=Paper=
{{Paper
|id=Vol-3308/paper3
|storemode=property
|title=Partial formal contexts with degrees
|pdfUrl=https://ceur-ws.org/Vol-3308/Paper03.pdf
|volume=Vol-3308
|authors=Francisco Pérez-Gámez,Pablo Cordero,Manuel Enciso,Ángel Mora,Manuel Ojeda-Aciego
|dblpUrl=https://dblp.org/rec/conf/cla/Perez-GamezCE0O22
}}
==Partial formal contexts with degrees==
https://ceur-ws.org/Vol-3308/Paper03.pdf
Partial formal contexts with degrees
Francisco Pérez-Gámez1,∗ , Pablo Cordero1 , Manuel Enciso1 , Ángel Mora1 and
Manuel Ojeda-Aciego1
1
University of Malaga, Bulevar Louis Pasteur 35, 29071 Málaga (Spain)
Abstract
Partial formal contexts are trivalued contexts that, besides allowing to establish whether a property is
satisfied or not, allow to represent situations in which there is ignorance about whether a property is
satisfied. This can be useful, not only for the cases in which the modeled phenomenon has intrinsically
unknown information, but also when summarizing information from a formal context by grouping
similar rows. In this paper, we prospect for its extension including degrees of knowledge.
Keywords
Implications, Unknown information, Formal concept analysis, Intuitionistic logic
1. Introduction
Trivalued logics are usually conceived as an extension of Classical Logic by adding an intermedi-
ate value to the set of Boolean truth-values {𝐹 , 𝑇 }. This extra value enriches the expressive power
of the logic and induces an order relation by considering an “intermediate” value to be located
between the other two values. This is the case of Łukasiewicz logic [1] or Kleene logic [2]. In
some sense, fuzzy logic also follows this idea, for it can be considered a generalisation which
introduces a set of (infinitely many) values between the two Boolean truth values. The main
difference between different 3-valued logics is the underlying meaning of the third value and,
specifically the meaning of its negation.
The usual interpretation of the truth value 𝐹 is “totally false”, whereas 𝑇 means “totally
true”. Then, the third truth value can be seen as an intermediate knowledge between these two
situations.
We interpret 𝑇 as “we do have information that shows it is true” and 𝐹 as “we do have
information showing that it is false”. Then, the third truth-value can be interpreted as “unknown”,
i.e. we do not have any information either about whether it is true or it is false. For instance, in
a health information system, the variable “being pregnant” matches this interpretation since the
unknown value collects the situation where we don’t have information at all (perhaps a test has
not yet been carried out, or we do not have access to its result). An exhaustive review of this
Published in Pablo Cordero, Ondrej Kridlo (Eds.): The 16𝑡ℎ International Conference on Concept Lattices and Their
Applications, CLA 2022, Tallinn, Estonia, June 20–22, 2022, Proceedings, pp. 35–44.
∗
Corresponding author.
Envelope-Open franciscoperezgamez@uma.es (F. Pérez-Gámez); pcordero@uma.es (P. Cordero); enciso@uma.es (M. Enciso);
amora@uma.es (Á. Mora); aciego@uma.es (M. Ojeda-Aciego)
Orcid 0000-0001-7518-1828 (F. Pérez-Gámez); 0000-0002-5506-6467 (P. Cordero); 0000-0002-0531-4055 (M. Enciso);
0000-0003-4548-8030 (Á. Mora); 0000-0002-6064-6984 (M. Ojeda-Aciego)
© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
topic can be found in [3]. In this paper we extend the research line in [4]. We present a graded
extension of the algebraic framework presented. With this extension, we defined the graded
partial formal contexts that are given by a triple (𝐺, 𝑀, 𝐼 ) where 𝐺 and 𝑀 are the sets with the
objects and the attributes (as usually) and 𝐼 ∶ 𝐺 × 𝑀 → [0, 1]2 expresses the degree to which we
know that object 𝑔 possesses attribute 𝑚 and the degree to which we know that object 𝑔 does
not possess attribute 𝑚. Finally, Galois connections are given to capture the concepts of the
graded formal context.
