Abstract
In this chapter we introduce the proof-theory of propositional hybrid logic. The chapter is structured as follows. In the first section of the chapter we sketch the basics of natural deduction systems and in the second section we introduce a natural deduction system for hybrid logic. In the third section we sketch the basics of Gentzen systems and in the fourth section we introduce a Gentzen system corresponding to the natural deduction system for hybrid logic. In the fifth section we give an axiom system for hybrid logic. The natural deduction system and the Gentzen system are taken from Braüner (2004a) whereas the axiom system is taken from Braüner (2006).
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Notes
- 1.
Instead of annotating assumptions and rule-instances with numbers, the discharging of parcels could have been recorded by placing a list of the undischarged parcels at every stage in the derivation. That is, instead of a formula φ in a derivation, we have a sequent \(\psi_{1}, \ldots, \psi_{m} \vdash \phi\) where the formulas in the list \(\psi_{1}, \ldots, \psi_{m}\) correspond to the undischarged parcels at the stage in question. Such sequents should not be confused with sequents in a Gentzen system.
- 2.
This was pointed out to the author by Balder ten Cate (personal communication).
- 3.
Labelled systems have the labelling machinery at the metalevel, whereas hybrid-logical systems have machinery with similar effect at the object level. A third option is chosen in Fitting (1972b) where a curious modal-logical axiom system is given in which labelling machinery is incorporated directly into the object language itself. In that system sequences of formulas of ordinary modal logic, delimited by a distinguished symbol *, are used as names for possible worlds. To be more specific, a sequence \(*\lozenge \phi_{1}, \ldots, \lozenge \phi_{n},\lozenge \phi_{n+1}*\) is used as the name of a world accessible from the world named by \(*\lozenge \phi_{1}, \ldots, \lozenge \phi_{n}*\) and in which the formula \(\phi_{n+1}\) is true, if there is one. It is allowed to form object language formulas by prefixing ordinary modal-logical formulas with such sequences. Intuitively, a prefixed formula \(*\lozenge \phi_{1}, \ldots, \lozenge \phi_{n}* \psi\) says that the formula ψ is true at the world named by the prefix. Prefixed formulas can be combined using the usual connectives of classical logic.
- 4.
Instead of using sets of formulas, it is possible to use multisets or lists of formulas. In some cases this is more convenient for combinatorial manipulation, but the cost is that rules have to be added to make the multisets or lists behave as sets: In the case of multisets, contraction rules have to be added (allowing formulas to be copied) and in the case of lists, exchange rules also have to be added (allowing formulas to be permuted).
- 5.
Note that this is different from natural deduction systems where the subformula property is guaranteed if a derivation has a certain form, namely if it is in normal form.
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Braüner, T. (2011). Proof-Theory of Propositional 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_2
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