Doxastic Attitudes as -Revision Policies

Alexandru Baltag Ben Rodenh¨auser Sonja Smets ILLC, University of Amsterdam

Abstract. While propositional doxastic atti- encodes what the recipient will do if an input tudes, like and belief, capture an from that source is received. agent’s about certain propositions, her Traces of this idea are found scattered attitudes towards sources of information express across the literature, sometimes formulated her opinion about the reliability (or trustwor- in terms of notions like “evidential reliabil- thiness) of those sources. If an agent trusts ity” or “epistemic trust”.1 In Bayesian Epis- a witness, then she will, within certain limits, temology, the reliability of a source is cap- tend to accept his testimony as veridical. But tured by weights or probabilities attached to if she considers the witness to be a notorious the new information, which determine differ- liar, she may come to believe the opposite of ent ways of processing it (e.g., Bayesian con- what he tells her. In this paper, we put such at- ditioning versus Jeffrey conditioning).2 In titudes towards sources (or dynamic (doxastic) Belief Revision theory, various methods for attitudes) center stage, and formalize them as (iterated) belief revision have been proposed belief-revision strategies: policies governing how that can be understood as corresponding to an agent changess her beliefs whenever new in- different attitudes to the incoming informa- formation from a certain (type of) source is re- tion.3 ceived. We present a semantic, qualitative mod- While most previous authors have focused elling of this notion and investigate its proper- on quantitative approaches formalizing de- ties. grees of acceptance or degrees of trust, we propose a qualitative-relational setting that allows us to model a much more general class Introduction of doxastic attitudes, including various forms of trust, distrust and “semi-trust”. We give This paper explores the idea that an agent’s a semantic formalization of these concepts, “information uptake” (i.e. what she does study the strength order between different with some new informational input) depends dynamic attitudes and the natural opera- substantially on her attitude towards the tions with them, and show how the standard source of information: her assessment of the propositional attitudes can be recovered as reliability of the source. Evidence obtained fixed points of dynamic attitudes. In the by direct observation, e.g., is normally con- 1 sidered to be more reliable than testimonial E.g., Spohn (2009) studies a variety of revision operations, parametrized by their “evidential force”, evidence, and testimonies from different wit- meant to capture the idea that information one ac- nesses may be differently assessed, depend- cepts comes in various degrees of “firmness”. And ing on the reliability of each witness. For- Lehrer and Wagner (1981) suggest to model the trust mally, we encode attitudes towards sources an agent places in another agent’s claims using a no- as strategies for belief change, applicable to tion of “epistemic weight”. 2Jeffrey (2004), Halpern (2003). any information received from a particu- 3Boutilier (1996), Spohn (1985, 2009), Nayak lar (type of) source. Such a strategy pre- (1994), Rott (2004, 2006), among others.

1 conclusion, we discuss a multi-agent exten- Going further, the agent may also possess sion of the setting, in which various proper- soft information, that is not absolutely cer- ties of communication acts, like honesty and tain but subject to revision, and that merely sincerity, can be studied. allows her to hierarchize the possibilities consistent with her hard information accord- ing to their subjective “plausibility”, but not 1 Background to exclude any of them. This relative hier- archy is represented by the relation ≤. We briefly review a fairly standard set- ting building on prior work in (semanti- Propositional Attitudes. Plausibility cally oriented) Belief Revision theory and orders allow us to capture a variety of “opin- Dynamic Epistemic Logic.4 The main no- ions” an agent might have about a given tions are plausibility orders, upgrades (i.e., proposition. A (doxastic) propositional order transformers) and propositional (dox- attitude is a function astic) attitudes.

