2. PHILOSOPHICAL REMARKS

Before describing existing formalisms that connect knowledge and obligation, let us spend some time discussing the subject of deontic epistemology and epistemic normativity from a purely philosophical point of view. Even if it is not our main goal here, we will briefly mention some philosophical works that fall within the domain of our inquiry. We are aware that the field is very broad, and the topics we have selected can very well be criticized for falling short of describing the field fully. For that reason, we advise the reader not to take our work as a full survey of the field, but rather as a subjective exposition of the field by working logicians. We focus our attention on those philosophical issues that raise some important questions for working logicians who may try to give a formal account of the topics in question.

In this section we first point out that knowledge and obligation appear in popular ethical treatises by pointing out a well-known example from the modern ethical literature. Then, we underline some aspects of obligations that bear some logical significance, such as normativity. This carries our discussion to the topic of beliefs, which is a central issue in epistemic and doxastic logic. Within the notion of belief, we concentrate on ‘doxastic voluntarism’. This debate is interesting as it carries some meta-logical value that most logicians have not yet broadly addressed.

We start by recalling a well-known ethical debate. In a famous thought-experiment, Peter Singer discusses the following scenario (Singer Reference Singer1993). Assume that on your way to your department at university, there is a small pond, and one day you see that a small child has fallen in it and is drowning. Clearly, no one can deny that you have an obligation to save the child even if it gets your clothes muddy and delays you from work. The principle behind such actions, Singer maintains, is this: ‘If it is in our power to prevent something very bad from happening, without thereby sacrificing anything of comparable moral significance, we ought to do it’. Even though this principle looks very plausible and reasonable, its generalizations are not always that easy to follow. For example, following Singer's example, extreme poverty is bad and it is in our power as affluent people to reduce it ‘without sacrificing anything of comparable moral significance’. Yet, we do not usually seem to spend sufficient effort to reduce the world's poverty even though, to some extent, it is in our power to reduce it.

Among many other objections to this thought-experiment discussed by Singer (Reference Singer1997) himself, one is of epistemic nature. Namely, in the pond example, we know that the child is about to drown. Thus, the knowledge of this situation triggers our action by imposing the obligation. On the other hand, once, say, poverty in Africa is considered, we usually do not know that a specific person in a specific town is in extreme poverty even though we consider it epistemically likely or hepistemically possible that, say, someone in Sudan is malnourished. Therefore, these two situations impose different types of obligations. The first obligation follows from a definitive knowledge of the event whereas in the second obligation, we do notpossess (full) knowledge of the situation. It is beyond our capacity to possess full knowledge of the problem of world's poverty. Thus, it may be claimed that we do have less obligation in the absence of the relevant knowledge. Another difference betweensaving the drowing child and saving some life in a poor country in Africa is that we know that by jumping into the pond we can save the child, whereas in the case of donating to a charity, we believe that it is highly likely that our contribution will help end suffering, yet we do not know. Peter Unger (Reference Unger1996) raises the very related issue that lack of full knowledge of the outcome of the action may create some hesitation in the implementation of the action.

Thus, our knowledge state influences which obligations we have. Another well-studied epistemic-deontic relation is the one between beliefs and permission. In (Pollock Reference Pollock1987), epistemic norms are defined as those norms which describe if it is epistemically permissible to hold various beliefs. A number of philosophers discussed different properties of norms. Some defined them in terms of whether they are rule based or not (Boghossian Reference Boghossian2008; Engel Reference Engel and Strawinski2007; Horwich Reference Horwich2008). Some identified themselves as internalist or externalist theorists (BonJour and Sosa Reference BonJour and Sosa2003; Feldman and Conee Reference Feldman, Conee and Kornblith2001; Kornblith Reference Kornblith2001; Pollock Reference Pollock1987; Sosa Reference Sosa, Greco and Sosa1999; Wright Reference Wright2008). Some philosophers take a rather radical position by maintaining that there cannot be any philosophical grounds for our familiar epistemic norms (Horwich Reference Horwich2008).

But before we adopt a certain theory of what epistemic norms are, it is important to ask oneself the question: To what extent are our beliefs voluntary? (Alston Reference Alston1988). In this respect, doxastic voluntarism is the view that our beliefs are voluntary and that we can control them (Feldman Reference Feldman1988).

