Normalization of IZF with Replacement

ZF is a well investigated impredicative constructive version of Zermelo-Fraenkel set theory. Using set terms, we axiomatize IZF with Replacement, which we call \izfr, along with its intensional counterpart \iizfr. We define a typed lambda calculus \li corresponding to proofs in \iizfr according to the Curry-Howard isomorphism principle. Using realizability for \iizfr, we show weak normalization of \li. We use normalization to prove the disjunction, numerical existence and term existence properties. An inner extensional model is used to show these properties, along with the set existence property, for full, extensional \izfr

Extended Initiality for Typed Abstract Syntax

Initial Semantics aims at interpreting the syntax associated to a signature as the initial object of some category of 'models', yielding induction and recursion principles for abstract syntax. Zsid\'o proves an initiality result for simply-typed syntax: given a signature S, the abstract syntax associated to S constitutes the initial object in a category of models of S in monads. However, the iteration principle her theorem provides only accounts for translations between two languages over a fixed set of object types. We generalize Zsid\'o's notion of model such that object types may vary, yielding a larger category, while preserving initiality of the syntax therein. Thus we obtain an extended initiality theorem for typed abstract syntax, in which translations between terms over different types can be specified via the associated category-theoretic iteration operator as an initial morphism. Our definitions ensure that translations specified via initiality are type-safe, i.e. compatible with the typing in the source and target language in the obvious sense. Our main example is given via the propositions-as-types paradigm: we specify propositions and inference rules of classical and intuitionistic propositional logics through their respective typed signatures. Afterwards we use the category--theoretic iteration operator to specify a double negation translation from the former to the latter. A second example is given by the signature of PCF. For this particular case, we formalize the theorem in the proof assistant Coq. Afterwards we specify, via the category-theoretic iteration operator, translations from PCF to the untyped lambda calculus

Implication functions in interval-valued fuzzy set theory

Interval-valued fuzzy set theory is an extension of fuzzy set theory in which the real, but unknown, membership degree is approximated by a closed interval of possible membership degrees. Since implications on the unit interval play an important role in fuzzy set theory, several authors have extended this notion to interval-valued fuzzy set theory. This chapter gives an overview of the results pertaining to implications in interval-valued fuzzy set theory. In particular, we describe several possibilities to represent such implications using implications on the unit interval, we give a characterization of the implications in interval-valued fuzzy set theory which satisfy the Smets-Magrez axioms, we discuss the solutions of a particular distributivity equation involving strict t-norms, we extend monoidal logic to the interval-valued fuzzy case and we give a soundness and completeness theorem which is similar to the one existing for monoidal logic, and finally we discuss some other constructions of implications in interval-valued fuzzy set theory