We study mathematical and algorithmic properties of Lambek's pregroups and illustrate them by the French noun phrase. An algorithm of complexity n3 to solve the reduction problem in an arbitrary free pregroup as well as recognition by a pregroup grammar is presented. This algorithm is then specified to run in linear time. A sufficient condition for a language fragment that makes the linear algorithm complete is given.

Doctors occasionally make diagnoses in strangers outside of formal medical settings by using the medical skill of visual inspection, such as noticing signs of melanoma or the symptoms of hyperthyroidism. This may cause considerable moral unease and doubts on the side of the diagnosing physician. Such encounters force physicians to consider whether or not to intervene by introducing themselves to the stranger and offering an unsolicited medical opinion despite the absence of a formal doctor-patient relationship. A small body of literature (...) has addressed the topic of the unsolicited medical opinion, often with a primary focus on practical advice. This article seeks to establish an ethical-theoretical basis for physicians' ethical obligation to offer an unsolicited medical opinion when they make a diagnosis by visual inspection in a stranger outside of the formal medical context. Using a utilitarian approach, it is argued that, if it is in the physicians' power to prevent a possible loss of well-being, without thereby sacrificing anything of equal value, physicians have an ethical obligation to intervene. (shrink)

Every pregroup grammar is shown to be strongly equivalent to one which uses basic types and left and right adjoints of basic types only. Therefore, a semantical interpretation is independent of the order of the associated logic. Lexical entries are read as expressions in a two sorted predicate logic with ∈ and functional symbols. The parsing of a sentence defines a substitution that combines the expressions associated to the individual words. The resulting variable free formula is the translation of the (...) sentence. It can be computed in time proportional to the parsing structure. Non-logical axioms are associated to certain words (relative pronouns, indefinite article, comparative determiners). Sample sentences are used to derive the characterizing formula of the DRS corresponding to the translation. (shrink)

Pregroup grammars have a cubic recognition algorithm. Here, we define a correct and complete recognition and parsing algorithm and give sufficient conditions for the algorithm to run in linear time. These conditions are satisfied by a large class of pregroup grammars, including grammars that handle coordinate structures and distant constituents.

ABSTRACT A new notion of model is presented which makes the Barcan formula and its converse hold in arbitrary frames without requiring constant, increasing or decreasing domains. Soundness and completeness of first order K is established for this class of new models. The failure of reasoning by substitution known as ? opacity ? is explained. An existenc predicate makes it possible to distinguish between actual and possible elements. The connections with the restricted Barcan formula are considered.

We give a formal interpretation of Martin-Löf's Constructive Theory of Types in Elementary Topos Theory which is presented as a formalised theory with intensional equality of objects. Types are interpreted as arrows and variables as sections of their types. This is necessary to model correctly the working of the assumption x ∈ A. Then intensional equality interprets equality of types. The normal form theorem which asserts that the interpretation of a type is intensional equal to the pullback of its “alignment” (...) along some “base” arrow relates this interpretation to categorical semantic of types. (shrink)

We show that vector space semantics and functional semantics in two-sorted first order logic are equivalent for pregroup grammars. We present an algorithm that translates functional expressions to vector expressions and vice-versa. The semantics is compositional, variable free and invariant under change of order or multiplicity. It includes the semantic vector models of Information Retrieval Systems and has an interior logic admitting a comprehension schema. A sentence is true in the interior logic if and only if the ‘usual’ first order (...) formula translating the sentence holds. The examples include negation, universal quantifiers and relative pronouns. (shrink)