In order to define a consequence relation, a stable notion of derivability is needed. Such a consequence relation can equivalently be expressed semantically in terms of a preferential semantics see Section Selection semantics. The selection semantics for minimal abnormality can be phrased as follows. We omit selection semantics for other strategies. A similar system to adaptive logics makes use of maximally consistent sets. In Makinson it was developed under the name default assumptions.
Conditional Entailment is another assumption-based approach Geffner and Pearl Just as in the adaptive logic approach, models are compared with respect to their abnormal parts. One important aspect that distinguishes conditional entailment from adaptive logics and default assumptions, is the fact that it implements the Specificity Principle. Models are then compared as follows:. This is best understood when put into action. Take the case with Tweety the penguin.
A consequence of expressing defeasible inferences via material implication in assumption-based approaches is that defeasible inferences are contrapositable. As a consequence, formalisms such as default logic have a more greedy style of applying default rules. We demonstrate this with conditional entailment. The reasoning follows a greedy policy in applying default rules: whenever a rule is applicable i. There is disagreement among scholars whether and when contraposition is a desirable property for defeasible inferences e.
In view of the fact that test subjects seem to perform very poorly in various paradigmatic reasoning tests e. In recent years this assessment has been criticized as the result of evaluating performances in tests against the standard of classical logic whereas other standards based on probabilistic considerations or on NMLs have been argued to be more appropriate. This resulted in the rise of a new probabilistic paradigm Oaksford and Chater , Pfeifer and Douven where probability theory provides a calculus for rational belief update. Although the program is sometimes phrased in decidedly anti-logicist terms, [ 7 ] logic is here usually understood as monotonic and deductive.
The relation to NML is less clear and it has been argued that there are close connections especially to probabilistic accounts of NML Over , Pfeifer and Kleiter Politzer and Bonnefon warn against the premature acceptance of the probabilistic paradigm in view of the rich variety of alternatives such as possibility measures, plausibility measures, etc. Another argument for the relevance of NML is advocated in Stenning and Van Lambalgen who distinguish between reasoning to and reasoning from an interpretation. When establishing a logical form agents choose—inter alia—a formal language, a semantics e.
Once a logical form is established, agents engage in lawlike rule-based inferences which are based on this form. It is argued that in the majority of cases in standard reasoning tasks, subjects use non-monontonic logical forms that are based on closed world assumptions. Non-monotonic logicians often state that their motivation stems from observing the defeasible structure of actual commonsense reasoning. Empirical studies have been explicitly cited as both inspiration for working on NMLs and as standards against which to evaluate NMLs. However, it has also been noted that logicians often rely too much on their own intuitions without critically assessing them against the background of empirical studies Pelletier and Elio Most studies support the descriptive adequacy of the rules of system P.
There are, however, some open or controversial issues. For instance, while some studies report results suggestive of the adequacy of weakened monotony principles such as Cautious Monotony Schurz , Neves et al. Specificity considerations play a role in the reasoning process of test subjects in Schurz , whereas according to Ford and Billington they do not. Benferhat et al. While the results were not fully conclusive they still suggest a preference for the former.
Their study makes use of the benchmark problems for defeasible reasoning proposed in Lifschitz It is, for instance, observed that the exceptional status of an object A with respect to some default is more likely to spread to other objects if they share properties with A that may play a role in explaining the exceptional status. For example, when confronted with a student club that violates the default that student clubs only allow for student members, subjects are more likely to ascribe this exceptional status also to another club if they learn that both clubs have been struggling to maintain minimum membership requirements.
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The question of the descriptive adequacy of NMLs to human reasoning is also related to questions concerning the nature and limits of cognitive modules in view of which agents are capable of logical reasoning. For instance, the question arises, whether such modules could be realized on a neurological level. There are three major issues connected with the development of logical frameworks that can adequately represent defeasible reasoning: i material adequacy; ii formal properties; and iii complexity.
A non-monotonic formalism is materially adequate to the extent to which it captures examples of defeasible reasoning and to the extent to which it has intuitive properties. The question of formal properties has to do with the degree to which the formalism gives rise to a consequence relation that satisfies desirable theoretic properties such as the above mentioned Reflexivity, Cut, and Cautious Monotony.
The third set of issues has to do with computational complexity of the most basic questions concerning the framework. There is a potential tension between i and ii : the desire to capture a broad range of intuitions can lead to ad hoc solutions that can sometimes undermine the desirable formal properties of the framework. In general, the development of NMLs and related formalisms has been driven, since its inception, by consideration i and has relied on a rich and well-chosen array of examples. Of course, there is some question as to whether any single framework can aspire to be universal in this respect.
More recently, researchers have started paying attention to consideration ii , looking at the extent to which NMLs have generated well-behaved relations of logical consequence. As Makinson points out, practitioners of the field have encountered mixed success.
In particular, one abstract property, Cautious Monotony, appears at the same time to be crucial and elusive for many of the frameworks to be found in the literature. This is a fact that is perhaps to be traced back, at least in part, to the above-mentioned tension between the requirement of material adequacy and the need to generate a well-behaved consequence relation. The complexity issue appears to be the most difficult among the ones that have been singled out. NMLs appear to be stubbornly intractable with respect to the corresponding problem for classical logic.
This is clear in the case of default logic, given the ubiquitous consistency checks. But besides consistency checks, there are other, often overlooked, sources of complexity that are purely combinatorial. Other forms of non-monotonic reasoning, besides default logic, are far from immune from these combinatorial roots of intractability. Although some important work has been done trying to make various non-monotonic formalisms more tractable, this is perhaps the problem on which progress has been slowest in coming.
Strasser ruhr-uni-bochum. Aldo Antonelli. Dealing with the dynamics of defeasible reasoning 2. Dealing with conflicts 2.
Non-monotonic formalisms 3. Non-monotonic logic and human reasoning 5.
Reasoning about Preference Dynamics
Dealing with the dynamics of defeasible reasoning Defeasible reasoning is dynamic in that it allows for a retraction of inferences. Free consequences.
Inevitable consequence. Dealing with conflicts A separate issue from the formal properties of a non-monotonic consequence relation, although one that is strictly intertwined with it, is the issue of how conflicts between potential defeasible conclusions are to be handled. For instance, our previous example featuring the bird Tweety is illustrated as follows: We use the following conventions: double arrows signify deductive or strict i. Take, for example, the situation in the following figure: On the left we see an inference graph with two conflicting arguments. This is illustrated in the following figure: The question is what defeasible conclusions are warranted on the basis of this body of knowledge, and in particular whether we should infer that Nixon is a pacifist or that he is not a pacifist.
Non-monotonic formalisms Pioneering work in the field of NMLs began with the realization that in order to give a mathematically precise characterization of defeasible reasoning CL is inadequate. This way LLL is significantly strengthened in a non-monotonic way. This is a simplified account compared to the more subtle logics presented in Batens There are numerous adaptive logics for other defeasible reasoning forms such as abductive reasoning, normative reasoning, belief revision, diagnosis, etc.
Various adaptive strategies come with marking mechanisms to handle more complicated cases such as the one in the following proof: 1. Non-monotonic logic and human reasoning In view of the fact that test subjects seem to perform very poorly in various paradigmatic reasoning tests e.
Conclusion There are three major issues connected with the development of logical frameworks that can adequately represent defeasible reasoning: i material adequacy; ii formal properties; and iii complexity.
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