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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
Representation of contradiction
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

Edge description

Style

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

Additional notes

O and Q stand for 'omnis' and 'quidam', respectively. Furthermore, - and > stand for 'est' and 'non est', respectively. (Cf. p. 48.)

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).
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