Linear Measurement Models-Axiomatizations and Axiomatizability

Area of Science:

• Measurement theory
• Mathematical psychology
• Theoretical computer science

Background:

• Ordinal data contexts require robust representation theorems.
• Existing models, like Scott's, provide a foundation for linear measurement.
• Generalizing these models is crucial for broader applicability.

Purpose of the Study:

• To examine axiomatizations and axiomatizability of linear and bilinear representations.
• To modify and generalize Scott's characterization of finite linear measurement models.
• To investigate the finite axiomatizability of finite linear measurement models.

Main Methods:

• Representational measurement theory
• Model-theoretic methods
• Axiomatization techniques

Main Results:

• New representation theorems are presented, generalizing Scott's work.
• These theorems utilize a finite number of axioms dependent on the data context size.
• Finite linear measurement models are demonstrated to be not finitely axiomatizable using first-order logic.

Conclusions:

• The study provides efficient, finite axiomatizations for specific measurement models.
• It highlights fundamental limitations in the first-order axiomatizability of finite linear measurement models.
• The findings advance the theoretical understanding of measurement and representation in mathematical contexts.