Knowledge-driven versus data-driven logics - IRIT

Knowledge-driven versus data-driven logics Didier DUBOIS*, Petr HAJEK** and Henri PRADE* * Institut de Recherche en Informatique de Toulouse (IRIT), Université Paul Sabatier CNRS, 118 route de Narbonne, 31062 Toulouse Cedex, France ** Institute of Computer Science, Academy of Sciences, 18207 Prague, Czech Republic

Abstract The starting point of this work is the gap between two distinct traditions in information engineering: knowledge representation and data-driven modelling. The first tradition emphasizes logic as a tool for representing beliefs held by an agent. The second tradition claims that the main source of knowledge is made of observed data, and generally does not use logic as a modelling tool. However, the emergence of fuzzy logic has blurred the boundaries between these two traditions by putting forward fuzzy rules as a Janus-faced tool that may represent knowledge, as well as approximate functions representing data. This paper lays bare logical foundations of data-driven reasoning whereby a set of formulas is understood as a set of observed facts rather than a set of beliefs. Several representation frameworks are considered from this point of view: classical logic, possibility theory, belief functions, epistemic logic, fuzzy rule-based systems. Mamdani's fuzzy rules are recovered as belonging to the data-driven view. In possibility theory a third set-function, different from possibility and necessity plays a key role in the data-driven view, and corresponds to a particular modality in epistemic logic. A bi-modal logic system is presented which handles both beliefs and observations, and for which a completeness theorem is given. Lastly, we point out that our results may shed new light in deontic logic and allow for a distinction between explicit and implicit permission that standard deontic modal logics do not often emphasize.

Key words: Epistemic Logic; Possibility Theory; Data-Driven Reasoning; Deontic Logic.

1 - Introduction Artificial Intelligence has put forward a subjective view of classical logic, that differs from the view developed in the first half of the century when logic was developed in connection with the problem of the foundations of mathematics. When Artificial Intelligence

Knowledge-driven versus data-driven logics (D. Dubois, P. Hájek & H. Prade)

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envisages a set of formulas, each formula represents a belief held by an agent. A set of formulas is then viewed as a belief base (Nebel, 1989, for instance). Each belief is understood as a constraint that delimits a set of possible worlds, described by means of interpretations of the language in which beliefs are expressed. Namely, this is the set of interpretations in which all formulas of the belief base are true, and this set is supposed to contain the "real world" insofar as the agent's beliefs are correct. In other words, the belief set represents an imprecise description of the real world. This semantic view is also the one developed by Gärdenfors (1988) in his theory of belief change, where the set of logical consequences of the belief base represents the full set of beliefs held by the agent (also called a belief set). In this view, each and every piece of information is a belief that stems from an agent, and adding a new belief to a belief base comes down to discarding worlds that become impossible. The more beliefs are available, the smaller the set of possible worlds and the more precise the information. However, many engineering sciences are data-driven rather than belief-driven. The available information is often under the form of data, each piece of data corresponding to an actually observed world. In contrast, each model of a belief base represent a potentially observable world only. Moreover, in the data-driven view, data are interpreted as examples, and are not necessarily mutually consistent in a logical sense since gathered data can be easily dissonant (i.e., observed values, or ranges of values, pertaining to the same phenomenon are not necessarily fully consistent). Moreover, the more numerous the data, the more possible worlds are observed. So, while accumulating beliefs discards possible worlds, accumulating data augments the set of possible worlds, as actually observed ones. To our knowledge there is no logical analysis of the data-driven view of information gathering. This paper aims at investigating this question in greater details, under the term "data-driven reasoning". In Section 2, various theories of belief-based reasoning are recalled, including their mutual relationships: classical logic, possibility theory, belief functions, and modal epistemic logic. In Section 3, data-driven reasoning is introduced in possibility theory, and a notion of "guaranteed possibility", already introduced in Dubois and Prade (1991a, 1992) and studied by Prade and Yager (1994) in greater details is laid bare. The corresponding set-function is anti-monotonic (w.r.t. set inclusion) and coincides with the so-called commonality function in the transferable belief model (Smets, 1988), based on Shafer (1976) belief functions. Section 4 presents the data-driven logic as a dualized version of the KD45 modal logic, and also as a dual view of classical logic. The connection with Mamdani's fuzzy rules is pointed out. Section 5 envisages reasoning with both beliefs and observations and studies the compatibility between a belief base and a base of observed facts pertaining to the same problem. Section 6 shows how to implement both belief-based reasoning and dual data-driven reasoning in a unique bimodal logic framework. The potential application of this system to deontic reasoning is outlined in Section 7.

2 - Theories of Belief


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The aim of this section is to survey several methods for modeling statements of the form "p is believed" by an agent, where p is a proposition. Here a proposition is identified to a subset of situations where the proposition holds. Let W be a set of situations (also called possible worlds) and A the set of situations where p is true. Then, when believing p, the agent implicitly considers that the situations where p is false are impossible. Usually a proposition can be expressed in a logic, which, most of the time, is classical logic. Namely, given a language L based on a finite set of propositional variables a, b, c,…, and denoting ¬, ∨, ∧ negation, disjunction and conjunction respectively, any proposition is represented as a formula ϕ in L where a, b, c are in L if ϕ is in L, ¬ϕ is in L if ϕ and ψ are in L, then ϕ ∧ ψ is in L ϕ → ψis short for ¬(ϕ ∧ ¬ψ), and ϕ ∨ ψ for ¬(¬ϕ ∧ ¬ψ). Formulas are denoted in the following by the Greek letters ϕ, ψ, or δ with possibly a subscript. Then the state space W is more or less coarsely represented by the set of interpretations Ω of L, each interpretation being a cluster of indistinguishable situations (or worlds). Ω is a partition of W whose granularity depends on the expressivity of the language L. Ω is assumed to be finite in the following. For simplicity, we shall identify W and Ω when it is possible. Moreover if ϕ is true in situation ω, we write ω . ϕ, and [ϕ] = {ω, ω . ϕ} is the set of models of ϕ.

