Dynamic belief modeling

Dynamic belief modeling Antonio Moreno, Ton Sales Departament de Llenguatges i Sistemes Informatics Universitat Politecnica de Catalunya Pau Gargallo, 5 08028-Barcelona E-mail: famoreno, [email protected] Abstract

The possible worlds model and its associated Kripkean semantics provide an intuitive semantics to epistemic logics, but they seem to commit us to model agents which are logically omniscient and perfect reasoners. In this article we show that this is not necessarily the case, if possible worlds are not considered as consistent descriptions of the real world. We propose to model the beliefs of an agent using analytic tableaux, and we suggest how beliefs can be analysed in a pure logic way, using (a modi ed version of) the classical analytic tableaux method. We also show a brief approach to a second dimension of analysis, the physical dimension, that will allow the user to perform tests in the real world and to add the results of these tests in the open tableaux of the logic analysis. Keywords: logics of knowledge and belief, logical omniscience, possible worlds, dynamic accessibility relation, analytic tableaux

1 Introduction Logics of knowledge and belief ([HALP92]) are tools used to analyse in a formal way the reasoning about knowledge performed by an intelligent agent. The semantic model traditionally adopted as a basis in these logics is the possible worlds model ([HINT62]). This model is based in the assumption that there is a set of possible states (or possible worlds) in which the agent can be in any moment; when the agent is in a possible world, there is a set of possible worlds which are compatible with the actual world, in the sense that the agent cannot distinguish these worlds from the actual one. Using the standard Kripke semantics ([KRIP63]) in the model of possible worlds, the agent believes a proposition p if and only if it is true in all the worlds that the agent cannot tell apart from the actual world. This semantics has been widely accepted1 because of its

1 Nevertheless, there are some authors who claim that the possible worlds model and its associated Kripke semantics rest in dubious metaphysical and epistemological assumptions; see e.g. [VARD86] and [BARW89].

1

naturalness and simplicity; nevertheless, adopting it seems to commit us to model agents which are 

logically omniscient, because they believe all tautologies (since all of them are true in



perfect reasoners, because they also believe all logical consequences of their beliefs (e.g.

every world), and

if an agent believes p and (p ) q), it means that these two propositions are true in all the worlds compatible with the actual world; therefore, q will also be true in all of these worlds, and the agent will also believe q).

Obviously there are many circumstances in which these conditions are unacceptable; that would be the case when the agent is supposed to be able to compute its knowledge or to take actions based on it. It is clearly a realistic model of neither human agents (which are not logically omniscient) nor computational agents (which have resource limitations which can prevent them from being perfect reasoners). There have been many approaches that have tried to solve these problems. Some of them are syntactic ([VARD86], [KONO86], [HINT86]) while others are semantic ([MONT70], [CRES72], [RANT75], [HINT75], [RESC79], [LEVE84], [FAGI85], [LAKE94]). In our opinion, none of these approaches solves the problems in a satisfactory way. The aim of our work is to analyse the real root of the problems and to propose a new approach from a somewhat dierent angle. We describe, in this new approach,  

how the beliefs of an agent can be analysed, and how the beliefs of an agent change dynamically as a consequence of this analysis.

2 Possible worlds and accessibility In our work we use a language L, which is a subset of the classical propositional modal logic of belief. This language is composed of a nite xed set of basic propositions (P ), two logic operators2 (:, _) and the modal operator B 3 . We consider the formulas of L generated by the following grammar (where p is any basic proposition): F =) B F j G G =) (:G) j (G _ G) j p With this grammar, the language is restricted to standard propositional formulas pre xed by a (possibly empty) sequence of modal operators4 (e.g. (:(p _ (q _ r))), B q, BB (p _ (q _ r))). In order to model agents which are neither logically omniscient nor perfect reasoners, you have to notice that the origin of these problems resides in the fact that possible worlds have always

2 Conjunction and implication can be de ned from negation and disjunction in the usual way. 3 We restrict ourselves for the moment to one single agent and propositional logic; in our future work we plan to include a modal operator Bi for each agent i and to work in predicate logic. 4 In our future work we also plan to allow any kind of formulas in this language, e.g. (p _ :B p).

