Comments on ‘The Logic of Conditional Doxastic Actions’

I first summarize Baltag and Smets’ contribution to this volume, and praise their work. Then I compare the anti-lexicographic plausibility update that they propose to a proposal by Aucher, as an illustration of the difference between a qualitative and a quantitive formulation of updates. I quote Spohn’s original work that is at the root of research in plausibility updates and of the notion of anti-lexicographic update. Some technical notes on different partial orders used in belief revision serve as a prelude to an observed relation between the qualitative and quantitative representation of structures. Finally I address Baltag and Smets’ analysis of the action of lying. In this commentary on ‘The Logic of Conditional Doxastic Actions’ I am in the delightful position of having the last word in an argument with Alexandru Baltag. This position is very hard to obtain. But because in this volume my commentary follows the chapter by Alexandru Baltag and Sonja Smets, any further elaborations and involutions will be out of reach to the readers of the volume. I am going to use this rare advantage to the limit. Having said that, I sent a preliminary version of these comments to Alexandru and Sonja for comments, and immediately received in response an email of about the same length as this submission. I am very grateful for Alexandru’s last words. Well, nearly last words. I made some further changes. Now, the work is done. 1 The logic of conditional doxastic actions In AGM belief revision a distinction is made between belief expansion and (proper) belief revision. Given a set of consistent beliefs, in belief expanKrzysztof R. Apt, Robert van Rooij (eds.). New Perspectives on Games and Interaction. Texts in Logic and Games 4, Amsterdam University Press 2008, pp. 33–44. 0044 0045 0046 0047 0048 0049 0050 0051 0052 0053 0054 0055 0056 0057 0058 0059 0060 0061 0062 0063 0064 0065 0066 0067 0068 0069 0070 0071 0072 0073 0074 0075 0076 0077 0078 0079 0080 0081 0082 0083 0084 0085 0086 34 H. van Ditmarsch sion new information can typically be added as such, without conflict with existing beliefs. But in belief revision, the incoming information is typically inconsistent with the prior beliefs. A long line of work in dynamic epistemic logic, prominently including the well-known framework of action models, also by Baltag but with different collaborators, can be seen as a generalization of belief expansion. Unlike AGM belief revision in its original formulation, this dynamic epistemic logic also models belief expansion for more than one agent, and what is known as higher-order belief change: given explicit operators for ‘the agent believes that’, self-reference to one’s own belief or to the beliefs of others can also be formalized. A problem in that line of research remained that the typical belief revision, i.e., how to process inconsistent new beliefs, cannot be modelled. Belief in factual information, for example, cannot be given up when confronted with new belief that is considered as acceptable evidence to the contrary. And this is not just impossible within the setting of knowledge, where one does not expect proper revision to be possible, because knowledge is truthful. It is also impossible for weaker epistemic notions. In this contribution to the volume, Alexandru Baltag and Sonja Smets introduce a dynamic epistemic framework in which belief revision in the proper sense is, after all, possible. Given a structure (an epistemic plausibility frame) wherein one does not merely have epistemic indistinguishability between states but also plausibility relations between states, one can define both knowledge and conditional belief operators. Unconditional belief is defined as belief that is conditional to the trivial state of information. (The trivial state of information is the epistemic equivalence class occupied by the agent, which is described by the formula ⊤.) In this setting belief revision is possible where the agent (unconditionally) believed some factual information p but after having been presented with convincing evidence to the contrary, changes his mind, and then believes the exact opposite. Just as for the relation between classic AGM expansion and dynamic epistemic logic, we now have again that this approach also models multi-agent and higher-order belief revision. The authors go much further, beyond that. In a multi-agent setting there are more complex forms of belief revision than revision with a publicly announced formula φ. That is merely