A non-probabilistic Approach to Inductive Prediction

Transcription

1 A non-probabilistic Approach to Inductive Prediction Lieven Haesaert Centre for Logic and Philosophy of Science Universiteit Gent, Belgium February 20, 2004 Abstract The underlying idea behind the adaptive logics of inductive generalization is that most inductive reasoning can be explicated by simple qualitative means. Therefore, those classical models are selected that are as uniform as possible with respect to a certain set of (empirical) data. This led to the question if the same idea of uniformity can be applied if no generalizations are derivable. It is clear that in this case one may be still interested to make some direct inductive predictions. The main problem with this kind of prediction is that we lack a decision theory for it. In the present paper we make some proposals to deal with this problem. Our purpose here is to get more control over the difficult aspects of inductive prediction. In order to do so, we will not proceed in a probabilistic context, but we will apply the idea of minimizing the abnormalities in uniform models, an idea that derives from the adaptive logic programm. 1 Aim of this paper In our [1], we have presented some adaptive logics for induction based on Classical Logic (henceforth: CL). The underlying idea of these adaptive logics of induction is that most inductive reasoning does not proceed in terms of probabilities, and cannot be explicated in terms of probabilities, but can be explicated by rather simple qualitative means. In that paper we presented for example the adaptive logic for inductive generalization IL +m : from a set of data and (possibly falsified) background knowledge, inductive generalizations are derived 1. In the same paper we also Research for this paper was supported by subventions from Ghent University and indirectly by the the Flemish Minister responsible for Science and Technology (contract BIL01/80). The author is greatly indebted to Diderik Batens for his very enlightening comments and discussions, his ideas and his help with the calculations made in this paper. 1 This is putted between quotes, because derived is meant here in a relative way. If a generalization is derived, this means it is derived with respect to a certain set of data. We will not use these quotes later 1

2 presented IL m, the logic of induction in which background knowledge is disregarded. Than, the underlying idea is the following: given a set of data, choose those models of the data that are maximally uniform the idea of course derives from Carnap s [7]. Now, to be more precise, the IL m -consequences of a set of data Γ are those generalizations that are verified by all minimally abnormal models of Γ, where Ab(M), the abnormal part of a CL-model M, is defined as the formulas of the form A A (in which A abbreviates the existential closure of A) that are verified by the model and a CL-model M of Γ is a minimally abnormal model of Γ if there is no CL-model M of Γ such that Ab(M ) Ab(M). Here, the preferred models of Γ are those CL-models that are as uniform as possible. Remark that the abnormal part of a model consists of all negations of the general axiom scheme A A. If this axiom scheme is added to CL, the uniform classical models are obtained. This axiom scheme is accepted for each instance case until and unless it is proven that it doesn t hold. The adaptive logic(s) of inductive generalization seem rather adequate and lead to a number of interesting insights on inductive generalization. This led to the question whether a similar idea, viz. uniformity, can be applied in the case of inductive predictions. Of course, inductive predictions can be derived from inductive generalizations, but what we are interested in here are predictions that seem natural even in cases where no inductive generalizations can be derived. So, for example, you better predict that the first person you will meet in the streets is not going to be a criminal and will kill you, even though you know that there are criminals around in the world and some of them are killers. To put the problem more technically, suppose the following, very simple, set of data is given: Γ = {P a Qa, P b Qb, P c Qc, P d Qd, P e} and that one is interested in predicting Qe. Of course, the generalization ( x)(p x Qx) is not IL m -derivable because of P d Qd; though it is reasonable to predict Qe. The main problem here is that we lack a positive test 2 for such an inference relation we do not have a real criterion to decide if the prediction holds. This lack of positive test is typical for consequence relations that are non-monotonic and it is clear that this is the case for these direct inductive predictions. If new data is gained and added to Γ, there may be good reasons to predict Qe for instance if 10 other objects are found to be P Q. At this point, adaptive logics appear on the boards, because they dispose of a dynamical proof format that is proven to be very suitable in explicating non-monotonic human reasoning processes. However, in this paper we will not focus on adaptive logics and the dynamical character of inductive prediction. 3 Instead, it is our present purpose to get more grip on some difficult aspects of direct inductive prediction, viz. how the inductive predictions may be justified (in a relative way). Let us here at once explain an important peculiarity of the direct inductive prediction we will consider in this paper. If the data justify the prediction that an object which is known to be P is Q, then it is right to predict so for any object known to be P. So, if the data justify the prediction that an object which is known to be a cube is red, then it is right to predict so for any object known to be a cube. This feature may be compared with the situation in a probabilistic 2 A standard reference of such matters is [6] 3 See [8] and [4] for philosophical introductions to the adaptive logic programme, and [3] and [5] for more technical ones. A constantly updated overview of the results can also be found at 2

