A formal approach to vague expressions with indexicals

In this paper, we offer a formal approach to the scantily investigated problem of vague expressions with indexicals, in particular including the spatial indexical `here' and the temporal indexical `now'. We present two versions of an adaptive fuzzy logic extended with an indexical, formally expressed by a modifier as a function that applies to predicative formulas. In the first version, such an operator is applied to non-vague predicates. The modified formulas may have a fuzzy truth value and fit into a Sorites paradox. We use adaptive fuzzy logics as a reasoning tool to address such a paradox. The modifier enables us to offer an adequate explication of the dynamic reasoning process. In the second version, a different result is obtained for an indexical applied to a formula with a possibly vague predicate, where the resulting modified formula has a crisp value and does not add up to a Sorites paradox

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

A repetition-free hypersequent calculus for first-order rational Pavelka logic

We present a hypersequent calculus \text{G}^3\text{\L}\forall for first-order infinite-valued {\L}ukasiewicz logic and for an extension of it, first-order rational Pavelka logic; the calculus is intended for bottom-up proof search. In \text{G}^3\text{\L}\forall, there are no structural rules, all the rules are invertible, and designations of multisets of formulas are not repeated in any premise of the rules. The calculus \text{G}^3\text{\L}\forall proves any sentence that is provable in at least one of the previously known hypersequent calculi for the given logics. We study proof-theoretic properties of \text{G}^3\text{\L}\forall and thereby provide foundations for proof search algorithms.Comment: 21 pages; corrected a misprint, added an appendix containing errata to a cited articl