Showing papers by "Thomas Vetterlein published in 2015"
TL;DR: The logic LAE q as discussed by the authors generalises classical propositional logic to the effect that conclusions can be drawn with a quantified imprecision and properties are modelled by subsets of a distance space and statements are of the form that one property implies another property within a certain limit of tolerance.
11 citations
TL;DR: In this paper, the authors consider tomonoids satisfying the following conditions: the multiplication is commutative; the monoidal identity is the top element; all nonempty suprema exist; and the multiplication distributes over arbitrary supremas.
Abstract: A totally ordered monoid, or tomonoid for short, is a monoid together with a translation-invariant (i.e., compatible) total order. We consider in this paper tomonoids fulfilling the following conditions: the multiplication is commutative; the monoidal identity is the top element; all nonempty suprema exist; and the multiplication distributes over arbitrary suprema. The real unit interval endowed with its natural order and a left-continuous t-norm is our motivating example. A t-norm is a binary operation used in fuzzy logic for the interpretation of the conjunction. Given a tomonoid of the indicated type, we consider the chain of its quotients induced by filters. The intention is to understand the tomonoid as the result of a linear construction process, leading from the coarsest quotient, which is the one-element tomonoid, up to the finest quotient, which is the tomonoid itself. Consecutive elements of this chain correspond to extensions by Archimedean tomonoids. If in this case the congruence classes are ...
10 citations
30 Jun 2015
TL;DR: This contribution considers the possibility of standardising the construction of fuzzy sets at least with regard to the borderline cases, and discusses a number of logic-based approaches to reasoning under vagueness.
Abstract: Fuzzy sets are a popular tool to model vague properties. It is, however, well-known that this model usually involves a good degree of arbitrariness. In this contribution we consider the possibility of standardising the construction of fuzzy sets at least with regard to the borderline cases. To this end, we identify a vague property with the sets of clearly positive cases and clearly negative cases and under the assumption that the universe of discourse is a metric space, we determine the fuzzy set by what can be described as a linear interpolation. Following this idea, we discuss a number of logic-based approaches to reasoning under vagueness.
7 citations