3 context: if the probability of tails is higher than.5, then it is justified to predict tails for any number of tosses in a row. A very different situation is obtained in the case of statistical predictions, such as the answer to the question How many of the next observed cubes will be red? This is a wholly different matter that is not considered in the present paper. Now, in order to get more control over the aspects of direct inductive predictions, we will not start from the traditional probabilistic context. Instead we will approach the matter in a rather qualitative way, viz. by applying the idea of uniformity. Then, a first important question that presents itself is the following: which form uniformity should take in order to deal with direct inductive prediction? 2 Maximal Uniformity As a first approach, inductive prediction presupposes that the world is as uniform as possible, and this means, in the unary case, that couples of objects that differ in some respect are minimized. The models we need here are those CL-models that are maximally uniform in view of a certain premise set. Now, let a CL-model be a couple D, v in which D is a set and v is a valuation (or interpretation) of primitive linguistic entities with respect to D as usual and let Γ be the set of data. Than, an abnormality of a CL-model M of Γ may be defined as a couple o 1, o 2 (o 1, o 2 D) such that, for some formula A(x), M = A(o 1 ) and M = A(o 2 ). So an abnormality occurs each time two objects differ in at least one aspect. The general idea now will be that the minimally abnormal (or maximally uniform) models of Γ are those CL-models of Γ that display the least number of abnormalities, which means the models are selected through a numerical comparison. Here we already remark that (and this is well known from approaches in terms of logical or personalistic probabilities), in measuring the abnormal part of a model, one has to take all formulas A into account, not just the primitive predicates and not just the strongest predicates definable in the language. On the other hand, it will be clear that there is no reason to count all abnormalities that belong to the same equivalence class. We refer to Section 5 for more details. Another remark concerns the defined form of abnormalities. In the logic of inductive generalization, the abnormalities are formulas of a certain form that are verified by a model and the preferred models of Γ are selected in view of subset-relations between their abnormal parts and those of other models of Γ. On the other hand, in the logic of inductive prediction, the abnormalities are defined as couples of elements of the domain (in general, couples of n-tuples of such elements) of which one member verifies and the other falsifies a formula A. The preferred models of Γ are selected in view of numerical relations between their abnormal parts and those of other models of Γ. A question that arises now is: how should we draw this numerical comparison? As we will illustrate, the degree of abnormality provoked by the predictions becomes higher if also tests are taken into account. But let us first consider the situation in which no tests are involved. 3

4 3 A mere Collection of Observations As a first approximation, we will start from the presupposition that the data are merely collected by the observer. By merely we mean that the observer did not perform some (scientific) action, such as a kind of test, but that he took the things as they came. The following sketch may be more clarifying. Let us consider a simplified universe, consisting of a set of objects, not all of which need to have names, and two unary predicates, P and Q. Suppose a large box is hanging from the ceiling, containing objects that either are spheres or cubes (say, P and P ) and are either red or black (say, Q or Q). So, in the box there will be red spheres (P Q), black spheres (P Q), red cubes ( P Q) and black cubes ( P Q). We can observe or estimate the number of objects in the box, but we cannot observe their shape or color. Two other boxes are hanging from the ceiling, a color box and a shape box. We can see the color of the objects in the color box (say because it has a little hole in each slot that can contain an object) and we can see the shape of the objects in the shape box (for example a shadow box). Now suppose a demon is moving between the different boxes, but we cannot see his moves. He now and then picks some objects from the large box and puts them into the color box or into the shape box, or throws them on the floor. The latter means we see both the color and the shape of the objects. The demon can also throw objects from the color box or from the shape box onto the floor. We ourselves can not undertake some action, but we can only observe colors or shapes and write it down. If the situation is analogous to the just described demon-episode, our observations may be summarized as follows: P n 1 n 2 n 3 P n 4 n 5 n 6?P n 7 n 8 n 9 The n i are numbers here, but we shall also use them to refer to the cells and to the cases (P α Qα,...) which they represent. So, n 1 different objects are known to be both P and Q, n 2 objects are known to be P and Q, n 3 objects are known to be P but are neither known to be Q nor known to be Q, etc..... So, to talk in terms of the demon-analogy, there are n 9 (completely undetermined) objects in the large box, n 3 spherical, n 6 cubic objects in the shape box, etc. In this scheme we can determine all abnormalities (of this simplified universe). For example, the objects in the n 1 -cell behave abnormal with respect to all objects that belong to the cells n 2, n 4, n 5, n 6 or n 8. Moreover, this scheme enables us to calculate the number of all abnormalities that are known until yet. We let it to the reader to check this. The cells n 1, n 2, n 4 and n 5 are central and represent the objects thrown on the floor, thus those that are fully known. The outer border cells contain the objects on which we want to make predictions. These objects are expressed by question marks. The purpose now is to move the objects belonging to cells with question marks into the four central inner cells in such a way that the number of abnormalities provoked by the model is as low as possible. 4

