逻辑、计算和博弈 (5).pdf

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1、1 LOGICAL DYNAMICS OF INFORMATION AND INTERACTION 2 TABLE OF CONTENTS Preface 1 Logical dynamics,agency,and intelligent interaction 2 Epistemic logic and semantic information 3 Dynamic logic of public observation 4 Multi-agent dynamic-epistemic logic 5 Dynamics of inference and awareness 6 Questions

2、 and issue management 7 Soft information,correction,and belief change 8 An encounter with probability 9 Preference statics and dynamics 10 Decisions,actions,and games 11 Processes over time 12 Epistemic group structure and collective agency 13 Logical dynamics in philosophy 14 Computation as convers

3、ation 15 Rational dynamics in game theory 16 Meeting cognitive realities 17 Conclusion Bibliography 121 Chapter 7 SOFT INFORMATION,CORRECTION,AND BELIEF CHANGE So far,we developed dynamic logics that deal with knowledge,inference,and questions,all based on information and truth.Now we want to look a

4、t another pervasive attitude that agents have toward information,namely,their beliefs.This chapter will show how belief change fits well with our dynamic framework,and develop some of its logical theory.131 This puts one more crucial aspect of rational agents in place:not their being right about eve

5、rything,but their being wrong,and subsequent acts of self-correction.7.1 From knowledge to belief as a trigger for actions While knowledge is important to agency,our actions are often driven by fallible beliefs.I am riding my bicycle this evening because I believe it will get me home,even though my

6、epistemic range includes worlds where the San Andreas Earthquake strikes.Decision theory is about choice and action on the basis of beliefs,as knowledge may not be available.Thus,our next step in the logical dynamics of rational agency is the study of beliefs,viewed as concretely as possible.Think o

7、f our scenarios so far.The cards have been dealt.I know that there are 52 of them,and I know their colors.But I have more fleeting beliefs about who holds which card,or about how the other agents will play.132 Hard versus soft information With this distinction in attitude comes a richer dynamics.A p

8、ublic announcement!P of a fact P was an event of hard information that changes irrevocably what I know.When I see the Ace of Spades played,I come to know that no one has it any more.This is the trigger that drove our dynamic epistemic logics in Chapters 3 and 4.Such events of hard information may al

9、so change our beliefs and we will find a complete logical system for this.But there are also events of soft information,affecting my beliefs without affecting my knowledge about the cards.I see you smile.This makes it more likely that you hold a trump card,but it does not rule out that you do not.To

10、 describe this,we will use worlds with plausibility orderings supporting dynamic updates.The tandem of jumping ahead and self-correction Here is what is most important to me in this chapter from the standpoint of rational agency.As acting agents,we are bound to form beliefs that go beyond the hard i

11、nformation we have.And this is not a concession to 131 Later on,we discuss how this relates to the alternative AGM style(Grdenfors&Rott 1995).132 Of course,I could even be wrong about the cards(perhaps the Devil added his visiting card)but this worry seems morbid,and not useful in investigating norm

12、al information flow.122 human frailty or to our mercurial nature.It is rather the essence of creativity,jumping ahead to conclusions we are not really entitled to,and basing our beliefs and actions on them.But there is another side to this coin,that I would dub our capacity for self-correction,or if

13、 you wish,for learning.We have an amazing capacity for standing up after we have fallen informationally,and to me,rationality is displayed at its best in intelligent responses to new evidence that contradicts what we thought so far.What new beliefs do we form,and what amended actions result?Chapter

14、1 saw this as a necessary pair of skills:jumping to conclusions(i.e.,beliefs)and correcting ourselves in times of trouble.And the hallmark of a rational agent is to be good at both:it is easy to prove one theorem after another,it is hard to revise your theory when it has come crashing down.So,in pur

15、suing the dynamic logics of this chapter,I am trying to chart this second skill.7.2 Static logic of knowledge and belief Knowledge and belief have been studied together ever since Plato proposed his equation of knowledge with justified true belief,and much of epistemology is still about finding an i

16、ngredient that would turn true belief into knowledge.Without attempting this here(see Chapter 13 for our thoughts),how can we put knowledge and belief side by side?Reinterpreting PAL One easy route reinterprets dynamic-epistemic logic so far.We read the earlier K-operators as beliefs,again as univer

