How to Square Knowledge and Belief (2012), p. 311
by Lenzen, Wolfgang

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Caption

Square of complex epistemic modalities

Logic

Aristotelian family

Classical Sigma-2

Boolean complexity

3

Number of labels per vertex (at most)

1

Uniqueness of the vertices up to logical equivalence

Yes

Errors in the diagram

No

Geometry

Shape

Rectangle (irregular)

Colinearity range

0

Coplanarity range

0

Cospatiality range

0

Representation of contradiction

By central symmetry

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

epistemic logic

Edge description

Style

Diagram is colored

No

Diagram is embellished

No

Additional notes

Given the equivalence of $C(a,p)$ with $\neg K(a, \neg K(a,p))$ (cf. p. 311), this diagram can equivalently be rewritten as follows:

$C(a,p)\ \ \ \ \ \ \ \ \ $ $C(a,\neg p)$
$\neg C(a,\neg p)\ \ \ $ $\neg C(a,p)$