253 research outputs found
Rejection in Łukasiewicz's and Słupecki's Sense
The idea of rejection originated by Aristotle. The notion of rejection
was introduced into formal logic by Łukasiewicz [20]. He applied it to
complete syntactic characterization of deductive systems using an axiomatic
method of rejection of propositions [22, 23]. The paper gives not only genesis,
but also development and generalization of the notion of rejection. It also
emphasizes the methodological approach to biaspectual axiomatic method of
characterization of deductive systems as acceptance (asserted) systems and
rejection (refutation) systems, introduced by Łukasiewicz and developed by
his student Słupecki, the pioneers of the method, which becomes relevant in
modern approaches to logic
Interval-valued algebras and fuzzy logics
In this chapter, we present a propositional calculus for several interval-valued fuzzy logics, i.e., logics having intervals as truth values. More precisely, the truth values are preferably subintervals of the unit interval. The idea behind it is that such an interval can model imprecise information. To compute the truth values of ‘p implies q’ and ‘p and q’, given the truth values of p and q, we use operations from residuated lattices. This truth-functional approach is similar to the methods developed for the well-studied fuzzy logics. Although the interpretation of the intervals as truth values expressing some kind of imprecision is a bit problematic, the purely mathematical study of the properties of interval-valued fuzzy logics and their algebraic semantics can be done without any problem. This study is the focus of this chapter
Lukasiewicz mu-Calculus
We consider state-based systems modelled as coalgebras whose type incorporates branching, and show that by suitably adapting the definition of coalgebraic bisimulation, one obtains a general and uniform account of the linear-time behaviour of a state in such a coalgebra. By moving away from a boolean universe of truth values, our approach can measure the extent to which a state in a system with branching is able to exhibit a particular linear-time behaviour. This instantiates to measuring the probability of a specific behaviour occurring in a probabilistic system, or measuring the minimal cost of exhibiting a specific behaviour in the case of weighted computations
- …