17、sal quantifiers over the accessible range,placing no constraints on the accessibility relation:just pointed arrows.One test for such an approach is that it must be possible for beliefs to be wrong:Example A mistaken belief.Consider the following model with two worlds that are epistemically accessibl

18、e to each other,but the pointed arrow is the only belief relation.Here,in the actual black world to the left,the proposition p is true,but the agent mistakenly believes that p:p p With this view of doxastic modalities(cf.Hintikka 1962),the machinery of DEL works exactly as before.But there is a prob

19、lem:Example,continued Consider a public announcement!p of the true fact p.The PAL result is the one-world model where p holds,with the inherited empty doxastic accessibility relation.But on the 123 universal quantifier reading of belief,this means the following:the agent believes that p,but also tha

20、t p,in fact B is true at such an end-point.In this way,agents who have their beliefs contradicted are shattered and start believing anything.Such a collapse is unworthy of a rational agent in the sense of Chapter 1,and hence we will change the semantics to allow for more intelligent responses.World

21、comparison by plausibility A richer view of belief follows the intuition that an agent believes the things that are true,not in all her epistemically accessible worlds,but only in those that are best or most relevant to her.I believe that my bicycle will get me home on time,even though I do not know

22、 that it will not suddenly disappear in a seismic chasm.But the worlds where it stays on the road are more plausible than those where it drops down,and among the former,those where it arrives on time are more plausible than those where it does not.Static models for this setting are easily defined:De

23、finition Epistemic-doxastic models.Epistemic-doxastic models are structures M=(W,iiI,i,siI,V)where the relations i stand for epistemic accessibility,and the i,s are ternary comparison relations for agents read as follows,x i,s y if,in world s,agent i considers y at least as plausible as x.Now episte

24、mic accessibility can be an equivalence relation again.Models like this occur in conditional logic,Shoham 1988 on preference relations in AI,and the graded models of Spohn 1988.One can impose several conditions on the plausibility relations,depending on their intuitive reading.Burgess 1981 has refle

25、xivity and transitivity,Lewis 1973 also imposes connectedness:for all worlds s,t,either s t or t s.The latter yields the well-known geometrical nested spheres for conditional logic.133 As with epistemic models,our dynamic analysis works largely independently from such design decisions,important thou

26、gh they may be.In particular,connected orders yield nice pictures of a line of equi-plausibility clusters,in which there are only three options for worlds s,t:either strict precedence s t or t s,or equiplausibility s t t s.But there are also settings that need a fourth option of incomparability:s t

27、t s.This happens when comparing worlds with conflicting criteria,as with some preference 133 The natural strict variant of these orderings is defined as follows:s t iff s t t s.124 logics in Chapter 9.Sometimes also,partially ordered graphs are just the mathematically more elegant approach(Andrka,Ry

28、an,Schobbens 2002;cf.Chapter 12).Languages and logics One can interpret many logical operators in this richer structure.In what follows,we choose intuitive maximality formulations for belief Bi.134 First of all,there is plain belief,whose modality is interpreted as follows:135 Definition Belief as t

29、ruth in the most plausible worlds.In epistemic-doxastic models,knowledge is interpreted as usual,while we put M,s|=Bi iff M,t|=for all worlds t that are maximal in the ordering xy.x i,s y.136 But absolute belief does not suffice.Reasoning about information flow and action involves conditional belief

30、.We write this as follows:B,with the intuitive reading that,conditional on,the agent believes that.This is close to conditional logic:Definition Conditional beliefs as plausibility conditionals.In epistemic-doxastic models,M,s|=B iff M,t|=for all worlds t that are maximal for the ordering xy.x i,s y

31、 in the set u|M,u|=.Absolute belief B is the special case BT.It can be shown that conditional belief is not definable in terms of absolute belief,so we have a genuine language extension.137 138 Digression on conditionals As with epistemic notions in Chapters 2,3,conditional beliefs pre-encode belief

32、s that we would have if we were to learn certain things.139 A formal 134 These must be modified in non-wellfounded models allowing infinite descent in the ordering.This issue is orthogonal to the main thrust of this chapter,and we will largely ignore it.135 For convenience,henceforth we drop subscri

33、pts where they do not add insight.136 Here we used lambda notation to denote relations,but plain x i,s y would serve,too.137 Likewise,the binary quantifier Most A are B is not definable in first-order logic extended with just a unary quantifier“Most objects in the universe are B”(cf.Peters&Westersth

