Constraints on Anaphoric Determiners

10.1007/s11168-011-9076-3Some constraints on functions from sets and relations to sets are studied. Such constraints are satisfied by anaphoric functions, that is functions denoted by anaphoric determiners. These constraints are generalisations of anaphor conditions known from the study of simpler cases of nominal anaphors. In addition a generalisation of the notion of conservativity as applied to anaphoric functions is proposed. Two classes of anaphoric determiners found in NLs are discussed as examples

A semantic constraint on binary determiners

A type quantifier F is symmetric iff F(X, X)(Y ) = F(Y, Y )(X). It is shown that quantifiers denoted by irreducible binary determiners in natural languages are both conservative and symmetric and not only conservative

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later

Bootstrapping Inductive and Coinductive Types in HasCASL

We discuss the treatment of initial datatypes and final process types in the wide-spectrum language HasCASL. In particular, we present specifications that illustrate how datatypes and process types arise as bootstrapped concepts using HasCASL's type class mechanism, and we describe constructions of types of finite and infinite trees that establish the conservativity of datatype and process type declarations adhering to certain reasonable formats. The latter amounts to modifying known constructions from HOL to avoid unique choice; in categorical terminology, this means that we establish that quasitoposes with an internal natural numbers object support initial algebras and final coalgebras for a range of polynomial functors, thereby partially generalising corresponding results from topos theory. Moreover, we present similar constructions in categories of internal complete partial orders in quasitoposes

Birational motives, II: Triangulated birational motives

We develop birational versions of Voevodsky's triangulated categories of motives over a field, and relate them with the pure birational motives studied in arXiv:0902.4902 [math.AG]. We also get an interpretation of unramified cohomology in this framework, leading to "higher derived functors of unramified cohomology".Comment: Compared to the initial version: previous Subsection 4.2 has been upgraded to Section 5; previous Lemmas 5.2.5 and 5.2.6 have been corrected to Proposition 6.2.5 and Lemma 6.2.6; at the referee's request, previous Appendix B and the proof of previous Proposition C.1.1 (now A.4.1) have been removed (please consult the initial version for them