## Internalizing Labelled Deduction (2000)

Venue: | Journal of Logic and Computation |

Citations: | 75 - 20 self |

### BibTeX

@ARTICLE{Blackburn00internalizinglabelled,

author = {Patrick Blackburn},

title = {Internalizing Labelled Deduction},

journal = {Journal of Logic and Computation},

year = {2000},

volume = {10},

pages = {2000}

}

### OpenURL

### Abstract

This paper shows how to internalize the Kripke satisfaction denition using the basic hybrid language, and explores the proof theoretic consequences of doing so. As we shall see, the basic hybrid language enables us to transfer classic Gabbay-style labelled deduction methods from the metalanguage to the object language, and to handle labelling discipline logically. This internalized approach to labelled deduction links neatly with the Gabbay-style rules now widely used in modal Hilbert-systems, enables completeness results for a wide range of rst-order denable frame classes to be obtained automatically, and extends to many richer languages. The paper discusses related work by Jerry Seligman and Miroslava Tzakova and concludes with some reections on the status of labelling in modal logic. 1 Introduction Modern modal logic revolves around the Kripke satisfaction relation: M;w ': This says that the model M satises (or forces, or supports) the modal formula ' at the state w in M....

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Citation Context ...luate" operators elsewhere. For example Prior [38] makes use of a T (s; ') construct in his second grade tense logic (this says, ' is true at the time s), the Situation Calculus of McCarthy and H=-=ayes [35]-=- makes use of the same idea, and such operators are characteristic of the Topological Logic of Rescher and Urquhart [40]. Moreover Allen [1] introduces a Holds(s; ') operator for temporal representati... |

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Citation Context ...: see [38, 13, 39, 36, 37, 26, 27, 28, 8, 9, 10]. 3 Internalizing labelled deduction We are now ready to internalize labelled deduction. Isrst do so using a Smullyanstyle unsigned tableau system (see =-=[46]-=-), which I will then reformulate as a sequent calculus. I doubt if it's possible to beat unsigned tableau systems for sheer simplicity. 6 To prove ' we test whether :' is a contradiction, and we do so... |

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Citation Context ...iminary list. 1 Internalizes the labelling mechanism The basic hybrid language obviously internalizes the apparatus of modal labelled deduction. Modal labelled deduction (which traces back to Fitting =-=[20]-=- and has been proposed as a systematic approach to deduction by Dov Gabbay; see in particular [21]) makes use of labels in the metalanguage to regulate the proof process. In such approaches, a modal f... |

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Citation Context ... use of re-entrancy marking tags in AVMs and other feature description formalisms (see, for example, [45]), the use of the A-box and the O operator (\one of") in description logics (see, for exam=-=ple, [19]-=-) and the Polish tradition of modal logics for rough sets (see, for example, [33]). Moreover, one recent approach to general modal proof theories (see [3]) combines modal languages with (parts of) the... |

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Citation Context ...ikely to be familiar from its metalinguistic role in labelled deduction, the reader has probably seen similar objectlevel \jump to the named state and evaluate" operators elsewhere. For example P=-=rior [38]-=- makes use of a T (s; ') construct in his second grade tense logic (this says, ' is true at the time s), the Situation Calculus of McCarthy and Hayes [35] makes use of the same idea, and such operator... |

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Citation Context ...f pspace-hardness is clear, for the basic hybrid language contains the satisability problem for ordinary propositional modal logic as a special case, and this problem is known to be pspace-hard (see [=-=3-=-4]). A pspace algorithm for basic hybrid satisability is given in [2]. So from a complexity-theoretic perspective, the basic hybrid language is oering us something for nothing | but what exactly? Esse... |

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Citation Context ...id on F and write Fs'. If ' 1 Other suggestive terms include: jump operator, situating operator, localizing operator, and supports operator. Another term is at operator, and i : ' is written @ i ' in =-=[8, 11, 12, 2]-=-. 4 is valid on all frames we call it a validity and writes'. These concepts are extended to sets of formulas in the obvious way. It should now be clear that the basic hybrid language internalizes the... |

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Citation Context ...is also true. For example, the Church-Rosser property is denable in the basic modal language by 32p ! 23p, but no pure formula denes this property. For further information on frame denability, see [23=-=-=-], [6], [8] and [2]. 7 denability examples, treating labels as formulas has important consequences for expressivity. We have not yet begun to bind labels (many of the formalisms mentioned above allow ... |

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Citation Context ...in the presence of nominals or bindable nominals see [41, 42, 43, 44, 8, 11, 12, 48, 2]. There are also papers on hybrid languages containing nominals but no satisfaction operators: see in particular =-=[37, 24, 25, 23, 6, 17, 1-=-8]; some of these papers use the universal modality (or the more powerful dierence operator) to simulate the eect of satisfaction operators, something Prior himself did. 5 Finally, there is a long tra... |

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Citation Context ...o true. For example, the Church-Rosser property is denable in the basic modal language by 32p ! 23p, but no pure formula denes this property. For further information on frame denability, see [23], [6], [8] and [2]. 7 denability examples, treating labels as formulas has important consequences for expressivity. We have not yet begun to bind labels (many of the formalisms mentioned above allow this)... |

