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1COGNITION AS INTERACTIONJohan van Benthem, University of Amsterdam & Stanford UniversityMarch 2005Abstract

Many cognitive activities are irreducibly social, involving interaction between severaldifferent agents. We look at some examples of this in linguistic communication and games,and show how logical methods provide exact models for the relevant information flow andworld change. Finally, we discuss possible connections in this arena between logico-computational approaches and experimental cognitive science.1Introduction: from lonesome cognitive performance to social skillsWhen King Pyrrhus of Epirus, one of the foremost well-educated generals of his age, hadcrossed over to Italy for his famous expedition, the first reconnaissance of a Roman campnear Tarentum dramatically changed his earlier perception of his enemies (Plutarch,\"Pyrrhus\", Penguin Classics, Harmondsworth, 1973):Their discipline, the arrangement of their watches, their orderly movements, and the planningof their camp all impressed and astonished him – and he remarked to the friend nearest him:\"These may be barbarians; but there is nothing barbarous about their discipline\".It is intelligent social life which often shows truly human cognitive abilities at their best andmost admirable. But textbook chapters in cognitive science mostly emphasize the apparatusthat is used by single agents: reasoning, perception, memory, or learning. And this empha-sis becomes even stronger under the influence of neuroscience, as the only obvious thingthat can be studied in a hard scientific manner are the brain processes inside individualbodies. Protagoras famously said that \"Man is the measure of all things\", and manyneuroscientists would even say that it\'s just her brain. By contrast, this very brief papermakes a plea for the irreducibly social side of cognition, as evidenced in the ways in whichpeople communicate and interact. Even in physics, many bodies in interaction can form onenew object, such as a solar system. This is true all the more when we have a meeting ofmany minds! Perhaps the simplest and yet most striking example of interactive cognitivebehaviour is language use in conversation. This will be our key example in what follows.22Questions, answers, and the spectrum of language useIn modern logic and semantics, the dynamics of language use has come into focus. Theindividual speech acts that take place can only be understood in terms of mutualinformation that agents have about each other, and more generally, the resulting er a simple conversation like:Q \"Is the KNAW on this canal?\"A \"Yes.\"The questioner Q conveys that she does not know the answer, and probably also, that shethinks it possible that the answerer A does know. After the answer has been given, Q hasnot just learnt the fact that this canal is indeed the location of the Royal Academy. She alsoknows that A knows that location, and even that A knows that she knows it now, and so y, the two agents have achieved common knowledge, to any depth of mutual , this simple episode of communicative language use lies on a much longer naturalchain of inter-locking cognitive abilities. First, before we can process the information ingiven statements, we need to understand what they say. This is seldom a purely one-agentmatter, but it rather involves a process of interpretation for reaching an equilibriumbetween speaker\'s and hearer\'s meaning. Bi-directional Optimality Theory describes part ofthis – but authors like Parikh 2002 and van Rooy 2002 have full-fledged game-theoreticaccounts of the crucial higher-level interactions here. Next, successful communicationrequires an account of the presuppositions and effects of various kinds of speech act. Justwhich information is passed exactly when we use a particular linguistic construction?

