On Reasoning with Ambiguities

On Reasoning with Ambiguities Uwe Reyle Institute for Computational Linguistics University of Stuttgart Azenbergstr.12, D-70174 Stuttgart, Germany e-mail: uwe@ims.uni-stuttgart.de Abstract The paper adresses the problem of reasoning with ambiguities. Semantic representations are presented that leave scope relations between quantifiers and/or other operators unspecified. Truth conditions are provided for these representations and different con- sequence relations are judged on the basis of intuitive correctness. Finally inference patterns are presented that operate directly on these underspecified struc- tures, i.e. do not rely on any translation into the set of their disambiguations. 1 Introduction Whenever we hear a sentence or read a text we build up mental representations in which some aspects of the meaning of the sentence or text are left underspe- cified. And if we accept what we have heard or read as true, then we will use these underspecified repre- sentations as premisses for arguments. The challenge is, therefore, to equip underspecified semantic repre- sentations with well-defined truth conditions and to formulate inference patterns for these representati- ons that follow the arguments that we judge as in- tuitively correct. Several proposals exist for the de- finition of the language, but only very few authors have addressed the problem of defining a logic of ambiguous reasoning. [8] considers lexical ambiguities and investigates structural properties of a number of consequence re- lations based on an abstract notion of coherency. It is not clear, however, how this approach could be extended to other kinds of ambiguities, especially quantifier scope ambiguities and ambiguities trigge- red by plural NPs. [1], [7] and [6] deal with ambigui- ties of the latter kind. They give construction rules and define truth conditions according to which an underspecified representation of an ambiguous sent- ence is true if one of its disambiguations is. The pro- blem of reasoning is adressed only in [5] and [7]. [5]'s inference schemata yield a very weak logic only; and [7]'s deductive component is too strong. Being weak and strong depends of course on the underlying con- sequence relation. Neither [5] nor [7] make any att- empt to systematically derive the consequence rela- tion that holds for reasoning with ambiguities on the basis of an empirical discussion of intuitively valid arguments. The present paper starts out with such a discussion in Section 2. Section 3 gives a brief introduction to the theory of UDRSs. It gives a sketch of the princip- les to construct UDRSs and shows how scope ambi- guities of quantifiers and negation are represented in an underspecified way. As the rules of inference pre- sented in [7] turn out to be sound also with respect to the consequence relation defined in Section 2 the- se rules (for the fragment without disjunction) will be discussed only briefly in Section 4. The change in the deduction system that is imposed by the new consequence relation comes with the rules of proof. Section 5 shows that it is no longer possible to use rules like Conditionalisation or Reductio ad Absur- dum when we deal with …

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