2.1 - Belief Sets in Classical Logic Let B = {ϕ1, …, ϕn} be a belief base (Nebel, 1989), namely a set of formulas ϕi that represent propositions believed by an agent. All situations ω ∈ Ω such that ∃ ϕi ∈ B, ω . ¬ϕi are considered impossible by the agent. To any belief base B can be associated the maximal set of possible situations where all beliefs are true, say (Gärdenfors, 1988) [B] = {ω, ∀ ϕi ∈ B, ω . ϕi} = Ái=1,n [ϕi], where [ϕi] is the set of models of ϕi. Adopting this semantic view, it is possible to enrich the belief base B with its classical consequences without altering the set of models. Namely let Cn(B) = {ϕ, B . ϕ} where B . ϕ means that [B] ⊆ [ϕ], i.e., all models of B satisfy ϕ. K = Cn(B) is called a belief set by Gärdenfors and includes implicit as well as explicit beliefs of the agent. Under this view, situations in [B] are, from the point of view of the agent candidates for being the real one, i.e., they are possible. However the term "possible" is to be understood in a weak sense because it is not required that situations of [B] be actually observed, be feasible. Situations in [B] are only not ruled out. However, this presupposes that the agent has expressed all of his beliefs, that is B represents all he knows. If there is a proposition ϕ


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he believes, but ϕ ∉ Cn(B), then [B] represents only a superset of the situations actually considered possible by the agent.


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2.2 - Possibility theory for belief representation It is convenient to describe the agent's beliefs about possible situations in terms of a possibility distribution, denoted π, which maps the set of situations Ω to the binary set {0,1}, with the following conventions (Dubois and Prade, 1988): π(ω) = 0 if the agent considers ω as impossible = 1 otherwise. The following inequality faithfully expresses that if π represents the set of situations the agent believes as possible, then he believes all propositions in B are true: π ≤ µ [B]

(1)

where µ[B] is the characteristic function of [B]. π is called the epistemic state of the agent. The above inequality expresses that the agent may come up with other believed propositions (Dubois and Prade, 1992). π is said to be an epistemic model of B. More generally, the modelling of beliefs can be refined if we allow the agent to express that some situations are more plausible than others. Then π maps the set Ω into a totally ordered scale S, which for simplicity, is identified to the unit interval [0,1]. The conventions remain like above, but the inequality π(ω1) > π(ω2) > 0 means that while neither of ω1 and ω2 are impossible, ω1 is more plausible than ω2. Moreover, situations ω such that π(ω) = 1 are called "normal" or "typical". In this approach the agent presupposes that the situation is normal, unless something exceptional occurs. The valued possibility distribution π induces two set functions over Ω namely, possibility measures and necessity measures defined respectively by ∀ A ⊆ Ω, ∏(A) = maxω∈A π(ω)

and

N(A) = minω∉A n(π(ω))

where n is an order-reversing map such that n(n(x)) = x, n(0) = 1, and n(1) = 0 (when taking S = [0,1], n(x) = 1 – x). ∏ and N can be defined for the whole language L, if we let ∏(ϕ) = ∏(A) iff [ϕ] = A. ∏(ϕ) expresses to what extent a proposition ϕ is unsurprising, since the smaller ∏(ϕ), the more surprising ϕ is; when ∏(ϕ) = 0, ϕ is impossible for the agent. Dually N(ϕ) = n(∏(¬ϕ)) expresses to what extent ϕ is believed by the agent since when N(ϕ) = 1, ∏(¬ϕ) = 0, so that the agent considers ¬ϕ as impossible (hence ϕ is certainly true). However any formula ϕ such that N(ϕ) > 0 is accepted by the agent. Indeed: Proposition 1: N(ϕ) > 0 iff ϕ is true in all the normal situations. Proof: Let π i = π(ωi) and {π 1 = 1 > π2 >… > πn > 0} = π(Ω) be the plausibility levels of the agent. Then


N(ϕ) > 0 ⇔ ∏(¬ ϕ) ≤ π2 ⇔ ∀ω, ω . ¬ϕ ⇒ π(ω) ≤ π 2 ⇔ ∀ω, π(ω) = 1 ⇒ ω . ϕ.

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It is easy to check that given a plausibility ordering π, the set of formulas {ϕ | N(ϕ) ≥ α} for α ∈ S, α > 0, is a belief set (closed under deduction). This is because N is monotonically increasing (ϕ . ψ ⇒ N(ϕ) ≤ N(ψ)) and conjunction decomposable (N(ϕ ∧ ψ) = min(N(ϕ), N(ψ))). Let Kα = {ϕ, N(ϕ) ≥ α} be the α-level belief set, and K = {ϕ, N(ϕ) > 0} be the belief set generated by π. It is easy to check that [Kα] = {ω, π(ω) > n(α)} and [K] = {ω, π(ω) = 1}. K is preordered by N since whenever N(ϕ) > N(ψ), ϕ is more believed that ψ. This preordering is called "epistemic entrenchment" by Gärdenfors (see Dubois and Prade, 1991b). Conversely, a given (pre)ordered belief base B = {(ϕi,α i), i = 1,m} can be interpreted as a set of constraints of the form N(ϕi) ≥ α i, i = 1,m on the set of possibility distributions. This is possibilistic logic (Dubois et al., 1994). This view leads to extending the notion of satisfiability from situations (ω . ϕ) to epistemic states π, whereby π . (ϕi,α i) ⇔ N(ϕi) ≥ α i. A preordered belief base induces a preordered set of plausible situations for the agent, by considering the principle of minimal specificity, which says that whenever a situation is not known as impossible, it should be supposed to be possible. First, a preordered belief base B = {(ϕi,α i), i = 1,m} leads to constrain the epistemic state π, since π is a epistemic model of B iff ∀i, π . (ϕi,α i). It leads to the following extension of the above inequality (1) π ≤ mini=1,m max(µ[ϕi], n(α i))

(2)

and the least specific epistemic state where beliefs in B hold is π* such that π*(ω) = min{n(α i), ω . ¬ϕi} (and =1 if ω . ϕi, ∀i). It may happen that N*(ϕj) > α j for some j, where N* is the necessity measure generated by π*. For instance, this happens if ϕi . ϕj and α j < α i in B.