5 4 Minimizing the abnormalities In the simplified universe we just presented, we can make the following inductive predictions: we can predict the color of an object known to be a sphere, or of an object known to be a cube (both of them are in the shape box), or of an object of which no property is determined as yet (because it is still in the large box). Similarly we can predict the shape of a red object, of a black object, or of an object of which the color is undetermined. Finally we can predict both the shape and color of an undetermined object. Sometimes we can verify our prediction, if the demon is kind enough to move the object to the right box or to the floor. Remark that all cells that contain objects that are not fully determined, will be emptied if it represents a CL-model all elements of the domain are determined with respect to both P and Q. Making a prediction means that an object of the outer border cells moves to either n 1, n 2, n 4 or n 5. It is worth to remark that they are actually putted in there in a conditional way: if new information is gained, there may be reasons to make another prediction, or it may be that a certain prediction is later falsified and should be withdrawn. However, the most interesting aspect of making predictions on partly or fully unknown objects is that they cause new abnormalities, by which the badness of its models, viz. the degree of uniformity, is increased. At the end, those predictions will be retained that together cause the least number of abnormalities. So, the choice of a prediction will depend on the set of all possible predictions, including the prediction itself. To clarify the whole situation, let us now consider a specific knowledge situation as summarized in the following table P P ?P and suppose that we want to predict the color of the object known to be spherical (the single object in the P?Q-cell). Some will predict that this object is Q because among the objects known to be P, 100 are known to be Q and only 3 are known to be Q. However, this reasoning is not really coherent. Indeed, each of the objects in the?p Q-cell, is either P or P, and if we follow the above reasoning, they are all going into the P Q-cell whereas the object from the?p Q should go in the P Q-cell. But then, there are 1003 objects with property P Q, which means that the object in the P?Q-cell should be moved to the P Q-cell. So, depending on the order in which this reasoning is applied, it leads to different results. One might hope to rescue the situation by first applying the very same reasoning to the objects in the?p?q-cell in order to locate them in the P Qcell. However, if there is, say, a 5 in the?p?q-cell, the inconsistency is not removed. Let us now have a look at the outcome obtained by the idea of minimizing the abnormalities in moving the outer border cells into the inner cells. So, we took all possibilities of combining the outer border cells with the four central cells into account and we calculated for each combination the abnormal part, 5

6 viz. the number of abnormalities, verified by the model. These calculations are complicated 64 combinations can be made but a computer programm may help here. We calculated the minimally abnormal model (which amounts to the maximally uniform model) and here is the outcome of the calculations: P P ?P P P 1 12 The table on the left side represents the knowledge situation, namely the data gathered by the observer. The table on the right side represents the situation after predicting all objects and is obtained by the idea of minimizing the abnormalities. In this case, one can see that the only object in the P?Q-cell moves to the P Q-cell. The 1000 objects from the?p Q-cell move to the same cell. Incidentally, in this case, where the objects of the different cells go, is independent of the value in the?p?q-cell (the fully undetermined objects), but it can be shown that this is not always so in other cases. 4 So, if we follow the above results to predict the partly or fully unknown objects, we obtain the least number of abnormalities. All other possibilities of moving the outer border cells into the four central cells, for example by moving the object in the P?Q-cell to the P Q-cell and leaving the other moves as they are, lead to models that produce a higher number of abnormalities, thus less uniformity. 5 Partial Analogies Should not be Neglected If we take a look on the above schemes, we may say that the underlying idea is roughly the following. As there are more P s than non-p s and more non-q s then Q s, the objects from the P?Q-cells and those from the?p Q-cells are concentrated in the P Q-cell. The objects from the P?Q-cell go to the P Q-cell, and those from the?p Q-cell to the P Q-cell this is as expected, as it results in the highest possible concentration. Finally, the objects from the?p?q-cell go to the cell that already contains the highest concentration, viz. the P Q-cell. Some will still find it somewhat strange that the only object from the P?Qcell moves to the P Q-cell, whereas it is known that most P are Q. But like already indicated, an approach in terms of the non-standard quantifier most does not work. One of the reasons is that in such an approach the effects of partial analogy do not weigh enough. Let us carefully explain what is meant. As we remarked before, in measuring the abnormal part of the model one has to take all formulas A into account and not just the primitive predicates and not just the strongest predicates definable in the language. It is right that P a Qa and (P b Qb) causes an abnormality, but if P b Qb should be known, then this abnormality should weigh heavier than in case P b Qb or P b Qb were known. We let it to the reader to check this. 4 Some examples are already known in which the result is dependent objects in the?p?qcell. 6

