Probabilistic squares and hexagons of opposition under coherence (2017), p. 292
by Pfeifer, Niki; Sanfilippo, Giuseppe

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Caption

Probabilistic hexagon of opposition $(A(x), E(x), I(x), O(x), U(x), Y(x))$ involving generalized quantifiers defined on the six sentence types with the threshold $x \in \, ]\frac{1}{2}, 1]$ (see Table 1 and Table 2). It provides a new interpretation of the hexagon of opposition, which we compose of the probabilistic square of opposition and the two additional vertices $U(x)$ (i.e., $A(x) \vee E(x)$; top) and $Y(x)$ (i.e., $I(x) \wedge O(x)$; bottom). In the extreme case when $x = 1$, we obtain our probabilistic version of the traditional hexagon of opposition (see also Fig. 2).

Logic

Aristotelian family

Jacoby-Sesmat-Blanché Sigma-3

Boolean complexity

3

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

Geometry

Shape

Hexagon (regular)

Colinearity range

0

Coplanarity range

0

Cospatiality range

0

Representation of contradiction

By central symmetry

Vertex description

Conceptual info

No

Mnemonic support (AEIO, purpurea ...)

Yes

Form

none

Label type

symbolic

Symbolic field

mathematics

,

logic

Contains partial formulas or symbols

Yes

Logical system

syllogistics

Mathematical branch

probability theory

Edge description

Style

Diagram is colored

Yes

Diagram is embellished

No

Tags

Boolean closed