an example of a doxastic action. More complex doxastic actions, where the action appears differently to each agent, are also conceivable. They present a very good example, namely the action where agent a is lying that φ. To a credulous agent b, this action will appear as a truthful announcement that φ. But not to a of course, who knows that she is lying. The general form of doxastic actions is like an epistemic plausibility model and is called an action plausibility model ; the difference is that instead of a valuation of atoms in each state of an 0087 0088 0089 0090 0091 0092 0093 0094 0095 0096 0097 0098 0099 0100 0101 0102 0103 0104 0105 0106 0107 0108 0109 0110 0111 0112 0113 0114 0115 0116 0117 0118 0119 0120 0121 0122 0123 0124 0125 0126 0127 0128 0129 Comments on ‘The Logic of Conditional Doxastic Actions’ 35 epistemic plausibility model we now have a precondition for the execution of each action (i.e., element of the domain) of an action plausibility model. The execution of such a doxastic action in a epistemic plausibility state is a restricted modal product, where I am tempted to say ‘as usual’, to avoid the obligation of having to explain this in detail. The only unusual aspect of this procedure is the very well-chosen mechanism to compute new plausibilities from given plausibities, called anti-lexicographic preorder relation. This says that plausibility among actions takes precedence over plausibility among states. It is the natural generalization of the implicit AGM principle that the revision formula takes precedence over the already believed formulas. Anti-lexicographic preorder prescribes that: a new state of affairs is more plausible than another new state of affairs, if it results from an action that is strictly more plausible than the action from which the other state results, or if the states result from equally plausible actions but the former state already was more plausible than the latter state before action execution. So far, this overview also describes the authors’ other publication (Baltag and Smets, 2006). A main focus of their underlying contribution is the interpretation of these results in terms of conditional reasoning and conditional doxastic action models. The conditional appearance maps of these conditional doxastic action models take the place of the plausibility relations among the actions in an action plausibility model. They motivate and justify in great detail various notions for conditional belief, and their interdependencies. A fabulous finale is a complete axiomatization with a reduction axiom that relates conditional belief after an action to conditional belief before that action. The technicalities of this logic with dynamic operators for conditional action execution may be hard to follow unless the reader is intimately familiar with the BMS action model framework, as these technical details are only somewhat summarily presented. In that case, just focus on this reduction axiom, the action-conditional-belief law, and the well-chosen examples given of its application. I can assure you, it’s all true. 2 Quality is better than quantity The authors claim that their approach is ‘in its spirit closer to qualitative logics than to approaches of a more quantitative flavour.’ This is a very well-considered phrasing. Let us see why they are right. One such approach of a more quantitative flavour is the proposal by Guillaume Aucher (2003). In terms of action plausibility models his proposal is a version of van Benthem’s soft update referred to in Example 5, but with a different recipe to compute new plausibilities. Aucher also employs structures with epistemic equivalence classes and plausibility relations, but the plausibility relations are derived from a finite total order of degrees of 0130 0131 0132 0133 0134 0135 0136 0137 0138 0139 0140 0141 0142 0143 0144 0145 0146 0147 0148 0149 0150 0151 0152 0153 0154 0155 0156 0157 0158 0159 0160 0161 0162 0163 0164 0165 0166 0167 0168 0169 0170 0171 0172 36 H. van Ditmarsch plausibility. If s is more plausible than s, its degree of plausibility is lower. The lowest degree of plausibility is 0. To each degree of plausibility corresponds a degree of belief, expressing that the believed proposition is at least as plausible as that. Unconditional belief is belief of degree 0. Given some model with domain S, Aucher’s ‘belief revision with a formula φ’ now amounts to the following. Whenever φ is true, subtract the minimum degree of the φ-states from the current degree. Otherwise, subtract the minimum degree of the ¬φ-states