Rohit Parikh

Parikh discusses recent developments in computer science, especially in the areas of program correctness, dynamic logic, and temporal logic. He then develops a meaning theory for a natural language versus the actual linguistic and nonlinguistic behavior of the members of a speech community, exploring three arguments concerned with the social character of language: Quine’s thesis of the indeterminacy of translation, Kripke’s skeptical paradox (derived from Wittgenstein) concerning the notion of following a rule, and John Searle’s Chinese room puzzle. Parikh argues that each of these apparently skeptical arguments is concretely realized in the programmer’s efforts to devise and apply a programming language. In thinking through the relation between high-level programming languages and the machine languages into which they must be translated in order to be implemented, computer scientists can offer significant insights into philosophical issues. A pragmatic approach to meaning theory in which uniformity in individual speakers’ usage of a communal language is not assumed, and in which there is no common underlying notion of truth—in which, ultimately, large portions of language are admitted which are not informational at all—is defended, with suggestions of applications to problems of vagueness.

This work is motivated by the perception that there is a gap between two communities. One community consists of mathematicians and logicians. The other community consists of social scientists and philosophers. The second community is keenly interested in the problems of society. But it lacks the tools to understand the way in which mathematical and logical tools help with these problems. The first community does have the technical tools, or has the expertise to easily adapt the existing tools to social problems. But they lack the interest. Thus important work on the problems of society is simply not being done. Thus I am appealing to people to address this lacuna.

We present a bimodal logic suitable for formalizing reasoning about points and sets, and also states of the world and views about them. The most natural interpretation of the logic is in subset spaces, and we obtain complete axiomatizations for the sentences which hold in these interpretations. In addition, we axiomatize the validities of the smaller class of topological spaces in a system we call topologic. We also prove decidability for these two systems. Our results on topologic relate early work of McKinsey on topological interpretations of S4 with recent work of Georgatos on topologic. Some of the results of this paper were presented at the 1992 conference on Theoretical Aspects of Reasoning about Knowledge.

It is well known that how an individual acts in a specific situation depends not only on her preferences (and her means) but also on what she believes. If she believes a restaurant is open, she will go to it, assuming she likes that restaurant and she has the means, e.g., transportation. But if she knows it is not open, then her liking becomes irrelevant.When someone wants some behavior to come about on a large scale he must arrange for the various individuals involved to have the information which they will need to carry out their own role in that behavior. If you put on a concert, you must also advertise.In some cases, if the interests of the individuals diverge from that of the planner, then he may also try to see to it that they lack certain information. No one wants his or her social security number to be common knowledge but everyone wants to know when the bus is coming. So the dance between knowledge and ignorance drives much social action.We will offer a formal account of the way this happens, using tools from epistemic logic, game theory and the analysis of algorithms. The simple answer to the question raised in the title is: No (This is an expository paper containing some recent ideas. (Pacuit E, Parikh R, Social interaction, knowledge, and social software. In: Goldin D, Smolka S, Wegner P (eds) Interactive computation: the new paradigm. Springer, New York, 2006) contains a nice survey. The results on campaigning are joint with Walter Dean (Dean W, Parikh R, The logic of campaigning. In Presented at logic and its applications, lecture notes in computer science, Volume 6521, 4th Indian Conference, ICLA 2011, Delhi, India, January 5–11, 2011). The material on influencing action by influencing knowledge is joint with Çağıl Taşdemír and Andreas Witzel (Int Game Theory Rev 15:1–28, 2013)).Turing was dealing with a relatively simple situation where he was interested in algorithms which took in numbers (or strings) as input and producing an output. Social algorithms deal with much more complex ‘inputs’ and the ‘output’ may be something more complex as well. And yet the project of developing a taxonomy of social algorithms seems both important and interesting.

We propose a new relevance sensitive model for representing and revising belief structures, which relies on a notion of partial language splitting and tolerates some amount of inconsistency while retaining classical logic. The model preserves an agent's ability to answer queries in a coherent way using Belnap's four-valued logic. Axioms analogous to the AGM axioms hold for this new model. The distinction between implicit and explicit beliefs is represented and psychologically plausible, computationally tractable procedures for query answering and belief..

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 examine the relationship between the justified true belief (JTB) account of knowledge and Plato’s theory about it as expounded in the Theaetetus. Considering Socrates’ remarks in the Theaetetus brings us to some concerns raised by Turing and to Wittgenstein’s famous comment explanations come to an end somewhere. We present two simple technical results which bear on the question. Finally, we look at the pragmatic aspects of knowledge attributions. In an Appendix we say a few words about Indian epistemology and Gettier problems.

Attempts to give a Logic or Semantics for vague predicates and to defuse the Sorites paradoxes have been largely a failure. We point out yet another problem with these predicates which has not been remarked on before,namely that different people do and must use these predicates in individually different ways. Thus even if there were a semantics for vague predicates, people would not be able to share it. To explain the occurrence nonetheless of these troublesome predicates in language, we propose a different approach based on asking the question, “How do these vague predicates help people to communicate with each other?” We show that in general, even though different people assign different extensions to vague predicates, they usually benefit from receiving information framed in terms of them