Leonardi.DB
a logical geometry project

Display Conventions for Octagons of Opposition (2024), p. 4
by Makinson, David

Copyright according to our policy

Caption

Buridan’s modal octagon pruned.

Logic

Aristotelian family
Buridan Sigma-4
Boolean complexity
6
Number of labels per vertex (at most)
1
Uniqueness of the vertices up to logical equivalence
Yes
Errors in the diagram
No

Geometry

Shape
Octagon (irregular)
Colinearity range
0
Coplanarity range
0
Cospatiality range
0
Representation of contradiction
By central symmetry

Vertex description

Conceptual info
No
Mnemonic support (AEIO, purpurea ...)
No
Form
dots
Label type
symbolic
Symbolic field
logic
Contains partial formulas or symbols
Yes
Logical system
modal syllogistics

Edge description

Contains definitions of relations
No
Form
solid lines
,
none
,
dashed lines
Has arrowheads
Yes
Overlap
No
Curved
No
Hooked
No
As wide as vertices
No
Contains text
No
Label type
none

Style

Diagram is colored
No
Diagram is embellished
No
Tags
composed operator duality

Additional notes

∀□¬\forall\Box\textcolor{white}{\neg} = all S are necessarily P = ∃x◊Sx∧∀x(◊Sx→□Px)\exists x \Diamond Sx \wedge \forall x (\Diamond Sx \to \Box Px)
∀◊¬\forall\Diamond\textcolor{white}{\neg} = all S possibly P = ∃x◊Sx∧∀x(◊Sx→◊Px)\exists x \Diamond Sx \wedge \forall x (\Diamond Sx \to \Diamond Px)
∃□¬\exists\Box\textcolor{white}{\neg} = some S are necessarily P = ∃x(◊Sx∧□Px)\exists x (\Diamond Sx \wedge \Box Px)
∃◊¬\exists\Diamond\textcolor{white}{\neg} = some S are possibly P = ∃x(◊Sx∧◊Px)\exists x (\Diamond Sx \wedge \Diamond Px)
∀□¬\forall\Box\neg = all S are necessarily not P = ∀x(◊Sx→□¬Px)\forall x (\Diamond Sx \to \Box \neg Px)
∀◊¬\forall\Diamond\neg = all S are possibly not P = ∀x(◊Sx→◊¬Px)\forall x (\Diamond Sx \to \Diamond\neg Px)
∃□¬\exists\Box\neg = some S are necessarily not P = ¬∃x◊Sx∨∃x(◊Sx∧□¬Px)\neg\exists x \Diamond Sx \vee\exists x (\Diamond Sx \wedge \Box\neg Px)
∃◊¬\exists\Diamond\neg = some S are possibly not P = ¬∃x◊Sx∨∃x(◊Sx∧◊¬Px)\neg\exists x \Diamond Sx \vee\exists x (\Diamond Sx \wedge \Diamond\neg Px)

Cf. p. 118 of this paper.
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