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Supplement to Temporal Logic

Supplement: Interdefinability of HS Modalities

  • In the semantics without point-intervals, the six modalities ⟨A⟩, ⟨B⟩, ⟨E⟩, ⟨¯A⟩, ⟨¯B⟩, ⟨¯E⟩ suffice to express all others, as shown by the following equivalences:

    ⟨L⟩φ≡⟨A⟩⟨A⟩φ,⟨¯L⟩φ≡⟨¯A⟩⟨¯A⟩φ,⟨D⟩φ≡⟨B⟩⟨E⟩φ,⟨¯D⟩φ≡⟨¯B⟩⟨¯E⟩φ,⟨O⟩φ≡⟨E⟩⟨¯B⟩φ,⟨¯O⟩φ≡⟨B⟩⟨¯E⟩φ.
  • In the semantics with point intervals, the four modalities ⟨B⟩, ⟨E⟩, ⟨¯B⟩, ⟨¯E⟩ suffice to express all others, as shown by the following equivalences: ⟨A⟩φ≡([E]⊥∧(φ∨⟨¯B⟩φ))∨⟨E⟩([E]⊥∧(φ∨⟨¯B⟩φ)),⟨¯A⟩φ≡([B]⊥∧(φ∨⟨¯E⟩φ))∨⟨B⟩([B]⊥∧(φ∨⟨¯E⟩φ)),⟨L⟩φ≡⟨A⟩(⟨E⟩⊤∧⟨A⟩φ),⟨¯L⟩φ≡⟨¯A⟩(⟨B⟩⊤∧⟨¯A⟩φ),⟨D⟩φ≡⟨B⟩⟨E⟩φ,⟨¯D⟩φ≡⟨¯B⟩⟨¯E⟩φ,⟨O⟩φ≡⟨E⟩(⟨E⟩⊤∧⟨¯B⟩φ),⟨¯O⟩φ≡⟨B⟩(⟨B⟩⊤∧⟨¯E⟩φ).

    Also, the modal constant π is definable in terms of ⟨B⟩ and ⟨E⟩, as [B]⊥ or [E]⊥, respectively.

Copyright © 2024 by
Valentin Goranko <valentin.goranko@philosophy.su.se>
Antje Rumberg <antje.rumberg@uni-tuebingen.de>

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