A : S,P 7−→ AS P Plausibility Orders. Fix a countable set Σ, called the set of all possible worlds (or that assigns a proposition AS P ⊆ S to each possible states of the world). A proposition given plausibility order S and proposition P . is a set of possible worlds. A plausibility or- Important examples of such attitudes are: der is a pair S = (S, ≤), where S ⊆ Σ is finite, and ≤⊆ S × S is a total preorder, i.e., • (Irrevocable) knowledge K, defined by a transitive, connected (and thus reflexive) KS P := {s ∈ S | S ⊆ P }. relation. The fact that s ≤ t indicates that state s • (Simple) belief B, defined by is at least as plausible as state t, from the BS P := {s ∈ S | best S ⊆ P }. perspective of our agent. We write bestS P • Strong belief Sb, defined by for the most plausible P -states in the order SbS P := {s ∈ S | P ∩ S 6= ∅ ∧ ∀t ∈ S, given by bestS P := {s ∈ P | ∀t ∈ P : s ≤ P ∀r 6∈ P : t < r}. t}. We write best S for bestS S. The set of all possible worlds Σ com- • Triviality >, defined by >S P := S. prises the totality of all possibilities that are consistent with some unchangeable, context- • Inconsistency ⊥, defined by ⊥S P := ∅. independent and time-independent informa- tion about the world. In addition, an agent Upgrades. We may now wonder how to may gain, over time, more information about represent belief changes in the setting given the world, thereby reducing the initial set of by plausibility orders. A (doxastic) upgrade possibilities to a smaller set, embodying her u is a function hard information, assumed to be absolutely certain, i.e., irrevocably known by the agent. u : S 7−→ Su This hard information is what the domain S of a plausibility order S captures. that takes a given plausibility order S = (S, ≤) to a plausibility order Su := (Su, ≤u), 4Settings of this kind, or close relatives, are dis- satisfying that Su ⊆ S. cussed, in various degrees of generality, by many au- The composition u · u0 of two upgrades u thors, cf., e.g., Spohn (1985), Grove (1988), Boutilier 0 u·u0 u u0 (1996), van Benthem (2007), Baltag and Smets and u is given as usual, by S = (S ) , (2008). i.e., “first apply u, then apply u0.”

2 Important examples of upgrades include 2 Dynamic Doxastic Atti- 5 the following. For each proposition P , tudes

• the update !P maps each plausibility or- A dynamic (doxastic) attitude τ is a function der S to the relativization of the or- der with P , i.e., all non-P -states are τ : P 7−→ τP deleted, everything else is kept the same. that maps each proposition P to an upgrade τP , satisfying • the radical upgrade ⇑ P makes all 1. SτP = Sτ(P ∩S), P -states more plausible than all non- P -states, leaving everything else un- 2. τP · τP = τP , changed. 3. if P ∈ {∅, Σ}, then τP ∈ {∅, id}. • the positive radical upgrade ⇑+ P is de- The first condition says that the result of fined exactly as radical upgrade ⇑ P , applying τP does not depend on the worlds except that, for any proposition P and satisfying P that are outside of S. The sec- order S such that P ∩ S = ∅, the re- ond condition says that dynamic attitudes sult of applying ⇑+ P to S is the empty are idempotent if their propositional argu- plausibility order. ment is kept fixed: receiving the very same semantic information one has just received • the conservative upgrade ↑ P promotes is redundant. The third condition says that the best P -states (i.e., the states in dynamic attitudes deal in a uniform manner bestS P ) to become the best states over- with information that is trivial or inconsis- all, leaving everything else unchanged. tent: such information uniformly leaves the order unaffected (τP = id) or deletes the • the positive conservative upgrade ↑+ P whole order (τP = ∅). is defined exactly as conservative up- We have said that we understand dynamic grade ↑ P , except that, for any proposi- attitudes as strategies for belief change. We tion P and order S such that P ∩S = ∅, can now spell out such a strategy from the + the result of applying ↑ P to S is the perspective of an agent as follows: empty plausibility order. “Whenever I receive the informa- • the semi-positive conservative upgrade tion that P from a source towards ↑∼+P adds the best P -states to the best which I have the attitude τ, I will states overall, leaving everything else apply the upgrade τP to my cur- unchanged. rent plausibility order.” Our above examples of upgrades readily • the null upgrade ∅ maps every plausi- translate to examples of dynamic attitudes: bility order to the empty plausibility or- der. • infallible trust ! maps each proposition P to the update !P ; • the trivial upgrade id maps every plau- sibility order to itself. • strong trust ⇑ (resp. strong positive trust ⇑+) maps each P to the radical upgrade + 5For a more thorough discussion of upgrades, see, ⇑ P (resp. positive radical upgrade ⇑ e.g., Baltag and Smets (2008). P );

