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2010 | OriginalPaper | Buchkapitel

5. Survey of Modal Logics

verfasst von : Dr. Andrzej Indrzejczak

Erschienen in: Natural Deduction, Hybrid Systems and Modal Logics

Verlag: Springer Netherlands

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Abstract

This Chapter provides a necessary background for studying applications of ND in modal logics. It is a collection of basic facts needed for understanding of the remaining chapters. Section 5.1. introduces propositional languages of multimodal logics and establishes notational conventions. After a presentation of general taxonomy of modal logics in Section 5.2. we characterize them axiomatically in the next section. Section 5.4. introduces relational semantics for different families of modal logics. Except standard Krikpe’s semantics it contains the basics of neighborhood semantics for weak modal logics. Some attention is paid to correspondence theory and some general schemata investigated later. After short section on completeness and decidability matters we finally present various kinds of first-order modal logic in Section 5.6.

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Fußnoten
1
In such case we read a formula \(\Box_a\varphi \)a knows that \(\varphi \) holds.
 
2
Cf. also [59] and [208].
 
3
Usually elements of this class are defined in a bit different (but equivalent) way as modal logics containing (K) and closed under (RG).
 
4
We follow Chellas [69] in naming conventions (although he used the name classical instead of congruent logics). Often different names are applied, particularly for regular logics, e.g. instead of R, C is used by Segerberg [246] and Fitting [93], and C2 by Lemmon [173].
 
5
The exception is Gasquet [106], where multimodal regular and monotonic logics are dealt with.
 
6
This result was proved by Gasquet and Herzig [107] and improved by Kracht and Wolter [167] and by Hansen [124].
 
7
Hansen [124] gives a couple of examples concerning interesting applications of monotonic logics.
 
8
Cf. e.g. [177, 116].
 
9
The names of axioms – with little exceptions – come from [117].
 
10
It may be of interest to consider also other operations on modalities. Resolution and labelled tableau systems for some logics of this kind are provided by De Nivelle, Schmidt and Hustadt [196, 242].
 
11
It does not mean that deduction theorem in weaker form does not hold for these logics. Many theorems of this kind were established by Perzanowski in [206, 207].
 
12
This is a particular exemplification of distinctions introduced in Chapter 1.
 
13
Detailed history of these early investigations may be found in [71].
 
14
One should note however that M is adequately characterized in terms of multimodal frames with countably many accessibility relations – cf. [93].
 
15
Of course, if we drop this condition from the definition of modal logic, we may characterize logics in terms of models – the content of every model is a modal logic in this sense (indeed normal logic). Such more general approach is presented e.g. in [112].
 
16
In fact, this condition is first-order definable in reflexive and transitive frames.
 
17
We omit frame consequence relation which is not recursively characterizable cf. e.g. [93].
 
18
By the way it is the only acceptable solution if we want to provide semantics for Lewis’ approach – in fact it must be strengthened to the effect that the intersection of two different domains is empty.
 
19
cf. [105].
 
20
Probably the first formulation is the famous Plutarchus’ story of Theseus’ boat where, through the centuries, every piece of wood and metal was replaced with a new one by grateful Athenians.
 
21
The literature on these questions is enormous; one may consult [96] for readable account.
 
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Metadaten
Titel
Survey of Modal Logics
verfasst von
Dr. Andrzej Indrzejczak
Copyright-Jahr
2010
Verlag
Springer Netherlands
DOI
https://doi.org/10.1007/978-90-481-8785-0_5