Let us briefly discuss a well-known argument from voluntarism. The so-called ‘Voluntarism Argument’ of Feldman has been endorsed by several philosophers like Plantinga and Alston (Alston Reference Alston1988; Feldman Reference Feldman1988; Plantinga Reference Plantinga and Austin1988).

1. 1. Doxastic voluntarism is false.

2. 2. If doxastic voluntarism is false, then no one has epistemic obligations.

3. 3. Therefore, no one has epistemic obligations.

At first sight, this argument seems sound and plausible: I cannot control believing that what I see in front of me is my computer screen, so I cannot be held responsible for this belief. However, several objections can be raised to this argument and some have been summarized in Feldman (Reference Feldman1988). For example, one could simply maintain that beliefs are voluntary and therefore disagree with (1). A more subtle reason for rejecting (1) has been proposed by Heil (Reference Heil1983) who claims that even though we may not be able to control our beliefs directly, we can control them indirectly by performing actions that lead us to alter the way we form beliefs. We agree with Feldman that there is a distinction between epistemic obligations to believe something (right now) and obligations to take actions to obtain evidence. Apart from rejecting (1), also (2) can and has been criticized. As Feldman (Reference Feldman1988) mentions, one could reject (2) by maintaining that obligations pertain to actions rather than belief states. Feldman himself endorses a different objection to the voluntarism argument. Namely, he rejects (2) by claiming that we can have obligations concerning involuntary beliefs in the same way as we can have a legal obligation to pay our mortgage even if our financial situation no longer allows for this. Thus, according to Feldman, the famous ‘Ought’ does not imply ‘Can’ any more, at least in the case of epistemic obligations.

In a fairly recent article, Weatherson (Reference Weatherson2007) discusses the voluntarism argument. Contrary to Feldman, Weatherson agrees with (2) but attempts to refute the argument by rejecting (1). He claims that there are two kinds of beliefs: perceptual and inferential ones. The perceptual beliefs are spontaneous and involuntary while the inferential beliefs, the ones that involve reasoning, are voluntary ‘in that we have the capacity to check them by paying greater heed to counter-possibilities’ [ibid]. Involuntary or perceptual beliefs, Weatherson argues, are best evaluated using externalist standards like reliability. Beliefs that involve reasoning on the other hand, are justified only when ‘well supported by reasons’.

Parikh (Reference Parikh2008) distinguishes lower level perceptual animal beliefs and higher level inferential beliefs and furthermore proposes a formal system to reason about such beliefs. Another significant separation of beliefs comes from Gendler. In her insightful exposition, Gendler (Reference Gendler2008) distinguishes two kinds of beliefs: belief and alief. The author cites several psychological experiments that she uses to address this distinction: people who are hesitant to drink a glass of juice with a completely sterilized dead cockroach in it or people who hesitate to wear a laundered shirt previously worn by a person they dislike. In such cases, as Gendler points out, ‘they believe that the items in question are harmless, they also alieve something very different’ (emphasis is hers), namely, that those objects are ‘filthy’ and those people felt that they should ‘stay away’ from them. This distinction accounts for the belief–action mismatch in some situations. In other words, we believe that the aforementioned glass of juice is harmless, but we alieve that it is filthy. The action of avoiding that glass of juice is caused by the alief in this case, not by the belief. This issue indirectly brings along the concept of alief revision. If we alieve p, and then learn not-p, what happens then? Gendler addresses this question, but does not provide a full answer akin to the famous discussions on belief revision.

An important aspect of epistemic normativity that we already briefly touched upon is its connection with actions. This debate revolves around issues such as the rationality criterion for actions, the relation between different levels of beliefs and actions and, moreover, the interaction between actions and knowledge. Alston (Reference Alston1988), when discussing the connection between deontology and epistemic justification, indicates that justification of actions is that they should not violate any relevant rules, laws, regulations etc. On the other hand, an action being permitted does not necessarily imply that ‘it was required or obligatory’ as it only means that ‘its negation was not required or obligatory’ which is indeed an application of the duality of the logical definition of the deontic modality. However, notice that the justification of beliefs differs heavily from that of actions. As argued by Alston, believing a proposition at a given time means that ‘the relevant rules or principles do not forbid [the agent's] believing’ that proposition at that given time. It should be remarked carefully that this does not entail any deontic obligation to believe. You may justify your belief in a god, but it does not mean that everyone is obliged to believe in a god.