2.3 - Belief Functions Note that in possibility theory, an agent's belief is modelled by a nested family of subsets of possible states {Aα = {ω, π(ω) ≥ α}, for α ∈ {π 1, …, π n}}. In the transferable belief model (Smets, 1988; Smets and Kennes, 1994), developed from Shafer (1976)'s theory of evidence, the agent's beliefs are modelled by a convex combination of subsets of possible states Ei ⊆ Ω. Each E i, called focal set, is a possible representation of what the agent believes. Ei is assigned a weight m i ∈ [0,1] which represents the extent to which Ei faithfully represents what the agent believes. Let F be the family of focal sets. The weights m i are such


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that ∀ Ei ∈ F, mi > 0, mi = 0 otherwise and ∑m i = 1, so that m i can be viewed as the probability that the agent's beliefs are described by Ei. Note that each Ei can be viewed as the set of models of a classical belief base Bi. The degree of belief of the agent in ϕ is understood as Bel(ϕ) = ∑i: Bi.ϕ mi and the degree of plausibility is Pl(ϕ) = 1 – Bel(¬ϕ) = ∑i: B∪{ϕ} consistent mi. This approach is in perfect agreement with Gärdenfors' view of modeling epistemic states by belief sets, although it is a randomized version. If the set of focal subsets F is nested, then Bel(ϕ ∧ ψ) = min(Bel(ϕ), Bel(ψ)) and the possibilistic representation of an epistemic state is recovered (with n(x) = 1 – x since belief functions require a numerical setting).

2.4 - Epistemic Logic Note that the possibilistic setting and the transferable belief models explicitly handle symbols representing that a proposition is believed (N, Bel) and symbols representing that a proposition is not disbelieved (∏, Pl). In classical logic, a proposition is not disbelieved (∏(ϕ) = 1) if B cannot prove ¬ϕ. A natural logical system for expressing one's beliefs and disbeliefs is the modal logic KD45 (Chellas, 1988; Voorbraak, 1993) called often the logic of belief. The language L is augmented with a modality t such that tϕ means "the agent believes ϕ". Call this language Lt. Axioms of KD45 are propositional tautologies K: D: 4: 5:

t(ϕ → ψ) → (tϕ → tψ) (distribution) tϕ → ¬t¬ϕ (consistency) tϕ → ttϕ (positive introspection) ¬tϕ → t¬tϕ (negative introspection).

Inference rules are modus ponens and necessitation (from ϕ deduce tϕ). The notion of a proof, provability and a theory are standard. The semantics is given by Kripke models of the following kind: M = (Ω, Ω+) where Ω is a set of interpretations of L and Ω+ is a non-empty subset of Ω (Ω = Ω + is not excluded). Ω+ is the set of all worlds the agent believes to be possible (believable worlds); namely Ω – Ω+ is the set of worlds which are judged to be impossible by the agent. Satisfaction is defined in the obvious way, in particular ω .M tϕ iff for each ω' ∈ Ω +, ω' .M ϕ (so that the satisfaction of tϕ does not depend on ω). It is easily seen that this semantics is sound, i.e., if KD45 ; ϕ then ϕ is true in each Kripke model (Ω, Ω +), i.e., ϕ is satisfied in each ω ∈ Ω for each Kripke model M = (Ω, Ω+). We also have strong completeness: for each theory T over KD45, T ;KD45 ϕ iff ϕ is true in each Kripke model of T (i.e., whenever all axioms of T are true in (Ω, Ω+) then so is ϕ).


The connection between classical logic and epistemic logic is given by the following

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Proposition 2: Let B be a belief base for propositional logic and tB = {tϕ: ϕ ∈ B}. Then B ; ϕ iff tB ;KD45 tϕ. Proof: If ϕ1, …, ϕn is a propositional proof of ϕ from B then show, by induction, that tB ;KD45 tϕi for all i (in fact, only K is to be used). Conversely, assume tB ;KD45 tϕ and let ω be an interpretation of L such that ω . B. Construct a one-element Kripke model M whose only possible and believable world is ω. Clearly M . tB (since B is true in all believable worlds); thus, by soundness, M . tϕ, which means that ω . ϕ. Thus B ; ϕ by the completeness of propositional calculus. y Hence Gärdenfors belief sets can be cast in the framework of epistemic logic. Remark 1: Note that the proof of Proposition 2 used only the axiom K and the fact that the one-element model constructed is a model of the logic in question; thus the Lemma holds for any modal logic containing K and admitting the above model. Remark 2: Note that we may define semantics of KD45 in more general ways: first we need not identify possible situations with the interpretations of atoms given by them, i.e., a Kripke model of KD45 has then the form (W, W +, .) where Ø ≠ W+ ⊆ W and . assigns to each w ∈ W an interpretation ωw of atoms of the propositional language. We shall mention this variant later. Still more general, an abstract Kripke model has the form M = (W, R, .) where R is a binary relation on W; w ; tϕ iff for each w' such that R(w,w'), w' . ϕ. A model (W, W+, .) corresponds to (W, R, .) where R = W × W+. But there are more general abstract Kripke models in which KD45 is true; in particular, if R is transitive, serial (∀w ∃w', wRw') and Euclidean (∀w, w', w"(wRw' and wRw" imply w'Rw") then M is a model of KD45. Remark 3: We mention in passing the logic S5 which results from KD45 by replacing D by the (stronger) axiom tϕ → ϕ. S5 is complete with respect to model (Ω, Ω+) of KD45 with Ω + = Ω (so that Ω+ is superfluous) as well as to models (W, R, .) where R is an equivalence.

3 - Modelling Observed Facts Using Guaranteed Possibility Instead of claiming that a proposition is necessarily true, an agent may declare that a proposition is possibly true. This type of statement may convey two distinct meanings: – the proposition is not incompatible with the agent's belief. In this case, it means that the converse proposition is not believed. If E denotes the set of situations not ruled out by the agent, then ϕ is possible means that E ⊆ [¬ϕ] is not true, that is, E ∩ [ϕ] ≠ Ø so that ϕ cannot be ruled out. This meaning has been considered above; – the agent may instead mean that all situations where the proposition is true are really possible. For instance, if he says "it is possible for you to sell your car at a price in the range [p 1,p2]" it means that any price in [p1,p2] is possible. "ϕ is possible" corresponds to the inclusion [ϕ] ⊆ E, where E is again the set of possible situations.