3 • minimal trust ↑ (resp. positive minimal our examples of dynamic attitudes so far, trust ↑+) maps each P to the conserva- except for neutrality, isolation and semi- tive upgrade ↑P (resp. positive conser- positive minimal trust, are weakly positive. vative upgrade ↑+ P ); Observe that (weakly) positive attitudes capture a type of uniform trust: the agent • semi-positive minimal trust ↑∼+ maps processes in the same (positive) way ev- each P to the semi-positive conservative ery information coming from this source. upgrade ↑∼+P ; In Section 5 we’ll model more realistic, • neutrality id maps each P to the iden- contextually-dependent forms of trust. tity upgrade id; Semi-Positive Attitudes. Many people • isolation maps each P to the null up- ∅ regard the weather forecast as a source of grade . ∅ positive, but inconclusive evidence. If the forecast predicts sun, they are still inclined Positive Attitudes. Within our frame- to take their umbrella; on the other hand, work, we can capture a notion of “accep- they also take a light shirt. The weather tance” or “trust” (as discussed in the intro- forecast leads them to drop their belief that duction) in the following way. An attitude τ it will rain, without their coming to be- is positive if it satisfies: lieve the opposite. This is an example of τP 1. P ∩ S 6= ∅ =⇒ S 6= ∅; a “semi-trusting” attitude, in-between trust and distrust. Formally, an attitude τ is τP 2. best S ⊆ P . semi-positive if

Essentially, having a positive attitude to- τP wards a source means that the agent will P ∩ S 6= ∅ =⇒ P ∩ best S 6= ∅. always come to believe any information re- In this way, we arrive at a class containing all ceived from that source, and moreover her the weakly positive attitudes, but also fur- new beliefs will be consistent whenever this ther interesting ones, such as semi-positive is possible (i.e. whenever the new informa- minimal trust ↑∼+. tion is consistent with her prior knowledge). Among the attitudes in our list of exam- ples above, the only positive ones are infalli- (Weakly, or semi-) negative attitudes. ble trust !, strong positive trust ⇑+ and min- These can be defined by dualizing the above imal positive trust ↑+. clauses (though a shorter definition will be given in Section 5). Weakly negative atti- Weakly Positive Attitudes. A dynamic tudes formalize “uniform distrust”, i.e., the attitude τ is weakly positive if source is generally mis-trusted, independent on the content of the new information, and τP τP P ∩S 6= ∅ =⇒ S 6= ∅∧best S ⊆ P. hence our agent essentially upgrades “to the contrary”. Essentially, having a weakly positive atti- tude towards a source means that, as long as the new information received from that 3 Fixed Points source is consistent with the agent’s prior knowledge, her new set of beliefs will be con- Intuitively, an upgrade τP is redundant, or sistent and include the new information. uninformative, in an order S if the order As the terminology suggests, all positive remains unchanged when applying τP , i.e., attitudes are weakly positive. In fact, all if SτP = S. Such “uninformativity” of a

4 given (dynamic-doxastic) attitude τ is itself need to “hear it again” from the second a propositional (doxastic) attitude τ. source. The strength relation on dynamic Formally, given a dynamic attitude τ, the attitudes matches the entailment relation on fixed point τ of τ is the propositional atti- their fixed points familiar from epistemic tude τ : S,P 7→ τ S P , defined by logic:

τP τ S P := {s ∈ S | S = S}. Proposition 2. For all dynamic attitudes σ and τ: Intuitively, τP holds whenever the agent has “already learned” everything she could learn σ ≤ τ ⇐⇒ ∀S∀P (σS P =⇒ τ S P ) . from a P -asserting, τ-trusted source. This notion allows us to link dynamic and propo- The strength order is bounded by iso- sitional attitudes in a perspicuous way: lation and neutrality, while infallible trust and minimal trust emerge as strongest, resp. Proposition 1. weakest weakly positive attitudes:

• The fixed point of infallible trust is Proposition 3. knowledge: ! = K. • For all dynamic attitudes τ: ∅ < τ < • The fixed point of strong positive trust id. is strong belief: ⇑+ = Sb. • For all weakly positive attitudes τ: ! ≤ • The fixed point of positive minimal trust τ ≤↑. is belief: ↑+ = B. • For all positive attitudes τ: ! ≤ τ ≤↑+. • The fixed point of neutrality is triviality: id = >. • For all semi-positive attitudes τ: ! ≤ τ ≤ ↑∼+. • The fixed point of isolation is inconsis- tency: = ⊥. ∅ 5 Operations with Attitudes

