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PreprintCC BY-SA 4.0arXiv

Modalities in non-classical variations of S4

  • Leonardo Pacheco
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Abstract

A classical result in modal logic states that S4 has $14$ modalities, that is, every sequence of negations, boxes, and diamonds is equivalent to one in a set of $14$ such sequences. We study analogous results for the non-classical analogues CS4, IS4, GS4, and GS4c of S4. First, we show that, while all these logics have finitely many {Box,Diamond}- and {¬,Box}-modalities, the logic CS4 has infinitely many {¬,Diamond}-modalities. Second, we show that IS4 and GS4 have finitely many {¬,Diamond}-modalities, but they have infinitely many {¬,Box,Diamond}-modalities. At last, we show that GS4c has finitely many {¬,Box,Diamond}-modalities.

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Cite

BibTeXRIS
BibTeX
@misc{pacheco2026modalities,
  title = {{Modalities in non-classical variations of $\mathsfS4$}},
  author = {Leonardo Pacheco},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00736},
  eprint = {2609.00736},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00736},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Modalities in non-classical variations of S4
AU  - Leonardo Pacheco
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00736
UR  - https://arxiv.org/abs/2609.00736
AB  - A classical result in modal logic states that S4 has $14$ modalities, that is, every sequence of negations, boxes, and diamonds is equivalent to one in a set of $14$ such sequences. We study analogous results for the non-classical analogues CS4, IS4, GS4, and GS4^c of S4. First, we show that, while all these logics have finitely many {Box,Diamond}- and {¬,Box}-modalities, the logic CS4 has infinitely many {¬,Diamond}-modalities. Second, we show that IS4 and GS4 have finitely many {¬,Diamond}-modalities, but they have infinitely many {¬,Box,Diamond}-modalities. At last, we show that GS4^c has finitely many {¬,Box,Diamond}-modalities.
ER  -