This can be much more sophisticated than the above rather elementary scenario. E.g., in aclassroom, if Q were a teacher, and A a student, we would not normally expect Q to beignorant of the answer, and we definitely do not expect her to have any illusions about the\'knowledgeability\' of A. Next, individual questions makes sense only when there is abroader intention behind them. This brings us to conversations with strategies for repeatedasking and answering, as well as just the right amount of revealing and hiding informationto achieve intended effects and reach goals. Conversations live at the level of strategicgames, and again they achieve some sort of game-theoretic equilibrium between agentsgiving and receiving information. But games are again just episodes in a longer stream oflinguistic behaviour, which involves accumulated memory over time. For instance, our ideasabout the reliability of conversation partners may encode a lot of past experience, as well asthe resulting expectations about the future. Thus, we get into the temporal analysis of long-term protocols and learning. And moving beyond the aggregation level and life-span ofsingle agents, we can even look at long-term linguistic practices in societies, or even3evolutionary processes of language change. These long-term phenomena transcend thescope of standard logic, and eventually involve the mathematics of dynamical own current interests lie mostly in the middle of this spectrum, viz. information update,belief revision, and games. Experience in this area has shown two things. First, there isenough substance to create exact theories – but also, such theories need to take their cuesfrom quite diverse disciplines, such as linguistics, philosophy, logic, computer science,economics, and cognitive psychology. The next section provides some concrete examplesof this confluence, all from a logician\'s perspective.3Dynamic epistemic logic for communicationCommunication involves update of information. To describe such processes in preciseterms, we need to have an account of both the information states of groups of agents, andthe basic and complex actions that can change one of these states into another. For thispurpose, logical techniques turn out quite mic logic To analyze the above question/answer episode, we use a well-knownsystem from the philosophical tradition, viz. epistemic logic – originally developed as a toolfor analyzing epistemological notions and arguments. In a self-explanatory notation, hereare some key features of our question-answer episode:Q asked a factual question \"P?\", A answered truthfully \"Yes\".For a truthful answer A must know that P: KAPA normal cooperative question then has the two presuppositions(a)

(b)

Q does not know if P:Q thinks it possible that A knows if P: ?KQP ∧ ?KQ?P (KAP

∨ KA?P) C(Q, A}PAfter the answer, P becomes common knowledge:

Information update Next, the updates that take place when speech acts occur change therelevant information models. E.g., here is a simple model for an initial situation where Pholds, Q does not know this (Figure 1): P Q ?P

Figure 1The line between the two worlds indicates the initial uncertainty of agent Q. The absence ofsuch a line for agent A indicates that the latter knows whether P is the case. This absenceis transparent for Q who therefore does know that A knows if P.4Next, the update corresponding to a public assertion P! by A that P holds (which is thelogical content of the above answer \"Yes\") removes the ?P-world to the right in this model,leaving us with only the situation depicted in Figure 2:

PFigure 2This new model shows pictorially how both agents know that P – and indeed, that P hasbecome common knowledge among them. As for the general process here, publicannouncement changes the current information model by world c-epistemic logic Epistemic logic describes what agents know at the staticintermediate stages of an informational process. But really, it is the state-changing actionsthemselves that seem of primary interest. To bring the latter into focus, one can borrow anidea from computer science, viz. the dynamic logic of programs, interpreted as actions thatchange states of some computing device. A combined dynamic-epistemic logic hasformulas with both epistemic operators and action modalities. These formulas areinterpreted in epistemic models M (i.e., diagrams as above) at particular worlds s, wherethey describe facts and properties of agents – as seen from the standpoint of s. Inparticular, the following dynamic action modality [P!] φ

describes what happens in theupdated model after something has been said: M, s |= [P!] φiff if M, s |=P, then M|P, s |=

φThe right-hand side of this truth condition says that

φ is true in model M at world s after anaction of true announcement of P has taken place to produce the model M|P. Morecomplex dynamic-epistemic formulas

φ then describe what agents know (or do not know)after an announcement action has taken place: [P!]Kiφ

