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3.1.3. Higher-Order Logic Higher-order logic extends the ontological commitment of first-order logic by assuming the the world consists of objects, subsets of objects, and relationships between them, in such a way that it is possible to reason about not just the objects but about the properties and subsets of objects specified by functions and predicates. Thus, in higher-order logic, variables range over predicates and functions as well as over individuals of the domain. This means that predicates and functions can be quantified, given as arguments of other predicates or functions, and be outputs of other predicates and functions as well. This makes higher-order logic very expressive, mathematically elegant, and permits a concise description of complex problems. But this increased complexity makes reasoning more difficult as the logic becomes undecidable, inconsistencies can be introduced, and proof systems also become incomplete. To alleviate these problems, the domain is usually constrained to have decidable or semi-decidable subclasses, a type discipline is used to avoid inconsistencies (e.g., Russell's paradox) by introducing hierarchies among the elements of the domain, and heuristics are developed to cope with incompleteness. With these amendments that retain its expressive advantages, higher-order logic has become very popular for reasoning about many types of problems and designing intelligent systems in areas such as hardware, software, protocols, and safety-critical systems (Gordon). 3.2. NONCLASSICAL LOGICSClassical deductive logic has several important limitations, among which are: (1) classical logic cannot express everything we want to say about the world, (2) there may be facts that have to be retracted when when new facts are derived, and (3) the typical language of logic cannot express uncertain knowledge. Nonmonotonic reasoning, probabilistic reasoning, induction, and abduction represent styles of logic that attempt to alleviate some of these limitations. 3.2.1. Nonmonotonic Reasoning A characteristic of classical logic is the monotonicity of the derived inferences. This means that deduced facts never contradict the premises nor previously derived facts. However, in many real-world situations this is not necessarily the case. Nonmonotonic reasoning is a logic-based formalism that sanctions inferences that do not deductively follow from a set of sentences stored in a knowledge base [open triangle]. For instance, we may have inference techniques, the application of which depends on certain sentences not being in [open triangle]. With such inference rules, if a new sentence is added to [open triangle], a previous inference may have to be retracted. There are many cases in which it is appropriate for an intelligent system to augment its knowledge base with new facts that do not deductively follow from the facts in the knowledge base. For example, circumstances may force immediate action from the system before acquiring all the required facts. Several techniques have been developed for nonmonotonic reasoning. Among these are the closed-world assumption, which allows us to infer the negation of an unprovable ground predicate (Reiter); predicate completion, in which a formula is computed and added to [open triangle] to restrict the domain of a selected predicate to just those that [open triangle] must satisfy the predicate (Clark); default reasoning, where typical properties of objects are described and then rules of inference called defaults are added to perform default inferences (Reiter); and circumscription, which is based on the idea of identifying minimal models of a knowledge base with respect to a selected predicate (McCarthy). Given a new sentence to be added to [open triangle], the number of sentences in [open triangle] that have to be retracted may be very big, if not infinite. This problem is called the qualification problem and is often cited as one of the reasons that a strict logic approach to AI will not work and has motivated much of the work in nonmonotonic reasoning.
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