Notes on intensional knowledge|conceptuality,

partiality, inconsistency and preference

Chris Nowak

Computer Science Department

The University of Adelaide

Email: [email protected]

1 Introduction

In this introduction we provide some fundamental notions of Formal Concept Analysis (FCA) and give an outline of the paper.

The FCA framework defines a context K to be a triple K(G;M; I), with objects G, attributes M , and an incidence relation I, where I : G ? M ?! f0; 1g is a total function. For Ga ? G and Ma ? M define G0 a = fm 2 M j

8g2Ga I(g; m) = 1g, and M a = fg 2 G j 8m2Ma I(g; m) = 1g. A pair (Ga; Ma),
where Ga ? G and Ma ? M , is a concept of K(G;M; I) if G0 a = Ma and

M a = Ga|the extent of the concept (Ga; Ma) is Ga, while its intent is Ma. See e.g. [GW96, DP90] for further details.
The considerations of the paper are outlined pictorially in Table 1.

conceptual intensional
z }| { z }| {

K0(G0; M0; I0) L0 T0 Dj= KDj= M L T j= A g semantic

% " ? = s"#c

P0 [ P0 3-valued D` HD` M L T ` A g syntactic

preference: (fTigi; O)

Table 1: Knowledge: conceptual, partial, semantic, syntactic, preferred.

Section 2 modifies CFA to allow 3-valued|or partial|incidence relation I0, and both (non-negated) attributes P0 and corresponding negated attributes P0 are used|hence M0 = P0 [ P0 is employed as a set of attributes. A resulting