psi$, for ‘after execution of epistemic action $(\mathsf {M},\mathsf {s})$, $\psi$’ (is true). An action model is like an epistemic model, only instead of a valuation it has preconditions and postconditions. The syntactic primitive $[\mathsf {M},\mathsf {s}]\psi$ may seem a peculiar mix of syntax and semantics, but there is a way to see this as a properly inductive construct in a two-typed language with both formulas and epistemic actions, because the preconditions and postconditions of these actions are again formulas. We now proceed with the technical details.

An action model$\mathsf {M}= (\mathsf {S},\mathsf {R},\mathsf {pre},\mathsf {post})$ for language ${\mathcal L}$ (assumed to be simultaneously defined with primitive construct $[\mathsf {M},\mathsf {s}]\varphi$, see above) consists of a domain $\mathsf {S}$ of actions, an accessibility function$\mathsf {R}: A\rightarrow {\mathcal P}(\mathsf {S}\times \mathsf {S})$, where each $\mathsf {R}(a)$, for which we write $\mathsf {R}_a$, is an accessibility relation, a precondition function$\mathsf {pre}: \mathsf {S}\rightarrow {\mathcal L}$, that assigns to each action its executability precondition, and postcondition function$\mathsf {post}: \mathsf {S}\rightarrow P\not \rightarrow {\mathcal L}$, where it is required that each $\mathsf {post}(\mathsf {s})$ only maps a finite (possibly empty) subset of all atoms to a formula. For $\mathsf {s}\in \mathsf {S}$, $(\mathsf {M},\mathsf {s})$ is an epistemic action.

The semantics of action model execution is as follows.

$$\begin{aligned} \begin{array}{lcl} M,s\models [\mathsf {M},\mathsf {s}] \psi&\text{ iff }&M,s\models \mathsf {pre}(\mathsf {s}) \text { implies } M\otimes \mathsf {M},(s,\mathsf {s}) \models \psi \end{array} \end{aligned}$$

where $M\otimes \mathsf {M}= (S', R', V')$ (known as update ofMwith$\mathsf {M}$, or as the result of executing$\mathsf {M}$inM) is such that $S' = \{(s,\mathsf {s}) \mid M,s\models ~\mathsf {pre}(\mathsf {s})\}$; $((s,\mathsf {s}),(t,\mathsf {t})) \in R_a$ iff $(s,t) \in R_a$ and $(\mathsf {s},\mathsf {t}) \in \mathsf {R}_a$; and $(s,\mathsf {s}) \in V'(p)$ iff $M,s\models \mathsf {post}(\mathsf {s})(p)$ for all $p$ in the domain of $\mathsf {post}$, and otherwise $(s,\mathsf {s}) \in V'(p)$ iff $s\in V(p)$.

A truthful public announcement of $\varphi$ corresponds to a singleton action model $\mathsf {M}= (\mathsf {S},\mathsf {R},\mathsf {pre},\mathsf {post})$ with $\mathsf {S}= \{\mathsf {s}\}$; $\mathsf {R}_a = \{(\mathsf {s},\mathsf {s})\}$ for all $a\in A$; $\mathsf {pre}(\mathsf {s}) = \varphi$ (and empty domain for postconditions). A believed public announcement of $\varphi$ corresponds to a two-point action model $\mathsf {M}= (\mathsf {S},\mathsf {R},\mathsf {pre},\mathsf {post})$ with $\mathsf {S}= \{\mathsf {s},\mathsf {t}\}$; $\mathsf {R}_a = \{(\mathsf {t},\mathsf {s}), (\mathsf {s},\mathsf {s})\}$ for all $a\in A$; $\mathsf {pre}(\mathsf {s}) = \varphi$ and $\mathsf {pre}(\mathsf {t}) = \lnot \varphi$ (and again the empty domain for postconditions). If the designated point is $\mathsf {t}$ it is a public lie and if the designated point is $\mathsf {s}$ it is a truthful (believed) announcement.

Appendix 2: Depicting knowledge and belief

To depict epistemic models, a pair of states in the accessibility relation is represented by an arrow between those states. There are visual conventions to simplify the display of multi-agent ${\mathcal {S}5}$ models and multi-agent ${\mathcal {KD}45}$ models. The conventions guarantee that, given a number of agents, any of the following three uniquely determines the two others: (1) the epistemic frame, (2) the fully displayed visualization, and (3) the simplified visualization.

In ${\mathcal {S}5}$ models all accessibility relations are equivalence relations, so that we can partition the domain in equivalence classes: sets of indistinguishable states. In the visualization we can therefore assume transitive, symmetric, and reflexive closure. Visually, indistinguishable states are connected by a path (transitive closure) of undirected edges (symmetric closure). Reflexive arrows are not drawn (reflexive closure). Singleton equivalence classes do not show in the visualization. But they are inferable as long as we know how many agents there are.

Examples are as follows. The simplified visualization is on the right and the visualization with all arrows of the accessibility relation is on the left.

In ${\mathcal {KD}45}$ models all accessibility relations are transitive, euclidean and serial. Therefore we can divide the domain in clusters. A cluster is a set of indistinguishable states (just as an equivalence class in ${\mathcal {S}5}$), but those states may also appear to be indistinguishable from the perspective of states outside the cluster (unlike in ${\mathcal {S}5}$), the so-called unreachable states. In unreachable states beliefs are incorrect. In the ${\mathcal {KD}45}$ visualization we can therefore assume transitive and euclidean closure, and (this is crucial) we can assume reflexive closure of states in a cluster. This is because a state in a cluster is a state that is reachable from another state. Reflexivity then follows from seriality and euclidicity: from $(x,y) \in R$ and once more) $(x,y) \in R$ follows with euclidicity that $(y,y) \in R)$. It is therefore not so different from the ${\mathcal {S}5}$ visualization. The only difference is that we also draw arrows from states outside a cluster to a (exactly one) cluster. From the source state of such an arrow is not only accessible the state in the cluster that is the endpoint of that arrow, but also all other states in that cluster.

Examples are as follows.

A good way to understand the visualization rules is to reproduce the full visualization on the left from the simplified visualization on the right. The role of singleton clusters takes some getting used to. For example, in the first ${\mathcal {KD}45}$ example, on the right: As there is an outgoing arrow from the left state, it is not a state in a cluster but an unreachable state. Therefore it has no loop in the completion, on the left. As there is an incoming arrow in the right state it is part of a cluster. This is therefore a singleton cluster. This state must have a loop in the completion, on the left. In the last ${\mathcal {KD}45}$ example, as there is an outgoing arrow for the solid relation from the left state, it is, just as in the first example, not a state in a cluster but an unreachable state. However, as there is no outgoing arrow for the dashed relation from the left state, we can infer that it is a singleton cluster and thus infer a loop for the dashed relation. And indeed, on the left, it is there. Etc.

The ${\mathcal {S}5}$ visualization is well-known. The ${\mathcal {KD}45}$ visualization is not well-known.