Diderik Batens and Joke Meheus. The adaptive logic of compatibility. Studia Logica, 66:327--348, 2000.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....ULL determines the standard of reasoning and specific deviations from this standard are minimized. They adapt themselves to specific violations of presuppositions of CL such as inconsistencies. The system I present in this paper is an ampliative logic like the more recent systems presented in [4], 16] 5] and [17] In ampliative logics, the standard of reasoning is determined by the LLL and specific extensions of this standard are maximized. The consequence set one obtains with an ampliative logic is thus richer than the one obtained with CL. It will easily be observed from the next ....

Diderik Batens and Joke Meheus. The adaptive logic of compatibility. Studia Logica, 66:327--348, 2000.

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Diderik Batens and Joke Meheus. The adaptive logic of compatibility. Studia Logica, 66:327--348, 2000.

.... in terms of adaptive logics (see, for instance, 4] and [11] A more recent development is to use the framework of adaptive logics to study ampliative forms of reasoning (see [15] At this moment, several ampliative adaptive logics are available examples include logics for compatibility ([9] and [17] enumerative induction ( 7] and [8] metaphorical reasoning ( 13] diagnostic reasoning ( 10] and causal reasoning ( 14] The logic T has several nice properties. One is that it provides us with a proof theory for default reasoning. The proof theory is dynamical, but warrants ....

D. Batens and J. Meheus. The adaptive logic of compatibility. Studia Logica, 66:327--348, 2000.

....All Right with respect to # # i# Ab (M) Ab (# # ) and define the consequence relation with respect to the selected models: Q ms X i# all Q models that are Simply All Right with respect to # # verify X. The logic Q is very similar to the logic of compatibility presented in [5]. Actually, this is not surprising. As a sentence A is compatible with a set of premises # i# # A, a question Q is informative with respect to a set # i# the negation of each member of dQ is compatible with #. The logic of compatibility presented in [5] enables one to derive #A from # # ....

....to the logic of compatibility presented in [5] Actually, this is not surprising. As a sentence A is compatible with a set of premises # i# # A, a question Q is informative with respect to a set # i# the negation of each member of dQ is compatible with #. The logic of compatibility presented in [5] enables one to derive #A from # # whenever A is compatible with #. This property is shared by Q . See [8] for a semantic characterization of Q . The semantics for Q may be characterized in di#erent ways, each of which is analogous to one of the characterizations of the Q ....

Diderik Batens and Joke Meheus. The adaptive logic of compatibility. Studia Logica, 66:327--348, 2000.

....upper limit logic determine a unique set of abnormalities. In this case, the set of abnormalities is a function of the lower limit logic and the upper limit logic. This caused some confusion which was only cleared up when the Ghent logic group started studying ampliative logics see for example [7]. Then it was realized that a lower limit logic may be combined with many di#erent sets of abnormalities to obtain the same upper limit logic. UCL is one of the many extensions of CL that fulfil all the traditional requirements. It is monotonic, structural, transitive, etc. The only trouble ....

Diderik Batens and Joke Meheus. The adaptive logic of compatibility. Studia Logica, 66:327--348, 2000.

....and let# be either #A##A # W #A W; #ML #A . In both cases, the upper limit is Triv the system in which A is logically equivalent to #A as well as to #A. However, the resulting adaptive logics are very di#erent see [13] for one involving the first set of abnormalities, and [14] for one involving the second set of abnormalities. A very important matter has to be brought up at this point. For all that was said before, the reader might think that there is a well defined set of formulas that need to behave abnormally in view of the premises. This is not the case. The ....

....is a minimal Dab consequence of a premise set. If this is the case, the Reliability and Minimal Abnormality strategies lead to the same result and coincide with what is called the Simple strategy: a formula behaves abnormally just in case the abnormality is derivable from the premise set see [14] , 24] and [ for examples. Several other strategies have been studied. Most of them were the result of characterizing an existing consequence relation by an adaptive logic. Examples may be found in [9] 12] 22] and [30] 3 Prioritized Adaptive Logics Let us start with combined adaptive ....

Diderik Batens and Joke Meheus. The adaptive logic of compatibility. Studia Logica, 66:327--348, 2000.

....proof may (in general) lead to the conclusion that Q is evoked by #. Another example concerns handling inconsistency. Consider the case in which a scientific (empirical or mathematical) theory T was meant to be consistent and was formulated with CL as its underlying logic, but turned out to See [18] for the adaptive logic of compatibility in the framework of CL. See [20] for adaptive logics that explicate several forms of reasoning that underly the search for explanations. be inconsistent. As we know from the literature, scientists do not just throw away such a theory. They reason ....