2. Preliminary definitions and results
2.1. An algebraic framework for unknown information
We consider the ∧-semilattice 3 = (3, ≤) where 3 = {+, −, ∘} and ≤ is the reflexive closure of
{(∘, +), (∘, −)} (see Fig. 1a). Given a universe 𝑈, a 3-set in 𝑈 is a mapping 𝐴 ∶ 𝑈 → 3 where, for
each 𝑢 ∈ 𝑈, 𝐴(𝑢) represents the knowledge about the membership of 𝑢 to 𝐴. Thus, + means that
𝑢 belongs to 𝐴 (we call it positive information), − means that 𝑢 does not belong to 𝐴 (we call it
negative information), and ∘ denotes the absence of information about the membership of 𝑢
(which is called unknown information). As usual, the set of 3-sets on 𝑈 inherits the ∧-semilattice
structure, denoted by 3𝑈 = (3𝑈 , ⊑), by considering the componentwise ordering:
𝐴 ⊑ 𝐵 iff 𝐴(𝑢) ≤ 𝐵(𝑢) for all 𝑢 ∈ 𝑈 .
Given a 3-set 𝐴, we call support of 𝐴 to the set Spp(𝐴) = {𝑢 ∈ 𝑈 ∣ 𝐴(𝑢) ≠ ∘}. We define also the
mappings Neg, Pos, Unk ∶ 3𝑈 → 2𝑈 where, for each 𝐴 ∈ 3𝑈 ,
Neg(𝐴) = {𝑢 ∈ 𝑈 ∣ 𝐴(𝑢) = −} = 𝐴−1 (−)
Pos(𝐴) = {𝑢 ∈ 𝑈 ∣ 𝐴(𝑢) = +} = 𝐴−1 (+)
Unk(𝐴) = {𝑢 ∈ 𝑈 ∣ 𝐴(𝑢) = ∘} = 𝐴−1 (∘)
An equivalent formalisation of 3-sets can be found in [3] where only the known information
(Positive or Negative) is given, and is represented as the so-called orthopairs by using sets (𝑃, 𝑁 )
with 𝑃 ∩ 𝑁 = ∅. When the support of a 3-set 𝐴 is finite, we express it as a sequence of elements
(with no delimiters). It can be seen as an ordered re-writing of the concatenation of the elements
of the orthopair, where the atoms in 𝑁 are overlined. For instance, 𝐴 = 𝑢1 𝑢5̄ 𝑢7 means that
𝐴(𝑢1 ) = 𝐴(𝑢7 ) = +, 𝐴(𝑢5 ) = −, and 𝐴(𝑥) = ∘ otherwise. In particular, when Spp(𝐴) = ∅, we
denote 𝐴 = 𝜀 (i.e. the empty sequence).
As mentioned above, we interpret a 3-valued set as the knowledge that we have about the
properties of certain object. Thus, we have a conjunctive interpretation on these sets and,
consequently, when we join two different 3-sets, we might find inconsistencies, that is, a
property can be found positive in one of the sets and negative in the other set. Then, in the
joined set, we have an inconsistent element. The sets 3 and 3𝑈 can be extended to model
this situation by introducing a fourth element, denoted 𝜄, which represents inconsistent or
contradictory information. In order to obtain a lattice with the order of information, this new
element will play the role of the (missing) maximum element of 3. This lattice of four elements
↗ (1, 1) ↖
↗𝜄↖
(1, 0) ↖ ↗ (0, 1)
+↖ ↗− +↖ ↗−
∘ ∘ (0, 0)
(a) ∧-semilattice 3 (b) lattice (4, ≤) (c) Boolean Algebra 2 × 2
Figure 1: Different structures for truth values
is denoted (4, ≤) and is shown in Fig. 1b; it is isomorphic to the Boolean algebra 2 × 2 (see
Fig. 1c). This algebraic structure is known as “information ordering” in the bilattice construction
introduced by Belnap [5].
In addition, 4𝑈 denotes the set of mappings 𝐴 ∶ 𝑈 → 4 or 4-sets, and we assume that 3𝑈 is a
subset of 4𝑈 . In the same way that we did for 3𝑈 , the order of 4 is componentwise extended to
4𝑈 . Note that the infimum of (4𝑈 , ≤) is 𝜀 (the constant mapping to ∘) and the supremum is the
4-set that maps any 𝑢 ∈ 𝑈 to 𝜄, which is called oxymoron and denoted by 𝜄.̇
The 4𝑈 -sets can be seen as paraconsistent orthopairs [6] but, since in our framework ex
contradictione quodlibet still holds, all the orthopairs containing a contradiction are identified
and 𝜄 ̇ is used as the canonical representative of this class. To formalise it, we define the following
closure operator:
𝐴 if 𝐴 ∈ 3𝑈 ,
𝒪 ∶ 4𝑈 → 4𝑈 being 𝒪(𝐴) = {
𝜄 ̇ otherwise.
