Rohit Parikh

Deontic Logic goes back to Ernst Mally’s 1926 work, Grundgesetze des Sollens: Elemente der Logik des Willens [Mally. E.: 1926, Grundgesetze des Sollens: Elemente der Logik des Willens, Leuschner & Lubensky, Graz], where he presented axioms for the notion ‘p ought to be the case’. Some difficulties were found in Mally’s axioms, and the field has much developed. Logic of Knowledge goes back to Hintikka’s work Knowledge and Belief [Hintikka, J.: 1962, Knowledge and Belief: An Introduction to the Logic of the Two Notions, Cornell University Press] in which he proposed formal logics of knowledge and belief. This field has also developed quite a great deal and is now the subject of the TARK conferences. However, there has been relatively little work combining the two notions of knowledge (belief) with the notion of obligation. (See, however, [Lomuscio, A. and Sergot, M.: 2003, Studia Logica 75 63–92; Moore, R. C.: 1990, In J. F. Allen, J. Hendler and A. Tate (eds.), Readings in Planning, Morgan Kaufmann Publishers, San Mateo, CA]) In this paper we point out that an agent’s obligations are often dependent on what the agent knows, and indeed one cannot reasonably be expected to respond to a problem if one is not aware of its existence. For instance, a doctor cannot be expected to treat a patient unless she is aware of the fact that he is sick, and this creates a secondary obligation on the patient or someone else to inform the doctor of his situation. In other words, many obligations are situation dependent, and only apply in the presence of the relevant information. Thus a case for combining Deontic Logic with the Logic of Knowledge is clear. We introduce the notion of knowledge based obligation and offer an S5, history based Kripke semantics to express this notion, as this semantics enables us to represent how information is transmitted among agents and how knowledge changes over time as a result of communications. We consider both the case of an absolute obligation (although dependent on information) as well as the (defeasible) notion of an obligation which may be over-ridden by more relevant information. For instance a physician who is about to inject a patient with drug d may find out that the patient is allergic to d and that she should use d′ instead. Dealing with the second kind of case requires a resort to non-monotonic reasoning and the notion of justified belief which is stronger than plain belief, but weaker than absolute knowledge in that it can be over-ridden. This notion of justified belief also creates a derived notion of default obligation where an agent has, as far as the agent knows, an obligation to do some action a. A dramatic application of this notion is our analysis of the Kitty Genovese case where, in 1964, a young woman was stabbed to death while 38 neighbours watched from their windows but did nothing. The reason was not indifference, but none of the neighbours had even a default obligation to act, even though, as a group, they did have an obligation to take some action to protect Kitty.

We present a method for relevance sensitive non-monotonic inference from belief sequences which incorporates insights pertaining to prioritized inference and relevance sensitive, inconsistency tolerant belief revision. Our model uses a finite, logically open sequence of propositional formulas as a representation for beliefs and defines a notion of inference from maxiconsistent subsets of formulas guided by two orderings: a temporal sequencing and an ordering based on relevance relations between the putative conclusion and formulas in the sequence. The relevance relations are ternary (using context as a parameter) as opposed to standard binary axiomatizations. The inference operation thus defined easily handles iterated revision by maintaining a revision history, blocks the derivation of inconsistent answers from a possibly inconsistent sequence and maintains the distinction between explicit and implicit beliefs. In doing so, it provides a finitely presented formalism and a plausible model of reasoning for automated agents.

The standard theory for belief revision provides an elegant and powerful framework for reasoning about how a rational agent should change its beliefs when confronted with new information. However, the agents considered are extremely idealized. Some recent models attempt to tackle the problem of plausible belief revision by adding structure to the belief bases and using nonstandard inference operations. One of the key ideas is that not all of an agent's beliefs are relevant for an operation of belief change.In this paper we incorporate the insights pertaining to local change and relevance sensitivity with the use of approximate inference relations. These approximate inference relations offer us partial solutions at any stage of the revision process. The quality of the approximations improves as we allow for more and more resources to be used. We are provided with upper and lower bounds to what would be obtained with the use of classical inference

Each of us has a story which we can think of as a set of beliefs, hopefully consistent. We make our decisions in view of our beliefs which may be probabilistic, in the general case, but simple yes or no as in this paper. Our beliefs are our envelope just as the shell of a tortoise is its envelope. Decision theory—or single agent game theory tells us when to make the best choice in a game of us against nature. But nature has no desire to further or frustrate our efforts. Nature is mysterious but not malign. Things change when there are other agents involved. Then the best thing for us to do will depend on what they do. And they will think the same. And we predict their actions in terms of what we think their beliefs are. But how do we coordinate with others whose beliefs are different? This paper addresses the issue of working together despite different beliefs.

Proof, Computation and Agency: Logic at the Crossroads provides an overview of modern logic and its relationship with other disciplines. As a highlight, several articles pursue an inspiring paradigm called 'social software', which studies patterns of social interaction using techniques from logic and computer science. The book also demonstrates how logic can join forces with game theory and social choice theory. A second main line is the logic-language-cognition connection, where the articles collected here bring several fresh perspectives. Finally, the book takes up Indian logic and its connections with epistemology and the philosophy of science, showing how these topics run naturally into each other.

In strategic situations, agents base actions on knowledge and beliefs. This includes knowledge about others’ strategies and preferences over strategy profiles, but also about other external factors. Bernheim and Pearce in 1984 independently defined the game theoretic solution concept of rationalizability, which is built on the premise that rational agents will only take actions that are the best response to some situation that they consider possible. This accounts for other agents’ rationality as well, limiting the strategies to which a particular agent must respond, enabling further elimination until the strategies stabilize. We seek to generalize rationalizability to account not only for actions, but knowledge of the world as well. This will enable us to examine the interplay between action based and knowledge based rationality. We give an account of what it means for an action to be rational relative to a particular state of affairs, and in turn relative to a state of knowledge. We present a class of games, Epistemic Messaging Games, with a communication stage that clarifies the epistemic state among the players prior to the players’ actions. We use a history based model, which frames individual knowledge in terms of local projections of a global history. With this framework, we give an account of rationalizability for subclasses of EMG.

We offer a probabilistic model of rational consequence relations (Lehmann and Magidor, 1990) by appealing to the extension of the classical Ramsey-Adams test proposed by Vann McGee in (McGee, 1994). Previous and influential models of nonmonotonic consequence relations have been produced in terms of the dynamics of expectations (Gärdenfors and Makinson, 1994; Gärdenfors, 1993).'Expectation' is a term of art in these models, which should not be confused with the notion of expected utility. The expectations of an agent are some form of belief weaker than absolute certainty. Our model offers a modified and extended version of an account of qualitative belief in terms of conditional probability, first presented in (van Fraassen, 1995). We use this model to relate probabilistic and qualitative models of non-monotonic relations in terms of expectations. In doing so we propose a probabilistic model of the notion of expectation. We provide characterization results both for logically finite languages and for logically infinite, but countable, languages. The latter case shows the relevance of the axiom of countable additivity for our probability functions. We show that a rational logic defined over a logically infinite language can only be fully characterized in terms of finitely additive conditional probability