Comparative Concept Similarity over Minspaces: Axiomatisation and Tableaux Calculus

Logics for Approximate Entailment in ordered universes of discourse

“…Then e preserves the Boolean operations and constants. Furthermore, we define S : W 2 → V as in (3), that is, S( δ ≈ , ε ≈ ) = max{c ∈ V | T δ > c ε}. We conclude as in the proof of Theorem 2.4 that (W, S) is a finite similarity space such that (4) holds.…”

“…Logics for spaces endowed with a metric or a more general distance function have been considered in a series of contributions by Kutz et al, see, e.g., [16,17]. Furthermore, logics on comparative similarity have been studied by Alenda et al, see, e.g., [1,2,3]. It is also worth mentioning that there are some connections with graded or fuzzy consequence relations as studied by Pavelka [21,20], Chakraborty [6,7], and Gerla [12] among others in the context of many-valued logics, since indeed, graded implications α > c β capture, at a syntactic (meta-)level, the idea of β being a consequence of α to the degree c. However, in the present context, α and β are classical propositions, not many-valued ones.…”

Weighted logics for artificial intelligence – an introductory discussion

“…The logic CSL [105] is based on a modal binary operator that is used for denoting the set of interpretations that are closer to p than to q. The underlying distance-based semantics can be restated in terms of preferential structures using a ternary relation expressing that all the points in a region z are at least as close to region x than to region y [2].…”

Logic of approximate entailment in quasimetric spaces

“…For two subsets A and B of a metric space, A B was supposed to contain all those elements whose distance to A is at most as large as their distance to B. The same connective has also been considered in the framework of Logics of Comparative Similarity; see, e.g., [1]. The presence of enabled us to overcome the problem of characterising the neighbourhood of the union of sets.…”

“…Details differ, but our modifications do not affect the overall picture. Our syntactical objects are of the form α d → β, (1) where α and β are formulas of CPL and d is a positive real number (possibly 0). The two CPL formulas are interpreted by subsets of a quasimetric space.…”