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In terms of the set-function ∏, the first view leads to write ∏(ϕ) = 1. In the second understanding of the statement "It is possible that ϕ is true", this set-function cannot be used. Another one called "guaranteed possibility", and denoted ∆, must be introduced as follows: ∆(A) = 1 if A is a subset of E = 0 otherwise, where A = [ϕ]. More generally, if the epistemic state is described by a possibility distribution π, the function ∆ is defined as: fl(A) = minω∈A π(ω), which estimates to what extent all the situations in A are actually possible according to what is known, i.e., any situation in A is at least possible at the degree fl(A) (Dubois and Prade, 1992). Clearly fl is a more demanding evaluation than ∏, i.e., fl ≤ ∏, since ∏ only estimates the existence of at least one situation in A compatible with the available knowledge, while the evaluation provided by fl concerns all the situations in A. Note also that fl and the certainty function N are unrelated, since N(A) estimates the certainty that the situation lies in A by checking the impossibility of all the situations out of A, while fl(A) > 0 considers the possibility of all the values in A. The dual measure, called 'potential necessity', or 'unguaranteed necessity' exists as ˜(A) = n(fl(äA)) = maxω∉A n(π(ω)) which estimates to what extent there exists at least one situation in the complement of A which has a low degree of possibility; this is a necessary condition for having ϕ = 'x ∈ A' somewhat certain (but in general far from being sufficient, except if äA has only one element). Obviously, N ≤ ˜. The duality relation between ∆ and ˜ expresses that the potential necessity of A corresponds to the absence of guaranteed possibility for äA. If we further have the information that there exists at least one situation in Ω which is impossible, i.e., ∃ ω ∈ Ω, π(ω) = 0, or equivalently n(π(ω)) = 1, the identity min(fl(A),fl(äA)) = 0 holds, and then fl(A) > 0 entails ˜(A) = 1. At the technical level, it is always possible to add an element to Ω if necessary in order to have n(π) normalized. fl and ˜ are monotonically decreasing set functions (in the wide sense) with respect to set inclusion; they contrast with ∏ and N which are monotonically increasing. Note that the four quantities ∏(A), N(A), fl(A) and ˜(A) are only weakly related since they are only constrained by max(∏(A), n(N(A))) = maxω∈Ω π(ω) (= 1 if π is normalized) min(fl(A), n(˜(A))) = minω∈Ω π(ω)(= 0 if n(π) is normalized). Moreover the following inequality holds when both π and n(π) are normalized


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max(N(A),fl(A)) ≤ min(∏(A),˜(A)). Note that this inequality agrees with the intuition: i) the higher the necessity of A, then the higher the possibility of A, i.e., the consistency of A with what is known, and the higher its potential certainty; ii) the higher the feasibility of all the values in A in the sense of fl, then the higher the possibility of A, and the higher the potential certainty of A. Although not related via an inequality, the functions N and ∆ are not totally independent, namely the following proposition holds Proposition 3: If A and B are subsets of Ω such that B – A ≠ Ø then ∆(B) ≤ n(N(A)). Proof: ∆(B) ≤ ∆(B – A) ≤ ∏(B – A) ≤ ∏(äA) = n(N(A)).

y

Corollary 4: ∆(A) ≤ n(N(äA)). Proof: This is just writing ∆(A) ≤ ∏(A).

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We can extend the above analysis to the case when the epistemic state is described by means of a belief function characterized by a set of focal elements {(Ei,mi), i = 1,p} then "guaranteed possibility" is naturally extended into the following: ∆(A) = ∑i: A⊆Ei mi. This set function is known in the belief function literature as the commonality function (Shafer, 1976) and often denoted Q. The meaning of this set-function has never been precisely described so far. The above analysis is filling this gap. The above proposition still holds, i.e., Q(B) ≤ 1 – Bel(A) whenever B – A ≠ Ø.

4 - The Logic of Accumulated Data A possibility distribution is not always specified as such, but often through the qualification of propositions that are modelled by subsets of Ω. Let A be an ordinary subset of Ω. Two kinds of specification exist, namely i) the agent believes that the actual situation x lies in A. It means that any value outside A is (completely) considered impossible, i.e., ∀ ω ∉ A, π(ω) = 0 and π is unspecified over A, or if we prefer "A is necessary for x" is translated by ∀ ω ∈ Ω, π(ω) ≤ µ A(ω)

(3)

where µ A is the {0,1}-valued characteristic function of A. This is like in (1). ii) A is a (completely) possible range of situations that the agent has observed. It means that ∀ ω ∈ A, π(ω) = 1 and π remains unspecified outside of A, or equivalently that


"A is possible for x" is translated by ∀ ω ∈ Ω, µ A(ω) ≤ π(ω).

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(4)

In the first kind of specification the agent expresses a constraint, delimiting a subset A which for sure contains the situation x; the larger A, the more imprecise the agent remains about the normal situation, the smaller A, the better the information. On the contrary, in the second kind of specification, the agent describes what he has observed, stating a range A of situations which are indeed possible, since encountered, without definitely rejecting any other situation. Then, the smaller A, the less informed the agent, since the fewer his observations. The larger A, the more has been observed by the agent. In the second case, the agent describes observed states of facts, or data, instead of beliefs. A set of formulas D = {δ1, …, δk} will be called a database if and only if δi ∈ D means that ∀ ω ∈ Ω, if ω . δi then ω is possible (that is, ∆(δi) = 1). In contrast, a belief base B gathers propositions ϕi such that ∀ ω ∈ Ω, if ω . ¬ϕi, then ω is impossible (i.e., N(ϕi) = 1). As a consequence, the set of models of a database D is defined as ]D[ = ªi=1,k [δi] = {ω, ∃ δi ∈ D, ω . δi}. ]D[ is the smallest set of situations which make all formulas δi possible; its characteristic function is the most specific possibility distribution such that ∆(δi) = 1, i = 1,k. This definition works opposite to the classical definitions of the set of models of a belief base in classical logic. Indeed ]D[ expresses a data accumulation process, while for a belief base B, [B] corresponds to an elimination process. Reasoning from a database D means deriving all formulas that describe clusters of observed situations. The above discussed semantic leads us to define specific symbols for inferring formulas with guaranteed possibility namely, at the semantic level, /∆ such that D /∆ ϕ iff [ϕ] ⊆ ]D[. In other words, at the syntactic level D :∆ ϕ ⇔ ϕ ; δ1 ∨ δ2 ∨… ∨ δk ⇔ {¬δ1, ¬δ2, …, ¬δk} ; ¬ϕ (in the classical sense). The symbol :∆ expresses a process of extraction of observed data from D. Note that the role of tautology T and contradiction are exchanged when going from a belief base to a database. Namely D :∆ ⊥ always holds since ⊥ expresses that nothing has been observed, so D :∆ ⊥ means that at worst nothing has been observed yet. On the contrary, D :∆ T means that all imaginable situations have been observed. The following counterpart to modus ponens rule reads {ϕ ∧ ψ, ¬ϕ} :∆ ψ. It means that if all situations where ϕ ∧ ψ hold are possible (have been observed), as well as all situations where ϕ does not hold, then all situations where ψ hold are possible (since the set of all situations where ϕ is false includes the set of all situations where ¬ϕ ∧ ψ holds).


