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Modal Logics -- RNmatrices vs. Nmatrices

Marcelo E. Coniglio, Paweł Pawłowski, Daniel Skurt

TL;DR

The paper investigates two non-deterministic semantic frameworks for modal logics—Nmatrices and RNmatrices—using the minimal logic $M$ and three normal extensions $MK$, $MKT$, and $MKT4$ as test cases. It builds four-valued Nmatrices from swap structures for $M$ and then derives corresponding MK, MKT, and MKT4 variants, including reduced value sets for the latter two. It then introduces restricted Nmatrices (RNmatrices) by restricting valuation sets to enforce modal axioms, showing that RNmatrices are generally stronger than Nmatrices and can handle a broader range of extensions, though they still face limitations with certain axioms and global rules relative to Kripke semantics. The paper concludes with a discussion of the scope, limits, and philosophical implications of these approaches, highlighting potential future work on decidability, hyperintensional logics, and connections to possible-world semantics.

Abstract

In this short paper we will discuss the similarities and differences between two semantic approaches to modal logics - non-deterministic semantics and restricted non-deterministic semantics. Generally speaking, both kinds of semantics are similar in the sense that they employ non-deterministic matrices as a starting point but differ significantly in the way extensions of the minimal modal logic M are constructed. Both kinds of semantics are many-valued and truth-values are typically expressed in terms of tuples of 0s and 1s, where each dimension of the tuple represents either truth/falsity, possibility/non-possibility, necessity/non-necessity etc. And while non-deterministic semantics for modal logic offers an intuitive interpretation of the truth-values and the concept of modality, with restricted non-deterministic semantics are more general in terms of providing extensions of M, including normal ones, in an uniform way. On the example of three modal logics, MK, MKT and MKT4, we will show the differences and similarities of those two approaches. Additionally, we will briefly discuss (current) restrictions of both approaches.

Modal Logics -- RNmatrices vs. Nmatrices

TL;DR

The paper investigates two non-deterministic semantic frameworks for modal logics—Nmatrices and RNmatrices—using the minimal logic and three normal extensions , , and as test cases. It builds four-valued Nmatrices from swap structures for and then derives corresponding MK, MKT, and MKT4 variants, including reduced value sets for the latter two. It then introduces restricted Nmatrices (RNmatrices) by restricting valuation sets to enforce modal axioms, showing that RNmatrices are generally stronger than Nmatrices and can handle a broader range of extensions, though they still face limitations with certain axioms and global rules relative to Kripke semantics. The paper concludes with a discussion of the scope, limits, and philosophical implications of these approaches, highlighting potential future work on decidability, hyperintensional logics, and connections to possible-world semantics.

Abstract

In this short paper we will discuss the similarities and differences between two semantic approaches to modal logics - non-deterministic semantics and restricted non-deterministic semantics. Generally speaking, both kinds of semantics are similar in the sense that they employ non-deterministic matrices as a starting point but differ significantly in the way extensions of the minimal modal logic M are constructed. Both kinds of semantics are many-valued and truth-values are typically expressed in terms of tuples of 0s and 1s, where each dimension of the tuple represents either truth/falsity, possibility/non-possibility, necessity/non-necessity etc. And while non-deterministic semantics for modal logic offers an intuitive interpretation of the truth-values and the concept of modality, with restricted non-deterministic semantics are more general in terms of providing extensions of M, including normal ones, in an uniform way. On the example of three modal logics, MK, MKT and MKT4, we will show the differences and similarities of those two approaches. Additionally, we will briefly discuss (current) restrictions of both approaches.
Paper Structure (14 sections, 5 theorems, 9 equations)

This paper contains 14 sections, 5 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. The minimal modal logic M
  3. Nmatrices for MK, MKT and MKT4
  4. MK
  5. MKT
  6. MKT4
  7. RNmatrices for MK, MKT and MKT4
  8. MK
  9. MKT
  10. MKT4
  11. Nmatrices vs. RNmatrices
  12. Normal modal logics
  13. Scope and limits of the methods
  14. Philosophical Remarks

Key Result

Theorem 1

For every $\Gamma \cup \{A\} \subseteq For(\Sigma)$ it holds: $\Gamma \vdash_{\mathcal{H}} A$ iff $\Gamma \vDash_{\mathcal{M}} A$.

Theorems & Definitions (8)

  • Theorem 1: Soundness and completeness of $\mathcal{H}$ w.r.t. $\mathcal{M}$
  • Theorem 2: Soundness and completeness of $\mathcal{H}_{\textsf{Ax}}$ w.r.t. the Nmatrix $\mathcal{M}_{\textsf{Ax}}$
  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Theorem 3: Soundness and completeness of $\mathcal{H}_{\textsf{K}}$
  • proof