Similarity Self/Ideal Index (SSI): A Feature-Based Approach to Modeling Psychological Well-Being
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsSee the atatchement
Comments for author File: 
Comments.pdf
Author Response
We sincerely thank the reviewer for their encouraging feedback and their positive assessment of the manuscript’s structure and rigor. We greatly appreciate the recognition of the novelty of combining fuzzy similarity spaces with Personal Construct Theory. We have carefully addressed all the minor comments raised, which we believe have significantly improved the clarity and presentation of the paper.
Point-by-Point Response:
1. Abstract: “a index” → “an index”.
Response: We have corrected this typographical error in the Abstract. It now reads "an index" (or "a similarity index" depending on the phrasing context) to ensure grammatical correctness.
2. Add citations when introducing semantic similarity space.
Response: We agree that this concept required proper theoretical grounding. We have added foundational references to the Introduction (Section 1.3), specifically citing Zadeh (1965), Zimmermann (2001), and Ruspini (1970). These citations contextualize the transition from discrete feature models to the continuous fuzzy similarity space used in our formalization.
3. Add a concise “Key Contributions” list in the Introduction.
Response: We have included a specific list of key contributions at the end of Section 1.3 (Feature-Based Approach to Modeling Psychological Well-Being). This list clearly outlines the three main innovations of the paper: the proposal of the SSI index, its formalization within a fuzzy similarity space, and the structural analysis of psychological resilience.
4. Include a comparison table summarizing SSI vs. traditional RGT metrics vs. fuzzy similarity metrics.
Response: We have added Table 1 in the Discussion section (Section 3). This table provides a detailed comparison of Traditional RepGrid Metrics, General Fuzzy Similarity, and the proposed SSI Index across several dimensions (Conceptual Model, Mathematical Basis, Symmetry, etc.). We also included a row summarizing the main limitations of each approach, as suggested by the comparative analysis.
Reviewer 2 Report
Comments and Suggestions for AuthorsPlease find the attached report.
Comments for author File: Comments.pdf
Author Response
We would like to express our sincere gratitude to the reviewer for the comprehensive and rigorous examination of our manuscript. We deeply value the time and effort dedicated to scrutinizing both the mathematical formalization and the theoretical underpinnings of our work. We have carefully considered every critical observation raised, and addressing these points has been instrumental in refining the model’s definitions and enhancing the overall coherence of the paper. As a result of this constructive feedback, we believe the revised manuscript now presents a significantly more robust and precise formulation of the SSI index. Please find below our detailed point-by-point response to each of the comments.
Response Comment 1: Theoretical Inconsistency: Discontinuity in Continuous Variables (Ref: Section 2.1, Definition 4)
We appreciate the reviewer's rigorous mathematical analysis regarding the discontinuity at the sign boundary. We agree that treating continuous variables with discrete set logic creates a singularity at zero. However, we respectfully argue that this property is not a flaw but a necessary feature of a model grounded in Personal Construct Psychology.
In PCP, constructs are inherently bipolar (dichotomous distinctions), not merely continuous traits. The transition from a score of (alignment with Pole A) to (alignment with Pole B) is not just a negligible numerical shift; it represents a qualitative change in meaning, the crossing of the semantic threshold from one opposing concept to the other. To "smooth" this transition using a fuzzy neighborhood would inaccurately imply that a subject who is "slightly hostile" is semantically equivalent to one who is "slightly friendly," obscuring the crucial fact that they are operating on opposite poles of the construct.
Therefore, we maintain the sign-based partition to preserve the semantic integrity of the construct system. To address the reviewer's concern and clarify this rationale for future readers, we have added Remark 2 following Definition 4, explicitly justifying the psychological necessity of this mathematical discontinuity.
Response Comment 2: Mathematical Singularity at di = 0 and si = 0 (Ref: Formula 8)
We appreciate the reviewer's detailed examination of the boundary conditions at . The reviewer expresses concern that excluding these points creates a "black hole" where a "neutral/balanced" state fails to contribute to the similarity index, potentially leading to an undefined value ().
We respectfully clarify that, within the specific methodological framework of the Weighted Implication Grid (WimpGrid) used here, a score of zero () represents indefinition or non-applicability, rather than a psychological "neutral midpoint" or "balance." Consequently, a construct where both the Self and Ideal are undefined () represents a semantic void: the construct is simply not relevant to the subject's self-concept in either state. Therefore, it is theoretically consistent that such a pair does not contribute to the magnitude of similarity ().
