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Leonardi.DB
a logical geometry project

# Was Lewis Carroll an Amazing Oppositional Geometer? (2014), p. 401 by Moretti, Alessio

### Caption

The Carrollian cube of 4-opposition (Greek, indicial and vernacular decoration).

### Logic

Aristotelian family
Moretti-Pellissier Sigma-4
Boolean complexity
4
Number of labels per vertex (at most)
1
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 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
predicate logic

### Style

Diagram is colored
Yes
Diagram is embellished
No
Tags
existential import

Consider the following partition:
1) A! (all x are y, and there is at least one x)
2) I $\wedge$ O (some x are y and some x are not y)
3) E! (no x are y, and there is at least one x)
4) there are no x

With the bitstrings based on this partition, the formulas of this diagram (in indicial notation) can be represented as follows:

1000 = $x_1y'_0$
0100 = $xy_1\ \dag \ xy'_1$
0010 = $x_1y_0$
0001 = $x_0$
1100 = $xy_1$
1010 = $x_1y_0\ § \ x_1y'_0$
1001 = $xy'_0$
0110 = $xy'_1$
0101 = $x_0\ § \ (xy_1\ \dag \ xy'_1)$
0011 = $xy_0$
1110 = $x_1$
1101 = $x_0y_1$
1011 = $xy_0 \ § \ xy'_0$
0111 = $x_0y'_1$

Note: $x'$ means not-$x$, $\dag$ means "and", $§$ means "or" (cf. footnote 5 on p. 387).