In their article, Hawthorne and Stanley (Reference Hawthorne and Stanley2008) discuss the relation between actions and epistemology. They propose a ‘reason-knowledge principle’ which states that ‘where one's choice is p-dependent, it is appropriate to treat the proposition that p as a reason for acting [if and only if] you know that [p]’. This principle does not solely apply to actions but does apply also to beliefs in such a way that we can treat the proposition p as a reason for believing in the proposition q only if we know that p, and p implies q.

As we will see, most of the existing formal frameworks deal with obligations to perform actions, rather than obligations or permissions to believe something. Formalizing these notions would give us a better understanding of their similarities and differences.

3. FORMALIZING KNOWLEDGE AND OBLIGATION

In the rest of this paper we focus on several formal frameworks that combine some epistemic and deontic logics. But before we go into details, we must pause for a moment and ask the question: why is it important to develop logics that combine agents' knowledge (or belief) on the on hand, and agents' obligations (or permissions) on the other?

Motivations for developing these formalisms are at least twofold. First, they provide us with a way of developing a deeper understanding of social situations and interactions. Several examples of important questions, some of which we already briefly touched upon in the introduction, are the following. When does agent i know that she ought to do action a? If she does not know that she ought to do a, is she obliged to do it anyway? Are obligations and blameworthiness always two sides of the same coin? Or can it be that because of my current beliefs, I cannot be blamed for not complying with an actual obligation? Also, in what cases ought she to know that she ought to do a? That is, in what cases is she obliged to obtain enough knowledge about her obligations? If agent i does a without knowing that it is her obligation to do so, does she fulfill her obligation nevertheless? How does direct interaction influence this picture: Suppose agents i and j are obliged to both perform a and j fails to satisfy his obligation, what happens to agent i's part of the obligation? In general, how can obligations and permissions be defined in situations where agents may (perhaps knowingly) fail to satisfy their obligations?

These and many other questions arise when we try to pinpoint the exact interactions between knowledge (and beliefs) on the one hand, and obligations (and permissions, blameworthiness) on the other. The different interactions between knowledge and obligations and their properties can be, and are to some extent, made precise in philosophical logic, and can thus provide us with a better understanding of the subject.

A second motivation for developing combined logical systems, is that they can help explicate important issues for both computer science applications and law. In distributed algorithms for example, it is useful to formally work with notions such as what an agent is allowed to know in a certain state and what action she is obliged to perform given her current knowledge. An example of a formal system in which exactly these issues are addressed is Lomuscio and Sergot (Reference Lomuscio and Sergot2003). In this work the authors develop so-called Deontic Interpreted Systems, a deontic extension of interpreted systems. Interpreted systems provide a general framework for reasoning about knowledge and communication in a multi-agent setting, and were introduced in Halpern and Moses (Reference Halpern and Moses1990). They have been used in computer science applications to describe agents's knowledge and their communication.

Lomuscio and Sergot's goal in their paper is to provide a grounded semantics for the deontic notions of the ideal functioning behavior of an agent, the knowledge that an agent is permitted to have, and that of the knowledge an agent has on the assumption that the components of the system are functioning correctly according to their protocols.

The main tool used to formalize these concepts is (local and global) states of violation and compliance. The notion of a local state in standard interpreted systems is used to represent the information available to the agent. In deontic interpreted systems, the set of local states is divided into allowed or correct (green) and disallowed or incorrect (red) states for each agent. Global states are tuples of local states for each agent. In this framework, the formula OK_i φ, ‘agent i ought to know that φ’, intuitively means the agent knows that φ in all the local states he is allowed to be in. More precisely, it means that an agent i should know φ, if in all global states in which his local state is a green state,he knows φ.

Also when defining computer security policies, combined notions of knowledge and obligations are important. In Glasgow et al. Reference Glasgow, Macewen and Panangaden1992, for example, a framework is developed to ‘specify and reason about security policies and to verify that the system adheres to such policies,’ which introduces epistemic, temporal and deontic modalities in an interactive fashion.