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Citation Context ...faction operators, something Prior himself did. 5 Finally, there is a long tradition of work which explores hybrid languages in which it is possible to bind nominals using the quantiers 8 and 9: see [=-=38, 13, 39, 36, 37, 26, 27, 28, 8, 9, 10]-=-. 3 Internalizing labelled deduction We are now ready to internalize labelled deduction. Isrst do so using a Smullyanstyle unsigned tableau system (see [46]), which I will then reformulate as a sequen... |

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Citation Context ...id on F and write Fs'. If ' 1 Other suggestive terms include: jump operator, situating operator, localizing operator, and supports operator. Another term is at operator, and i : ' is written @ i ' in =-=[8, 11, 12, 2]-=-. 4 is valid on all frames we call it a validity and writes'. These concepts are extended to sets of formulas in the obvious way. It should now be clear that the basic hybrid language internalizes the... |

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Citation Context ...faction operators, something Prior himself did. 5 Finally, there is a long tradition of work which explores hybrid languages in which it is possible to bind nominals using the quantiers 8 and 9: see [=-=38, 13, 39, 36, 37, 26, 27, 28, 8, 9, 10]-=-. 3 Internalizing labelled deduction We are now ready to internalize labelled deduction. Isrst do so using a Smullyanstyle unsigned tableau system (see [46]), which I will then reformulate as a sequen... |

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Citation Context ...] :s : t : ' :t : ' [: : ] That is, given a pair of nested satisfaction operators, the outermost can be discarded. Now for the most interesting rules; those dealing with the modalities: s : 3' s : 3a =-=[3]-=- :s : 3' s : 3t :t : ' [:3] a : ' s : 2' s : 3t t : ' [2] :s : 2' s : 3a [:2] :a : ' 9 Note that the 3-rule and :2-rules are stated using the metavariable a. This means that when we apply these rules ... |

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Citation Context ... the Topological Logic of Rescher and Urquhart [40]. Moreover Allen [1] introduces a Holds(s; ') operator for temporal representation in AI (this says that ' holds at the interval named s), Cresswell =-=[-=-16] makes use of satisfaction operators for applications in natural language semantics, van Benthem [4] introduces a goto s operator, and Buvac, Buvac and Mason [14] have explored similar ideas to den... |

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Citation Context ...ds for Situation Theory (see [41, 42, 43]). In more recent work, Seligman (see [44]) has investigated what I call mixed systems, an idea that has been independently explored by Miroslava Tzakova (see =-=[48]-=-). Mixed methods are an interesting alternative to full internalization, and I shall discuss them in Section 8. Two other recent approaches are worth noting: Demri's sequent systems for nominal tense ... |

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Citation Context ...(for example, the use of state descriptions in Carnap [15]) giving way to moresexible accounts (notably the relativized semantics of Kripke [31, 32], and a decade later the general frames of Thomason =-=[47]-=-). In fact most of the semantical freedom we currently enjoy stems either from the Kripkean parameter (the idea that by making the transition relation R explicit and varying its properties we can cont... |

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Citation Context ...out proofs in either system without diculty. 2 Internalized labelled deduction does not originate in this paper: it traces back to Jerry Seligman's work on deduction methods for Situation Theory (see =-=[41, 42, 43]-=-). In more recent work, Seligman (see [44]) has investigated what I call mixed systems, an idea that has been independently explored by Miroslava Tzakova (see [48]). Mixed methods are an interesting a... |

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Citation Context ... (see, for example, [45]), the use of the A-box and the O operator (\one of") in description logics (see, for example, [19]) and the Polish tradition of modal logics for rough sets (see, for exam=-=ple, [33]-=-). Moreover, one recent approach to general modal proof theories (see [3]) combines modal languages with (parts of) their correspondence languages, thereby allowing the object language access to label... |

19 | Nominal Tense Logic and Other Sorted Intensional Frameworks - Blackburn - 1990 |

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Citation Context ...faction operators, something Prior himself did. 5 Finally, there is a long tradition of work which explores hybrid languages in which it is possible to bind nominals using the quantiers 8 and 9: see [=-=38, 13, 39, 36, 37, 26, 27, 28, 8, 9, 10]-=-. 3 Internalizing labelled deduction We are now ready to internalize labelled deduction. Isrst do so using a Smullyanstyle unsigned tableau system (see [46]), which I will then reformulate as a sequen... |

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Citation Context ...lization, and I shall discuss them in Section 8. Two other recent approaches are worth noting: Demri's sequent systems for nominal tense logic (see [17]), and the Demri and Gore display calculi (see [=-=18-=-]) for nominal tense logic enriched with the dierence operator. These papers don't make direct use of satisfaction operators, but many of the underlying ideas are similar to those discussed here. Also... |

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Citation Context ...ator for temporal representation in AI (this says that ' holds at the interval named s), Cresswell [16] makes use of satisfaction operators for applications in natural language semantics, van Benthem =-=-=-[4] introduces a goto s operator, and Buvac, Buvac and Mason [14] have explored similar ideas to dene logics of context. And I doubt if this is even close to being an exhaustive list: satisfaction ope... |