[P!]CGφ

agent i knows that

φ

after P has been announcedafter P has been announced,

φ

is common knowledgeIn this manner, we get a logical system that describes the effects of communication. Its setof valid principles is known to be simply axiomatizable, and it is even decidable by meansof a mechanical algorithm that could run – in principle – on an ordinary ant advances in this paradigm of dynamic-epistemic logic have been made in the1990s by Plaza, Gerbrandy, van Ditmarsch, and others – with Baltag, Moss & Solecki1998 as a particularly seminal contribution. We refer to van Benthem 2002, 2005 for asurvey of the state of the art plus a list of open problems.5For our purpose here, perhaps the main thing to observe is the meeting of academiccultures in just one dynamic-epistemic formula describing effects of communication: [P!]KiφThe systematic study of speech acts like P! was initiated in the philosophy of language byAustin and Searle. The knowledge operator K goes back to Hintikka\'s work inphilosophical logic and epistemology. And the dynamic logic operator [] comes from thetradition of Hoare, Pratt, and many others in computer science and mathematics. Finally,the study of communication by such means seems more like a topic in the social , humanities, natural sciences, and social sciences meet in one single locus. C.P. Snowdeplored the chasm between the \'Two Cultures\', but they still do meet in unexpected ts: new issues Let us explore this interdisciplinary meeting point in a bit moredetail. A logical system provides a lense for looking at phenomena that may not have beenvisible clearly before. For instance, what is the general effect of a speech act like making apublic assertion P? It seems plausible that this must always produce common knowledge,reflected in the dynamic-epistemic formula [P!]CGPBut the latter principle is problematic as true logical law of communication. Back at thestart of analytical philosophy, G.E. Moore formulated his notorious true-but-infelicitoussentences such as\"P, but I don\'t believe that P\",which do not seem appropriate in communication. In particular, given some minimalassumptions in epistemic logic, one can never know the analogous logical proposition P &?KP. This phenomenon made its way from philosophy to mathematics in puzzles, such asthe famous Muddy Children and its ilk: \'hats\', cheating housewives, ... In a given group ofmuddy and clean children, each child can see the others, but not its own forehead. Nowtheir father tells them publicly that at least one child is dirty. In the ensuing process ofquestioning which child knows its status, all say first that they do not know if they havemud on their foreheads. But as this question and answer process is repeated, a last roundoccurs where everybody has figured out who are dirty. For instance, in a group with twodirty children and one clean one, the dirty ones can figure out their status after one round:If I were clean, the one dirty child I see would have seen only clean children around her,and so she would have known that she was dirty at once. But she did not. So I must be

dirty, too!” This reasoning is symmetric for both muddy children – so both know in the