7 On the other hand, there is no reason to count all abnormalities. Thus, if o 1, o 2 is abnormal with respect to A, then it is abnormal with respect to A, etc., and moreover o 2, o 1 is abnormal with respect to A, with respect to A, etc. So we took the following provisional decision: we introduce equivalence classes of formulas, pick a preferred simplest form A for each of them, and count the abnormalities with respect to A provided A is not of the form B. We admit, however, that there is need for further study in these quarters. At this point, it is interesting to see what happens if for example only the strongest definable predicates of the language are taken into account. Here the partial analogies are neglected in our calculation of the abnormal part of models. The result is then quite different, viz. as follows: P P ?P P P So, as may be seen, here the partly or fully unknown objects are simply concentrated in such a way that abnormalities with respect to the strongest definable predicates are minimized, and the effects of partial analogy (here with respect to P and with respect to Q) are not taken into account. The result is quite different from the former result and some will even contend that it is a better result, because now the only object of the P?Q-cell moves to the P Q-cell. However, the following example will illustrate more clearly why partial analogies should not be neglected. Suppose we dispose of the following knowledge situation: P P 1 1 1?P If just the strongest definable predicates are taken into account, the only minimally abnormal model is: P 14 1 P 1 3 As the objects in the P?Q-cell and in the?p Q-cell are concentrated in the P Q-cell this gives the maximum concentration for them in view of the two objects that were already present in that cell, the objects in the P?Q-cell and in the?p Q-cell are concentrated in the P Q-cell. The objects from the?p?q-cell go to the P Q-cell (the one that has to comprise the highest number of objects anyway). However, this result is not adequate. Indeed, the amount of the P Qobjects, the P Q-objects and the P Q-objects is in this specific knowledge 7

8 situation similar, viz. 1. So, there seems no good reason to predict the Q?P - objects to be P. Moreover, as there are more P s than P s, it seems better predicted to be P. The same reasoning applies for the object in the P?Q-cell: as there are more Q s than not Q s, it seems better predicted to be Q. If we also take into account abnormalities with respect to P and with respect to Q, we obtain the following result: P 14 2 P 2 1 Now, the objects in the P?Q-cell and that in the?p Q-cell move to different cells, which is more adequate than to concentrate them together in the P Q-cell. It means that if the data are collected in the way described in the demon-episode, we cannot make an appeal on our normal meaning (or intuitions) about the most. This all means that reasoning in terms of the most is not really coherent. But like we will show now, the whole situation is getting totally different if we also take into account the effects of tests. 6 Tests Let us reconsider the first example: P P ?P P P On the approach discussed so far, the only minimally abnormal model is the table on the right side and so one should predict that the object that was known to be P?Q is P Q. Nevertheless, some will still call the above results into question and they will argue that the fact that 103 P s have been tested on Q-hood and that 100 of them turned out to be Q is an argument to predict that the object that was known to be P?Q is P Q. However, this argument is confused. In view of the way in which we represented our knowledge before the boxes and our increase of knowledge by the moves of the demon we have not tested any P s on Q-hood. All knowledge considered so far was the result of accidental matters, viz. the moves of the demon. Although, it is indeed correct that we are able to test objects of a kind on some properties the previous representation of our knowledge situation was a simplification and that the outcomes of such tests play a role in our predictions. It is easy to refashion the knowledge representation in terms of the demonepisode in such a way that tests are taken into account as well. Now we should be given the possibility to pick objects from the large box and move them to either the color box or the shape box or throw them onto the floor. In that way we can test the fully undetermined objects on their color or on their shape or 8