The codomain of 𝒪 will be denoted 3𝑈̇ ; it consists of the elements in 3𝑈 , the consistent
orthopairs, together with the oxymoron 𝜄.̇ It is a closure system in (4𝑈 , ≤) and, therefore, it is
a ∧-subsemilattice of (4𝑈 , ≤) (but not a sublattice) and (3𝑈̇ , ⊑) is a complete lattice (see Fig. 2).
Since the infimum operation coincides in both structures, they will be denoted by the same
symbol ∧, however the supremum in (4𝑈 , ≤) will be denoted by ∨, whereas in (3𝑈̇ , ⊑) will be
denoted by ⊔. Thus, for all {𝐴𝑗 ∶ 𝑗 ∈ 𝐽 } ⊆ 3𝑈̇ , we have that
⨆ 𝐴𝑗 = 𝒪(⋁ 𝐴𝑗 )
𝑗∈𝐽 𝑗∈𝐽
and, in particular,
⨆ 𝐴𝑗 ≠ 𝜄 ̇ implies ⨆ 𝐴𝑗 = ⋁ 𝐴𝑗 . (1)
𝑗∈𝐽 𝑗∈𝐽 𝑗∈𝐽
The maximal sets of 3𝑈 are called full sets, and the set of all of them is denoted by Full(𝑈 ).
They are the super-atoms of (3𝑈̇ , ⊑) and coincide with those orthopairs (𝑃, 𝑁 ) such that 𝑃 ∪𝑁 = 𝑈
and 𝑃 ∩ 𝑁 = ∅.
We extend the mappings Neg, Pos, Unk ∶ 3𝑈̇ → 2𝑈 by considering Pos(𝜄)̇ = Neg(𝜄)̇ = 𝑈,
Unk(𝜄)̇ = ∅.
Proposition 1 ([4]). For any 𝐴, 𝐵 ∈ 3𝑈̇ , we have that
1. 𝐴 ⊑ 𝐵 if and only if Neg(𝐴) ⊆ Neg(𝐵) and Pos(𝐴) ⊆ Pos(𝐵).
↗↗ 𝜄 ̇ ↖↖
𝑢1 𝑢2 𝑢1 𝑢2 ↖ 𝑢1 𝑢2 𝑢1 𝑢2 ↖ ↗ 𝑢1 𝑢2↖ 𝑢1 𝑢2
↑ ↖ ↗ ↗ 𝑢1 𝑢2↖ 𝑢1 𝑢2
↗ ↑ ↑ ↖ ↗ ↗ ↑
𝑢1 ↖ 𝑢2 𝑢 ↗ 𝑢1 𝑢1 ↖ 𝑢2 𝑢 ↗ 𝑢1
↖ ↗ 2 ↖ ↗ 2
𝜀 𝜀
{𝑢 ,𝑢 }
(a) The ∧-semilattice 3{𝑢1 ,𝑢2 } (b) The lattice 3̇ 1 2
Figure 2: Lattices from the set {𝑢1 , 𝑢2 }
2. 𝐴 ∈ Full(𝑈 ) if and only if Unk(𝐴) = ∅, or equivalently Pos(𝐴) ∪ Neg(𝐴) = 𝑈 and
Pos(𝐴) ∩ Neg(𝐴) = ∅.
3. The mappings Pos and Neg restricted to Full(𝑈 ) are bijections in 2𝑈 .
Finally, we define the operation ( ) ∶ 3̇𝑈 → 3𝑈̇ , which we call opposite, such that 𝜄 ̇ = 𝜄 ̇ and, for
all 𝐴 ∈ 3𝑈 , and 𝑢 ∈ 𝑈,
+ if 𝐴(𝑢) = −
𝐴(𝑢) = { − if 𝐴(𝑢) = +
∘ if 𝐴(𝑢) = ∘
Thus, Neg(𝐴) = Pos(𝐴), Pos(𝐴) = Neg(𝐴), and Unk(𝐴) = Unk(𝐴).