been assumed to be consistent5 , describable with models (in the logical sense of the word). If this condition holds both problems are dicult to avoid, because tautologies are true in every model (which implies logical omniscience) and the set of propositions which are true in a model is (by rst-order completeness) closed by modus ponens and the ordinary deductive laws (which implies perfect reasoning). We de ne a world as a set of formulas of L. Furthermore, we do not impose any restriction on this set (it does not have to be consistent, it may be not deductively closed, it may fail to contain some tautologies, etc.). Intuitively, we envision a world as any situation that the agent may consider as real. It can be a situation that has happened to the agent, or maybe something it6 has imagined. With this approach to possible worlds maybe it would be more appropriate to call them conceivable or imaginable worlds7 , bearing always in mind that they can be logically inconsistent or physically unrealizable. With this vision of worlds and the logic analysis of beliefs that we suggest in the following section, the problems of logical omniscience and perfect reasoning vanish. This approach to possible worlds has been dismissed many times in the past, especially for philosophical reasons (because it does not seem a very acceptable idea to allow worlds which are logically inconsistent). Nevertheless, there are many examples of worlds which are inconsistent, not physically realizable, but perfectly conceivable. For instance we could take dreams, or unrealizable designs (many pictures from Escher are a great example of this kind of situations, see e.g. [HOFS80]). There are even more speci c examples in the history of mathematical logic, e.g. the set theory of Frege. This theory is well de ned and is perfectly comprehensible, but it is inconsistent, as Russell proved later with his famous paradox. Another possible justi cation of inconsistent worlds is given by Shoham in [SHOH91]; he suggests that the agent can be located in dierent contexts, and each context can limit the access of the agent to a particular subset of the world (in the same way as views limit the access of a user to a certain part of a database). Using contexts it is possible to construct a non trivial theory of inconsistent worlds, because the subsets accessible in each context could be consistent while the whole world is inconsistent. The set of formulas that de nes a world w can be divided in two subsets:  Belief formulas (w): formulas of w that are pre xed by the modal operator. ( )=f j B  wg

BF w



Propositional formulas (w): standard propositional formulas of w.

( )=w - BF (w)

PF w

Note that, for any world w, BF (w) and P F (w) are unrelated (because the set of formulas that de nes w is unrestricted). Furthermore, if they were equal, that would be an unstable situation, as suggested by Smullyan in [SMUL86].

5 This problem was pointed out by Hintikka twenty years ago, in [HINT75], where he already proposed to allow epistemic possible worlds that are not necessarily logically possible (hence the term "[logically] impossible [epistemically] possible worlds" coined in that article). 6 In this article we always use the pronoun it to refer to the agent. 7 It would even be better to call them conceivable situations, in the spirit of Barwise and Perry ([BARW83]), because we do not require them to describe every aspect of the world, but maybe just a small portion of it.

We are going to use the standard Kripke semantics ([KRIP63]) for modal logics of knowledge and belief: a formula  is believed by the agent in world w if and only if  appears in all the doxastic alternatives to w (all the worlds related to w by the accessibility relation R). In the classical approaches the accessibility relation between worlds is predetermined, so the set of beliefs is constant. This fact implies that these approaches are unsuitable to model changing beliefs. In our framework we want to model the evolution of the agent's beliefs through time. This evolution can be due to internal logic analysis of the agent or to incorporation of data derived from tests performed in the environment in which the agent is located. The former possibility is explored in this article, while the latter will be discussed in future work. In our model the set of beliefs of an agent in a world changes in time, because the accessibility relation between worlds changes. The agent generates a sequence of accessibility relations; each of them (via the standard Kripke semantics) de nes a dierent set of beliefs. We will describe the evolution of the beliefs of the agent in a world with a small example. Assume that we want to study the beliefs of the agent in a world we . World we , as de ned above, is just a set of formulas of L, e.g. we

 f P, :Q, (R _ S),

, (:R _ (S _ P)) g

BR B

This set can be divided into BF (we ) and P F (we ), as described previously: ( ) = f R, (:R _ (S _ P)) g ( ) = f P, :Q, (R _ S) g

BF we

P F we

Using the set BF (we ), the agent can generate the initial accessibility relation, R0 , applying the standard Kripke semantics: 8w0 