....such that # is a singleton. In such cases, the Reliability and Minimal Abnormality strategies lead to the same result and coincide with what is called the Simple strategy: a formula behaves abnormally just in case the abnormality is derivable from the premise set. Examples may be found in [37] and [18]. Several other strategies have been studied, but seem to have a less general import. Most of them were the result of characterizing an existing consequence relation by an adaptive logic. Examples may be found in [14] 17] 29] and [49] A di#erent way to characterize most flat adaptive logics ....

Diderik Batens and Joke Meheus. The adaptive logic of compatibility. Studia Logica, 66:327--348, 2000.

....(they can handle logically abnormal theories for instance, inconsistent ones) In [41] it was shown, however, that exactly the same kind of formal techniques can be used to explicate ampliative forms of reasoning. Meanwhile, formal) results are available on adaptive logics for compatibility ([17]) pragmatic truth ( 48] the closed world assumption and negation as failure ( 69] diagnostic reasoning ( 70] and [21] inductive generalizations ( 14] and [16] abduction and inference to the best explanation ( 54] and [53] and the analysis of metaphors ( 32] and [33] An informal ....

Diderik Batens and Joke Meheus. The adaptive logic of compatibility. Studia Logica, 66:327--348, 2000.

....This distinction is mainly introduced for pragmatic reasons. Inconsistency adaptive logics (see [4] and many other places) are corrective: a possibly inconsistent set of premises is interpreted as consistently as possible. Examples of ampliative adaptive logics are the logic of compatibility from [11] or the logics of induction from [9] if one disregards the background generalizations) Remark that C 1 . Cm #ULL #, where is defined by # # A. This formulation presupposes the presence of a disjunction that behaves in a standard way. It is possible to get around this, see, ....

Diderik Batens and Joke Meheus. The adaptive logic of compatibility. Studia Logica, 66:327--348, 2000.

....the Ghent group are intended for contexts in which CL is the standard of deduction, an adaptive logic is classified as ampliative if it has CL as its lower limit logic. Examples of ampliative adaptive logics are logics for inductive generalization (see, for instance, 6] for compatibility (see [7]) and for abduction (see, for instance, 14] and [12] It is typical of these logics that they lead to a consequence set that is richer than the one obtained by CL (it also contains, for instance, all inductive generalizations that can sensibly be drawn from the premises) If the lower limit ....

....necessary in view of the premises . To define the selection criterion, I first define the abnormal part of a Q model: ##A M verifies ##A . and the set of abnormalities that are unavoidable in view of # # : The logic Q is very similar to the logic for compatibility presented in [7]. Actually, this is not surprising. As a sentence A is compatible with a set of premises # i# # A, a question Q is informative with respect to a set # i# the negation of each member of dQ is compatible with #. The logic for compatibility presented in [7] enables one to derive #A from # # ....

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Diderik Batens and Joke Meheus. The adaptive logic of compatibility. Studia Logica, 66:327--348, 2000.

....reasoning. Generalizations that are inductively derived from the set of premises, #, should be compatible with #. A further requirement on inductively derived statements is that they should be jointly compatible with #. The latter requirement is the harder one. The logic of compatibility see [BM00] provides us with the set of all statements that are compatible with #. The problem of induction is, in its simplest guise, to narrow down this set in such a way that the second requirement is fulfilled. And yet, as I now shall explain, this problem is easy to solve. Consider an (extremely ....

....considered the dynamic proof theory of LI. This proof theory is extremely important, as it enables us to explicate actual inductive reasoning humans reach conclusions by finite sequences of steps. A logical 20 This definition presupposes that nothing is compatible with an inconsistent set see [BM00], also for an alternative. 21 Cn L (#) A # #L A as usually. 13 semantics serves di#erent purposes. Among other things, it provides insights in the conceptual machinery. Such insights increase our understanding of a logic, even if they are not directly relevant for the computational ....

Diderik Batens and Joke Meheus. The adaptive logic of compatibility. Studia Logica, 66:327--348, 2000.

....# # . Statements derivable from the position of some participant may be said to be (at least) implicitly a#rmed during the discussion. The consistent core of # # comprises those statements that are at least implicitly a#rmed and are moreover compatible with the position of all participants see [6] for two logics of compatibility. The consistent core may be seen as the statements that all participants agree about in the discussion. Agreement is here meant in a weaker (and more useful) sense than the one expressed by # # # S5 #A. Some people may complain that, if A belongs to the ....

....all participants. 7 A more thorough analysis may reveal the distinction between agreement and irrelevance, but if the statement is indeed irrelevant, it would not have much import for the outcome of the discussion anyway. 8 I mean a predicative model that allows for #a = b # ##a = b. See [6] or [13] for details. 9 I shall need several kinds of abnormal parts of models and sets of premises. They will be distinguished by superscripts and subscripts the choice of which is obvious from the context. 6 all participants. Thus # # #AJ #A i# A belongs to the consistent core of # # . If # ....

