Equivalences Among Polarity Algorithms

The concept of polarity is pervasive in natural language. It relates syntax, semantics and pragmatics narrowly (Giannakidou, in: Maienborn, von Heusinger, Portner (eds.), Semantics: an international handbook of natural language meaning, De Gruyter Mouton, Berlin, 2011; Israel in The grammar of polarity: pragmatics, sensitivity, and the logic of scales, Cambridge studies in linguistics, Cambridge University Press, Cambridge, 2014), it refers to items of many syntactic categories such as nouns, verbs and adverbs. Neutral polarity items appear in affirmative and negative sentences, negative polarity items cannot appear in affirmative sentences, and positive polarity items cannot appear in negative sentences. A way of reasoning in Natural Language is through Natural Logic (van Benthem in Essays in logical semantics, vol. 29 of Studies in linguistics and philosophy, Reidel, Dordrecht, 1986; Language in action: categories, lambdas, and dynamic logic, vol. 130 of Studies in logic, Elsevier, Amsterdam, 1991). This logic is based on the concept of polarity in order to make the meaning of a sentence weaker o stronger without changing its truth value. There exist many proposals to compute polarity in the Natural Logic context, the most widely known are the ones by: van Benthem (1986, 1991), Sánchez-Valencia (Studies on natural logic and categorial grammar, Ph.D. thesis, Universiteit van Amsterdam, 1991), Dowty (Proceedings of the 4th conference on semantics and theoretical linguistics, Cornel University, CLC Publications, Rochester, 1994), and van Eijck (in: ten Cate, Zeevat (eds.), 6th international Tbilisi symposium on logic, language, and computation, Batumi, Georgia, Springer, 2007). If Natural Logic is going to be used, as an inferential mechanism between text fragments, in Natural Language Processing applications such as text summarization, question answering, and information extraction, it is a priority to know what the existing relationship among the aforementioned algorithms is; for example, to implement the most general. We show in this paper the equivalence among the analyzed algorithms, filling a gap in Natural Logic research, particularly in computing polarity, and the soundness of their algorithms.