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Leonardi.DB
a logical geometry project # A Cube of Opposition for Predicate Logic (2020), p. 108 by Nilsson, Jørgen Fischer ### Caption

The cube clad for Aristotelian modal logic

### Logic

Aristotelian family
Buridan Sigma-4
Boolean complexity
6
Number of labels per vertex (at most)
2
Equivalence between (some) labels of the same vertex
Yes
Analogy between (some) labels of the same vertex
No
Uniqueness of the vertices up to logical equivalence
Yes
Errors in the diagram
No

### Geometry

Shape
Cube (regular)
Colinearity range
0
Coplanarity range
0
Cospatiality range
0
By some other geometric feature

### Vertex description

Conceptual info
No
Mnemonic support (AEIO, purpurea ...)
No
Form
none
Label type
symbolic
Symbolic field
logic
Contains partial formulas or symbols
Yes
Logical system
modal syllogistics

### Style

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

Let $C,D$ be unary predicates.

$\forall\Box$ stands for $\forall x( Cx \to \Box Dx)$
$\forall\Diamond$ stands for $\forall x( Cx \to \Diamond Dx)$
$\exists\Box$ stands for $\exists x( Cx \wedge \Box Dx)$
$\exists\Diamond$ stands for $\exists x( Cx \wedge \Diamond Dx)$

$\forall\Box\neg$ stands for $\forall x( Cx \to \Box \neg Dx)$
$\neg\exists\Diamond$ stands for $\neg\exists x( Cx \wedge \Diamond Dx)$

$\forall\Diamond\neg$ stands for $\forall x( Cx \to \Diamond \neg Dx)$
$\neg\exists\Box$ stands for $\neg\exists x( Cx \wedge \Box Dx)$

$\exists\Box\neg$ stands for $\exists x( Cx \wedge \Box\neg Dx)$
$\neg\forall\Diamond$ stands for $\neg\forall x( Cx \to \Diamond Dx)$

$\exists\Diamond\neg$ stands for $\exists x( Cx \wedge \Diamond\neg Dx)$
$\neg\forall\Box$ stands for $\neg\forall x( Cx \to \Box Dx)$

(Based on pp. 104-105.)

Note that Nilsson remains silent on the issue of ampliation, which would involve replacing all occurrences of $Cx$ with $\Diamond Cx$ in the formulas above.