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Note that while the most usual normal form for a belief base is a conjunction of clauses, the most natural form for a database is a disjunction of phrases (= conjunctions). Indeed a belief often takes the form of a rule whereas a piece of data rather presents itself as a conjunction of facts. Let us now consider the cases of qualification where the possibility or the certainty is not complete but corresponds to an intermediary level α in the plausibility scale [0,1]. It leads to the two following generalizations of (3) and (4) i) The statement "the agent accepts at least at degree α that the situation is in A", will be interpreted as "any situation outside A is considered at most possible at the complementary degree, namely n(α)", i.e., ∀ ω ∉ A, π(ω) ≤ n(α), which leads to "A is α-certain for x" is translated by ∀ ω ∈ Ω, π(ω) ≤ max(µA(ω), n(α)). This explains the semantics for a weighted set of beliefs (Section 2). ii) The statement "A is a possible range of situations at least at the degree α" will be understood as ∀ ω ∈ A, π(ω) ≥ α, which leads to "A is α-possible for x" is translated by ∀ ω ∈ Ω, min(µA(ω), α) ≤ π x(ω). It can be checked that this is equivalent to fl(A) ≥ α. Note that for α = 1, µ A ≤ π is recovered. When α decreases from 1 to 0, our knowledge evolves from the certainty that A is the minimal range of possible situations to a total lack of information whatsoever. This kind of possibility-qualification goes back to Zadeh (1978) and Sanchez (1978). Now, a weighted database will be of the form D = {(δi,ρ i), i = 1,k} where (δi,ρ i) means ∆(δi) ≥ ρ i, ρ i ∈ S. The semantics of D will be described in terms of the most specific distribution where all formulas δi are possible with degree ρ i. Namely if ∆ derives from π, ∀i, ∆(δi) ≥ ρ i ⇔ ∀i, π(ω) ≥ min(µ[δi](ω), ρ i) ⇔ π(ω) ≥ maxi=1,k min(µ[δi](ω), ρ i) = π*(ω). At this point, it is noticeable to compare the above equations pertaining to a data-driven view, with the equation (2) π(ω) ≤ mini=1,m max(µ[ϕi](ω), n(α i)) = π*(ω) that derives from a belief base. Interpreting the weights α i and ρ i as a degree of membership in a fuzzy set Ci, and remembering the possibilistic approach to fuzzy rules in fuzzy logic (Dubois and Prade, 1991a, 1992, 1996), a fuzzy certainty rule "the more Ci is true, the more ϕi is believed" translates into the inequality π(ω) ≤ max(µ[ϕi](ω), n(µCi(ω))) (Dienes-Rescher implication)


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while the possibility rule "the more Ci is true, the more δi is possible" is expressed by: π(ω) ≥ min(µ[δi](ω), µCi(ω)).(Mamdani conjunction-like rule) This may contribute to an analysis of the fuzzy relational modeling of fuzzy rules in fuzzy control (Mamdani, 1977). In fuzzy controllers, a rule is often represented by the conjunction of its antecedent and its conclusion, and not using logical implication. Moreover consequences are computed as weighted unions of the fuzzy sets appearing in the conclusion parts of the rules. Modelling a fuzzy rule-based system as a disjunction of conjunctions makes full sense in the scope of the data-driven logic whereby formulas in a database are combined disjunctively, and elementary formulas are conjunctions of atoms. A similar analysis of fuzzy control in a dual version of possibility theory is also proposed by Weisbrod (1995). One may venture to suggest that systems engineers view fuzzy rules as a form of observed data. Extracting information from a database D can be achieved using an adaptation of the resolution principle to sets of formulas in conjunctive normal form (disjunction of phrases): Proposition 5 (Dubois and Prade, 1992): The following inference pattern holds: ∆(ϕ ∧ ψ) ≥ α ∆(¬ϕ ∧ ξ) ≥ β ———————— ∆(ψ ∧ ξ) ≥ min(α,β) Proof: ∆(ψ ∧ ξ) = ∆((ψ ∧ ξ ∧ ϕ) ∨ (ψ ∧ ξ ∧ ¬ϕ)) = min(∆(ψ ∧ ξ ∧ ϕ), ∆(ψ ∧ ξ ∧ ¬ϕ)) ≥ min(∆(ψ ∧ ϕ), ∆(ξ ∧ ¬ϕ)) = min(α,β).

y

Lastly, it is interesting to develop the data-driven counterpart of the KD45 system, introducing a new modality ∆. Considering a Kripke model M = (Ω,Ω*) and define modal satisfiability of ∆ϕ as follows ω .M ∆ϕ ⇔ ∀ ω' ∈ Ω s.t. ω' . ϕ, ω' ∈ Ω* holds. It contrasts with tϕ where ω .M tϕ iff ∀ω', ω' ∈Ω+ implies ω' . ϕ. A counterpart to the KD45 system called KD45d can be defined as follows: Tautologies: K': D': 4': 5':

;ϕ whenever ϕ is a tautology ;∆(ϕ ∧ ¬ψ) → (∆(¬ϕ) → ∆(¬ψ)) ;∆ϕ → ¬∆(¬ϕ) ;∆¬∆ϕ → ∆(ϕ) ;∆∆ϕ → ¬∆ϕ


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and as inference rules, modus ponens and from ;ϕ deduce ;∆¬ϕ. The latter rule expresses that ∆⊥ is a tautology. This system is straightforwardly obtained from KD45 changing tϕ into ∆¬ϕ. It is an easy exercise to show that KD45 d is sound and complete with respect to Kripke models of the form (Ω, Ω*) where Ω * is a proper subset of Ω (particularly, Ω* may be empty). Of course, it is easy to check that if D is a database then D :∆ ϕ iff {∆δi, δi ∈ D} ; ∆ϕ in KD45d.

