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Capturing dual team properties with inclusion atoms

Matilda Häggblom

Abstract

We introduce propositional team-based logics expressively complete for (quasi) downward and (quasi) upward closed properties in a syntactically dual way, by using variants of the inclusion atom. In particular, the variants of the primitive inclusion atoms used in the (quasi) upward closed setting have equivalent formulas using variants of the might modality. The duality is visible in the logics' normal forms, mirroring the duality between the (quasi) upward and downward closed settings, where the quasi variants take special care of the empty and full team. Furthermore, we defined sound and complete natural deduction systems for each logic.

Capturing dual team properties with inclusion atoms

Abstract

We introduce propositional team-based logics expressively complete for (quasi) downward and (quasi) upward closed properties in a syntactically dual way, by using variants of the inclusion atom. In particular, the variants of the primitive inclusion atoms used in the (quasi) upward closed setting have equivalent formulas using variants of the might modality. The duality is visible in the logics' normal forms, mirroring the duality between the (quasi) upward and downward closed settings, where the quasi variants take special care of the empty and full team. Furthermore, we defined sound and complete natural deduction systems for each logic.
Paper Structure (8 sections, 21 theorems, 27 equations, 4 tables)

This paper contains 8 sections, 21 theorems, 27 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Upward and downward closed logics
  3. Quasi upward closed logic
  4. Upward closed logic
  5. Quasi downward closed logic
  6. Downward closed logic
  7. Symmetry and might modalities
  8. Conclusion and future work

Key Result

Proposition 2.2

The logic $\mathcal{L}_{qu}$ is quasi upward closed, $\mathcal{L}_u$ is upward closed, $\mathcal{L}_{qd}$ is quasi downward closed, and $\mathcal{L}_d$ is downward closed.

Theorems & Definitions (43)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.6
  • proof
  • Theorem 2.7
  • ...and 33 more