Regarding the mathematical stability: while the set definitions () formally categorize these zeros into the distinctive sets, they do not distort the index because the magnitude functions (Definitions 6 and 7) rely on absolute values ( and ). Since , these attributes contribute exactly zero to the denominator. They do not act as discrepancies; they are effectively "silent," as intended.
The singularity () mentioned by the reviewer is only possible in the trivial and clinically non-existent case where an individual has no defined constructs (i.e., the entire grid is zero).
To prevent confusion and formalize this constraint, we have added a new Remark 4 following Definition 7, explicitly stating that the zero value denotes non-applicability.
Response Comment 3: Model Incompleteness: Undefined Parameters α and β (Ref: Formula 8)
The reviewer rightfully points out that leaving and as undefined free parameters compromises the reproducibility of the index as a standardized metric.
We would like to clarify two key aspects of our approach:
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Structural Diagnosis vs. Scalar Metric: The primary contribution of this work is not merely to output a single similarity score, but to analyze the SSI canonical surface (as detailed in Property 3 and Figure 5). The topology of this surface (governed by the structural coefficients and ) acts as a diagnostic map of the system's resilience, providing clinical insight regardless of the specific point chosen.
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Standardization: To address the need for reproducibility in comparative settings, we have introduced Remark 5. We propose the default values of . This specific weighting aligns the SSI with the mathematical logic of the Sørensen-Dice coefficient, which balances the magnitude of shared attributes against the average magnitude of the distinctive sets, preventing discrepancies from disproportionately skewing the index in standard assessments.
Finally, as discussed in Section 3.3 ("Limitations and Future Research Directions"), we are currently developing a probabilistic framework to empirically estimate these parameters for individual subjects. However, for the purpose of the current theoretical presentation, the default of ensures the model is fully defined and reproducible.
Response Comment 4: Methodological Circularity and Self-Citation (Ref: WimpGrid and PB Space)
We appreciate the opportunity to clarify the relationship between the proposed index and the elicitation instruments. We contend that there is a fundamental distinction between the mathematical model (SSI) and the data source (WimpGrid).
The validity of the SSI is not predicated on the WimpGrid structure, but on the algebraic properties of the feature-based comparison defined in Section 2. The WimpGrid was employed in this manuscript merely as an illustrative provider of the input vectors and . As we have clarified in the newly added Remark 1, the SSI is mathematically self-contained and instrument-agnostic. It can be immediately applied to standard Repertory Grid data by verifying that the input scores are normalized to the domain, a process we have now explicitly defined in the text.
Furthermore, regarding the concern about "PB Space," we emphasize that the core SSI formulation (Eq. 8) does not utilize PB Space metrics. References to PB Space are restricted to the Discussion section solely to contextualize future possibilities for hierarchical weighting. Therefore, the calculation of the SSI as presented is fully reproducible using standard psychometric data and is independent of the authors' other methodologies.
Response Comment 5: Background Context Enhancement
We are grateful for the reviewer's insightful suggestion to broaden the contextualization of our graph-structured formalism. We agree that the logic of deriving system properties from structural constraints has strong parallels in other scientific domains, specifically in systems engineering and network analysis.
Following the reviewer's recommendation, we have integrated a reference to Wang et al. (2026) in the Introduction (Section 1.1). We now explicitly draw a parallel between the global reliable diagnosis frameworks used in computer system sciences and our proposed method for diagnosing the structural stability of the self-system. This addition enriches the manuscript by highlighting that the mathematical modeling of diagnosability—whether in digital networks or psychological systems—relies on analogous structural principles.
Response Comment 6: Dimensional Analysis and Quadratic Decay (Ref: Formula 5)
The reviewer raises a profound theoretical point regarding the choice of the decay function, correctly noting that Shepard’s Universal Law suggests exponential decay for stimulus generalization. We have chosen to retain the quadratic decay () because it better reflects the psychological processing of self-evaluative judgments, as opposed to pure sensory discrimination. Psychologically, small distances from the ideal are not typically experienced as a loss of congruence; rather, they are cognitively assimilated as falling within the valid range of the ideal pole. The quadratic function captures this phenomenon through its vanishing derivative near zero, which creates a "plateau" of maximum similarity. This shape mathematically formalizes the concept of semantic tolerance, where minor fluctuations are treated as negligible. This approach is consistent with the standard formulation of membership functions in Fuzzy Set Theory, specifically the quadratic -functions described by Zimmermann (2001), which are explicitly designed to model graded concepts that possess a stable core of full membership before similarity begins to decline. We have added Remark 3 to the manuscript to ground this decision in both this psychological reality and its corresponding fuzzy mathematical formalization.
Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsI recommend the acceptance, thank you!
