Probabilistic squares and hexagons of opposition under coherence (2017), p. 283
by Pfeifer, Niki; Sanfilippo, Giuseppe

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Caption

Traditional and probabilistic square of opposition defined on the four classical sentence types $A$, $E$, $I$, $O$ and their relations in between. The probabilistic semantics of the basic sentence types involving the predicate term $P$ and the subject term $S$ is interpreted by a suitable probability assessment on the conditional event $P|S$ (see Table 1). For the relations see Definition 9.

Logic

Aristotelian family

Classical Sigma-2

Boolean complexity

3

Number of labels per vertex (at most)

3

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

Square (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

linguistic

,

symbolic

Language

English

Lexical field

syllogistics

Contains partial sentences or single words

No

Contains abbreviations

Yes

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