5 - Putting Data and Beliefs Together In usual contexts, an agent possesses beliefs, and has also observed facts. So, information provided by an agent will take the form of two sets of formulas B and D, respectively a belief base and a database. However there must be some coherence between D and B in the following sense: data in D cannot violate beliefs in B. For instance, consider the situation when throwing a die. The set of beliefs may consist of propositions like: a die has six facets, a die is cubic, there is a single die, etc. On the contrary, the database may contain the results of observing facets; clearly D cannot contain a 'seven' without contradicting our beliefs about the game in question. This type of situation is typically encountered in the field of information systems: a database stores items that describe cases. However not all cases can be entered into the database. There are so-called integrity constraints that, if violated by a case, prevent the user from storing this case. All stored cases satisfy the integrity constraints. One may think of D as a set of cases, economically described by a set of formulas, and B as a set of integrity constraints. More formally, B and D contradict each other if and only if ∃ ω ∈ ]D[ such that ω ∉ [B]: Definition 6: A belief base B and a database D are coherent if and only if ]D[ ⊆ [B]. Let π* and π* be the characteristic functions of [B] and ]D[ respectively. They respectively act as upper and lower bounds on the possibility distribution π that represents the agent's epistemic state π, i.e., π* ≥ π ≥ π *. The agents knows at least π* and at most π *. This idea is similar to the modal logic of Levesque (1990) which tries to model pieces of knowledge like "all I know is ϕ". However in Levesque's calculus, this statement is modeled by means of an equality (π = µ[ϕ]) rather than the inequality π ≥ µ [ϕ]. At the syntactic level, proving the coherence of a database D = {δ1, …, δk} with a belief base B = {ϕ1, …, ϕm} comes down to proving that ◊ i=1,k δi ; Âi=1,m ϕj


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or equivalently using the deduction theorem, that each formula ϕi ∨ ¬δj is a theorem of propositional logic. Indeed ◊ j=1,k δj ; Âi=1,m ϕi iff ;(Âj=1,k ¬δj) ∨ (Âi=1,m ϕi). In database terminology the truth of ϕi ∨ ¬δj means that the set of cases encompassed by δj satisfies the (integrity) constraint ϕi. When the possibility distributions π* and π * are no longer binary, the coherence condition translates into the inequality π* ≥ π*. It leads to the following inequalities between the set functions ∆* induced by π * and N* induced by π*: ∆* (ϕ) ≤ ∆(ϕ) ≤ n(N(¬ϕ)) ≤ n(N*(¬ϕ)) where ∆ and N derive from π, π* ≥ π ≥ π*, and using Corollary 4. It means that we cannot have ϕ such that ∆*(ϕ) ≥ α and N*(¬ϕ) ≥ β with n(β) < α. Let D and B be a weighted database and a weighted belief base respectively. Then D and B are incoherent as soon as ∃ψ such that D :∆ (ψ,α) and B ; (¬ψ,β) with n(β) < α. In terms of level-cuts, it comes down to checking the coherence (in the sense of Definition 6) of the following pairs of classical data and belief bases: (Dα, Bn(α), α ∈ [0,1)) where Dα = {δj, (δj, α j) ∈ D, α j > α}, Bn(α) = {ϕi, (ϕi,β i) ∈ B, β i ≥ n(α)}. It is easy to check that if ∀ϕ, ∆*(ϕ) ≤ n(N*(¬ϕ)), where ∆*(ϕ) = max{α, D :∆ (ϕ,α)} and N*(ϕ) = max{α, B ; (ϕ,α)} then D and B are coherent in the sense that π * ≤ π*. Indeed it means that ∀ϕ, minω.ϕ π * (ω) ≤ ∏*(ϕ) = maxω.ϕ π*(ω). Just take ϕ such that [ϕ] = {ω}. In the modal setting it is also possible to put both beliefs and data together, using two modalities t and ∆ where t satisfies the axioms of KD45, and ∆ satisfies the axiom of the system KD45 d given in Section 4. At the semantic level, let Ω* and Ω* be distinguished subsets of Ω where Ω * ⊆ Ω*. Ω* forms the set of situations that have been actually observed (possible situations) and Ω* forms the set of situations that are not impossible. We interpret the modal symbols as follows ;tϕ iff Ω* ⊆ [ϕ] ;∆ϕ iff [ϕ] ⊆ Ω* and the consistency between t and ∆ is expressed by the axiom ;t¬ϕ → ¬∆ϕ which is the modal translation of N*(äA) = 1 implies ∆ *(A) = 0. The following are examples of valid formulas: ϕ if ϕ is a propositional tautology t(¬ϕ ∨ ψ) → (¬tϕ ∨ tψ) ; ∆(ϕ ∧ ¬ψ) → (∆¬ϕ → ∆¬ψ)


tϕ → ¬t¬ϕ tϕ → ttϕ ¬tϕ → t¬tϕ t¬ϕ → ¬∆ϕ

; ; ; ;

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∆ϕ → ¬∆¬ϕ ∆ϕ → ∆¬∆ϕ ¬∆ϕ → ∆∆ϕ ∆ϕ → t∆ϕ; ;tϕ → ∆¬tϕ.

The following inference rules are modus ponens, and from ;ϕ deduce ;tϕ and ;∆¬ϕ. This bimodal system can also be encoded by means of two necessity-like modalities t+ and t– where t+ϕ ≡ tϕ and t–ϕ ≡ ∆¬ϕ, that correspond to start with two subsets Ω+ and Ω – on Ω, such that Ω+ = Ω*, Ω– = ÄΩ*, that is Ω+ ∪ Ω– = Ω. t+ and t– are usual KD45 necessity modalities with respect to Kripke models of the form (Ω, Ω+, .) and (Ω, Ω –, .) respectively. Soundness and completeness of this bimodal logic is proved in the next section.

6 - A Completeness Theorem Language: The language contains propositional variables, connectives, two necessities t+, t–; dual possibilities: ◊+ is ¬t+¬, ◊– is ¬t–¬. We let u, v vary over {+, –}. Semantics: We consider Kripke models K = (Ω, Ω+, Ω –, .) where Ω = Ω+ ∪ Ω–, Ω + ≠ Ø ≠ Ω–, the satisfiability relation . assigns to each propositional variable p and each world ω the value 1 (denoted: ω . p) or 0 (ω ° p). Elements of Ω+ are not impossible, elements of Ω– are not observed yet. Ω = Ω+ ∪ Ω– means that any world ω is either not impossible or not observed or both. . extends to all formulas; for connectives using truth tables, for t+ we define ω . t+ϕ iff (∀ ω ∈ Ω+)(ω . ϕ), similarly for Ω–, t–. We may write .t+ϕ since ω . t+ϕ does not depend on ω. Note that similar bimodal logics were investigated by (Goranko, 1990); he works with Kripke models of the form K = (Ω, R+, R –, .) where R+ is a binary relation on Ω and R– is its complement in Ω × Ω with the usual Kripke semantics, i.e., ω. t+ϕ iff all ω' such that ω R + ω' satisfy ω' . ϕ and similarly for R–. This applies to our approach only if Ω+ ∩ Ω – = Ø, i.e., all potentially observable situations have been observed, since then, given Ω+ and Ω–, R+ = Ω × Ω + and R– = Ω × Ω –, and thus R– is Ω2 – R+. (Thanks are due to Luis Fariñas del Cerro for calling our attention to Goranko (1990).) Derived Modalities ∆, ∇: We let ∆+ϕ be t–¬ϕ, ∆–ϕ be t+¬ϕ; ∇ uϕ is ¬∆ u¬ϕ for u being + or –. Let [ϕ] = {ω | ω . ϕ}; then ;t+ϕ iff Ω+ ⊆ [ϕ], ;t–ϕ iff Ω– ⊆ [ϕ], ;◊+ϕ iff Ω+ ∩ [ϕ] ≠ Ø, ;◊–ϕ iff Ω– ∩ [ϕ] ≠ Ø, ;∆+ϕ iff [ϕ] ⊆ Ω+,


