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Leonardi.DB
a logical geometry project

Metalogical Decorations of Logical Diagrams (2016), p. 269 by Demey, Lorenz; Smessaert, Hans

Caption

Seuren’s relations as decorations of a an ‘unconnected-4’ hexagon (no constraints) and b a weak JSB hexagon (constraint: satisfiability of the first argument)

Logic

Aristotelian family
Degenerate Sigma-3 with Unconnectedness 4
Boolean complexity
5
Number of labels per vertex (at most)
1
Uniqueness of the vertices up to logical equivalence
Yes
Errors in the diagram
No

Geometry

Shape
Hexagon (irregular)
Colinearity range
0
Coplanarity range
0
Cospatiality range
0
By central symmetry

Vertex description

Conceptual info
No
Mnemonic support (AEIO, purpurea ...)
No
Form
none
Label type
symbolic
Symbolic field
metalogic
Contains partial formulas or symbols
No

Style

Diagram is colored
No
Diagram is embellished
No
Tags
Leuven

The 'fifth' anchor formula, which makes the difference between Boolean complexities 4 and 5, is $(\textit{RI} \cup \textit{NI}) \cap (\textit{CD} \cup \textit{C} \cup \textit{SC} \cup \textit{BI} \cup \textit{LI} \cup \textit{NI}) \cap (\textit{SC} \cup \textit{NCD})$. If $\varphi$ is a tautology and $\psi$ is contingent, then $(\varphi,\psi)$ indeed stand in the relation $(\textit{RI} \cup \textit{NI}) \cap (\textit{CD} \cup \textit{C} \cup \textit{SC} \cup \textit{BI} \cup \textit{LI} \cup \textit{NI}) \cap (\textit{SC} \cup \textit{NCD})$, so the latter is not empty, and hence, the Boolean complexity of this diagram is indeed 5 (rather than 4). Also cf. Footnote 34 on p. 268.