In law, it is also extremely useful to formally connect the notions of knowledge and obligations. If agent i unknowingly commits the crime b, that is, performs an act that is not permissible, is she still to be blamed for her act? How does this situation change if b is an act that is explicitly ruled out by a law? Isn't agent i supposed to know the law? In notions like knowingly, purposefully and recklessly performing a criminal act as well as the notion of liability are formalized.

Thus, there are many interesting interactions between knowledge and obligation that are worth formalizing using deontic-epistemic logics. Both the logic of knowledge and deontic logics have been extensively studied, and solid frameworks are well established. For example, the logic called S5 is a commonly used modal logic to represent knowledge. In this system knowledge is assumed to be reflexive (if an agent knows something, then it must be true), satisfy positive introspection (if an agent knows that something is true, he also knows that he knows it), and negative introspection (agents know what they don't know). A standard system of deontic logic is so-called ‘Standard Deontic logic’ or simply SDL. It is a standard modal logic satisfying, among other contraints, normalcy (if both a conditional and its antecedent are is obligatory, then so is its consequent) and necessitation (if something is derivable, then the claim that that thing is obligatory is also derivable).

The formal interaction between the two systems, however, has not been studied extensively. The most straightforward way to combine epistemic and deontic (modal) logic would be to simply add all the axioms of knowledge or belief (S5, or a weaker versionwithout negative introspection called S4) to the standard deontic axioms of standard deontic logic (SDL). This yields a strong logic but as it turns out, also a logic that fails to capture many of the subtleties that relate knowledge and obligation. A striking example showing that simply stacking the two logics together results in apparent contradictions is Åqvist's paradox of the knower. We will discuss the paradox and some of its implications in the next section.

Our aim in this paper to highlight some of the main ideas and high-level connections between different approaches. Therefore, we do not go into much technical detail but rather introduce formalisms only to clarify concepts. For more details on the formal systems, e.g., their syntax, semantics and axiomatization, we refer the reader to the original works.

4. THE PARADOX OF THE KNOWER

An important paradox of deontic logic is Åqvist's paradox. In his seminal work, Åqvist (Reference Åqvist1967) discussed his famous paradox of epistemic obligation. This paradox, also called the paradox of the knower, can be summarized as follows. Consider the following three sentences (from Hawthorne and Stanley 2008):

1. (1) If there is a fire, Gladys (a firefighter) ought to know it. (p → OK_p)

2. (2) There is a fire. (p)

3. (3) There ought not to be a fire. (O¬p)

If it is assumed that, on the one hand, knowledge implies truth (Kp → p), and moreover, that the standard epistemic modality ought (O) is normal (O(p → q) →(Op → Oq)) and satisfies necessitation (If ⊢p then ⊢Op), then both Op and O¬p can be derived from (1 – 3). The straightforward derivation of the paradox is as follows.

Thus, there ought to be a fire and there ought not to be a fire, which, according to Åqvist, is absurd. The problem is that OK p →O p is valid. This means that, if I ought to know something, it ought to be the case. Clearly, Åqvist and many others argue, this formula cannot be accepted as a valid principle of a formal account of ‘ought to know’.

The paradox of the knower has sparked a lot of research on epistemic obligations and several systems of deontic logic were developed to account for this problem (e.g. Feldman Reference Feldman1990; Goble Reference Goble1990; Hulstijn Reference Hulstijn2008), each focussing on different aspects of the paradox and discussing different solutions.

For example, Feldman (Reference Feldman1990) points out the temporal aspect of the paradox. If p is settled true at time t, then O¬p can no longer be true at the same time (assuming that ought implies can). Thus, Feldman argues, making the temporal aspect explicit in the framework solves the paradox. Moreover, Feldman argues that obligations are personal and that facts about one person's obligations cannot entail facts about another's. In the current example: How can it be possible that from Gladys's obligation to know about the fire, we derive Nature's obligation for there to be a fire? Feldman makes these ideas explicit by introducing obligation operators with two subscripts: one for time and one for agents. Given these insights we can no longer derive Op and the paradox seems to be circumvented.