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;∆–ϕ iff [ϕ] ⊆ Ω–, ;∇ +ϕ iff [¬ϕ] ∩ Ω– ≠ Ø, ;∇ –ϕ iff [¬ϕ] ∩ Ω+ ≠ Ø. We finally define tϕ to be t+ϕ ∧ t–ϕ; clearly, ;tϕ iff [ϕ] = Ω. Axioms and Rules of BBL (Bimodal Belief Logic): (1) (2) (3) (4)

tu(ϕ → ψ) → (tuϕ → tuψ) tuϕ ≡ tvtuϕ ◊uϕ ≡ tv◊uϕ tϕ → ϕ.

Modus ponens (from ϕ, ϕ → ψ infer ψ), necessitations: from ϕ infer t+ϕ, t–ϕ. Soundness is clear. (Caution: if we allowed Ω+ = Ø then the implication ← in (3) above would be violated, similarly for Ω– = Ø.) Lemma 7: (1) ;ϕ ≡ ψ implies ;tuϕ ≡ tuψ (2) ;tu(ϕ ∨ tvψ) ≡ tuϕ ∨ tvψ and similarly for conjunction instead of ∨. Proof: (1) From axiom (1). (2) From ;ϕ → (ϕ ∨ tvψ) we get ;tuϕ → tu(ϕ ∨ tvψ); furthermore, ;tvψ → (ϕ ∨ tvψ), ;tutvψ → tu(ϕ ∨ tvψ), but tutvψ is equivalent to tvψ, thus we have proved the implication ⇐. Conversely, ;tu(ϕ ∨ tvψ) ⇒ ;tu(◊v¬ψ → ϕ) ⇒ ;(tu◊v¬ψ → tuϕ) ⇒ ;(◊v¬ψ → tuϕ) ⇒ ;(tvψ ∨ tuϕ) . A similar proof applies for conjunction. Note: These are very standard proofs; we present them as examples for the reader's convenience. y Corollary 8: Each formula is logically equivalent to a formula without nested modalities. Completeness theorem 9: BBL is complete with respect to the given semantics, i.e., for each formula ϕ, BBL ; ϕ if for each model K = (Ω, Ω+, Ω–, .) and each ω ∈ Ω, ω . ϕ. The rest of this section elaborates on the proof; assume BBL % ϕ* for some ϕ* and find a K = (Ω, Ω +, Ω, .) and ω ∈ Ω such that ω° ϕ*. Note that the model we shall construct is often called the canonical model. Let T0 be the set of all formulas provable in BBL; a theory will be any set T ⊇ T0 of formulas. T ; ϕ means that ϕ is provable from T using only modus ponens (theories will correspond to formulas true in a possible world). T is consistent if T % false. T is complete when T ; ϕ or T ; ¬ϕ for each ϕ. T is (deductively) closed if T ; ϕ implies ϕ ∈ T for each ϕ. Α formula is boxed if it is of the form t+ϕ, t–ϕ, ◊+ϕ, ◊–ϕ for some formula ϕ. By the usual


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completion procedure, construct an arbitrary (but fixed) theory êT0 ⊇ T0 with the following properties: (i) ê T0 results from T0 by adding only boxed formulas, (ii) êT0 decides each boxed formula (i.e., êT0 ; tuϕ or êT0 ; ◊u¬ϕ for each ϕ), (iii) êT0 ∪ {¬ϕ*} is consistent. (Boxed formulas from êT0 will be true in the model constructed; note that a boxed formula is either true in all worlds or in no world of a given model). Lemma 10: For u = + or –, if êT0 ; ◊uϕ then the theory êT0 ∪ {ϕ} ∪ {tuψ → ψ | ψ arbitrary} is consistent. Proof: Suppose not, then there are ξ1… ξn ∈ êT0 and some ψ1… ψm such that ¬ϕ has a modus ponens proof from Âi = 1, n ξ i ∧ Âj = 1, m (tuψj → ψj); write it Âi = 1, n ξi ∧ Âj = 1, m (tuψj → ψj) ; ¬ϕ . Since here we have deduction theorem (no generalization used!) we get ;(Âi = 1, n ξi ∧ Âj = 1, m (tuψj → ψj)) → ¬ϕ . Now generalize by tu, after obvious transformations we get ;(Âi = 1, n tuξi ∧ Âj = 1, m tu(tuψi → ψi)) → tu¬ϕ. Besides BBL proves tuξi for each ξi ∈ T0, proves ξi → tuξi for each boxed ξi , thus êT0 proves tuξi for each ξi ∈ êT0 (either since ξi ∈ T0 or since ξi is boxed ) and also êT0 proves tu(tuψ → ψ) (since tu(tuψ → ψ) is provably equivalent to tuψ → tuψ, cf. Lemma 7). This means that êT0 proves the antecedent of the last displayed formula and hence its consequent, tu¬ϕ, a contradiction. This completes the proof of the lemma. y Proof of completeness theorem 9 continued: Now consider the set T of all complete extensions of êT0. Let T+ = {T ∈ T | (∀ψ)(T ; t+ψ → ψ)} T– = {T ∈ T | (∀ψ)(T ; t–ψ → ψ)}. Observe that T = T+ ∪ T– (since if T ; t+ϕ ∧ ¬ϕ, T ; t–ψ ∧ ¬ψ, then T ; t+(ϕ ∨ ψ) ∧ t–(ϕ ∨ ψ) ∧ ¬(ϕ ∨ ψ), or equivalently T ; t(ϕ ∨ ψ) ∧ ¬(ϕ ∨ ψ) i.e., T is contradictory). Observe that each T ∈ T uniquely determines the corresponding interpretation ωof the language (ω(p) = 1 iff T ; p). Let Ω be the set of all these interpretations. Each ω ∈ Ω uniquely determines the corresponding T since T extends êT0 so that ω decides each boxed formula. (Each ω ∈ Ω results from a theory T ∈ T and T ⊇ êT0 , whereêT0 is the fixed theory deciding each boxed formula. Each formula is a Boolean combination of atoms and boxed formulas; thus knowing which atoms and which boxed formulas are true/false, we know the truth/falsity of each formula. This is why ω determines back its T uniquely). Denote Tω the theory given by ω and Ω u = {ω | Tω ∈ Tu}. Note that Ω+ ≠ Ø ≠ Ω– thanks to our Lemma 10. Indeed, since êT0 clearly proves ◊+true (where true is any tautology), by Lemma 10, êT0 ∪ {t+ψ → ψ | ψ arbitrary} is consistent and hence has a consistent complete extension; this extension is clearly in T+ and the corresponding ω is in Ω+. Then there is a ω* ∈ Ω such that T ω* ; ¬ϕ*. Finally define, for ω ∈ Ω and p a propositional variable, ω . p iff Tω ; p.


