Abstract
A new construction of a certain conceptual space is presented. Elements of this conceptual space correspond to (and serve as code for) concept elements of reality, which potentially comprise an infinite number of qualities. This construction of a conceptual space solves a problem stated by Dietz and his co-authors in 2013 in the context of Voronoi diagrams. The fractal construction of the conceptual space is that this problem simply does not pose itself. The concept of convexity is discussed in this new conceptual space. Moreover, the meaning of convexity is discussed in full generality, for example when space is deprived of it, its substitutes for concept domains are considered.
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Notes
For a different than ours application of metalanguage in linguistic semantics see Wierzbicka (1972).
All mathematical terminology will be defined and illustrated by elementary examples in the respective paragraphs (in which they appear).
The emphasis in the text in italic font was made by the author of the statement.
The procedure for completing metric space is described in the Rudin textbook (1976).
A relation \(\rho \) on a set S is an equivalence relation if
-
(i)
for every \(a\in S,\)\(a\rho \,a\) (reflexivity),
-
(ii)
for every \(a,b\in S\) if \(a\,\rho \,b\) then \(b\,\rho \,a\) (symmetry),
-
(iii)
for eery \(a,b,c\in S\) if \(a\,\rho \,b\) and \(b\,\rho \,c\) then \(a\,\rho \,c\) (transitivity).
-
(i)
Clearly, if \(\gamma \) is a geodesic ray in X then \(\gamma (n)\) converges to infinity.
A metric space is said to be proper if all closed balls with a centre x and radius r\(B(x,r)=\{y\in X:\;d(x,y)\le r\}\) are compact.
It seems that the construction of the \(p+1\)-ary Cantor set, which is a generalization of the standard construction of the 3-ary Cantor set does not appear in literature, or at least the author did not encounter such generalization.
A set in a topological space \((X,\mathcal O)\) is nowhere dense if its closure has empty interior. Equivalently, a nowhere dense set in X is a set that is not dense in any nonempty open set from \(\mathcal O.\)
Let us remind that for a subset A of a metric space (X, d) the diameter of A is defined as \({{\mathrm{diam}}}A=\sup _{x,y\in A}d(x,y).\)
The topological space is called connected if it cannot be represented as a sum of two disjoint closed.
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The authors wishes to thank an anonymous referees for remarks that improved the overall presentation of the results.
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Urban, R., Mróz, S. A Class of Conceptual Spaces Consisting of Boundaries of Infinite p-Ary Trees. J of Log Lang and Inf 28, 73–95 (2019). https://doi.org/10.1007/s10849-018-9273-7
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DOI: https://doi.org/10.1007/s10849-018-9273-7