W

, (we R0 w0) , f R, (:R _ (S _ P)) g  w0

World w0 is accessible (through R0 ) from world we if and only if every proposition which is believed by the agent in we appears in w0 . Moreover, we restrict the beliefs of the agent in a world w in the following way: they have to be exactly all the propositions which appear in all its doxastic alternatives. That is, there can be no proposition common to all the accessible worlds from w which is not a belief in w. The ontology of possible worlds just de ned is represented8 in gure 1. World we is R0-related to all the worlds in class w. w = f w1 , w2 , : : :, wn g, where wi = f R, (:R _ (S _ P)) g [
i . According to the above de nitions, every proposition believed in world we appears in all the worlds w1 , w2 , ..., wn ; furthermore, there is no other proposition in the intersection of all these worlds, so

T
=; i

8 Each world w (or class of worlds w) is represented by a rectangle, that contains on the left side P F (w) and (in an inner rectangle inside each world, on the right side) BF (w). Each subscripted
represents a set of propositions.

R

P

:Q R_S

R (:R_(S_P))

R0

-

(:R_(S_P))


i

we

w

Figure 1: World we and its doxastic alternatives The initial accessibility relation, R0, determines the initial set of beliefs of the agent in we , Bel0 (we ), and the initial list of disjunctive beliefs of the agent in we , DL0 (we ):  

( ) = f R, (:R _ (S _ P)) g DL0(we ) = [ (:R _ (S _ P)) ] Bel0 we

The sequence of accessibility relations (R0, R1, R2 , : : :) is generated using the disjunctions included in DL0, DL1 , etc. The recursive de nition of the sequence of accessibility relations is the following: 

Ri

, i1 8w,w0

, (w Ri w0 ) , (w Ri?1 w0) ^ (((:i 6  Beli?1 (w)) ^ (i  w0)) _ ((: i 6  Beli?1 (w)) ^ ( i  w0 )))  W

, where (i _ i) is the i-th element of the list DLi?1 (w). This de nition says that the Ri-accessible worlds are the subset of the previously accessible worlds that contain one of the members of the i-th disjunction of DLi?1 (w), as long as this member is not the negation of a current belief. This fact implies that, if the agent believes :P and Q in a certain point, and then analyses (P _ :Q) to generate the next accessibility relation, then the set of accessible worlds will become empty, and the agent will stop believing its initial set of beliefs9 . The syntactic counterpart of this generation of accesibility relations is the use of the splitting rule in the analytic tableaux method, as will be shown in section 3. As you can see in the de nition, the agent needs to use the rst element of DL0 (we), (:R _ (S _ P)), to generate the R1-accessible worlds from we . The formal expression is the following10 : 9 This is the process discovered with much fracas by the Pythagoreans, usually called reductio ad

absurdum. 10This is the result if the simpli cation :(:R) = R is automatically applied.

8w0

, (we R1 w0) , (we R0 w0) ^ (((R 6  Bel0 (we )) ^ (:R  w0 )) _ ((:(S _ P) 6  Bel0 (we)) ^ ((S _ P)  w0 )))  W

Note that (R 6  Bel0 (we )) is false and (:(S _ P) 6  Bel0 (we )) is true, so the previous expression is just asserting that we is R1-related to those worlds of class w that contain (S _ P). Assume that class w is divided in three subclasses, as shown in gure 2:   

: worlds that contain :R. w2 : worlds that contain (S _ P). w3 : w - (w1 [ w2 ), i.e. worlds that do not contain neither :R nor (S _ P). w1

x

x

w1

x x xx xx x x x x x x x x x

x

xx x x x

w2

w3 w

Figure 2: Division of w in three subclasses. was R0-related to all the worlds in class w, but is R1 -related only to the worlds in class . = f w21 , w22 , : : :, w2m g, where w2i = f R, (:R _ (S _ P)), (S _ P) g [
2i . This restriction of the accessible worlds is shown in gure 3. The change of the accessibility relation implies a (possible) change in the beliefs of the agent. The beliefs in we at this point would be: we

w2 w2

 

( ) = f R, (:R _ (S _ P)), (S _ P) g DL1(we ) = [ (:R _ (S _ P)) , (S _ P)]

Bel1 we

w1 R0 P

:Q R_S

R

:R_(S_P)

R1

> -

R

:R_(S_P) S_P
2i

w2

we

w3 w

Figure 3: Reduction of doxastic alternatives Notice that the restriction of the doxastic alternatives to we has caused the addition of a new belief, (S _ P) (which is also included at the end of the list of disjunctive beliefs). This formula is a new belief because it was not included in all the worlds in w, but it appears in all the worlds in w2 . If the agent wants to pursue the analysis of its beliefs, it can generate the next accessibility relation, R2. Using the previous de nition, the disjunction used to generate R2 from R1 is the second element of DL1 , which is (S _ P). The formal expression in this case is the following: 8w0

, (we R2 w0) , (we R1 w0) ^ (((:S 6  Bel1 (we)) ^ (S  w0 )) _ ((:P 6  Bel1 (we)) ^ (P  w0)))  W

This formula expresses the idea11 that we is R2-related to those worlds of class w2 that contain S and/or P. We can have a more intuitive interpretation of this restriction if we consider the following division of w2 in three subclasses: 

w2

1:

worlds that contain S.