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Lemma 11: For each formula ϕ and each ω ∈ Ω, ω . ϕ iff Tω ; ϕ. Proof: By induction on the complexity of ϕ. For ϕ atomic: by definition; for connectives: trivial. We handle the modalities. If .t+ϕ then for each ω ∈ Ω+, ω . ϕ which, by induction assumption, implies Tω ; ϕ; but then êT0 ; t+ϕ (since otherwise we would have êT0 ; ◊+¬ϕ, thus by Lemma 10 there would exist a ω ∈ Ω + such that Tω ; ¬ϕ, a contradiction). But since each T ω extends êT0, for each ω ∈ Ω, we get Tω ; t+ϕ. This proves the first part of our inductive step, namely that if t+ϕ is true in our model then each Tω proves t+ϕ. To prove the converse assume T ω ; t+ϕ for some ω (hence êT0 ; t+ϕ and Tω ; t+ϕ for all ω). Now if ω ∈ Ω+ then T ω ; t+ϕ → ϕ, hence for such ω we get T ω ; ϕ, hence ω .ϕ for all ω ∈ Ω + by the induction assumption and hence t+ϕ is true (independent) of ω) by the definition of satisfaction. y Consequently, K = (Ω, Ω +, Ω–, .) is the desired model of BBL: ω* ∈ Ω is such that Tω* ; ¬ϕ*, hence ω* . ¬ϕ*.

7 - Towards an Application to Deontic Logic In possibility theory, the ∆ set function is characterized by the decomposability property ∆(A ∨ B) = min(∆(A), ∆(B)) which in the non-graded setting of our bimodal logic writes ;∆(ϕ ∨ ψ) ≡ ∆ϕ ∧ ∆ψ. This is a consequence of KD45d axioms. It is has been already mentioned (Dubois and Prade, 1992) that ∆ might be interpreted in terms of permission. Namely, if all the worlds ω such that ω . ϕ ∨ ψ are permitted so should be all the world ω such that ω . ϕ and all the worlds such that ω . ψ. However, it should be pointed that this does not correspond to the usual view of permission in deontic logic where ϕ is permitted iff ¬ϕ is not obligatory (or equivalently iff ϕ is not forbidden). In a von Wright-type system (see, e.g., Meyer and Wieringa, 1993), this leads to have ;P(ϕ ∨ ψ) ≡ P(ϕ) ∨ P(ψ) where P denotes the permission modality. This contrasts with the so-called "free choice permission" reading of P(ϕ ∨ ψ) (Kamp, 1973), which corresponds to the above characteristic property satisfied by ∆.


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With the view of ∆ functions and ∆ modalities we have advocated, our bimodal logic could be understood in the following way: – Ω* is the set of worlds which are known from past observations as indeed permitted situations, so ∆ϕ means that all the worlds in ϕ are explicitly permitted; its complement Ω– = ÄΩ* is the set of worlds which are at least potentially forbidden. The condition Ω* ≠ Ω means that not everything is explicitly permitted; – ÄΩ* is the set of worlds which are known as forbidden by regulations while Ω* = Ω+ is the set of worlds which are at least implicitly permitted. The condition Ω* ≠ Ø means that at least some situation is implicitly permitted. Clearly Ω* ⊆ Ω* and Ω* – Ω* forms the set of implicitly permitted, not explicitly permitted (since not encountered yet) situations, typically the type of situation which, once observed, may be judged contrary to the spirit of the law and lays bare the incompleteness of the set of regulations. In this view, the set of formulas B stands for such regulations, and its sets of models is Ω*, while the set of formulas D stores a collection of explicitly permitted situations, which can be viewed as the jurisprudence memory. Recently, Liau (1997) has developed a graded modal logic calculus based on a deontic logic proposed by Meyer (1988) and he also exploits the ∆ set function of possibility theory, oriented towards the choice of acts in the face of graded permissions.

8 - Conclusion This paper is a first attempt at formalizing the idea that an agent does not only handle beliefs that restrict how the world is supposed to be, but also experience consisting of things that the agent observed in the world. The latter generates a form of modality that is in some sense dual to the usual possibility modality for reasoning about beliefs although this duality is different from the usual one between possibility and necessity. It also suggests a dual theory of possibility where pieces of information represent lower bounds on possibility distributions, and a maximal specificity principle is applied in order to make the best of the available information. In that case, only what has been observed is declared to be possible, while in usual possibilistic reasoning (Zadeh, 1979) involving flexible constraints, a principle of minimal specificity is at work where everything that is not ruled out by beliefs is declared possible. These dual forms of possibilistic reasoning have also been noticed and described in some detail by Weisbrod (1995) at the semantic level. We have also emphasized the potential attractiveness of the data-driven logic put together with the knowledge-driven logic, as a setting for information systems involving constraints which stored data must satisfy, and also as a logical framework for deontic reasoning involving obligations, implicit permissions, and explicit permissions.


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Lastly, we have not considered, in this paper, the case of repeated observations, namely we model in the same way a piece of data regardless of whether it has been observed once or several times. Hence our approach is very crude from an operational point of view. Accounting for the number of times a piece of data is observed would lead us to connect datadriven reasoning with statistical modeling, a topic for further research.

Acknowledgements: Petr Hájek acknowledges partial support by the ground No. A1030601 of the Grant Agency of the Academy of Science of the Czech Republic. All three authors work is partially supported by a France-Czech (CNRS-Czech Academy of Sciences) agreement No. 4008.

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