Rough sets (RS)proved a thriving realm with successes inn many fields of ML and AI. In this note, we expand RS to RM - rough mereology which provides a measurable degree of uncertainty to those areas.

Given a raw knowledge in the form of a data table/a decision system, one is facing two possible venues. One, to treat the system as closed, i.e., its universe does not admit new objects, or, to the contrary, its universe is open on admittance of new objects. In particular, one may obtain new objects whose sets of values of features are new to the system. In this case the problem is to assign a decision value to any such new object. This problem is somehow resolved in the rough set theory, e.g., on the basis of similarity of the value set of a new object to value sets of objects already assigned a decision value. It is crucial for online learning when each new object must have a predicted decision value.\ There is a vast literature on various methods for decision prediction for new yet unseen object. The approach we propose is founded in the theory of rough mereology and it requires a theory of sets/concepts, and, we root our theory in classical set theory of Syllogistic within which we recall the theory of parts known as Mereology. Then, we recall our theory of Rough Mereology along with the theory of weight assignment to the Tarski algebra of Mereology.\ This allows us to introduce the notion of a part to a degree. Once we have defined basics of Mereology and rough Mereology, we recall our theory of weight assignment to elements of the Boolean algebra within Mereology and this allows us to define the relation of parts to the degree and we apply this notion in a procedure to select a decision for new yet unseen objects.\ In selecting a plausible candidate which would pass its decision value to the new object, we employ the notion of Vapnik - Chervonenkis dimension in order to select at the first stage the candidate with the largest VC-dimension of the family of its $\varepsilon$-components for some choice of $\varepsilon$.

We propose a novel technique for proving the consistency of large, complex and heterogeneous theories for which ‘standard’ automated reasoning methods are considered insufficient. In particular, we exemplify the applicability of the method by establishing the consistency of the foundational ontology DOLCE, a large, first-order ontology. The approach we advocate constructs a global model for a theory, in our case DOLCE, built from smaller models of subtheories together with amalgamability properties between such models. The proof proceeds by (i) hand-crafting a so-called architectural specification of DOLCE which reflects the way models of the theory can be built, (ii) an automated verification of the amalgamability conditions, and (iii) a (partially automated) series of relative consistency proofs.

In this paper, I defend a multiplicative approach that distinguishes statues from amounts of matter, political entities from physical ones, qua entities (e.g. John qua Alitalia passenger) from players (e.g. John), etc. I develop a theory of levels which is based on the primitive notions of level, parthood, and grounding (a kind of existential dependence) and that is used to characterize more specific relations like constitution, inherence, and abstraction. I neither aim to propose a `definitive' theory of levels nor to commit to their ontological or conceptual nature. Hence, the adjective `ontological' used in the title does not qualify the nature of the entities that belong to levels but the way the notion of level is characterized, i.e. in terms of general and philosophically well-founded notions. By keeping away from a purely realist attitude, I can then discuss the adequacy of some alternative first-order theories to account for three puzzling scenarios.

We determine the implicit assumptions and the structure of the Single and Double Cross Calculi within Euclidean geometry, and use these results to guide the construction of analogous calculi in mereogeometry.  The systems thus obtained have strong semantic and deductive similarities with the Euclidean-based Cross Calculi although they rely on a different geometry. This fact suggests that putting too much emphasis on usual classification of qualitative spaces may hide important commonalities among spaces living in different classes.