from the current degree and add one. This ensures that at least one φ-state will get degree 0, and thus factual information φ will be unconditionally believed after revision, as required. For an example, consider one agent a only and a domain consisting of four states 0, 1, 2, 3 comprising a single equivalence class (all four states are considered possible by the agent) and such that 0 ≤a 1 ≤a 2 ≤a 3. In this initial epistemic plausibility structure, the degree of each state is its name. First, suppose that factual information p is true in state 3 only (the valuation of p is {3}). According to the recipe above, the result is the order 3 ≤a 0 ≤a 1 ≤a 2. How come? Write dg(s) for the old degree of state s and dg(s) for its new degree, after revision. Then dg(3) = dg(3)− Min{dg(s) | s |= p} = 3−3 = 0. Whereas dg(1) = dg(1)−Min{dg(s) | s 6|= p}+ 1 = 1− 0 + 1 = 2. Etcetera. So far so good. Now for some other examples, demonstrating issues with such quantitatively formulated proposals for belief revision. Suppose that, instead, p is true in states 1 and 2. We now obtain that 1 ≤a 2 ≃a 0 ≤ 3. As a result of this revision, states 2 and 0 have become equally plausible. As a side effect of the revision, such ‘loss of plausibility information’ may be considered less desirable. Finally, suppose that p was already believed: suppose that p is true in states 0 and 1. We then get 0 ≤a 1 ≃a 2 ≤ 3. This is also strange: instead of reinforcing belief in p, the ¬p-states have become more plausible instead! This example demonstrates some issues with a quantified formulation of belief revision. Of course Aucher is aware of all these issues. See Aucher’s PhD thesis (2008) for a quite novel way to perform higher-order and multiagent belief revision, based on plausibility relations among sets of formulas describing the structure in which the revision is executed. 3 Belief revision known as maximal-Spohn When I first heard from Baltag and Smets’ work on plausibility reasoning my first response was: “But this has all been done already! It’s maximal-Spohn belief revision!” After some heavy internal combustion, I told Alexandru, who has his own response cycle, and this is all a long time ago. At the time I thought to remember specific phrasing in Spohn’s well-known ‘Ordinal Conditional Functions’ (1988). But I never got down to be precise about 0173 0174 0175 0176 0177 0178 0179 0180 0181 0182 0183 0184 0185 0186 0187 0188 0189 0190 0191 0192 0193 0194 0195 0196 0197 0198 0199 0200 0201 0202 0203 0204 0205 0206 0207 0208 0209 0210 0211 0212 0213 0214 0215 Comments on ‘The Logic of Conditional Doxastic Actions’ 37 this source of their work. Now I am. This section of my comments can be seen as yet another footnote to the extensive references and motivation of the authors’ contribution. In their Example 5 they explain that for revision with single formulas all the following amount to more or less the same: antilexicographic update, lexicographic update, soft public update. To this list we can add yet another term: what Hans Rott (in his presentations and in Rott, 2006) and I call ‘maximal-Spohn belief revision’. Spohn introduces the ‘simple conditional functions’ (SCF) and the equivalent notion of ‘well-ordered partitions’ (WOP) to represent the extent of disbelief in propositions. In terms of Baltag and Smets, a WOP defines a totally ordered plausibility relation on the domain. Spohn then observes that such WOPs (plausibility relations) need to be updated when confronted with incoming new information in the form of a proposition A. In our terms A is the denotation of some revision formula φ. He then proceeds to discuss some specific plausibility updates. His presentation is based on ordinals α, β, γ, . . . that label sets Eα, Eβ , Eγ , . . . of equally plausible states (all the Eα-states are more plausible than all the Eβ-states, etc.). For a simplifying example, consider a partition of a domain W into a well-ordered partition E0, E1, . . . , E6. The set E0 are the most believed states, etc. Assume that a proposition A has non-empty intersection with E4 and E5. Thus, the most plausible A-states are found in E4. If we now also read ‘state x is less plausible than state y’ for ‘world x is more disbelieved than world y’ we are ready for an original quote from (Spohn, 1988), explaining two different ways to adjust E1, . . . , E6 relative to A. A clear sign of a great writer is that one can take his work out of context but that it remains immediately intelligible. A first proposal might be this: It seems plausible to assume