2.2. Formal concept analysis for unknown information
We start the extension of the FCA framework by presenting the notion of partial formal context,
introduced by Ganter in [7], which is defined as a triple ℙ = (𝐺, 𝑀, 𝐼) being 𝐺 and 𝑀 non-empty
sets and 𝐼 ∶ 𝐺 × 𝑀 → 3. We call the elements of 𝐺 and 𝑀 objects and attributes respectively.
Given 𝑔 ∈ 𝐺 and 𝑚 ∈ 𝑀, the assignment 𝐼 (𝑔, 𝑚) = + means that the object 𝑔 has the attribute
𝑚 ; 𝐼 (𝑔, 𝑚) = − means that the object 𝑔 has not the attribute 𝑚; and 𝐼 (𝑔, 𝑚) = ∘ means that we
do not know whether the object 𝑔 has the attribute 𝑚 or not. As in the classical case, these
contexts are shown as tables (see Figure 3, for instance).
ℙ 𝑚1 𝑚2 𝑚3
𝑔1 + ∘ −
𝑔2 ∘ ∘ +
𝑔3 − − ∘
Figure 3: Partial formal context ℙ
A partial formal context ℙ = (𝐺, 𝑀, 𝐼 ) can induce the following (classical) formal contexts:
• 𝕂+ + + −1 +
ℙ = (𝐺, 𝑀, 𝐼 ) where 𝐼 = 𝐼 (+), that is 𝑔𝐼 𝑚 iff 𝐼 (𝑔, 𝑚) = +.
Their derivation operators are denoted by the symbol ⊕, that is, for all 𝑋 ⊆ 𝐺 and 𝑌 ⊆ 𝑀
𝑋 ⊕ = ⋂ 𝑔𝐼 + ( ) = {𝑚 ∈ 𝑀 ∣ 𝑔𝐼 + 𝑚, ∀𝑔 ∈ 𝑋 }
𝑔∈𝑋
𝑌 = ⋂ ( )𝐼 + 𝑚 = {𝑔 ∈ 𝐺 ∣ 𝑔𝐼 + 𝑚, ∀𝑚 ∈ 𝑌 }
⊕
𝑚∈𝑌
• 𝕂− − − −1
ℙ = (𝐺, 𝑀, 𝐼 ) where 𝐼 = 𝐼 (−) and their derivation operators are denoted by the
symbol ⊖ and defined in a similar way.
We use these formal contexts to define the derivation operators in the partial formal context
as follows. The classical derivation operator is generalised by defining ( )↑ ∶ 2𝐺 → 3𝑀̇ and
↓ 𝑀 𝐺
( ) ∶ 3̇ → 2 as
𝑋 ↑ = ⋀ 𝐼 (𝑔, ), and 𝑌 ↓ = Pos(𝑌 )⊕ ∩ Neg(𝑌 )⊖
𝑔∈𝑋
A pair (𝐴, 𝐵) ∈ 2𝐺 × 3𝑀
̇ is said to be a (formal) concept if 𝐴↑ = 𝐵 and 𝐵↓ = 𝐴. As in the classical
case, the concepts can be hierarchically ordered as
(𝐴1 , 𝐵1 ) ≤ (𝐴2 , 𝐵2 ) if and only if 𝐴1 ⊆ 𝐴2 (or, equivalently, iff 𝐵2 ⊑ 𝐵1 ).
Theorem 1 ([4]). Let ℙ = (𝐺, 𝑀, 𝐼 ) be a partial formal context. The couple (↑, ↓) is a Galois
connection between the lattices (2𝐺 , ⊆) and (3𝑀 ̇ , ⊑) and, therefore, concepts are fix-points of the
Galois connection and, if 𝔅⋆ (ℙ) is the set of all concepts, the pair (𝔅⋆ (ℙ), ≤) is a complete lattice.
In [4] we showed that a classical formal context can be seen as a partial formal context, and
partial formal contexts without any unknown information were called total formal context. We
also proved the existence of a classical formal context whose concept lattice is isomorphic to
that of our partial formal context. Conversely, given a formal context, we proved the existence
of a partial formal context whose concept lattice is isomorphic to that of the classic formal
context.
