Abstract
Relating logics are examples of non-classical logics that make it possible to seriously consider various non-logical relationships that occur between sentences. The main idea behind such logics is that logical values is not the only thing that matters when we consider truth conditions for formulas built by propositional connectives. What should be considered as another important factor is a content relationship or some other relations, such as causality and temporal successions which hold between sentences or between the states of affairs that are usually expressed by means of those sentences. Hence, the factor is in fact intensional. In the article, we analyze relating logics and make a philosophical introduction to this subject; we introduce a relating language and its general semantic framework. It is a short introduction to the field of relating logics. Next, we describe some special cases of relating logics from a semantic point of view, and finally, as a decision procedure we introduce adequate tableau systems for those logics.
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Notes
- 1.
Epstein in [3, pp. 61–84, 115–143] considers relating connectives on the grounds of the following logics: relatedness logics \(\mathrm {S}\) and \(\mathrm {R}\), dependence logics \(\mathrm {D}\), \(\mathrm {dD}\), \(\mathrm {Eq}\) and \(\mathrm {DPC}\).
- 2.
More about the ideas of relating semantics (generalizations, historical issues, applications etc.) can be read in [9].
- 3.
The letter
in notation of relating connectives comes from the Polish words ‘
iązać’ (verb) and ‘
iążący’ (adjective) which might be translated as ‘relate’ and ‘relating’. - 4.
It is possible to express in the considered language purely intensional functors. For instance, we can examine a functor defined in the following way
. We can easily check that 
iff 
. Let us also notice that it is possible to consider unary relating functors. For instance, a kind of a relating negation (the relating counterpart of Boolean negation) could be defined as 
, where \(\perp \) is interpreted in the standard way. These issues, however, require a separate analysis that goes beyond the scope of this article. - 5.
Surely, if \(\mathbf {Q}= \emptyset \), then \(\models _{\mathbf {Q}}A\) and \(\varSigma \models _{\mathbf {Q}}A\), for all \(\varSigma \cup \{A\} \subseteq \mathsf {For}\). But we would not like to discuss the trivial logic here. The very similar situation is in Definition 5.
- 6.
The name of logic \(\mathrm {W}\) comes from the Polish words ‘
iązać’ (verb) and ‘
iążący’ (adjective) which might be translated as ‘relate’ and ‘relating’ (cf. footnote 3). - 7.
We write ‘heuristically’, since precise definitions of these distinctions have not been worked out yet.
in notation of relating connectives comes from the Polish words ‘
iązać’ (verb) and ‘
iążący’ (adjective) which might be translated as ‘relate’ and ‘relating’.
. We can easily check that 
iff 
. Let us also notice that it is possible to consider unary relating functors. For instance, a kind of a relating negation (the relating counterpart of Boolean negation) could be defined as 
, where 
iązać’ (verb) and ‘
iążący’ (adjective) which might be translated as ‘relate’ and ‘relating’ (cf. footnote 3).References
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Acknowledgements
The research of Tomasz Jarmużek presented in the following article was financed by the National Science Centre, Poland, grant No.: UMO-2015/19/B/HS1/02478. While the research of Mateusz Klonowski presented in the following article was financed by the National Science Centre, Poland, grant No.: UMO-2015/19/N/HS1/02401. The authors also would like to thank the anonymous referee for all his remarks and interesting suggestions.
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Jarmużek, T., Klonowski, M. (2021). Some Intensional Logics Defined by Relating Semantics and Tableau Systems. In: Giordani, A., Malinowski, J. (eds) Logic in High Definition. Trends in Logic, vol 56. Springer, Cham. https://doi.org/10.1007/978-3-030-53487-5_3
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