that, after information A is accepted, all the possible worlds in A are less disbelieved than the worlds in A (where A is the relative complement W \ A of A). Further, it seems reasonable to assume that, by getting information only about A, the ordering of disbelief of the worlds within A remains unchanged, and likewise for the worlds in A. (Spohn, 1988, pp. 112–113) (. . . ) the assumption that, after getting informed about A, all worlds in A are more disbelieved than all worlds in A seems too strong. Certainly, the first member, i.e. the net content of the new WOP, must be a subset of A; thus at least some worlds in A must get less disbelieved than the worlds in A. But it is utterly questionable whether even the most disbelieved world in A should get less disbelieved than even the least disbelieved world in A; this could be effected at best by the most certain information. This last consideration suggests a second proposal. Perhaps one 0216 0217 0218 0219 0220 0221 0222 0223 0224 0225 0226 0227 0228 0229 0230 0231 0232 0233 0234 0235 0236 0237 0238 0239 0240 0241 0242 0243 0244 0245 0246 0247 0248 0249 0250 0251 0252 0253 0254 0255 0256 0257 0258 38 H. van Ditmarsch should put only the least disbelieved and not all worlds in A at the top of the new WOP (. . . ). (Spohn, 1988, pp. 113–114) The first proposal has become known as maximal-Spohn. The second proposal has become known as minimal-Spohn. Applied to the example partition the results are as follows; on purpose I use a formatting that is very similar to the displayed formulas on pages 113 and 114 in (Spohn, 1988). Further down in the sequence means less plausible. E4 ∩A,E5 ∩A,E0, E1, E2, E3, E4 ∩A,E5 ∩A maximal-Spohn E4 ∩A,E0, E1, E2, E3, E4 ∩A,E5 minimal-Spohn In maximal-Spohn, as in antilexicographic update, the A-states now come first, respecting the already existing plausibility distinctions amongA-states, so that we start with E4 ∩ A,E5 ∩ A. The order among the non-A-states also remains the same (whether intersecting with A or not), thus we end with E0, E1, E2, E3, E4 ∩A,E5 ∩A. In minimal-Spohn, the states in E5 are not affected by proposition A; only the equivalence class containing most plausible A-states is split in two, and only those most plausible A-states, namely E4 ∩A, are shifted to the front of the line. These are now the most plausible states in the domain, such that A is now (in terms of Baltag and Smets again) unconditionally believed. Aucher’s plausibility update (Aucher, 2003), that we discussed in the previous section, implements a particular kind of ‘minimal-Spohn’ that also employs Spohn’s ordinal conditional functions. We do not wish to discuss those here—things are quantitative enough as it is, already. Aucher’s is not as minimal as it can be, e.g., I demonstrated the side-effect of merging plausibilities. It would be interesting to see a truly qualitative form of plausibility update that amounts to minimality in the Spohn-sense, or at least to something less maximal than anti-lexicographic update but equally intuitively convincing; but I do not know of one. 4 Well-preorders In epistemic plausibility frames (S,∼a,≤a)a∈A the epistemic indistinguishability relations are equivalence relations and the plausibility relations are required to be well-preorders, i.e., reflexive and transitive relations where every non-empty subset has minimal elements. The non-empty-subset requirement ensures that something non-trivial is always conditionally believed. I will clarify some technical points concerning these primitives, to illustrate their richness for modelling purposes. On the meaning of minimal. A well-order is a total order where every non-empty subset has a least element, so by analogy a well-preorder should indeed be, as Baltag and Smets propose, a pre-order where every non-empty 0259 0260 0261 0262 0263 0264 0265 0266 0267 0268 0269 0270 0271 0272 0273 0274 0275 0276 0277 0278 0279 0280 0281 0282 0283 0284 0285 0286 0287 0288 0289 0290 0291 0292 0293 0294 0295 0296 0297 0298 0299 0300 0301 Comments on ‘The Logic of Conditional Doxastic Actions’ 39 subset has minimal elements. So far so good. Right? Wrong! Because we haven’t seen the definition of minimal yet. I thought I did not need one. For me, an element is minimal in an ordered set if nothing is smaller : Min(≤a, S) := {s | t ∈ S and t ≤a s implies t ≃a s}. Also, their first occurrence of ‘minimal’ before that definition was in a familiar setting, ‘something like a well-order’, so that I did not bother about the precise meaning. And accidentally the definition of