For more details for the algebraic background, we refer to [8, 9], and for Formal Concept
Analysis, we refer to [10, 11].
3. Graded extension of the algebraic framework
In this section, we extend the previous algebraic framework to a graded one following the idea
of intuitionistic logic. Hereinafter, we consider a residuated lattice I = ([0, 1], ∧, ∨, ⊗, →, 0, 1).
We generalise the lattice 4 by considering the residuated lattice I2 = I × I where the ordering
and the operations are componentwise extended. Thus, for a universal set 𝑈, the 4-sets are
extended by using grades as I2 -sets, i.e. mappings from 𝑈 to I2 . Equivalently, a I2 -set 𝑋 ∶ 𝑈 → I2
can be seen as a graded paraconsistent orthopair (𝑋 + , 𝑋 − ) such that 𝑋 + , 𝑋 − ∶ 𝑈 → I and
𝑋 (𝑢) = (𝑋 + (𝑢), 𝑋 − (𝑢)) where 𝑋 + (𝑢) means the degree in which we know that 𝑢 belongs to 𝑋 and
𝑋 − (𝑢) means the degree in which we know that 𝑢 does not belong to 𝑋.
The set of I2 -sets will be denoted by (I2 )𝑈 . The operations can be extended from I2 to (I2 )𝑈
as usual:
(𝑋 ⋆ 𝑌 )(𝑢) = 𝑋 (𝑢) ⋆ 𝑌 (𝑢) for all 𝑢 ∈ 𝑈 and all ⋆ ∈ {∧, ∨, ⊗, →}.
We also consider the order relation defined as
𝑋1 ⊑ 𝑋2 iff 𝑋1 (𝑢) ≤ 𝑋2 (𝑢) for all 𝑢 ∈ 𝑈,
or, equivalently,
𝑋1 ⊑ 𝑋2 iff 𝑋1+ (𝑢) ≤ 𝑋2+ (𝑢) and 𝑋1− (𝑢) ≤ 𝑋2− (𝑢) for all 𝑢 ∈ 𝑈.
𝑈
It is easy to see that (I2 ) = ((I2 )𝑈 , ∧, ∨, ⊗, →, 𝜀, 𝜄)̇ is also a residuated lattice where 𝜀 and 𝜄 ̇
are the I2 -sets such that 𝜀(𝑢) = (0, 0) and 𝜄(𝑢) ̇ = (1, 1) for all 𝑢 ∈ 𝑈. The I2 -set 𝜀 means that we
have absolutely no knowledge about it, whereas the set 𝜄 ̇ denotes total contradiction (we have
knowledge that all elements belong to the set and, at the same time, do not belong to it).
This leads us to distinguish between consistent and inconsistent sets. We will say that a
I2 -set 𝑋 on 𝑈 is consistent if 𝑋 + (𝑢) ⊗ 𝑋 − (𝑢) = 0 for all 𝑢 ∈ 𝑈, i.e. its range is contained in
ℂ = {(𝑥1 , 𝑥2 ) ∈ I2 ∶ 𝑥1 ⊗ 𝑥2 = 0}. Contrariwise, 𝑋 is inconsistent if there exists 𝑢 ∈ 𝑈 such that
𝑋 + (𝑢) ⊗ 𝑋 − (𝑢) > 0.
The notion of consistent set is a generalization of the well-known Atanassov intuitionistic
fuzzy set [12], which is a mapping 𝑋 ∶ 𝑈 → [0, 1]2 such that, for all 𝑢 ∈ 𝑈, if 𝑋 (𝑢) = (𝛼, 𝛽) then
𝛼 + 𝛽 ≤ 1. It is the particular case in which the Łukasiewicz product is considered because
0 = 𝛼 ⊗ 𝛽 = max{0, 𝛼 + 𝛽 − 1} is equivalent to 𝛼 + 𝛽 ≤ 1.
Obviously, ℂ is a ∧-subsemilattice of I2 that generalises 3, in the same way that I2 generalises
4. The set of consistent sets on 𝑈, denoted ℂ𝑈 , with the componentwise ordering is also a
∧-semilattice that we will be denoted by ℂ𝑈 .
