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Doxastic Attitudes As Belief-Revision Policies

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Doxastic Attitudes As Belief-Revision Policies Doxastic Attitudes as Belief-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 knowledge and belief, capture an from that source is received. agent's opinion 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 := fs 2 S j S ⊆ P g: 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 := fs 2 S j best S ⊆ P g: perspective of our agent. We write bestS P • Strong belief Sb, defined by for the most plausible P -states in the order SbS P := fs 2 S j P \ S 6= ? ^ 8t 2 S, given by bestS P := fs 2 P j 8t 2 P : s ≤ P 8r 62 P : t < rg: tg: 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 2 f?; Σg, then τP 2 f?; idg. • 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.
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