Diderik Batens and Joke Meheus. The adaptive logic of compatibility. Studia Logica, 66:327--348, 2000.

....A(#) Another di#erence is that, in LA, abduced hypotheses are not rejected when they are falsified. As we shall see, both properties are needed for a reconstruction of Kuipers notion of empirical progress. 5 4 At the moment, formal) results are available on adaptive logics for compatibility ([7]) for pragmatic truth ( 15] for the closed world assumption and negation as failure ( 20] for diagnostic reasoning ( 21] and [9] for induction ( 5] for abduction ( 8] 16] and [18] for question generation ( 17] and for the analysis of metaphors ( 10] and [11] An informal ....

....that A is necessary (#A) If A is compatible with #, then A is true in some model of #, but not necessarily in all of them. In line with all this, it seems sensible to consider the members of # as necessarily true, and the sentences that are compatible with # as possibly true. This idea is used in [7] to design an adaptive logic for compatibility that is called COM. COM is based on S5, and is defined with respect to sets of premises of the form # # = #A A # # . As is shown in [7] COM has the interesting property that # # #COM #A if and only if # ## CL A, and hence, if and only ....

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Diderik Batens and Joke Meheus. The adaptive logic of compatibility. Studia Logica, 66:327--348, 2000.

....ones) An important new development concerns the design of ampliative adaptive logics (see [19] and [21] for an informal introduction) These logics are meant to handle various forms of ampliative reasoning. 4 At the moment, formal) results are available on adaptive logics for compatibility ([7]) pragmatic truth ( 26] the closed world assumption and negation as failure ( 39] diagnostic reasoning ( 40] and 2 It is traditionally accepted that inconsistent theories are not and cannot be used to generate meaningful explanations. Case studies indicate, however, that scientists ....

....problem can easily be solved, however, by moving to a modal approach. That A is true in some CL model of # can naturally be expressed as A is possible (#A) and that A is true in all of them as A is impossible (#A) or, in other words, A is necessary (#A) This modal approach is used in [7] to design an adaptive logic for compatibility that is called COM. COM is based on S5, and is defined with respect to sets of the form # # = #A A # # . As is shown in [7] COM has the interesting property that # # #COM #A i# # ## CL A, and hence, i# A is compatible with #. Note ....

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Diderik Batens and Joke Meheus. The adaptive logic of compatibility. Studia Logica, 66:327--348, 2000.

....ampliative adaptive logic, the standard of reasoning is determined by the lower limit logic; specific extensions of this standard (that are considered desirable within the application context at issue) are maximized. 14 Examples of ampliative adaptive logics are the logic of compatibility (see [8]) logics of diagnosis (see [9] and logics of abduction (see [18] and [19] The semantics of an adaptive logic is obtained in the following way. Let al.. be an adaptive logic, and let LL and UL be its lower limit logic and its upper limit logic. The AL models of a set of premises # are obtained ....

....there is no # # # # such that Dab(# # ) is a Dab consequence of #. I first define the abnormal part of a model: Definition 5 Ab(M) df A # F a M verifies #(#A # A) The set of formulas that are unreliable with respect to # is defined by: 15 This S5 semantics was first presented in [8]. 14 Definition 6 U(#) # # Dab(#) is a minimal Dab consequence of # . DR entailment is defined in terms of the reliable models of #: Definition 7 A S5 model M of # is reliable i# Ab(M) # U(#) Definition 8 Where A # W, # = DR A i# all reliable S5 models of # verify A. Note ....

Diderik Batens and Joke Meheus. The adaptive logic of compatibility. Studia Logica, 66:327--348, 2000.

....all dynamic reasoning patterns are non monotonic. In [6] for instance, Batens shows that the pure logic of relevant implication can be characterized by a dynamic proof theory. 3 Readers familiar with adaptive logics will see that I rely on insights and proof techniques from [2] 3] 4] and [9]. The proofs of Lemma 2 and Theorem 14 rely on proof techniques first presented in [3] the proof of Theorem on a technique from [4] 3 a whole variety of non classical logics was designed. Most of these are obtained by simply dropping some CL presuppositions and by restricting the inference ....

....Priest s LP m is best viewed as an ampliative logic. For Priest, the standard of reasoning is the monotonic paraconsistent logic LP, but it is sensible to presuppose consistency unless and until proven otherwise see [28] Other examples of ampliative logics are logics of compatibility (see [9]) of metaphorical reasoning (see [16] of diagnosis (see [10] of abduction (see [26] and of inference to the best explanation (see [25] The semantics of an adaptive logic is obtained by selecting a subset of the models of the lower limit logic. The selection is determined by the adaptive ....

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Diderik Batens and Joke Meheus. The adaptive logic of compatibility. Studia Logica, 66:327--348, 2000.

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