Following the same scheme as in the non-graded case, we consider equivalent all the incon-
sistent sets and we identify them with 𝜄.̇ Thus, we define the closure operator
𝑋 if 𝑋 ∈ ℂ𝑈 ,
𝒪 ∶ (I2 )𝑈 → (I2 )𝑈 where 𝒪(𝑋 ) = {
𝜄̇ otherwise.
and denote by ℂ̇ 𝑈 its range that is ℂ𝑈 ∪ {𝜄}.̇ Since this set is a closure system, we have that it is a
𝑈
∧-subsemilattice of (I2 )𝑈 , and also a complete lattice that we denote by ℂ̇ . To distinguish the
𝑈
supremum of (I2 )𝑈 from that of ℂ̇ we will use the symbols ∨ and ⊔ respectively. Thus, given
{𝑋𝑗 ∶ 𝑗 ∈ 𝐽 } ⊆ ℂ̇ 𝑈 ,
⨆ 𝑋𝑗 = 𝒪(⋁ 𝑋𝑗 )
𝑗∈𝐽 𝑗∈𝐽
4. Graded partial formal contexts
We begin by extending the notion of partial formal context. A graded partial formal context is
a triple ℙ = (𝐺, 𝑀, 𝐼) where 𝐺 and 𝑀 are sets whose elements are called objects and attributes
respectively, and 𝐼 is an ℂ-set on 𝐺 × 𝑀, i.e. 𝐼 ∶ 𝐺 × 𝑀 → ℂ. Thus, for each (𝑔, 𝑚) ∈ 𝐺 × 𝑀, the
degree 𝐼 + (𝑔, 𝑚) is those in which we know that 𝑔 has the attribute 𝑚 and 𝐼 − (𝑔, 𝑚) is the degree
in which we know that 𝑔 does not have the attribute 𝑚. As usual, eventually we use a currifying
process and, given 𝑔 ∈ 𝐺 and 𝑚 ∈ 𝑀, we consider the ℂ-sets 𝐼 (𝑔, .) ∈ ℂ𝑀 and 𝐼 (., 𝑚) ∈ ℂ𝐺 defined
as:
𝐼 (𝑔, .)(𝑥) = 𝐼 (𝑔, 𝑥) for all 𝑥 ∈ 𝑀; 𝐼 (., 𝑚)(𝑥) = 𝐼 (𝑥, 𝑚) for all 𝑥 ∈ 𝐺.
A first approach to extend the results given for partial formal contexts is to consider the
following Galois connection.
Theorem 2. Given a graded partial formal context ℙ = (𝐺, 𝑀, 𝐼 ), the derivation operators
( )↑ ∶ 2𝐺 → ℂ̇ 𝑀 and ( )↓ ∶ ℂ̇ 𝑀 → 2𝐺 defined as
𝑋 ↑ = ⋀ 𝐼 (𝑔, ) and 𝑌 ↓ = {𝑔 ∈ 𝐺 ∶ 𝑌 ⊑ 𝐼 (𝑔, )}
𝑔∈𝑋
𝑀
form a Galois connection between the lattices 2𝐺 and ℂ̇ .
The concept lattice is defined in the usual way from the above Galois connection. Its minimum
element is (∅, 𝜄)̇ and, for any other formal concept (𝑋 , 𝑌 ), if 𝑌 (𝑚) = (𝛼, 𝛽), the value 𝛼 is a degree
in which it is known that all the objects in 𝑋 have the attribute 𝑚, whereas 𝛽 is a degree
in which it is known that all the objects in 𝑋 have not the attribute 𝑚 (rephrasing in more
mathematical terms, 𝛼 and 𝛽 are, respectively, lower bounds of the degrees of each object of 𝑋
having, respectively not having, the attribute 𝑚).
To have a more general framework, a second attempt could be done by considering fuzzy
sets of objects. To do so, we must introduce some additional notation.