In a different paper on the paradox of the knower, a formal notion of knowledge-whether is distinguished from knowledge-that (Hulstijn Reference Hulstijn2008). The main point of the framework is that if, in the paradox, ‘know that’ (K _t) is replaced with ‘know whether’ (K _w), the paradox does not arise. It still holds that ‘if there is a fire and Gladys knows whether there is a fire then Gladys knows that there is a fire’.

and by the same reasoning as before we can derive that

but because we do not have Op we cannot derive OK _tp and thus Op and according to the reasoning in the paper the paradox is ‘stopped’. This is for the simple reason that knowing whether φ does not imply that φ is true.

In the next section we discuss several frameworks that do not try to formalize what it means to have an obligation to know, but rather, how knowledge influences which obligations arise.

5. KNOWLEDGE BASED OBLIGATION

One approach to formalizing obligations under uncertainty was developed by Horty (Reference Horty2001). In line with Unger's argument mentioned in the introduction, Horty's deontic theory of agency models personal obligations to do something in situations in which outcomes of actions are uncertain. Basically, an agent has an obligation to perform a particular action a if, no matter what the other agents choose to do, all the possible results of performing a will be at least as good as the results of performing any other action.

In a recent work, Horty's decision theoretic approach has been generalized by using some standard techniques from game theory (Olde Loohuis Reference Loohuis, He, Horty and Pacuit2009). The basic idea in that paper is that obligations should not be defined irrespective of all the possible moves of other players, but rather, that obligations should be defined in a world in which other agents are assumed to act responsibly or rationally. Of course, assuming rationality of other players is only one way of taking other agents into account. The framework presented can be extended to include all kinds of ‘types’ of players, not necessarily responsible ones.

In their (2006) paper Pacuit, Parikh and Cogan (PPC henceforth) also formalize obligations to act under uncertainty. Even though their model accounts for uncertainty of outcomes of actions, their main focus is on uncertainty of the circumstances. In order to choose responsibly, they argue, one needs sufficient knowledge about the circumstances. A motivating example that is used is the following.

A formal system combining epistemic and deontic logic is proposed that can be used to study situations like the example above: situations in which obligations are circumstantial and depend on an agent's knowledge. This system builds on the history based semantics of Parikh and Ramanujam (Reference Parikh and Ramanujam2003) which can be summarized as follows.

Moments in time form a tree-like Kripke model: the past is deterministic, the future branches. Each (infinite) branch of the tree is called a history (H) and each history is assigned a real numbered value, a utility to the world.

In this model events happen but each agent observes only some events. This is made precise by a distinction between global histories and local histories. A global history H includes all the (relevant) events that have taken place. An agent i's local history h, on the other hand, contains only those events from H that agent i has actually observed. For events that the agent does not observe, he only notices the passing of time, modeled using non-informative clock ticks.

We denote the global histories up to and including moment m by H _m. At moment m, two histories H, H ^′ are said to be indistinguishable for agent i, H _m ~_iH′_m, if and only if the i's corresponding local histories h _m and h′_m are the same.

In the PPC framework, actions are also events. An action a can be performed at a finite history and yields a set a(H) of global extensions of H. Formally,

where ⊆ is the initial segment relation and Ha denotes the finite history H appended with event a. Thus, a(H) is the set of infinite histories that start with the finite history H, followed by action a, and then followed by some other (infinite) sequence of actions. Also, note that in PPC's framework agents move sequentially.

In order to formalize obligations, a notion of H-good histories G(H) is introduced. G(H) is defined as the set of extensions of the finite history H with the highest possible value. Thus, the set of H-good histories is actually the set of H-optimal ones. Given this notion of H-goodness, an action a is defined to be good if and only if G(H) ⊆ a(H), i.e. every H-good history involves performing a. Note that this does not imply that performing a guarantees an H-good future.

Finally, the PPC notion of obligation is as follows:

Thus, at moment m, agent i is obliged to perform a iff a is a good action and i knows this.

It is clear that this notion of knowledge based obligation eliminates Uma's obligation to treat Sam in Example 1. The reason is that Uma does not know that Sam is ill. Assuming that treating Sam when he is ill is optimal, it follows since Sam is ill, treating him is good. However, Uma does not know this and hence has no obligation (as yet) to treat him.

