Incidence calculus is a probabilistic logic which possesses both numerical and symbolic approaches. However, Liu in [5] pointed out that the original incidence calculus had some drawbacks and she established a generalized incidence calculus theory (GICT) based on Łukasiewicz’s threevalued logic to improve it. In a GICT, an incidence function is defined to relate each proposition φ in the axioms...

In this paper we develop a 2-valued reduction of many-valued logics, into 2-valued multi-modal logics. Such an approach is based on the contextualization of many-valued logics with the introduction of higher-order Herbrand interpretation types, where we explicitly introduce the coexistence of a set of algebraic truth values of original many-valued logic, transformed as parameters (or possible w...

The higher-order types of Herbrand interpretations for databases arise often in practice when we have to deal with uncertain or imprecise information, or in constrained databases. They are a consequence of the introduction of a kind of the observational equivalence for the hidden information in the ordinary Herbrand interpretations. It means that the logics for such databases are particular fun...

On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. The two identities are intimately connected with the classical circle and divisor problems, respectively. For each of Ramanujan’s identities, there are three possible interpretations for the double series. In two earlier papers, t...