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Normatively Determined Propositions

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Diagrammatic Representation and Inference (Diagrams 2022)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13462))

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Abstract

In the present work we provide a logical analysis of normatively determined and non-determined propositions. The normative status of these propositions depends on their relation with another proposition, here named reference proposition. Using a formal language that includes a monadic operator of obligation, we define eight dyadic operators that represent various notions of “being normatively (non-)determined”; then, we group them into two families, each forming an Aristotelian square of opposition. Finally, we show how the two resulting squares can be combined to form an Aristotelian cube of opposition.

Matteo Pascucci was supported by the Štefan Schwarz Fund for the project “A fine-grained analysis of Hohfeldian concepts” (2020–2023) and by the VEGA project no. 2/0125/22 “Responsibility and modal logic”. The article results from a joint work of the two authors.

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Notes

  1. 1.

    More precisely, “being determined” is a dyadic notion of non-contingency. An axiomatic characterization of dyadic non-contingency has been recently proposed by Pizzi [13]. For more on the logic of (non-)contingency, see Cresswell [2].

  2. 2.

    Since “being normatively determined” is a bilateral notion, in examples (1)-(3), if a proposition B is normatively (non-)determined by a proposition A, then so is \(\lnot B\). For instance, in example (2) both the proposition that one pays via a bank transfer and the proposition that one does not pay via a bank transfer are normatively determined by the reference proposition that one pays in advance, since the latter excludes one of the two alternatives and forces the other. Furthermore, we highlight that our analysis covers also cases of (non-)determination with respect to forbidden propositions, as long as one defines “A is forbidden” as “\(\lnot A\) is obligatory” (\(\square \lnot A\)).

  3. 3.

    In normal modal systems \(\triangledown (A,B)\) boils down to \(\lozenge (A \wedge B) \wedge \lozenge (A \wedge \lnot B)\).

  4. 4.

    We assume familiarity with the meaning of the Aristotelian relations at issue. For details, see Pizzi [11, 12].

  5. 5.

    For details, see Åqvist [1]. For Aristotelian squares built on non-normal modal systems, see Demey [3].

  6. 6.

    As observed by Pizzi [13], the notion of absolute (non-)determination may be defined in terms of dyadic (non-)determination by replacing the reference proposition with a tautology \(\top \). For instance: \({\vartriangle }(\top ,B)\).

  7. 7.

    Thus, a semiaristotelian \(\mathbf {S}\)-square is a square whose edges are associated with some of the relations holding between the edges of an Aristotelian \(\mathbf {S}\)-square.

References

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Authors and Affiliations

  1. Department of Analytic Philosophy, Institute of Philosophy, Slovak Academy of Sciences, v.v.i., Bratislava, Slovakia

    Matteo Pascucci

  2. Emeritus, University of Siena, Siena, Italy

    Claudio E. A. Pizzi

Authors
  1. Matteo Pascucci
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  2. Claudio E. A. Pizzi
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Correspondence to Matteo Pascucci .

Editor information

Editors and Affiliations

  1. CNRS, Ecole Normale Supérieure, Paris, France

    Valeria Giardino

  2. Lancaster University in Leipzig, Leipzig, Germany

    Sven Linker

  3. West Chester University, West Chester, PA, USA

    Richard Burns

  4. Universitá di Bologna, Bologna, Italy

    Francesco Bellucci

  5. Université Bourgogne Franche-Comté, Dijon, France

    Jean-Michel Boucheix

  6. Universidade Federal Fluminense, Niterói, Brazil

    Petrucio Viana

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Pascucci, M., Pizzi, C.E.A. (2022). Normatively Determined Propositions. In: Giardino, V., Linker, S., Burns, R., Bellucci, F., Boucheix, JM., Viana, P. (eds) Diagrammatic Representation and Inference. Diagrams 2022. Lecture Notes in Computer Science(), vol 13462. Springer, Cham. https://doi.org/10.1007/978-3-031-15146-0_6

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