Was Lewis Carroll an Amazing Oppositional Geometer? (2014), p. 403
by Moretti, Alessio
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Caption
- The twelve orange segments of independence let emerge three squares of a new kind.
- Aristotelian family
- Degenerate Sigma-3 with Unconnectedness 12
- Boolean complexity
- 4
- Number of labels per vertex (at most)
- 2
- Equivalence between (some) labels of the same vertex
- No
- Analogy between (some) labels of the same vertex
- No
- Uniqueness of the vertices up to logical equivalence
- Yes
- Errors in the diagram
- No
- Shape
- Octahedron (regular)
- Colinearity range
- 0
- Coplanarity range
- 0
- Cospatiality range
- 0
- Representation of contradiction
- By central symmetry
Logic
Geometry
- Conceptual info
- No
- Mnemonic support (AEIO, purpurea ...)
- No
- Form
- dots
- Label type
- linguistic ,
- symbolic
- Language
- English
- Lexical field
- syllogistics
- Contains partial sentences or single words
- No
- Contains abbreviations
- Yes
- Symbolic field
- logic
- Contains partial formulas or symbols
- Yes
- Logical system
- predicate logic
Vertex description
Edge description
- Diagram is colored
- Yes
- Diagram is embellished
- No
- Tags
- existential import
Style
Additional notes
- 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).
With the bitstrings based on this partition, the formulas of this diagram (in vernacular notation) can be represented as follows:
1000 = all x are y
0100 = some x are y and some x are y'
0010 = all x are y'
0001 = there are no x
1100 = some x are y
1010 = all x are y or all [x] are y'
1001 = no x are y'
0110 = some x are y'
0101 = not all x are y nor are all y' [not all x are y and not all x are y']
0011 = no x are y
1110 = some x exist
1101 = not all x are y'
1011 = no x are y or none [i.e. no x] are y'
0111 = not all x are y