11Notice that (:S 6  Bel1 (we)) is true and (:P 6  Bel1 (we )) is also true.

 

worlds that contain P. w2 : worlds that do not contain neither S nor P. 3 w2

2:

, that was R1-related to the worlds in w2 , is only R2-related to the worlds of the subclasses and w22 , but not to those in w23 . This situation is depicted in gure 4.

we

w2

1

w1 R0

P :Q

R_S

-

R1

R (:R_(S_P))

R2

(S_P)

-

1

R2

we

j

R (:R_(S_P)) S_P S


21i

w21

R (:R_(S_P)) S_P P


22i

w22

w23

w2

w 3 w

Figure 4: Further reduction of doxastic alternatives Updating the set of beliefs (and the list of disjunctive beliefs), we would obtain the following result:  

( ) = f R, (:R _ (S _ P)), (S _ P) g DL2(we ) = [ (:R _ (S _ P)) , (S _ P)]

Bel2 we

Notice that the set of beliefs has not changed, because there is no formula common to all the -accessible worlds which was not already common to all the R1 -accessible worlds from we . As a result, the list of disjunctive beliefs has not changed either. This list contains only two elements, and the agent would need a third disjunction to generate R3 from R2 and this disjunction, so this is the end of the (purely logic) analysis that the agent can perform in we .

R2

3 Belief analysis With this vision of worlds in mind, we propose a two-dimensional analysis of the agent's beliefs. These dimensions are called logic and physical. In the rest of this article we explore the former; we describe how to analyse the beliefs in a pure logic way, using a modi ed version of the classical method of the analytic tableaux. It is also shown how, as a consequence of our previous de nition of beliefs, this logic analysis modi es them dynamically. This logic analysis is an automatic way of generating the sequence of accessibility relations that has been described previously. In a future work the physical dimension will be introduced. In this new kind of analysis the agent will perform some experiments or observations in the real world, and will incorporate the results of these tests in the tableaux where the logic analysis is being made. It will also be explained how the logic analysis suggests the experiments to be made in the physical analysis in order to increase the beliefs of the agent. 3.1

Logic analysis of the beliefs

We propose to use the method of the analytic tableaux (see, e.g. [SMUL68]) to perform the logic analysis of the beliefs of the agent. We will describe this analysis with the example used in the previous section: we

 f P, :Q, (R _ S),

, (:R _ (S _ P)) g

BR B

We begin the analysis with a tableau T0, which contains the initial set of beliefs of the agent in we : T0



( ) = f R, (:R _ (S _ P)) g

BF we

In light of previous considerations, we can say that T0 not only represents BF (we ); it can also be seen as a (very probably partial) description of all the worlds initially connected to we , because all the beliefs in we are true in all of these worlds. Thus, a tableau represents a class of worlds: all the worlds in which the propositions contained in the tableau are true. T0

represents w = fw j (we R0 w)g

The syntactic counterpart of the generation of the accessibility relation R1 from R0 is the application of the splitting rule12 of the method of the analytic tableaux to the formula (:R _ (S _ P)). After the application of this rule, we obtain the situation shown in gure 5.

12Recall that the splitting rule applied to a disjunction ( _ ) generates two subtableaux, one containing  and the other one containing .

