Peirce grammar (Q5949345)

In this article the author presents Peirce grammars as an alternative basis for linguistic analyses of natural languages. A Peirce grammar is a context-free grammar with a Peirce algebra as its semantics. A Peirce grammar is defined as a pair \(\langle G,F\rangle \) with \(G\) a context-free grammar and \(F\) a (partial) function \(F : P_G\to\) \(\Omega\), where \(P_G\) is the set of productions of \(G\) and \(\Omega \) is the Peircean clone, i.e. the set of all operations that can be defined in terms of the fundamental operations of a Peirce algebra. In order to function as a grammar for natural languages, various constraints have to be imposed on \(F,\) like, e.g., that the right hand part of a production \(p\) must have at least as many symbols as \(F(p).\) Denotations of terminal symbols are specified by a model for \(G,\) which is a non-empty set \(U\) together with a (partial) mapping \(v\) from the terminal vocabulary into some Peirce algebra over \(U,\) a Peirce algebra being a two-sorted algebra that combines a Boolean algebra with a relational algebra. The author claims Peirce grammars to be superior to other algebraic approaches to natural language analysis, like the Boolean semantics of Keenan for instance, because Peirce algebras have an equational theory that admits to capture many inferences in natural language that are intuitively valid in terms of the surface structure of sentences. Derivations of well-formed expressions (of \(G\)) generate a semantic tree with respect to the model structure in which each node is labeled by a category symbol and a denotation. This is illustrated by means of various examples involving a variety of anaphoric pronouns.