4 The Strength Order We present here four natural operations with dynamic doxastic attitudes. It is natural to compare propositional atti- tudes to each other, e.g., knowledge K is 1. Opposite. The opposite τ ¬ of an at- stronger than belief B, since knowledge en- titude τ is given by τ ¬P := τ(¬P ). Using tails belief. this notion, we obtain attitudes like mini- We introduce an analogue notion of mal negative distrust ↑−:= (↑+)¬ (the op- strength for dynamic attitudes. Given dy- posite of positive minimal trust), strong dis- namic attitudes σ and τ, the attitude σ is at trust ⇑¬ (the opposite of strong trust) etc. least as strong as τ (notation: σ ≤ τ) iff the More generally, (weakly, or semi-) negative following holds: attitudes can now be defined as the attitudes ¬ σ ≤ τ ⇐⇒ ∀P (σP · τP = σP ) . τ of the form τ = σ , for some (weakly, or semi-) positive attitude σ. Intuitively, this relation of strength captures that a source towards which the agent has 2. Contextual Mixtures. Whether a the attitude σ “subsumes” a source towards source is trusted (in a certain way, as given which the agent has the attitude τ: after by a particular weakly positive attitude) the first source tells you that P , you don’t may depend on the topic of conversation.

5 E.g., a certain source may be trusted if she Conclusions and Future Work is making mathematical statements, but not on other topics. We capture such finer dis- Our setting can be extended to an analy- tinctions as follows. A context is a set of sis of information exchange among commu- propositions. Given two dynamic attitudes nicating agents. In a communication setting, σ and τ and a context Γ, the mixture σΓτ of sources of information are speakers making σ and τ w.r.t. Γ is given by assertions, and attitudes towards sources be- ( come attitudes of hearers towards speakers. σP P ∈ Γ To capture such scenarios, we introduce σΓτP := τP P 6∈ Γ plausibility frames, which are just the the natural multi-agent versions of our notion In this way, we can “mix” positive and neg- of plausibility order.6 For a set of agents ative attitudes, or weakly positive and semi- A, a plausibility frame (S, ∼a, ≤a)a∈A, is a positive ones etc. set of states S ⊆ Σ, together with an agent- indexed family of equivalence relations ∼a 3. Restrictions. A special case of a mix- and relations ≤a such that ≤a respects ∼a ture is the restriction of an attitude τ to a (i.e. s ≤a t implies s ∼a t), and the restric- context Γ, given by tion of ≤a to each ∼a-equivalence class is a plausibility order (in the sense of Section 1). τ|Γ := τΓ∅ We represent the agents’ mutual attitudes towards each other at a given state by means 4. Implicative Closure. If a speaker, of a trust graph, whose nodes are agents, and when asked whether she will come to a party, whose edges from any node a to any other says I have to work, we typically understand node b are labeled with the attitude of agent that she will not come: for if both were a towards agent b. E.g. in the trust graph true—that she has to work, and that she will depicted below, agent a has an attitude of come—, it would have been more common to minimal trust ↑ towards agent c: simply say I will come to the party. We capture such “Gricean” phenomena as follows. An implicative context is a strict c partial order < on ℘(Σ), the set of proposi- tions. If Q < P , then we interpret this as: ↑ ⇑− “if P and Q are the case, then it is more ↑ ↑ common to say Q than P ”. Using the no- tion of implicative closure of a proposition P (w.r.t. an implicative context <), given by b ⇑ a \ ∼+ P < := P \ Q, ↑ Q

6 The interest of this multi-agent version de- aimed at getting the hearer to adopt what rives from the fact that it now becomes pos- the speaker to be the “correct” at- sible to investigate properties of assertions titude towards the issue in question. made by a speaker, building on the work In on-going work, we develop this setting of the previous sections. For instance, an into a general study of the formal epistemol- assertion of P is sincere if the speaker be- ogy of testimony, persuasion and trust. lieves that P . The assertion is called hon- est if the fixed point of the hearer’s dynamic References attitude towards the speaker is the same as the speaker’s to- Alexandru Baltag and Sonja Smets. A qualita- wards P : this means that the hearer’s atti- tive theory of dynamic interactive belief revi- tude towards the speaker is matched by the sion. Texts in Logic and Games, 3, 2008. speaker’s attitude to her assertion. If, e.g., Craig Boutilier. Iterated revision and minimal the hearer’s attitude towards the speaker is + change of conditional beliefs. Journal of Philo- positive minimal trust ↑ , honesty requires sophical Logic, 25(3):263–305, 1996. that the speaker actually believes that P (since the fixed point of ↑+ is belief). Adam Grove. Two modellings for theory change. One interesting observation is that dis- Journal of Philosophical Logic, 17(2):157–170, 1988. trust induces a tension between sincerity and honesty. For example, consider the reply Joseph Halpern. Reasoning about Uncertainty. given by Brad Pitt to Wired magazine about MIT Press, 2003. people lying (in their online dating profile) Richard Jeffrey. Subjective Probability, The Real on how much money they earn: Thing. Cambridge University Press, 2004.

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