second round. The third child knows it is clean one round later, after they announced that.6Thus, the last ignorance assertion in the sequence makes its own negation into commonknowledge. Fagin et al. 1995 give important applications of the reasoning in such puzzlesto understanding computational processes with distributed agents which exchange infor-mation. In dynamic-epistemic logic, the point is that any true announcement (P & ?KP)!with a factual assertion P makes the left-hand conjunct P into common knowledge, therebyinvalidating the right-hand conjunct ?KP. This phenomenon is ubiquitous. Van Benthem2004B shows how the same difficulty underlies the \'Fitch Paradox\' of verification, whichsays that the apparently plausible principle that \"Every true proposition is learnable\" cannotbe maintained consistently. In logic itself, this raises the technical issues just whichpropositions P achieve their own common knowledge when truly announced. See vanBenthem 2002 for some partial results – but the general question is still icative update in general The preceding example shows how even something assimple as making a public announcement can hide unexpected subtleties. But of course,there are many further types of communication. We seem to be very good in providingpartial information, making sure that only the right people know what we want them toknow, while others do not. This brings in issues of security, hiding, lying, and all thosemore subtle skills that lubricate civilized social life. Moreover, this information is not justconveyed by speech acts. We also get it by mere observation, or by inference from what wealready know. All these sources can occur intertwined, with switches from one to the I want to know if paracetamol is sold close to the Academy, I can either try to reason itout inside my head, using the KNAW documentation at my disposal, or I can perceptuallyinspect houses on the Kloveniersburgwal, or I can resort to communication, and ask somereliable-looking person in the vicinity. Richer systems of dynamic-epistemic logic existwhich can handle at least communication and perception at the same time, includingpartiality and hiding, by using update with event models. Indeed, this was the maininnovation due to Baltag, Moss & Solecki 1998. And once we have such more subtlesystems in place, we can start looking at the total area of human communicative behaviour,asking which tasks are harder than others – and where are the natural boundaries ofcomplexity separating one practice form triangle: logic, cognition, and computationIn the approach advocated here,studying information and communication involves three main ingredients. Logic suppliesthe theory, cognitive reality supplies the phenomena that we are interested in. But there isalways a third party involved, viz. the role of computation of some sort. This influence isboth theoretical: witness our use of ideas from dynamic logics of programs – but it is alsopractical. We communicate with machines perhaps more often nowadays than with fellow7humans, and man-created virtual reality is all around us: just as visionary computerscientists had predicted way back when \'single-minded\' traditional hardware was still theonly game in town (Licklider 1965). To us, this triangle affair between logical theory,experimental facts, and computational design is not an annoying case of diversity. It ratherreflects a deep and fruitful insight. The very same phenomenon may manifest itself eitherin the models of a logical theory, or be embodied in human cognition, or in the design ofsome computational process running on a machine. There is no more difficulty under-standing this diversity, and unity, than there is in grasping the concept of the Holy Trinity.4From single actions to strategic gamesThe individual information-bearing moves described by dynamic epistemic logics naturallylead to the next step in the ladder of cognitive phenomena. For instance, humans are goodat turning any given practice into a sort of ation games Here is an example. Any topic of discourse, and any conversationspace of admissible assertions about it, leads to information games, where players mustsay something non-trivial at every turn, and the goal is to be the first to know the truesituation. More complex examples are found in parlour games like \"Clue\" (van Ditmarsch2000) which have been designed so that the game remains enjoyable to play, though nottoo hard to get stuck in sheer theory and dynamic epistemic logic To describe agents\' behaviour in this richersetting where conversation has a goal, and participants are aware of this, we must turn togame theory. The optimal thing to say depends on what we have heard before, plus ourcurrent plan. This requires strategies on players\' part, and the best outcomes for all areoften well-described by the usual Nash equilibria. Though these equilibria may becomputed by general techniques (including probabilistic mixed strategies), most of theiroutcome values are unknown for even simple information games. But even when available,game values are global features at best of the process. Dynamic-epistemic logic adds thefine-structure of players\' information as they are observing events in the course of thegame, while trying to plan their next revision In a game setting, however, update of information is no longer the onlyrelevant process. Players must also be able to revise their beliefs when confronted withevidence contradicting their expectations so far. To see this, consider a game of the famousCentipede type. Two players can play either \'Down\' or \'Across\' (Figure 3):8A 0, 1 E 3, 0 A 2, 43, 3Figure 3The value of an outcome for E is indicated first in these pairs, that for A follows. In thisscenario, the standard recommendation of game theory proceeds by a line of reasoningcalled \'Backward Induction\':We analyze the behaviour of A at the end, and then work back toward the initial node. It is tohis advantage there to play Down (this gives him 4 instead of 3). But since E knows this, shewill play Down at her move, since this gives her 3 instead of 2. But then, knowing this again, atthe start of the game, A should play Down – because this gives him 1 instead of reasoning, though it sounds compelling, has the surprising effect that, with thisoutcome, both players are worse off than they would have been if the game had proceededtoward the far right. In longer Centipede games, the difference in pay-off can be will this initial opt-out happen, and does Backward Induction apply? There is apresupposition involved here. Standard rational agents always choose the action that is totheir own greatest advantage. But this is not the only possible kind of player, and E mayhave other beliefs about the sort of agent she is dealing with, such as\'A is stupid, or generous, or adventurous…\'In particular, even when E starts with a standard expectation of ruthless \'rationality\' – afterA plays across, it seems likely that E will change her beliefs about A. As with informationupdate, the best current logical theories of belief revision come from computer science, ing in AI (G?rdenfors & Rott 1995). But there are also connections with beliefrevision as studied in cognitive science (Castelfranchi 2004).Diversity of agents In the richer game-theoretic setting, many further questions arise forlogical analysis of communication and interaction that have a strong cognitive flavour.

