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Leonardi.DB
a logical geometry project # Ploucquet's ''Refutation'' of the Traditional Square of Opposition (2008), p. 57 by Lenzen, Wolfgang ### Caption

“True” logical relations between the formulas of Ploucquet’s logic.

### Logic

Aristotelian family
Buridan Sigma-4
Boolean complexity
5
Number of labels per vertex (at most)
1
Analogy between (some) labels of the same vertex
No
Uniqueness of the vertices up to logical equivalence
No
Errors in the diagram
No

### Geometry

Shape
Three Dimensional Shape (irregular)
Colinearity range
0–1
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
No
Logical system
syllogistics

### Style

Diagram is colored
No
Diagram is embellished
No
Tags
quantification of the predicate

Note: the identification of this diagram as a Buridan sigma-4 (of Boolean complexity 5) depends on an interpretation of the formulas O(S) - O(P) and Q(S) > Q(P) that is different from the one given in the article. For example, we interpret O(S) - O(P) as follows: $\forall x \forall y (Sx \to (Sy \to x=y))$, whereas Lenzen interprets this formula as the conjunction of $\forall x(Sx \to Px)$ and $\forall x(Px \to Sx)$ (cf. p. 56). Furthermore, note that on our interpretation, $\alpha$ is actually equivalent to O(S) - O(P), and \beta is actually equivalent to Q(S) > Q(P).