minimal was not just after that first textual occurrence but even on the next page. So I had skipped that. This was unwise. For Baltag and Smets, an element is minimal in an ordered set if everything is bigger : Min(≤a, S) = {s | t ∈ S implies s ≤a t}. This is a powerful device, particularly as for partial orders some elements may not be related. The constraint that all non-empty sets have minima in their sense applies to two-element sets and thus enforces that such relations are connected orders (as explained in Footnote 4 of Baltag and Smets’ text). So every well-preorder is a connected order. On connected orders the two definitions of minimal (Min and Min) coincide. We can further observe that the quotient relation ≤a/≃a is a total order, and that it is also a wellorder. Given a non-empty subset S of ≤a/≃a, there is a non-empty subset S of ≤a such that S ′′ \ ≃a = S . The ≃a-equivalence class of the minimal elements of S is the least element of S. The well-preorders of the authors are sometimes known as templated orders (Meyer et al., 2000). All this corresponds to Grove systems of spheres, as the authors rightly state. Partial orders in belief revision. Partial orders that are not connected or not well-ordered according to the authors’ definition do occur in belief revision settings. From now on I will only use ‘minimal’ in the standard sense. Given one agent a, consider the frame consisting of five states {0, 1, 2, 3, 4}, all epistemically indistinguishable, and such that the relation ≤a is the transitive and reflexive closure of 0 ≤a 1 ≤ 4 and 0 ≤a 2 ≤a 3 ≤a 4. It is a partial order, and every non-empty subset has minima. The reader can easily check this, for example, Min(≤a, {0, 1, 2, 3, 4}) = {0}, Min(≤a, {1, 2, 3}) = {1, 2} = Min(≤a, {1, 2}), and so on. If neither s ≤a t nor t ≤a s, states s and t are called incomparable. States 1 and 2 are incomparable, as are 1 and 3. Consider the symmetric closure Sy(≤a) of a plausibility relation ≤a that is a partial order and where every non-empty subset has minima. Given a state s, we call a state t ∼a s plausible iff (s, t) ∈ Sy(≤a). Conditionalization 0302 0303 0304 0305 0306 0307 0308 0309 0310 0311 0312 0313 0314 0315 0316 0317 0318 0319 0320 0321 0322 0323 0324 0325 0326 0327 0328 0329 0330 0331 0332 0333 0334 0335 0336 0337 0338 0339 0340 0341 0342 0343 0344 40 H. van Ditmarsch to implausible but epistemically possible states is clearly problematic. So as long as all states in an equivalence class are plausible regardless of the actual state in that class, we are out of trouble. This requirement states that the epistemic indistinguishability relation ∼a must be a refinement of Sy(≤a), or, differently said, that ∼a ∩ Sy(≤a) = ∼a. Incomparable states in a partial order can be ‘compared in a way’ after all. Define s ≡a t iff for all u ∈ S, u ≤a s iff u ≤a t. Let’s say that the agent is indifferent between s and t in that case. Clearly, equally plausible states are indifferent: ≃a ⊆ ≡a. But the agent is also indifferent between the incomparable states 1 and 2 in the above example. The quotient relation ≤a/≡a is a total order. In belief contraction this identification of incomparable objects in a preorder typically occurs between sets of formulas, not between semantic objects. See work on epistemic entrenchment involving templated orders, e.g., (Meyer et al., 2000). Qualitative to quantitative. As already quoted by me above, the authors consider their approach ‘in its spirit closer to qualitative logics than to approaches of a more quantitative flavour.’ Well-chosen wording, because— as the authors know—in its nature their approach is fairly quantitative after all. Let us see why. From a preorder where all non-empty subsets have minimal elements we can create degrees of plausibility as follows. Given that all sets have minimal elements, we give the ≤a-minimal states of the entire domain S degree of plausibility 0. This set is non-empty. Now the entire domain minus the set of states with degree 0 also has a non-empty set of minimal elements. Again, this set exists. These are the states of degree 1. And so on. Write Degree(≤a) for the set of states of degree i. We now have: Degree(≤a) := Min(≤a, S) Degree(≤a) := Min(≤a, S \ ⋃ j=0..k Degree (≤a)) Note the relation with the total order ≤a/≡a introduced above. Of course, I entirely agree that a qualitative presentation of an epistemic plausibility framework is to be preferred over a quantitative representation. And—this is once again Alexandru Baltag providing an essential comment to the preliminary