More formally, there is a finite history H′_m that Uma cannot distinguish from the actual history H _m in which Sam is not ill. In extensions of this history, treating Sam is not optimal, i.e. for H′ ∈ treat(H′ _m), H′ ∉ G(H′ _m). Thus, (∀H′) (H _m ~_iH′ _m and H′ ∈ G(H′_m) implies H′ ∉ treat(H′ _m)) is not satisfied and Uma does not have the obligation to treat Sam.

The PPC-framework is also applied to an actual situation called the Kitty Genovese Murder, where a New York City woman was stabbed to death in 1964. Thirty-eight people saw the murder but only after thirty-five minutes did somebody call the police. Why? The informal analysis PPC give is the following. ‘The people who saw Kitty being killed did not have default knowledge that they had the obligation to help her. They all knew that the good histories were ones in which someone called the police, but not all these histories were ones where they themselves were the caller. Someone else could be the caller’.

Apart from the absolute notion of knowledge based obligation, also a weaker concept of default obligation is introduced in PPC. Intuitively, an agent has a default obligation to perform action a if all maximal histories that an agent considers plausible are ones in which a is performed.

In a (2007) follow up work by Pacuit, history based structures and temporal logics have been compared and connected.Footnote ¹

6. KNOWINGLY DOING

While PPC focus on the way knowledge states imply certain obligations and how changes in an agent's information state lead to changes in his obligations, Broersen focuses on the epistemic conditions in which an agent can comply with or violate an obligation. Broersen (Reference Broersen2008) presents a logical study of the interaction between ‘ought-to-do’ and ‘knowingly’ or ‘consciously’ doing. Broersen introduces epistemic modalities within a deontic ‘Seeing To It That’ or STIT logic. STIT logic, a modal logic of action reports, was originally proposed in a series of papers by Belnap, Bartha and Horty (see Belnap et al. Reference Belnap, Perloff and Xu2001 for an overview). The central concept of the logical system is that of ‘seeing to it that’, or, ensuring that something holds by performing an action. Technically, the main modality of this logic, [i stit : φ] denotes that the agent i sees to it that φ is true.

Broersen (Reference Broersen2008) uses a STIT framework of actions with, in addition, an indistinguishability equivalence relation for each agent between history/moment indices. Actions, in Broersen's model, take effect at the next moment in time. The concept of knowingly doing is defined as follows. An agent knowingly does something if its action ‘holds’ for all the history/state indices in the epistemic equivalence set containing the actual history/state index. Within this epistemic model action model, Broersen introduces the following STIT operator [i xstit] φ with the intuitive interpretation that agent i sees to it that in the next state φ holds. A personal epistemic ought operator O[i xstit]φ is defined as follows:

where V is a ‘violation constant denoting that a violation occurs’. Also, ¬[i xstit]φ expresses that i does not see to it that φ and thus allows a choice with possible outcome ¬φ. The intuition behind the above definition is that agent i ought to see to it that φ if and only if, by not seeing to it, he allows a violation.

According to Broersen, simply seeing to it that φ is not enough to comply with the obligation to see to it that φ. Instead, Broersen proposes, an agent should perform the action knowingly in order to avoid violation. Formally:

Finally, Broersen discusses a third variant of the obligation operator, one that involves not only knowledge about the action, but also knowledge about the obligation. Awareness of an obligation is associated with awareness of the act of bringing about a violation in case the agent does not comply in the following way:

7. DEONTIC DYNAMIC EPISTEMOLOGY

So far, we have discussed various frameworks that capture the idea that agents's obligations depend on their knowledge of the circumstances. Therefore, based on this assumption, if an agent's knowledge changes, his obligations can change as well. This brings us to a dynamic epistemic approach to deontic logic where we discuss how knowledge updates effect permissions.

