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Relating Semantics as Fine-Grained Semantics for Intensional Logics

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Logic in High Definition

Part of the book series: Trends in Logic ((TREN,volume 56))


This text has a programmatic and introductory character. In the paper, we outline a fine-grained semantics for intensional logics. The fundamental idea of the semantics is that the logical value of a given complex proposition is the result of two things: a valuation of propositional variables supplemented with a valuation of relation between the main components of this complex proposition. The latter thing is a formal representation of intensionality that emerges from the connection of several simpler propositions into one more complex proposition. In the first part of the paper, we present some linguistic motivations for the semantics. Later, we propose a very general, multi-valued view on relating semantics, and, in a more detailed way, we consider its two-valued specification, referring also to its historical applications and origin. A further generalization is made when we combine relating semantics with possible world semantics in the subsequent part. The paper concludes with a proposal of defining intensional operators as secondary notions that are based on relating connectives. By dint of the proposal, we can control the behavior of the operators by changing properties of semantic structures for the relating connectives that we use in the definitions.

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  1. 1.

    For the more extensive explanation and introduction of the notions given below see [7].

  2. 2.

    More about the issue of connections between both consequence relations can be found in [7].

  3. 3.

    In [4] multi-relating models similar to models (2\(*\)), but only two-valued, were suggested.

  4. 4.

    The question about necessary conditions is a crucial part of the research on the foundations of relating semantics that we already conduct. Both problems — sufficient and necessary conditions — are two sides of the correspondence theory we propose. However, these issues need further, less programmatic articles. And we have already some solutions to the problems.

  5. 5.

    It may sound strange to talk about power of expression in the context of semantics, because this term is usually used in reference to a syntax, to some formal language. Here, we intensionally refer it to semantics, as we think that also logical semantics can be compared in respect with which logical systems can be determined by those logical semantics. Maybe a better term would be power of determining.

  6. 6.

    Our approach is similar to some interpretations of obligation operator in deontic logic, where \(\textsf {O}\) is treated as a secondary logical notion defined by an implication and a specific constant (see [1, 2, 5, 13]). However, here we use the relating implication.

  7. 7.

    For example, non-validity of \(\textsf {K}\)-distribution might be helpful in avoiding Fitch’s paradox.

  8. 8.

    It has to be mentioned that some activities in this scope have been performed in an attempt to adapt relatedness logic to first-order logic [12]. However, as in the context of propositional logic, this is an interesting, but a special case of relating logic only with regard to the connective of implication and content-related problem.


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The research presented in the following article was financed by the National Science Centre, Poland, grant No.: UMO-2015/19/B/HS1/02478. However, the author would also like to express his gratitude to the doctoral students under his supervision, especially to Mateusz Klonowski, for the discussions, inspirations and for the joint exploration of the field of relating logics.

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Correspondence to Tomasz Jarmużek .

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Jarmużek, T. (2021). Relating Semantics as Fine-Grained Semantics for Intensional Logics. In: Giordani, A., Malinowski, J. (eds) Logic in High Definition. Trends in Logic, vol 56. Springer, Cham.

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