In a natural way, from a graded partial formal context ℙ = (𝐺, 𝑀, 𝐼 ), two fuzzy formal contexts,
𝕂+ + − −
ℙ = (𝐺, 𝑀, 𝐼 ) and 𝕂ℙ = (𝐺, 𝑀, 𝐼 ), can be induced. The derivation operators in 𝕂ℙ and 𝕂ℙ
+ −
𝐺
will be denoted by the symbols ⊕ and ⊖ respectively. Thus, given 𝑋 ∈ [0, 1] , the fuzzy sets
𝑋 ⊕ , 𝑋 ⊖ ∈ [0, 1]𝑀 are defined as
𝑋 ⊕ (𝑚) = ⋀ (𝑋 (𝑔) → 𝐼 + (𝑔, 𝑚)) and 𝑋 ⊖ (𝑚) = ⋀ (𝑋 (𝑔) → 𝐼 − (𝑔, 𝑚))
𝑔∈𝐺 𝑔∈𝐺
and, given 𝑌 ∈ [0, 1]𝑀 , the fuzzy sets 𝑌 ⊕ , 𝑌 ⊖ ∈ [0, 1]𝐺 are defined as
𝑌 ⊕ (𝑔) = ⋀ (𝑌 (𝑚) → 𝐼 + (𝑔, 𝑚)) and 𝑌 ⊖ (𝑔) = ⋀ (𝑌 (𝑚) → 𝐼 − (𝑔, 𝑚))
𝑚∈𝑀 𝑚∈𝑀
We use these formal contexts to define the derivation operators from a partial formal context
as follows.
Theorem 3. Given a graded partial formal context ℙ = (𝐺, 𝑀, 𝐼 ), the derivation operators
( ) ∶ [0, 1]𝐺 → (I2 )𝑀 and ( ) ∶ (I2 )𝑀 → [0, 1]𝐺 defined as
𝑋 = (𝑋 ⊕ , 𝑋 ⊖ ) and 𝑌 = (𝑌 + )⊕ ∩ (𝑌 − )⊖
𝑀
form a Galois connection between the lattices I𝐺 and I2 .
From this theorem, which is the direct extension of Theorem 1, we can define formal concepts
as fix pairs of the Galois connection and study the corresponding concept lattice. However, the
main difference between this theorem and the previous ones is that 𝑋 could be inconsistent.
5. Conclusions and further works
Partial formal contexts are useful to work with formal contexts that have been obtained via a
granularisation process from a (classical) formal context, as described in [7]. This framework
for the management of formal contexts in which there is missing or unknown information has
been extended by considering graded knowledge about whether a property holds. Thus, we
consider the degree in which is known that an object has certain property and, on the other
hand, the degree in which it is known that this object does not have the property. We conclude
with a pair of theorems that establish Galois connections and from which we can generalise the
concept lattice and all the machinery of FCA.
A first approach to this extension is provided by Theorem 2 where the induced concepts,
leaving aside the case of the minimum concept, are couples (𝑋 , 𝑌 ) being 𝑋 a classical set and 𝑌
a consistent ℂ-set of attributes.
The Galois connection established in Theorem 3 induces concepts (𝑋 , 𝑌 ) where 𝑋 is fuzzy set
of objects and 𝑌 is a I2 -set of attribute. Nevertheless, contrariwise to the previous cases (those
obtained from Theorems 1 and 2), the I2 -set 𝑌 could be inconsistent. As a consequence, a further
refinement can be made to identify any concepts inconsistent with the pair (𝜄↓̇ , 𝜄),̇ in the same
way that the closure operator 𝒪 does. The problem of finding alternative Galois connection
𝑀
between the lattices I𝐺 and ℂ̇ that will allow to avoid the computation of inconsistent concepts
is left as future work.
Further work in the short term is to situate the results obtained from Theorem 2 in the
framework of the pattern structures introduced by Ganter and Kuznetsov [13], as well as the
results arising from Theorem 3 with the works of Konecny [14] and Dubois et al [15].
The consistency issue is specially relevant when we work with attribute implications, where
ex contradictione quodlibet principle must hold. In [4], not only the concept lattice of a partial
formal context was introduced, but also the notion of attribute implication, and an axiomatic
system proved to be sound and complete. Another interesting direction for future work will
be to extend the previous study of implications to the framework defined from Theorem 2 by
considering implications where premise and conclusion are ℂ-sets ̇ of attributes.
Acknowledgments
Partially supported by the Spanish Ministry of Science, Innovation, and Universities (MCIU),
State Agency of Research (AEI), Junta de Andalucía (JA), Universidad de Málaga (UMA) and
European Regional Development Fund (FEDER) through the projects PGC2018-095869-B-I00
and TIN2017-89023-P (MCIU/AEI/FEDER), and UMA2018-FEDERJA-001 (JA/UMA/FEDER).
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