In particular, agents clearly differ in their powers of reasoning and observation, theirpropensities in belief revision (eager or conservative), and the sheer cleverness of theirstrategies. Thus, any society of agents will show diversity along many dimensions, andlogical systems should account for this. For a first attempt, cf. van Benthem & Liu 2004,who compare ideal Turing machines with finite automata in their role as informationupdaters. As one striking instance, when we encounter new agents, we need to determinetheir type, both as to processing capacities and as to general behaviour. Simple quickconclusions are not always the right ones here. E.g., Axelrod 1984 studied the simplefinite-automaton strategy like Tit-for-Tat, which\' cooperates\' if you did so in the preceding9round, and \'defects\' if that\'s what you did there. Axelrod showed how, in social encounters,Tit-for-Tat wins out against much more sophisticated forms of punishing and rewardingpast behaviour. Likewise, sophisticated analyses of interactive computation using linearlogic (Abramsky 1998) use general Copy-Cat strategies which copy another player\'s movefrom one game to another. When suitably composed, these basic routines create amazinglyeffective mathematical and computational behaviour in very complex games. The resultingscenarios can be hard to compute in general, but movies provide interesting tanding 1990s classics like The Matrix or Memento requires a sort of modelingwhere game theory and logic have not yet caught -term processes Finally, even games as normally understood are just episodes on alarger time-scale. In the total arena of life, finite terminating processes co-exist with onesthat are for all practical purposes infinite, such as the operating system of my computer, orthe general conventions of the society that our behaviour lives in. Understanding thisbroader setting requires an integration of dynamic epistemic logic and game theory withthe mathematical theory of dynamical systems. This is happening already in modernevolutionary game theory, and I am sure that logic will follow suit. Cf. the epistemictemporal logic systems of Fagin et al. 1995, Parikh & Ramanujam 2003, Belnap et al.2001. Thus, serious mathematical models and logical calculi live along the whole spectrumof cognitive phenomena mentioned in Section 2.5Toward cognitive realityThe preceding account of logics for communication and games may have shown that thereare many interesting structures to be discovered in the area of social cognitive heless, it is fair to say, looking at the relevant literature in logic, linguistics,philosophy and computer science, that much of the reference made there to \'real cognition\'is mere rhetoric! Even the most innovative authors on the logic side show little inclinationto really go out and be confronted with the experimental evidence. But this is not becausesuch contacts would be useless. In fact, if we were to go out and see whether humans arereally \'so good\' at communicative subtlety, or even prior to that: how they really do things –we might be in for many creative reason for a persistent armchair syndrom are philosophical preconceptions, such asthe celebrated \'anti-psychologism\' of Frege and his followers, which has been adopted bymainstream philosophers and logicians ever since. This allows them to theorize, use theattractions of real-life examples (and sell papers by making a few eye-catching claims inthat direction) – but, just at the moment when confrontation with reality threatens: tocomfortably retreat to a more normative or theoretical position. Personally, I find this10strategy more and more empty, and almost intellectually dishonest. But much worse thandishonesty: it has become boring, and the time seems ripe to actually confront all of theabove with experimental facts. Most of what was said in the preceding two sectionssuggest experimental questions, such as:What do people get out of various types of assertion?What do their conversation plans really look like?How do they cope with diversity of agents?and on another note, do they really feel \'complexity barriers\' when jumping from onepractice to another (say from honest reporting to lying) as predicted by computationaltheory? Experimental game theorists have already made their move toward cognitivepsychology here: semanticists, philosophers, and logicians might er, much of the relevant experimental material lies close at hand. Just watch yourstudents play information games on their GSMs (Muddy Children-like puzzles often comeas accessories with them), or let people play known games in various degrees ofcomplexity by manipulating rules, such as in the move from Chess to Kriegsspiel, whereone does not know the positions of the Opponent\'s pieces. After lectures on dynamicepistemic logic, I often get intriguing responses