The Algebra of Opposition (and Universal Logic Interpretations)

Diaconescu, Răzvan

doi:10.1007/978-3-319-10193-4_6

Cited by 2 publications

(7 citation statements)

References 12 publications

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“…(a) A strong Jacoby-Sesmat-Blanché hexagon for statements of the form R(ϕ, ϕ) (with R ∈ ℘ ∪ (OG)), and (b) its reformulation using more familiar terminology More importantly, however, because of this correspondence, the JSB hexagon in Figure 11(a) can be reformulated as the more familiar JSB hexagon in Figure 11(b). This metalogical hexagon was first studied by Béziau in [4, Paragraph 3.2.4] and [5], and later also by Diaconescu [33]. What we have shown here is that this hexagon can be seen as (a special instance of) a subdiagram of the Aristotelian RDH for OG that was introduced in the previous subsection.…”

Section: Béziau's Hexagon For Tautology and Related Metalogical Notionsmentioning

confidence: 59%

“…This well-known hexagon [4,5,33] can thus not only be seen as (a special instance) of a subdiagram of the Aristotelian RDH for OG (as was shown in Subsection 4.2), but also as (a special instance) of a subdiagram of the Aristotelian RDH for IG.…”

Section: From Opposition To Implication Decorations Of Aristotelian Dmentioning

confidence: 99%

“…In exactly the same way, it can be shown that SC ∪NCD and RI ∪ NI are subcontrary to each other, and that there are subalternations from BI ∪ LI to SC ∪ NCD and from CD ∪ C to RI ∪ NI . On the assumption that their first argument is satisfiable, Löbner's relations thus end up constituting a classical Aristotelian square, which is shown in Figure 24( 33 and is defined as follows:…”

Section: Aristotelian Diagrams For the Aristotelian Relationsmentioning

confidence: 99%

“…In the last two or three years, however, authors such as Béziau and Seuren have also started investigating Aristotelian diagrams that are decorated with meta-level notions such as tautology, contradiction, satisfiability, contingency, strong and weak interpretations of (sub)contrariety, etc. [4,5,33,72]. As will be explained later in this paper, this transition from object-to metalogical decorations of Aristotelian diagrams is quite important, since it sheds some interesting new light on the ways in which the Aristotelian relations are usually defined.…”

Section: Introductionmentioning

confidence: 99%

“…Secondly, we will discuss the connections between the new metalogical diagrams presented in this paper and those that were previously studied. For example, we will show that the metalogical squares and hexagons studied by Béziau [4,5], Diaconescu [33] and Seuren [72] can be seen as subdiagrams or specific instances of the diagrams presented here. Thirdly, because of these connections, the paper also offers a unifying perspective on metalogical diagrams.…”

Section: Introductionmentioning

confidence: 99%

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Metalogical Decorations of Logical Diagrams

Demey

Smessaert

2016

Log. Univers.

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Abstract. In recent years, a number of authors have started studying Aristotelian diagrams containing metalogical notions, such as tautology, contradiction, satisfiability, contingency, strong and weak interpretations of (sub)contrariety, etc. The present paper is a contribution to this line of research, and its main aims are both to extend and to deepen our understanding of metalogical diagrams. As for extensions, we not only study several metalogical decorations of larger and less widely known Aristotelian diagrams, but also consider metalogical decorations of another type of logical diagrams, viz. duality diagrams. At a more fundamental level, we present a unifying perspective which sheds new light on the connections between new and existing metalogical diagrams, as well as between object-and metalogical diagrams. Overall, the paper studies two types of logical diagrams (viz. Aristotelian and duality diagrams) and four kinds of metalogical decorations (viz. those based on the opposition, implication, Aristotelian and duality relations).Mathematics Subject Classification (2010). Primary 03A10, 03B45; Secondary 03B65, 03B80.

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Section: Béziau's Hexagon For Tautology and Related Metalogical Notionsmentioning

confidence: 59%

Section: From Opposition To Implication Decorations Of Aristotelian Dmentioning

confidence: 99%

Section: Aristotelian Diagrams For the Aristotelian Relationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Metalogical Decorations of Logical Diagrams

Demey

Smessaert

2016

Log. Univers.

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Aristotelian diagrams for semantic and syntactic consequence

Demey

2018

Synthese

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Several authors have recently studied Aristotelian diagrams for various metatheoretical notions from logic, such as tautology, satisfiability, and the Aristotelian relations themselves. However, all these metalogical Aristotelian diagrams focus on the semantic (model-theoretical) perspective on logical consequence, thus ignoring the complementary, and equally important, syntactic (proof-theoretical) perspective. In this paper, I propose an explanation for this discrepancy, by arguing that the metalogical square of opposition for semantic consequence exhibits a natural analogy to the wellknown square of opposition for the categorical statements from syllogistics, but that this analogy breaks down once we move from semantic to syntactic consequence. I then show that despite this difficulty, one can indeed construct metalogical Aristotelian diagrams from a syntactic perspective, which have their own, equally elegant characterization in terms of the categorical statements. Finally, I construct several metalogical Aristotelian diagrams that incorporate both semantic and syntactic consequence (and their interaction), and study how they are influenced by the underlying logical system's soundness and/or completeness. All of this provides further support for the methodological/heuristic perspective on Aristotelian diagrams, which holds that the main use of these diagrams lies in facilitating analogies and comparisons between prima facie unrelated domains of investigation. has been a surge of interest in Aristotelian diagrams at the metalogical level, which contain statements or notions belonging to the metatheory of a given logical system. For example,

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The Algebra of Opposition (and Universal Logic Interpretations)

Cited by 2 publications

References 12 publications

Metalogical Decorations of Logical Diagrams

Metalogical Decorations of Logical Diagrams

Aristotelian diagrams for semantic and syntactic consequence

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