R (:R

_

(S_P))



R

R (:R

T0 R

_ (S_P))* :R

(:R

T1

_

(S_P))*

S_P

T2

Figure 5: Splitting rule applied to (:R _ (S _ P)) We can wonder now which class of worlds is represented by T1 or T2. Recall our previous division of the class w in three subclasses, w1 , w2 and w3 . It is clear that the tableau T1 represents the class of worlds w1 , and that T2 represents w2 . The worlds in class w3 are not even considered by the analytic tableaux method, because it looks for models of the initial set of formulas, and it is not possible to have a model of the set f R, (:R _ (S _ P)) g in which both :R and (S _ P) fail to be true. Of course, there can be worlds which contain both of them; these worlds would be in the intersection of w1 and w2 , as previously shown in gure 2. In the analytic tableaux method, those tableaux that contain an atom and its negation are closed, and thus dismissed from the logic analysis (because these tableaux cannot represent a model of the initial set of formulas, which is the aim of the method). In the logic analysis, the agent can consider these worlds as logically impossible, not physically realizable, and can eliminate them from the analysis closing the tableau that represents this kind of worlds. In our example, T1 contains both R and :R, so the agent can consider the class of worlds represented by this tableau ( w1 ) as logically impossible13 and can get rid of these worlds closing T1 and dismissing the branch that contains it from the logic analysis. The aspect of the tableaux tree in this point is shown in gure 6. The eect of analysing proposition (:R _ (S _ P)) in T0 is shown in gure 7, where you can notice that the semantic counterpart to the syntactic application of one of the splitting rules of the method of the analytic tableaux is the substitution of the accessibility to a class of worlds by the accessibility to subclasses of it (as shown in gure 3). The resulting tableaux (one, in this example) are more speci c than the original one (logically more detailed), because they always contain at least the same formulas than the original tableau. Following the example, now the agent would analyse the formula (S _ P) in T2, and would obtain the situation shown in gure 8. 13With our de nition of worlds, the worlds that contain an atom and its negation are not impossible, so

it would be more appropriate to designate them with another expression; they are conceivable, imaginable but physically unrealizable, incoherent, logically inconsistent, non-classical, non-standard, etc.

R (:R

_

(S_P))

?

T0

R (:R

_

(S_P))*

S_P

T2

Figure 6: Tableaux tree after closing T1 Recalling our previous division of w2 in three subclasses, we can notice that T3 represents w2 and T4 represents w2 . The worlds in class w2 are no longer considered, because they 1 2 3 do not contain neither S nor P. The eect of the analysis of (S _ P) is shown in gure 9. This analysis can be seen as the syntactic counterpart of the generation of the accessibility relation R2 from R1, as shown in gure 4. 3.2

Introduction of doubts in the analytic tableaux

In this logic analysis of beliefs we use the classical analytic tableaux method, with the following modi cations:  After applying a splitting rule, the analysed proposition is maintained in the resulting tableaux, it is not deleted. We just add a tag to the proposition (shown with an asterisk in the gures) to mark that it has already been analysed. Notice that, in the example shown in gure 5, if (:R _ (S _ P)) would not have been included in T1 and T2, then the set of propositions common to these tableaux would have been fRg and, therefore, the beliefs of the agent would have decreased from fR,:R _ (S _ P)g (before starting the analysis) to fRg. If analysed propositions are not eliminated during the logic analysis, then the set of beliefs of the agent cannot decrease in this process.  We do not allow the user to add to an open tableau any tautology; otherwise, we could not avoid the problem of logical omniscience. This prohibition is based on the idea that a tableau is just a mere list of the beliefs of the agent in a certain moment, and it cannot be modi ed with no apparent reason.  There is only one exception to the previous rule: we allow free introduction of instances of the axiom of the excluded middle, (p _ :p), being p any atomic proposition14 . 14In our future work we intend to eliminate this restriction to atomic propositions.

x

R (:R

_

(S_P))

?

.......