version of this commentary—although this comparison can be made on the structural level, the language of conditional doxastic logic is apparently not expressive enough to define degrees of belief, that use the above order. This matter is explained in their related publication (Baltag and Smets, 2006). But with that result one can wonder if a weaker structural framework, more qualitative in nature, would already have sufficed to obtain the same logical result. It seems to me that the quest for the nature and the spirit of qualitative belief revision has not yet been ended. Other frameworks for belief revision, 0345 0346 0347 0348 0349 0350 0351 0352 0353 0354 0355 0356 0357 0358 0359 0360 0361 0362 0363 0364 0365 0366 0367 0368 0369 0370 0371 0372 0373 0374 0375 0376 0377 0378 0379 0380 0381 0382 0383 0384 0385 0386 0387 Comments on ‘The Logic of Conditional Doxastic Actions’ 41 such as the referenced work by Fenrong Liu, her PhD thesis (2008), and my own work (van Ditmarsch, 2005) (where the non-empty minima requirement is only for the entire domain, thus allowing the real number interval [0, 1]), sometimes employ other partial orders and basic assumptions and may also contribute to this quest. 5 This is a lie The authors’ analysis of “lying about φ” involves an action plausibility model consisting of two actions Liea(φ) and Truea(φ) with preconditions ¬Kaφ and Kaφ respectively. These actions can be distinguished by the lying agent, the speaker a, but are indistinguishable for the target, the listener b. Further, b considers it more plausible that a speaks the truth, than not: Truea(φ) ≤b Liea(φ). So ‘agent a lies about φ’ means that a announces that φ is true, thus suggesting that she knows that φ is true, although a does in fact not know that. For convenience I am presenting this action as a dynamic operator that is part of the language (which can be justified as the authors do in Section 5). In the authors’ subsequent analysis it is explained how the contextual appearance of an action may also determines its meaning, both the context of states wherein the action may be executed and the context of states resulting from the action’s execution. Again, ‘lying about φ’ makes for a fine example. If the listener b already knows that φ is false, the act of lying does not appear to b as the truth that φ, but as a lie that φ. I have two observations to this analysis. Lying and bluffing. I think that the precondition of a ‘lying that φ’ is not ignorance of the truth, but knowledge to the contrary: the precondition of the action Liea(φ) should not be ¬Kaφ but Ka¬φ. If the precondition is ¬Kaφ instead, I call this bluffing, not lying. As I am not a native speaker of English, and neither are the authors, this seems to be as good a moment as any to consult a dictionary (Merriam-Webster). To bluff is “to cause to believe what is untrue.” Whereas to lie is “to make a statement one knows to be untrue.” It is further informative to read that the etymology for ‘bluff’ gives “probably from the Dutch bluffen, for ‘to boast’, ‘to play a kind of card game.” It is of course typical Anglo-Saxon prejudice that all bad things concerning short-changing, scrooging, boasting, diseases, and unfair play (‘Dutch book’) are called Dutch. But let’s not pursue that matter further. Given the action Truea(φ), that expresses the for b more plausible alternative, I think that its precondition Kaφ properly expresses the part ‘to cause to believe what is untrue’. On the other hand, given that the action Liea(φ) that is considered less plausible by b, the precondition Ka¬φ seems to express accurately ‘to make a statement one knows to be untrue,’ and this condition is stronger than the precondition ¬Kaφ suggested by 0388 0389 0390 0391 0392 0393 0394 0395 0396 0397 0398 0399 0400 0401 0402 0403 0404 0405 0406 0407 0408 0409 0410 0411 0412 0413 0414 0415 0416 0417 0418 0419 0420 0421 0422 0423 0424 0425 0426 0427 0428 0429 0430 42 H. van Ditmarsch Baltag and Smets. When I proposed this commentary to the authors, Alexandru Baltag came with an interesting response: the precondition ¬Kaφ of action Liea(φ) also involves ‘to make a statement one knows to be untrue’, namely the statement ‘I know φ’. In fact, a knows that she does not know φ. This is true. But a bit further-fetched, if I may. For me, the prototypical example of a lie remains the situation where, way back in time, my mother asks me if I washed my hands before dinner and I say: “Yes.” Whereas when my grandfather held up his arm, with a closed fist obscuring a rubber (for Americans: eraser) and asked me: “What have I got in my hand?” and I then respond “A marble!” he never accused me of being a liar. Or did he? I’d like to investigate these matters further. I am unaware of much work on lying in dynamic epistemics. For a setting involving only belief and not knowledge, and public but not truthful announcements, see (van Ditmarsch et al., 2008). Is contextual appearance relevant? I question the need for contextual appearances of actions. I make my point by resorting to lying, again. The authors say that the precondition of a lie is ¬Kaφ but that, if the listener b already knows that φ is false, the act of lying no longer appears to b as the truth that φ, but as a lie that φ. I would be more inclined to strengthen the precondition for lying about φ from ¬Kaφ to ¬Kaφ∧¬Kb¬φ. In which case there is no need for this contextual precondition. Combining this with the previous I therefore think that the precondition of Liea(φ) should be Ka¬φ ∧ ¬Kb¬φ rather than ¬Kaφ. And this is only the beginning of a more and more fine-grained analysis of lying, not the end. For example, it is reasonable to expect that the speaker is aware of the listener’s ignorance about φ. That makes yet another precondition, namely Ka¬Kb¬φ. A distinction between knowledge and belief may also be important to model lying. The typical convention is to assume common belief that the speaker is knowledgeable about φ but the listener not, although in fact the speaker knows (or at least believes) the opposite of φ; so we get Ka¬φ ∧ CBab(Bb(Kaφ ∨Ka¬φ) ∧ ¬Kbφ ∧ ¬Kb¬φ) where CB is the common belief operator. We cannot replace common belief by common knowledge in this expression. Then it would be inconsistent. (We can also replace all other K-operators in this expression by Boperators.) There are also truly multi-agent scenarios involving lying, where only the addressee is unaware of the truth about φ but other listeners in the audience may have a different communicative stance. If this is only the beginning and not the end, why should there be an end at all? It is in fact unclear (as Alexandru Baltag also mentioned in response to reading a version of these comments) if by incorporating more 0431 0432 0433 0434 0435 0436 0437 0438 0439 0440 0441 0442 0443 0444 0445 0446 0447 0448 0449 0450 0451 0452 0453 0454 0455 0456 0457 0458 0459 0460 0461 0462 0463 0464 0465 0466 0467 0468 0469 0470 0471 0472 0473 Comments on ‘The Logic of Conditional Doxastic Actions’ 43 and more ‘context’ we finally have taken all possible contexts into account.Maybe there will always turn up yet another scenario that we might alsowant to incorporate in the precondition of lying. On the other hand—meagain trying to have to last word—it seems that by employing infinitaryoperators in preconditions such as common knowledge and common belief,as above, we can already pretty well take any kind of envisaged variationinto account. So my current bet is that the preconditions of contextualappearances (not the postconditional aspect) can be eliminated altogether.I am detracting myself, and the reader. So let me stop here. Does thisshow that the authors’ analysis of lying is flawed? Not at all! In fact it isvery well chosen, as it is a very rich speech act with many hidden aspectsthat are crying aloud for analysis, and the authors’ framework of doxasticactions is the obvious and very suitable formalization for such an analysis.Also, different arguments than the above can be put forward, in supportof ¬Kaφ as precondition of Liea(φ) instead of my preferred Ka¬φ. Letme therefore conclude by complimenting the authors again on their richcontribution, and hope for more from this productive duo. ReferencesAucher, G. (2003). A Combined System for Update Logic and Belief Re-vision. Master’s thesis, ILLC, University of Amsterdam, Amsterdam, TheNetherlands. ILLC Publications MoL-2003-03.Aucher, G. (2008). Perspectives on Belief and Change. PhD thesis, Uni-versity of Otago, New Zealand & Institut de Recherche en Informatique deToulouse, France.Baltag, A. & Smets, S. (2006). Dynamic Belief Revision over Multi-AgentPlausibility Models. In Bonanno, G., van der Hoek, W., & Wooldridge,M., eds., Proceedings of the 7th Conference on Logic and the Foundationsof Game and Decision Theory (LOFT’06), pp. 11–24.van Ditmarsch, H. (2005). Prolegomena to Dynamic Logic for Belief Revi-sion. Synthese, 147(2):229–275.van Ditmarsch, H., van Eijck, J., Sietsma, F., & Wang, Y. (2008). On theLogic of Lying. Under submission.Liu, F. (2008). 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