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understanding of the semantics of nominals, and in particular of their dual nature as referential and quantificational expressions? Answer: Nominals are doubly dual, in a sense to made made precise in this presentation. 2 Phase quantification Löbner suggested that the emergence of a square of logical duality between the semantic values of natural language expressions is a major empirical touchstone to ascertain their quantificational nature [Löb87]; and van Benthem, while noting that the ubiquity of the square of duality may be the sign of a semantic invariant possibly rooted in some cognitive universal, highlighted its heuristic value for research on quantification inasmuch as "it suggests a systematic point of view from which to search for comparative facts" [vanBent91](p.23). Given the issues raised by (*), it is of note that the square of duality in (2) is different from the classical square of oppositions in (1): The difference lies in the fact that duality, inner negation and outer negation are third order concepts, while compatibility, contrariness and implication are second order concepts. (1) p contraries q (2) Q inner negation Q~ subalternes contrad r s compatibles subalternes outer negation inner negation There are instantiations of the square of oppositions without corresponding squares of duality, and vice-versa ([Löb87],p.56 for discussion). Although the two squares are logically independent, the empirical emergence of a square of oppositions, such as the one in (*), naturally raises the question about the possible existence of an associated square of duality. We will argue that the answer to this question is affirmative and that it provides also an answer to question (A) above. Before this result may be worked out, some analytical tools are to be introduced first. We will resort to the notion of phase quantification, which was introduced in [Löb87] to study the semantics of aspectual adverbials and was shown there to be extended to characterize quantification in general. For the sake of concreteness, consider a diagrammatic display of the semantics of these adverbials: ¬Q t t t t dual outer negation ¬Q~ P ~P P ~P ~P P ~P P no_longer'(P) still'(P) not_yet'(P) already'(P) Very briefly, phase quantification requires the following ingredients: (i) a partial order over the domain of quantification; (ii) a property P defining a positive phase in a sequence of two opposite phases; (iii) a parameter point t; and (iv) the starting point of the relevant semiphase given the presupposition about the linear order between P and ~P phases. 2
For aspectual adverbials, (i) the order is the time axis; (ii) P denotes the instants where the proposition containing the adverbial holds; (iii) t is the reference time of the utterance; (iv) the starting point s(R,t) is the infimum of the set of the closest predecessors of t which form an uninterrupted sequence in R — the adverbials no longer and still bear the presupposition that phase P precedes phase ~P (~P.P for the other two adverbials). These adverbials express the following quantifiers: (3) still': λP.every’(λx.(S(P,t)
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SemPrag03.Progr.pdf - Institut für Linguistik/Germanistik - Universität ...

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