Argumentation frameworks (AF) play important roles in artificial intelligence. This paper is an exploration in establishing semantics of fuzzy AFs by fuzzy sets. There are many ways to characterize the semantics of fuzzy AFs. In this paper, our work is based on the assumption that some inconsistency of the system is permitted. Firstly, we formalize the conflict-freeness with a consistency degree x and the acceptability with a consistency degree y. Various types of extensions are then defined in a way similar to Dung's approach. The conflict-freeness and acceptability can be seen as an interpretation of the corresponding notion in Janssen's work. Formally, we add the conflict-freeness into the admissible extensions and the preferred extensions. We also introduce the complete extensions and the grounded extensions. Moreover, some basic properties are proven, such as the Fundamental Lemma, the algorithm of the grounded extension, etc. At last, it is proven to be consistent with Dung's original semantics in crisp AFs.

The main results are about the groups of the negations on the unit square, which is considered as a bilattice. It is proven that all the automorphisms on it form a group; the set, containing the monotonic isomorphisms and the strict negations of the first (or the second or the third) kind, with the operator “composition,” is a groupG2(orG3orG4, correspondingly). All these four kinds of mappings form a groupG5. And all the groupsGi,i=2,3,4are normal subgroups ofG5. Moreover, forG5, a generator set is given, which consists of all the involutive negations of the second kind and the standard negation of the first kind. As a subset of the unit square, the interval-valued set is also studied. Two groups are found: one group consists of all the isomorphisms onLI, and the other group contains all the isomorphisms and all the strict negations onLI, which keep the diagonal. Moreover, the former is a normal subgroup of the latter. And all the involutive negations on the interval-valued set form a generator set of the latter group.

Conventionally, probability in probabilistic argumentation frameworks (PAFs) is treated as meta-information attached to arguments or attack relations. Such probability often relies on external sources, which may in fact be unreliable. This paper proposes a novel perspective. It directly integrates probability into the content of arguments. This allows probabilities themselves to be subject to critical scrutiny. This approach introduces a new binary relation called challenge. Unlike traditional attack relations that target the acceptability of arguments, this relation modifies the assigned probability of an argument. It thus extends the expressive capabilities of PAFs. Based on this concept, we develop probabilistic argumentation frameworks with challenge relation (PAFCs) via the constellation method. To simplify the structure of possible worlds while preserving PAFC information, we further define a probabilistic attack normal form for PAFCs. Finally, we demonstrate the practical applicability of PAFCs in trust modeling and highlight their enhanced capacity to capture nuanced probabilistic argumentation scenarios.