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Citation Context ...thods are an interesting alternative to full internalization, and I shall discuss them in Section 8. Two other recent approaches are worth noting: Demri's sequent systems for nominal tense logic (see =-=[1-=-7]), and the Demri and Gore display calculi (see [18]) for nominal tense logic enriched with the dierence operator. These papers don't make direct use of satisfaction operators, but many of the underl... |

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Citation Context ... of nested diamonds. 8 One exception to this is the work of the Soa school who make use of a rule called COV in modal languages containing nominals; for further discussion of COV see [36], [23], and [=-=9]-=-. 14 And now 3L rule is staring us in the face. First, let's get rid of of the ` symbols, turn the material implication arrow ! into the sequent arrow !, and expand our 3 a abbreviation. This yields: ... |

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Citation Context |

12 | Determinism and looping - Gargov, Passy - 1988 |

10 | A cut-free sequent calculus for elementary situated reasoning
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Citation Context ...out proofs in either system without diculty. 2 Internalized labelled deduction does not originate in this paper: it traces back to Jerry Seligman's work on deduction methods for Situation Theory (see =-=[41, 42, 43]-=-). In more recent work, Seligman (see [44]) has investigated what I call mixed systems, an idea that has been independently explored by Miroslava Tzakova (see [48]). Mixed methods are an interesting a... |

9 |
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Citation Context |

9 |
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Citation Context ...r for some time. As well as the hybrid logic and labelled deductive system traditions, there is the use of re-entrancy marking tags in AVMs and other feature description formalisms (see, for example, =-=[45]), th-=-e use of the A-box and the O operator (\one of") in description logics (see, for example, [19]) and the Polish tradition of modal logics for rough sets (see, for example, [33]). Moreover, one rec... |

6 |
An Irre Lemma
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(Show Context)
Citation Context ...rovides a transparent link between labelled deduction methods and the Gabbay-style proof rules that have played an increasingly important role in modal Hilbert systems over the last twenty years (see =-=[22]-=-). As I shall show, in the basic hybrid language the basic Gabbay-style proof rule simply is the crucial sequent rule for internalized labelled deduction. Indeed, this observation was the starting poi... |

6 |
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Citation Context |

5 |
Hybrid languages and temporal logic (full version
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Citation Context ...ms has been an important motivation for introducing nominals, and a variety of analogous results have been proved for Hilbert systems in various hybrid languages; see [13], [37], [25], [23], [11] and =-=[12-=-]. 21 Pure is globally satised in M, and every state in F is the denotation of some nominal in S under V . Then FsPure. Proof Suppose for the sake of a contradiction that F 6sPure. Then for some stand... |

5 | Situated consequence for elementary situation theory
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(Show Context)
Citation Context ...out proofs in either system without diculty. 2 Internalized labelled deduction does not originate in this paper: it traces back to Jerry Seligman's work on deduction methods for Situation Theory (see =-=[41, 42, 43]-=-). In more recent work, Seligman (see [44]) has investigated what I call mixed systems, an idea that has been independently explored by Miroslava Tzakova (see [48]). Mixed methods are an interesting a... |

4 | Proof theory for contextual reasoning - Seligman - 1998 |

3 |
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(Show Context)
Citation Context ...ogic enriched with the dierence operator. These papers don't make direct use of satisfaction operators, but many of the underlying ideas are similar to those discussed here. Also relevant is Hasle's [=-=29]-=- HOL implementation of Prior's third grade tense logic. 2 Internalizing Kripke semantics Let's start by recalling the syntax and semantics of ordinary propositional modal logic. Given a set of proposi... |

2 |
Towards a general theory of knowledge and action
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(Show Context)
Citation Context ... the time s), the Situation Calculus of McCarthy and Hayes [35] makes use of the same idea, and such operators are characteristic of the Topological Logic of Rescher and Urquhart [40]. Moreover Allen =-=[1]-=- introduces a Holds(s; ') operator for temporal representation in AI (this says that ' holds at the interval named s), Cresswell [16] makes use of satisfaction operators for applications in natural la... |

2 |
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Citation Context ...rs, and that is what modal logic is all about. The history of modal logic is essentially the story of relatively rigid forms of semantic analysis (for example, the use of state descriptions in Carnap =-=[15]-=-) giving way to moresexible accounts (notably the relativized semantics of Kripke [31, 32], and a decade later the general frames of Thomason [47]). In fact most of the semantical freedom we currently... |

1 |
and T.Tinchev
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- 1987
(Show Context)
Citation Context ...in the presence of nominals or bindable nominals see [41, 42, 43, 44, 8, 11, 12, 48, 2]. There are also papers on hybrid languages containing nominals but no satisfaction operators: see in particular =-=[37, 24, 25, 23, 6, 17, 1-=-8]; some of these papers use the universal modality (or the more powerful dierence operator) to simulate the eect of satisfaction operators, something Prior himself did. 5 Finally, there is a long tra... |