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On natural logic and feasible reasoning
Reinhard Muskens
The notion of entailment plays a core role in semantics. With some exaggeration we could say that natural language semantics is the study of natural language entailment and its friends natural language synonymy and consistency. Semantics in this view is the study of natural logic, the theory of speakers' judgements about what is valid. But unlike the relation of logical consequence in any of the usual artificial languages, entailment in ordinary language seems to be a `layered' notion. Some valid inferences are drawn automatically and without any difficulty, others, no less valid, require a concious effort. This suggests that it is worthwile to investigate logics which are a) based on a natural language like syntax and b) whose entailment relations are incomplete although they capture a core of `easy' inferences. Such logics have been studied by Victor Sanchez Valencia, who isolates a (Boolean) algebraic core in natural language reasoning. This algebraic core is also manifest in generalized quantifier theory. Sanchez develops a proof system that is strongly tied to one syntactic framework, the Lambek calculus.In this paper I intend to build on a number of Sanchez' insights but will present a calculus that is largely independent from the choice of an underlying syntax. I will proceed in three steps. First I will show how any fragment of syntax that admits of a Montague-like interpretation can also be given an interpretation in such a way that no non-trivial entailments hold. The target of translation consists of a set of logical terms that are still very close to the original sentences. Secondly, it will be shown how we can strengthen this trivial logic with the help of meaning postulates, so that we get a standard Montague-like interpretation. Thirdly, I will give a simple calculus that is sound with respect to the interpretation given in terms of meaning postulates, but not complete. The calculus in an obvious sense can be said to capture a Boolean core of natural language. It will be compared with other logics of low complexity, such as the syllogistic and terminological logics.
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