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Boolean proportions

Christian Antić

TL;DR

The paper applies an abstract algebraic framework for analogical proportions to the boolean domain, exploring how boolean proportions behave under constants and logical operators. It defines a formal boolean domain, justifications, and a maximality criterion to determine when a proportion holds, then compares the results to Klein82 and Miclet09. The main finding is that the framework reproduces Klein's proportions in boolean domains with negation or no constants and Miclet's in other boolean domains, while full propositional logic reduces to the base boolean structure; negation and XOR interactions yield explicit equality/inequality detections and robust proportional properties. This demonstrates that a single, general framework can subsume multiple established boolean proportion models and provides a foundation for extending to boolean function and finite set proportions.

Abstract

The author has recently introduced an abstract algebraic framework of analogical proportions within the general setting of universal algebra. This paper studies analogical proportions in the boolean domain consisting of two elements 0 and 1 within his framework. It turns out that our notion of boolean proportions coincides with two prominent models from the literature in different settings. This means that we can capture two separate modellings of boolean proportions within a single framework which is mathematically appealing and provides further evidence for the robustness and applicability of the general framework.

Boolean proportions

TL;DR

The paper applies an abstract algebraic framework for analogical proportions to the boolean domain, exploring how boolean proportions behave under constants and logical operators. It defines a formal boolean domain, justifications, and a maximality criterion to determine when a proportion holds, then compares the results to Klein82 and Miclet09. The main finding is that the framework reproduces Klein's proportions in boolean domains with negation or no constants and Miclet's in other boolean domains, while full propositional logic reduces to the base boolean structure; negation and XOR interactions yield explicit equality/inequality detections and robust proportional properties. This demonstrates that a single, general framework can subsume multiple established boolean proportion models and provides a foundation for extending to boolean function and finite set proportions.

Abstract

The author has recently introduced an abstract algebraic framework of analogical proportions within the general setting of universal algebra. This paper studies analogical proportions in the boolean domain consisting of two elements 0 and 1 within his framework. It turns out that our notion of boolean proportions coincides with two prominent models from the literature in different settings. This means that we can capture two separate modellings of boolean proportions within a single framework which is mathematically appealing and provides further evidence for the robustness and applicability of the general framework.

Paper Structure

This paper contains 9 sections, 6 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Constants
  4. Negation
  5. Exclusive or
  6. Propositional Logic
  7. Comparison
  8. Future Work
  9. Conclusion

Key Result

Theorem 2.6

In case $t$ is injective in $\mathfrak B$, we have In this case, we call $t^\mathfrak B(c)$ a functional solution of $a: t^\mathfrak B(a)::c:x$ in $\mathfrak B$ characteristically justified by $x\to t(x)$.

Theorems & Definitions (16)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6: Functional Proportion Theorem
  • Theorem 2.7
  • proof
  • Theorem 3.1
  • proof : Proof of t:B
  • ...and 6 more