In a nutshell, dynamic epistemic logic is a field within epistemic logic which formalizes knowledge changes and updates in multi-agent settings. Within the field of dynamic epistemic logic, there are several different methods to achieve such updates. In this section, we focus on ‘public announcement logic’ – a logic where agents' knowledge changes by external truthful announcements (Plaza Reference Plaza, Emrich, Pfeifer, Hadzikadic and Ras1989; Gerbrandy Reference Gerbrandy1999; van Ditmarsch et al. Reference van Ditmarsch, van der Hoek and Kooi2007). In the framework of public announcement logic (PAL, henceforth), a truthful external agent makesan announcement, and all the agents hear the announcement, believe it, and consequently update their knowledge based on this announcement. PAL framework has successfully been used to analyze games (especially the backward induction scheme) (Van Benthem and Gheerbrant Reference van Benthem and Gheerbrant2010; Baskent forthcoming; Fagin et al. 1995).

The language of PAL introduces a special modality [·] (and its dual 〈·〉) for public announcements. PAL updates the epistemic model by ‘state elimination’. Namely, after an announcement of, say, φ the possible worlds that do not satisfy φ are eliminated from the model. Consequently, the epistemic accessibility relation and the valuation are updated as well.

A recent framework purposed by Balbiani, van Ditmarsch and Seban captures not only the obligations to act in general, but rather the more subtle notion of ‘permissions to say’ within the context of PAL (2009a, 2009b). They extend the basic language of PAL by adding a new permissibility modality P(ψ, φ) which reads ‘after ψ has been publicly announced, it is permitted to say φ’. We call their system of public announcements with permissions PPAL.

Card games present an intuitive case study for PPAL (Balbiani et al. Reference Balbiani, van Ditmarsch and Seban2009a, Reference Balbiani, van Ditmarsch and Seban2009b). In card games, say, poker or the French game ‘la belote’, players make announcements during the game play. However, they are not allowed to make any announcement. Rules of the game allow or disallow the players to make some certain announcements.

Let us now discuss the deontic permissibility modality P(ψ, φ) which satisfies the monotonicity principle:

The monotonicity principle intuitively says that after ‘two logically equivalent announcements, the same formulas are permitted to be announced, and moreover if two propositions are logically equivalent, they are permitted to be said in the same way’ (Balbiani 2009b).

We refrain from going into more technical details, and refer the reader to the aforementioned work to see the formal aspects of PPAL such as its full axiomatization and completeness. However, here we mention one major similarity of PPAL to other public announcement logics. As common in the variations of PAL, the announcement operator with the permission modality in its scope is reducible to the basic language of epistemic logic by the reduction axioms using the following scheme:

The given reduction axiom simplifies the complex formula with public announcement and permission operators, and reduces it to a formula with simpler syntax. By this method, every PPAL formula can be re-written as a PAL formula which we know to be complete.

In conclusion, PPAL presents a precise, intuitive and effective framework to discuss a certain class of games where rules prohibit or allow some moves, announcements or declarations.

8. DESCRIPTIVE OBligations

Another approach to (dynamic) epistemic deontic logic focuses on prescriptive and descriptive obligations to analyze the ‘distinction and relation between prescriptive and descriptive obligations’ (Aucher et al. Reference Aucher, Boella and van der Torre2010). An example to clarify the distinction between two such obligations can be helpful.

The logical system Aucher et al. introduced is based on the system that Castañeda developed, and reflects his idea of a two-sorted system with distinguished atoms for grounds and foci [12]. For Castañeda, grounds are the ‘circumstances that originate, or surround, obligations’ and are propositions that can be true or false. Foci, on the other hand, are neither true or false, but are ‘the core contents of commands’.

The language has two sets of atoms: one for those propositions which cannot on their own be the foci of deontic operators and one for those propositions which can. The first set of atoms is called propositions while the second is called practitions. We refer the reader to the original work for the details of the system, including a rather involved axiomatization (Aucher et al. Reference Aucher, Boella and van der Torre2010).

The given language can be extended by epistemic and deontic dynamic operators, yielding a rather complex syntax and semantics. In such extended systems, moreover, authors discuss several cases where norms can be changed, and how such a system would work in a multi-agent setting.

This work by Aucher is one of the instances where philosophical concerns are addressed within formal logic, and are presented by rather involved technical language with axiomatization. Their system is significantly more complex than what Castañeda has envisioned, making it more expressive and but at the same time more complex.

