Logic of Typical and Atypical Instances of a Concept—A Mathematical Model

Round 1

Reviewer 1 Report

This is a valuable paper from several points of view: it provides a mathematical model of a logic of determination of objects, but is open enough to cover complex properties of concepts and objects which are usually and incorrectly ignored. A conceptual 'space' is defined -  a quasi-topology - that is more applicable than para-consistent or non-monotopic logics. While the authors remain solidly in the deductive domain, they, correctly in my view, attack standard set concepts to include a-typicalinstances and entities "almost belonging or not belonging entirely". A new categorization of objects and ontologies is possible. I strongly recommend publication of this paper.

My only criticisms are that the arguments remain embedded in a substantially abstract world of semantic entities or their mathematical equivalents. For the categorization of ontologies, from a natural philosophical standpoint it is, I think, fair to ask what ontologies and what are the philosophical limitations of the selection that the authors have made. 

In other words, opening the paper to 'what might lie beyond' could make a good paper even better.     

Author Response

Thanking you for your comment, we agree with your statement that the complexity of today's world imposes such complex ontology modeling. The mathematical structure of quasi-topology is only a model a little finer than others. These advantages lie in the fact that from this model one can build automatic systems for the analysis of ontologies. We continue to work on the possible applications of this model.

Reviewer 2 Report

I really enjoyed this manuscript and got many insights. I can empathize with authors' research approach, especially, the representation of the "concept" and the "object" by a same graph at the same time.

In general, firstly, syntax and semantics are presented separately, next, a connection of the two are considered, however, this differentiation is intrinsically arbitrary. In this point, the simultaneous representation of the "concept" and the "object" is reasonable and much worth.

I think that this worthy approach of the research can reach the further.

1) The above and below graphical structures of f and tau-f can be analyzed and/or comparable by formal concept analysis, for instance.

2) Related to this, the authors considered about the concepts and the objects in Fig. 3, Fig. 4, and on lines of 373-384. However, in my opinion, this analysis seems to be using not the relation between the concept and the object, but semantics of our normal language.

3) From the same reason as 1), I think that this approach can figure out a dynamical feature of concepts and/or objects, by a way that reciprocal influences of the two are considered.

4) In the part below the line 433, the authors mentioned the relations between the concepts and typical / atypical objects, however it will be more clear by using / examining the relations between the concepts and the objects. It will be still better using an example and a figure.

Belows are some typos that I guess.

L66: Logcal -> logical
L107: vue -> view
L223: some spaces after DELTA -> a space
L227: Or [BOLD?] -> Or [NORMAL]
L234: some spaces after DELTA -> a space
L311: m_2(m_1(p) -> m_2(m_1(p))
L311: m_1(m_2(q) -> m_1(m_2(q))
L319: 2,in black -> 2, in black
Figure 3: To-br-an-irregular-inhabi -> To-be-an-irregular-inhabi

Overall, I could study much through the manuscript. I would like to appreciate the authors.

Author Response

Thank you for your remarks. I try to answer your questions :

1) Yes, the LDO can be compared with FCA. At least two points of LDO are not described by FCA : FCA does not give a logical status to the concept and the object. FCA does not deal with the problem of typicality / atypicality.

2) The logical status of the concept (an operator) and of the object (an operand) (in Curry sense) is encoded in natural languages by some grammatical categories.

3) There is not a duality between concepts and objects. It this true that in the cognition one « implies » the other and conversely. But the double « implication » can not be interpreted as an equivalence. And this fact is stressed by the type theory.

4) Sometimes a mathematical model erases some cognitive traits in the benefit of others.

We conceived a same graph for the « quasi-duality » concepts – objects in order to have a easier automatic processing that we intend to do further.