from members of the audience telling mehow they play modified information games, \'complexifying\', e.g., Clue to make it moreinteresting. Some variations seem to work there, others do not. E.g., adding a fourthguessing category of Motive in addition to Room, Person, and Murder Weapon is reportedas easy – whereas allowing some \'cheating moves\' makes the game almost impossible tosuccessfully complete. It would be up to socially minded cognitive scientists to use thisspontaneous evidence, and find out…6ConclusionThis paper has tried to make a case that interactive social cognitive phenomena areimportant – and also, that they are endowed with a logical-computational structure richenough to make for interesting scientific analysis. But perhaps, the sheer social stance byitself is worth emphasizing. I am always amazed by the \'individualist\' bias in cognitivestudies. We look at language in terms of individual competence, even though there areusually Speaker and Hearer interacting, and even though the real language skill to be taughtis successful communication, not writing grammatically correct sentences in your privatediary. And while we are at it, take that teaching itself. We model single agents forminghypotheses about a grammar or some other model for an observed string of phenomena.11But again, this is just a one-dimensional projection of the most striking two-agent settingfor teaching, being that of a Student with a Teacher. Our models should emphasize thelatter, and then specialize to the former. Finally, logical systems emphasize lonesomereasoners, or even crystalline proofs where all traces of human activity have been washedoff with formaline. But surely, the proto-typical logical activity is argumentation, whichhappens in an interactive social next main point has been the Trinity of logic, cognition, and computation. I reallybelieve this is also the way to go in cognitive studies generally. First, not accidentally, thesethree areas often undergo the same intellectual movements. Notably, the social perspectiveemerged in logic and computer science around the same time in the 1980s. But perhaps astronger argument for the juxtaposition is the emergence of deep and surprising newinsights. Here is an example from dynamic epistemic logic again. Puzzles like MuddyChildren or complex conversational strategies involve complex behaviour that can bedescribed as standard program constructions IF THEN ELSE, WHILE DO telling peopleto say and ask certain things until certain effects are reached. Now Miller & Moss 2003have shown that the complexity of this sort of reasoning on top of the logic of publicannouncement is undecidable. This seems a purely negative result, but their method ofproof is quite interesting. The authors show how to encode the behaviour of arbitraryTuring machines (and hence, correlated undecidable issues like the Halting Problem) interms of planning conversational strategies for achieving specified epistemic goals as towho is to know what at the end. Thus, taking a positive view of the result, we see thatComputation and conversation have equal processing power!Insights like this allow us to look at familiar phenomena in unexpected ways. Computingmachines communicate, but communicating humans also engage in complex computationalprocesses. Heaven knows, we might even be able to tap those computational resources!This commonality is reinforced by the earlier observation that we are already living in, andcoping with, mixed human-machine societies, involving diversity of agents of manydifferent types. Indeed, our ability to do so constitutes one of the most amazing cognitiveskills that we have – and one that should One final attractive feature of the social aspect of cognition is that it is all around us. Thismay be a bit hard to see at first. In science we are used to the idea that some things are toofar away to see with the naked eye, and astronomers have to send their rockets and otherexpensive machines out to do the exploring for them. But social cognition may be tooclose to our skins to recognize its importance and structure straightaway. But once we do,we see that our whole world is already a gigantic Cognitive Lab. We do not need multi-12billion dollar machines to force elementary particles into accelerating loops that reveal theirpotential and limitations. Such things happen automatically. Just think of the Internet,surely a multi-billion dollar machine, created for free, and all those agents playing all sortsof differential informative games using their emails, and playing their cc and bcc buttonsfor achieving complex epistemic effects! So, the cognitive experiments are already we must go and read off the results…ReferencesAbramsky, Samson. 1998. From Computation to Interaction, towards a science of

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