Normatively Determined Propositions

Pascucci, Matteo; Pizzi, Claudio

doi:10.1007/978-3-031-15146-0_6

Cited by 2 publications

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“…This article provides a novel and general framework for assisted reasoning with Aristotelian diagrams, based on a systematic interaction between logic and graph theory (Section 2). We illustrate how this framework can be used to represent two problems of normative determination [7]. The first problem (RP1) consists in checking whether the truth of some statements in a set X determines the truth-value of all statements in a set Y in light of a normative theory Γ.…”

Section: Introductionmentioning

confidence: 99%

Assisted Normative Reasoning with Aristotelian Diagrams

Hanauer,

Novotná,

Pascucci

2023

Frontiers in Artificial Intelligence and Applications

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We design a framework for assisted normative reasoning based on Aristotelian diagrams and algorithmic graph theory which can be employed to address heterogeneous tasks of deductive reasoning. Here we focus on two problems of normative determination: we show that the algorithms used to address these problems are computationally efficient and their operations are traceable by humans. Finally, we discuss an application of our framework to a scenario regulated by the GDPR.

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Section: Introductionmentioning

confidence: 99%

Assisted Normative Reasoning with Aristotelian Diagrams

Hanauer,

Novotná,

Pascucci

2023

Frontiers in Artificial Intelligence and Applications

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Varieties of Cubes of Opposition

Pizzi

2024

Log. Univers.

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The objects called cubes of opposition have been presented in the literature in discordant ways. The aim of the paper is to offer a survey of such various kinds of cubes and evaluate their relation with an object, here called “Aristotelian cube”, which consists of two Aristotelian squares and four squares which are semiaristotelian, i.e. are such that their vertices are linked by some so-called Aristotelian relation. Two paradigm cases of Aristotelian squares are provided by propositions written in the language of the logic of consequential implication, whose distinctive feature is the validity of two formulas, A $$\rightarrow $$ → B $$\supset \lnot $$ ⊃ ¬ (A $$\rightarrow $$ → $$\lnot $$ ¬ B) and A $$\rightarrow $$ → B $$\supset \lnot $$ ⊃ ¬ ($$\lnot $$ ¬ A $$\rightarrow $$ → B), expressing two different forms of contrariety. Part of section 1 is devoted to define the notions of rotation and of r-Aristotelian square, i.e. a square resulting from some rotation of an Aristotelian square. In section 2 this notion is extended to the one of a r-Aristotelian cube, i.e. of a cube resulting from some rotation of some square of an Aristotelian cube. This notion is used in the sequel to analyze various cubes of oppositions which can be found in the literature: (1) the one used by W. Lenzen to reconstruct Caramuel’s Octagon; (2) the one used by D. Luzeaux to represent the implicative relation among S5-modalities; (3) the one introduced by D. Dubois to represent the relations between quantified propositions containing positive predicates and their negations; (4) the one called Moretti cube. None of such cubes is strictly speaking Aristotelian but each of them may be proved to be r-Aristotelian. Section 5 discusses the assertion that Dubois cube was anticipated in a paper published by Reichenbach in 1952. Actually Dubois’ construction was anticipated by the so-called Johnson–Keynes cube, while the Reichenbach cube, unlike Dubois cube, is an instance of an Aristotelian cube in the sense defined in this paper. The dominance of such notion is confirmed by J.F. Nilsson’s cube, representing relations between propositions with nested quantifiers, and also by a cube introduced by S. Read to treat quantifiers with existential import. A cube similar to Read’s cube, introduced by Chatti and Schang, is shown to be r-Aristotelian. In section 6 the author remarks that the logic of the formulas occurring in the cubes of Chatti–Schang and Read have the drawback of not satsfying the law of Identity. He then proposes a definition of non-standard quantifiers which satisfies Identity, are independent of existential assumptions and such that their interrelations are represented by an Aristotelian cube.

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Normatively Determined Propositions

Cited by 2 publications

References 10 publications

Assisted Normative Reasoning with Aristotelian Diagrams

Assisted Normative Reasoning with Aristotelian Diagrams

Varieties of Cubes of Opposition

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