T0

1. ... . . ... (:R _ (S_P))* . . R

S_P

T2

x

w1

x x xx xx x x x x x x x x x

x

xx x x x

w2

w3 w

Figure 7: Semantic eect of the logic analysis of (:R _ (S _ P)) The possibility of adding instances of the axiom of the excluded middle in the analytic tableaux is a classical idea in the tradition of model theory, as described e.g. in [BELL77]. This idea is also very related to the interrogative moves of the interrogative model of inquiry proposed by Hintikka ([HINT86], [HINT87], [HINT88], [HINT92]). He proposes to model arguments with plays of the interrogative game. Deductions performed in the argument are modeled with deductive moves (via the application of analytic tableaux rules) in the interrogative model, and new data introduced in the argument nd their counterpart in interrogative moves. One kind of interrogative move allows the player of the interrogative game to add (some) instances of the axiom of the excluded middle in the left side of a tableau15 . More speci cally, (p _ :p) can be added to a subtableaux if and only if p is a subsentence of a formula of the subtableaux or a substitution instance of a subformula of a formula in the subtableaux with respect to names that appear in the subtableaux. For the time being, we are restricting ourselves to the introduction of instances of (p _ :p) in which p is only an atomic proposition of the language. The introduction of doubts is also related to the physical dimension, to the tests performed by the agent in the real world in order to increase its knowledge. Suppose the agent doubts of the validity of a general law such as All birds y (i.e. (8xBird(x) ) F lies(x))). It adds to its set of beliefs the formula ((8xBird(x) ) F lies(x)) _ (:8xBird(x) ) F lies(x))) and, when it analyses this disjunction, it will have access to two classes of worlds: in the rst class the law will hold, while the second will contain (:8xBird(x) ) F lies(x)), from which by normal tableau processing the agent would get Bird(a) and :F lies(a) (for some (undetermined) a). In 15Hintikka uses the classical Beth-style analytic tableaux, as described in [BETH55].

R (:R

_

(S_P))

T0

? R (:R

_

(S_P))*

S_P



R

T2

R (:R

R

_ (S_P))* (S_P)* S

(:R

_ (S_P))* (S_P)*

T3

P

T4

Figure 8: Analysis of (S _ P) this point, the agent can see that it can increase its beliefs if it can dismiss this class of worlds; if the agent looks actively in the real world for an individual a such that Bird(a) and :F lies(a) and it cannot nd it, then it can conclude that an individual with these properties does not exist; when this conclusion is nally reached, the agent can declare this class of worlds physically impossible, and so the agent will have grounds not to consider it any more as a conceivable, realizable alternative. Then the tableau that represents this class of worlds would be closed and dismissed from the analysis. In that moment all the worlds accessible by the agent would contain (8xBird(x) ) F lies(x)) so, by de nition, the agent would believe this law. Notice the dierence with the beliefs obtained in the pure logic analysis; this new belief is provisory and open to falsi cation, in case the agent later nds an individual with the desired properties. Thus, as Hintikka suggested in [HINT86], doubt can be understood as the beginning of a dynamical process that, by reducing the number of conceivable worlds that the agent considers, reinforces one side of the doubt over the other, sets the conditions to verify it (and falsify the other) and tendentially gives credence to it. Such a process can be made compatible, in a natural way, with e.g. falsi cation strategies in the Popperian philosophy of science. It can also explain why we nally know things; they simply make themselves present, not directly (through the senses, as it were) but indirectly, through their involvement in a reinforcement/disabling process which eliminates possible worlds (that are then seen as \impossible") and so reinforces -as belief, now turned knowledge- what active experience has indirectly but forcefully shown.

w1

........................... :_ _ . _ . .
21 R ........... i . . (:R _ (S_P)) . . T0 :_ _ . ? . .............. . _ . R . . .
22i (:R _ (S_P))* . . . S_P .. . . . .. R T2 . . .. R R . (:R _ (S_P))* (:R _ (S_P))*. R

R (S P) S P S

w21

R

R (S P) S P P

(S_P)* S

T3

(S_P)* P

T4

w22

w23

w2

w3 w

Figure 9: Semantical eect of the analysis of (S _ P) This analysis not only matches some dynamical models of concept formation in Psychology or Arti cial Intelligence, but it additionally suggests a simpler approach to the \justi cation" and \truth-tracking" concepts as the philosophers' missing ingredient for knowledge; it is certainly more akin to Barwise and Perry's idea ([BARW83]) that knowledge is \successful belief" than to standard epistemological traditions16.

4 Summary With the vision of possible worlds and the logic analysis of beliefs that we suggest, agents are neither logically omniscient nor perfect reasoners. They are not logically omniscient because the introduction of tautologies in the tableaux is forbidden. They are not perfect reasoners because their beliefs depend on the level of logic analysis that they have reached; if an agent believes (p ) q) and p, it will not belief q until it analyses the proposition (p ) q) and closes the tableaux 16Although the idea of \successful belief" in [BARW83] is of a probabilistic nature.

containing p and :p. In a future work we will show how this analysis (and especially the introduction of instances of the axiom of the excluded middle) can suggest the agent what kind of experiences it can perform in the real world in order to increase even more its beliefs. The result of these experiences will also be included in all open tableaux, so we will combine the pure logic analysis (via analytic tableaux) with a physical dimension (with data derived from reality).

References

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