9. RESPONSIBILITIES AS DYNAMIC EPISTEMOLOGY

Lima et al. (Reference Lima, Royakkers and Dignum2010) develop a logic for reasoning about responsibility that is closely related in spirit to Broersen's ‘knowingly doing’. Lima et al. consider the problem of task allocation in multi-agent systems. One way to allocate tasks to agents is by assigning obligations to them. From his obligations an agent can then infer which actions are forbidden, permitted or obligatory. For example, from ‘it is obligatory for agent i that φ’ the agent i can infer that ‘it is obligatoryfor i to execute action a’. However, Lima et al. argue that just assigning obligations is not enough to adequately guide an agent's decisions. They give the following illustrative example:

In this example Alice does not have enough information to decide what to do to fulfill her obligations. And thus, Lima et al. argue, she cannot be held responsible for keeping the balances nonnegative. However, if Alice has the possibility of asking Bob which bank account he is going to use she becomes responsible again. Note that contrary to Pacuit et al. (Reference Pacuit, Parikh and Cogan2006) she still has the obligation to keep the balances positive but cannot act in a responsible way. Thus, in line with Lima's argument, obligation and responsibility are not necessarily two sides of the same coin: it may be that an agent has the obligation to perform an action, or to achieve a state, but because of lack of knowledge she cannot be held responsible for failing to satisfy her obligation. Lima et al. make this distinction explicit by defining two distinct types of responsibilities.

In their paper, Lima et al. propose an extension of PDL that they call Coalitional Epistemic Dynamic Logic. They introduce group actions δ|_G meaning roughly that group G jointly performs action δ. Given δ|_G, [δ|_G : ensures] φ means that if a group G executes δ, they ensure that φ will be the case, and 〈G〉φ that group G is able to ensure φ by some action. Given these operators the authors are able to define obligations and furthermore responsibility: it is obligatory for an agent i that φ holds if the occurrence of ¬φ may result in a violation. Technically, O _i φ if the empty group of agents (〈∅〉) has an action that can ensure that a violation occurs if ¬φ.

Here, vio_i is a special atom with the meaning that the agent i is in violation.

Using these concepts, the authors define two types of responsibilities: forward-looking and backward-looking responsibilities. A forward-looking responsibility, also called task responsibility, means the following. An agent i is forward-looking responsible for an outcome φ if she is held responsible for accomplishing the task of achieving φ. In their words: ‘φ must be achieved by i, and agent i will be held accountable if it does not happen.’ (Lima et al. Reference Lima, Royakkers and Dignum2010). Formally, they propose the following inductive definition of forward-looking responsibility: R _i¹ φ : = O _i φ, and

which means that agent i is forward-looking responsible for φ (R _i^ωφ) if and only if it is obligatory for i that φ holds (O _iφ) for every reachable state. Moreover, the agent also needs to know how to ensure φ(_δ∈ΔK _i [δ|_i : ensures]φ), and cannot lose this knowledge in the future. Thus, it is obligatory for i to keep the knowledge of how to ensure φ.

An agent i is backward-looking responsible for the (bad) outcome φ if he is to be blamed for this outcome. This, according to Lima et al., can only be the case if the agent has had enough knowledge about the situation at hand. Going back to example 4, Alice does not have enough knowledge to be backward-looking responsible for not transferring money from account 2 to 1. We refer to the original paper for a formal treatment of this concept.

The work of Lima et al. is significant in the sense that they formalize task allocation issues by using an extended version of propositional dynamic logic and then use the very system to express responsibilities in a natural way. Their complex operators enable us to represent various deontic situations in this system.

Before concluding, we would like make a remark on the use of ‘violations’ as atomic propositions that is common in the literature (see Lima et al. Reference Lima, Royakkers and Dignum2010: example 7). Whether we are talking about history based models, STIT frames, plain Kripke models or interpreted systems, violations are very useful for defining obligation and responsibilities. However, at the same time they ‘hide’ many of the complex relations between knowledge and obligations. Violations depend not only on the propositional truth values of a state but depend on the structure of the mode, for example uncertainty relations, as well. When does a violation hold at a moment in history, or at a state in a Kripke model? If a violation is a consequence of an action, how can it be a property of a state as well?