Abstract
In this chapter we introduce intuitionistic hybrid logic and its proof-theory. Intuitionistic hybrid logic is hybrid modal logic over an intuitionistic logic basis instead of a classical logical basis. The chapter is structured as follows. In the first section of the chapter we introduce intuitionistic hybrid logic (this is taken from Braüner and de Paiva (2006)). In the second section we introduce a natural deduction system for intuitionistic hybrid logic (taken from Braüner and de Paiva (2006)) and in the third and fourth sections we introduce axiom systems for intuitionistic and paraconsistent hybrid logic (taken from Braüner and de Paiva (2006)). In the last section we discuss certain other work, namely a Curry-Howard interpretation of intuitionistic hybrid logic.
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Notes
- 1.
The fact that in the intuitionistic semantics based on Definition 8.1, nominals are interpreted using a family \(\{ \sim_{w} \}_{w \in W}\) of equivalence relations, not identity, seems to imply that in an equivalent many-valued semantics, nominals should be allowed to take on arbitrary truth-values, not just top and bottom.
- 2.
In the model for intuitionistic hybrid logic, and in the model for intuitionistic first-order logic as well, there is a set D w for each state of knowledge w (subject to the monotonicity requirement that \(D_{w} \subseteq D_{v}\) whenever \(w \leq v\)). One might instead consider having a constant set like in the constant domain semantics for first-order modal logic, cf. in Section 6.1.1. In Görnemann (1971) such a constant semantics for intuitionistic first-order logic was axiomatized by adding to an axiom system for the usual semantics the axiom \(\forall a (\phi \vee \psi) \rightarrow (\phi \vee \forall a \psi)\), where the variable a does not occur in φ. It would be interesting to investigate whether such a constant version of intuitionistic hybrid logic could be given (whatever proof-theoretic machinery is chosen, it is not clear whether the completeness proof of Section 8.2.3 can be modified as appropriate).
- 3.
It turns out that the set of truth-values equipped with both of the orderings ⊆ and ≤ constitutes what is called a bilattice. See Fitting (2006a) where it is demonstrated that bilattices naturally generalize a number of truth-value spaces. It would be interesting to investigate whether the results of the present section can be generalized to other bilattices than the described four-valued bilattice.
- 4.
For a paper also having the aim being of providing a foundation for distributed functional programming languages, but using a natural deduction system for intuitionistic \(\textsf{S5}\) without satisfaction operators, see Murphy et al. (2004).
- 5.
Announced by D. Galmiche (personal communication).
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Braüner, T. (2011). Intuitionistic Hybrid Logic. In: Hybrid Logic and its Proof-Theory. Applied Logic Series, vol 37. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0002-4_8
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