9 even simultaneously on both. In the same way, we may pick objects from the color box and from the shape box and move them onto the floor. Tests actually have two effects on our predictions. On the one hand, by doing tests we force some objects from cells that are described by expressions containing question marks to cells described by expressions that contain less question marks. Thus by testing the P -hood objects in the?p Q-cell, the number in that cell will decrease and the number in the P Q-cell and/or in the P Q-cell will increase. There is a second effect of tests. Suppose that the figures in the P Q-cell and in the P Q-cell were obtained as follows: two objects in each cell are known to be there by accident, whereas the others are there as the result from tests. In other words, we have tested 99 objects from the P?Q-cell on Q-hood and the result was, for instance, that 98 turned out to be Q and 1 turned out to be Q. If this is the situation, we have a further reason to predict that an object in the P?Q-cell is Q: if we would test it, we would expect it to be Q. So, roughly we may say that there is a difference in abnormalities that are originating from tests and abnormalities originating from mere observation. These originating from tests weigh heavier. The main problem with these new abnormalities is to define them with respect to the model. Let us offer a tiresome but simple solution in the next section. 7 Defining the new abnormalities We extend the language of our simplified universe until now only consisting of two predicates P and Q in such a way that we may discriminate between the data gained from tests and the data that are given by the demon. For any predicate P, we introduce the predicates P and P to denote the objects that were fully undetermined and turned out to be P, respectively P as the result of a test for P -hood. For any two predicates P and Q we introduce the predicates P ; Q and P ; Q. The former is used as an expression for the objects that were known to be P, were tested on Q-hood and turned out to be Q, whereas the latter is seen as an expression for the objects that were fully undetermined, were tested on P -hood and turned out to be P, and next were tested on Q-hood and turned out to be Q. By using the same reasoning, there will as well expressions such as P ; Q, P ; Q, Q; P, Q; P etc. Finally we also introduce P Q to denote the objects that were tested simultaneously on P -hood and Q-hood and turned out to be both P and Q. The same holds for P Q, etc. Now we extend the assignment function v to the effect way that v(p ) v(p ) and v( P ) D v(p ). It follows that v(p ; Q) (v(p ) v(q)), v(q; P ) (v(q) v(p )), v(q; P ) (v(q) v(p )), v(p Q) (v(q) v(p )), etc. At the end, we require that v(p ; Q) v(q; P ) = (that is, we will not test P -objects on Q-hood, if we already knew the object was Q and turned out to be P in a test on P -hood) v(p ; Q) v(p ; Q) =, v(p ; Q) v(p Q) =, etc. It is tiresome to express all the independencies, but it is all extremely simple and systematic. As will be shown, the introduction of these new predicates enables us to take into account the effects of tests by making inductive predictions. As desired, we may now define new abnormalities that will have their influence by the 9

10 calculation of the minimal abnormal model. Any object now that is known to be P as well as Q is not only abnormal with respect to the P, Q, P Q, etc., but also with respect to the objects that are P, Q, P ; Q, etc. Remark here that the way in which we introduced the tested predicates, provides that there can be no abnormalities such as (P a Qa) (P b; Qb) and (P a Qa) ( Qb; P b) or (P a Qa) (P b; Qb) and (P a Qa) (P b; Qb) at the same time. But the nice effect of the introduction of new predicates is that we now can integrate the effects tests have on inductive prediction. Indeed, new abnormalities will evolve now and as will be shown in the next section, taking into account these abnormalities leads to interesting results. 8 Taking into account the new abnormalities Let us return to the first situation again: P P ?P The point now to be considered is the weight of the new abnormalities that arise by the introduction of tests. Suppose the numbers in the P Q-cell and in the P Q-cell were obtained as follows: 99 of them are the result of testing P -objects on Q-hood, 98 turned out to be Q en 1 to be Q, whereas two objects in each cell are known there by accident. The test outcome may be summarized as follows: P 98 1 To take into account the effects of this test, we again rely on Carnap s [7] to set the weight of an abnormality equal to the number of objects tested in the relevant test. So, analogous as in the case of partial analogies, it is right that P a Qa and P b Qb causes an abnormality, but if P b; Qb is known, then this weighs heavier as in the case only P b Qb is known. It is obvious now that the outcome of the present test changes the weight of abnormality caused by predicting that the object in the P?Q-slot is Q. As only one new abnormality is created by this prediction, the prediction increases the badness of its models by 99 1 = 99. If we should predict that the P?Qobject is Q, 98 new abnormalities are created and it increases the badness of its models by = We have good reasons to expect now that the minimal abnormal model of this set of data will verify that it is right to predict that the object in the P?Q-cell belongs to Q-hood. However, the calculation of the minimally abnormal model of a set of data comprising tested predicates is even more complex than in the case no tests are available: we do not only have to take into account the partial analogies now, but we have to take the tested predicates into account as well. Here again a computer programm may help. So we calculated the minimally abnormal model and we obtained the following result: 10

