Topologies and Sheaves Appeared as Syntax and Semantics of Natural Language.

In a sheaf-theoretic framework, we describe the process of interpretation of a text written in some unspecified natural language, say in English. We consider only texts written for human understanding, those we call admissible. A meaning of a part of a text is accepted as the communicative content grasped in a reading process following the reader’s interpretive initiative formalized by the term sense. For the meaningfulness correlative with an idealized reader’s linguistic competence, the set of all meaningful parts of an admissible text is stable under arbitrary unions and finite intersections, and hence it defines a topology that we call phonocentric. We interpret syntactic notions in terms of topology and order; it is a kind of topological formal syntax. The connectedness and the T0 -separability of such a phonocentric topology are linguistic univer- sals. According to a particular sense of reading, we assign to each meaningful fragment of a given text the set of all its meanings those may be grasped in all possible readings in this sense. This way, to any sense of reading, we assign a sheaf of fragmentary meanings. All such sheaves constitute a category, in terms of which we develop a sheaf-theoretic formal semantics. It allows us to generalize Frege’s compositionality and contextuality principles related with the Frege duality between the category of all sheaves of fragmentary meanings and the category of all bundles of contextual meanings. The acceptance of one of these principles implies the acceptance of the other. This Frege duality gives rise to a representation of fragmentary meanings by continuous functions. Finally, we develop a kind of dynamic semantics that describes how the interpretation proceeds as a stepwise extension of a meaning representa- tion function from the initial meaningful fragment to the whole interpreted text.

Authors

• Oleg Prosorov

Keywords

• Sense
• Meaning
• Phonocentric Topology
• Linguistic Universals
• Sheaf of Fragmentary Meanings
• Compositionality Principle
• Contextuality Principle
• Bundle of Contextual Meanings
• Frege Duality
• Mathematics
• Linguistics: Aspects of Interaction 2012” (PhML-2012)
• held on May 22–25
• 2012 at the Euler International Mathematical Institute