34、l 2005).138 One can also interpret richer languages on epistemic-doxastic models.E.g.,maximality suggests a binary relation best defined as“t is maximal in xy.s xy”.One can introduce a modality for this,defining conditional belief as best.Dynamic extensions of our language will come below.139 Static

35、 pre-encoding versus dynamics.A conditional belief B does not quite say that we would believe if we learnt that.For an act of learning changes the current model M,and hence the truth value of might change,as modalities in now range over fewer worlds in M|.Similar things happened with epistemic state

36、ments after communication in Chapter 3 and also in logic in 125 analogy is this.A conditional says that is true in the closest worlds where is true,along some comparison order on worlds.This is exactly the above clause.Thus,results from conditional logic apply.For instance,on reflexive transitive pl

37、ausibility models,we have this completeness theorem(Burgess 1981,Veltman 1985):Theorem The logic of B is axiomatized by the laws of propositional logic plus obvious transcriptions of the following principles of conditional logic:(a),(b)implies ,(c),imply ,(d),imply(),(e),imply().Epistemic-doxastic l

38、ogics In line with the general approach in this book,we do not pursue completeness theorems for static logics of knowledge and belief.But for greater ease,this chapter makes one simplification that reflects in the logic.Epistemic accessibility will be an equivalence relation,and plausibility a pre-o

39、rder over the equivalence classes,the same as viewed from any world inside the class.This makes the following axiom valid:B KB Epistemic-Doxastic Introspection While this is a debatable assumption,it helps focus on the core ideas of the dynamics.7.3 Belief change under hard information Our first dyn

40、amic logic of belief revision puts together the logic PAL with our static models for conditional belief,following the same methodology as earlier chapters.We will move fast,as the general points of Chapters 2,3 apply,and indeed thrive here.A complete axiomatic system For a start,we must locate the k

41、ey recursion axiom for the new beliefs,something that can be done easily,using update pictures as before:Fact The following formula is valid for beliefs after hard information:!PB (P BP(!P).This is like the PAL recursion axiom for knowledge under announcement.But note the conditional belief in the c

42、onsequent,that does not reduce to a conditional absolute belief B(P .Still,to keep the language in harmony,this is not enough.We need to know,not general.The relativized quantifier in“All mothers have daughters”does not say that,if we relativize to the subset of mothers,all of them have daughters wh

43、o are mothers themselves.126 just which beliefs are formed after new information,but which conditional beliefs.140 What is the recursion principle for change in conditional beliefs under hard information?There might be a regress here toward conditional conditional beliefs,but in fact,we have:Theorem

44、 The logic of conditional belief under public announcements is axiomatized completely by(a)any complete static logic for the model class chosen,(b)the PAL recursion axioms for atomic facts and Boolean operations,(c)the following new recursion axiom for conditional beliefs:!PB (P B P!P !P).Proof Firs

45、t we check the soundness of the new axiom.On the left hand side,it says that in the new model(M|P,s),is true in the best-worlds.With the usual precondition for true announcement,on the right-hand side,it says that in M,s,the best worlds that are P now and will become after announcing that P,will als

46、o become after announcing P.This is indeed equivalent.The remainder of the proof is our earlier stepwise reduction analysis,noting that the above axiom is recursive,pushing announcement modalities inside.To get a joint version with knowledge,we just combine with the PAL axioms.Clarifying the Ramsey

47、Test Our dynamic logic sharpens up the Ramsey Test that says:“A conditional proposition A B is true,if,after adding A to your current stock of beliefs,the minimal consistent revision implies B.”In our perspective,this is ambiguous,as B need no longer say the same thing after the revision.That is why

48、 our recursion axiom carefully distinguishes between formulas before update and what happens to them after:!P.Even so,there is an interesting special case of factual propositions without modal operators(cf.Chapter 3),that do not change their truth value under announcement.In that case,with Q,R factu

49、al propositions,the above recursion axioms read as follows:!PBQ (P BPQ),!PBRQ (P BPR Q)141 140 This is overlooked in classical belief revision theory,which says only how new absolute beliefs are formed.One gets stuck in one round,as the new state does not pre-encode what happens in the next round.Th

50、is so-called Iteration Problem cannot arise in a systematic logical set-up.141 For some paradoxes in combining the Ramsey test with belief revision,cf.Grdenfors 1988.The nice thing of a logic approach is that every law we formulate is automatically sound.127 Belief change under hard update is not ye