11 P P 11 2 As expected, the object only known to be P moves to the P Q-cell. So taking into account the outcome of the test of 99 P s with respect to Q-hood has dramatic consequences. It is not only sufficient to predict that the object in the P?Q-slot is Q, but also to predict that objects?p?q-slot are both P and Q, and that objects in the P?Q-slot have property Q. One might suspect that the dramatic change is largely due to the fact that there are so many (viz ) objects in the?p?q-cell. This, however, is not the case, even if there is no object in that cell, but the other figures are the same (including those of the test), then the outcome is similar, viz. P P ?P P P 11 2 Let us remark that in some cases the number of P s tested on Q-hood does not have a sufficiently large effect. One then should first test more P s on Q- hood, as expected. As the example reveals, this may require that one first tests undetermined objects on P -hood, and next tests those found to be P on Q-hood. This is exactly as it should be. If you want to reliably predict the behavior of blackbirds, first locate blackbirds and next observe their behavior. To finish it is worth noticing that the introduction of tests does not free our knowledge from bias. Thus the demon may have a bias for placing red objects in the shape box, and this will have an effect on the representative character of our color tests on cubes and spheres. This, however, is all right as long as our color tests on cubes have only effects on our color predictions about objects we know to be cubes these have been moved to the shape box by the same demon. Moreover, some further demons may be present in the different boxes, and influence the properties of objects we pick from them. Thus there may be a demon in the color box that moves the objects in such a way that, whenever we pick a red object, it is more likely to be a cube. Again, if there is such a demon, its presence will likewise influence our tests and the correctness of our predictions. 9 In Conclusion Further research will focus on making the criteria determining prediction as simple as possible. This is why we are trying for example to reduce these criteria to the abnormalities caused by prediction. As was clarified in this paper, predictions increase the amount of abnormalities verified by a model. It will depend on the degree of abnormality a prediction provokes, if it will be accepted or not. The main problem is that one single prediction is dependent on the set of all possible predictions. Another interesting aspect that was not considered in this paper are the dynamics by which inductive predictions are characterized. It should be clear 11

12 to the reader now that the predictions change if the set of data changes or if new insights in the data are obtained. That is why we are trying to construct a suitable dynamic proof theory for inductive prediction, for which we will proceed in the context of adaptive logics. Finally, it may be argued that the present approach is not probabilistic, but still rather complex. However, the obtained results are adequate and shed some more light on the very complicated matter of inductive prediction. 5 References [1] Diderik Batens and Lieven Haesaert, (2001) On Classical adaptive logis of induction, in: Logique et Analyse, vol , pp (appeared 2003) [2] Diderik Batens, (1998), Inconsistency-adaptive logics, in: Or lowska E., ed., Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa, Springer Verlag, Heidelberg, New York, pp [3] Diderik Batens, (2000), A survey of inconsistency-adaptive logics, in: Batens D., Mortensen C., Priest G., Van Bendegem J.P., eds., Frontiers of Paraconsistent Logic, Kings College Publications, London, pp [4] Diderik Batens, Extending the Realm of Logic. The adaptive logic program, to appear. [5] Diderik Batens, The need for adaptive logics in epistemology, to appear [6] George S. Boolos and Richard J. Jeffrey, (1989), Computability and Logic, third edition, Cambridge University Press [7] Rudolf Carnap, (1952), The Continuum of Inductive Methods, University of Chicago Press, Chicago [8] Diderik Batens, (1997), Inconsistencies and beyond. A logical-philosophical discussion, in: Revue Internationale de Philosophie, vol. 51, pp Unpublished papers in the reference section are available at the internet address 12