A three-valued logic approach to partially known formal concepts 1

Abstract

Formal concept analysis, originally proposed by Wille, is a mathematical tool to analyse and represent data in the form of complete formal context. However, in situations with incomplete information, one only has partial knowledge about a concept, recently, a common conceptual framework of the notions of interval sets and incomplete formal contexts for representing partially-known concepts were presented. In this study, we examine and reinterpret the existing studies on partially known concepts by means of three-valued logics. By treating an incomplete formal context as a three-valued formal context and considering the one-to-one correspondence between interval sets and three-valued mappings, we investigate the condition under which the four types of partially known concepts can be generated by using three-valued implication operators. Moreover, we also evaluate the role of three-valued logic in characterizing attribute implications. A sufficient and necessary condition for computing the true value of an implication correctly in the sense of Kriple semantics is provided.

Keywords

Formal context partially-known concept three-valued logic attribute implication

1 Introduction

Formal concept analysis has been introduced by Wille and developed by Ganter and Wille ([1]) as an effective method to data mining and knowledge discovery. It is formulated based on the notion of a formal context, which is a binary relation between a set of objects and a set of properties of attributes. The binary relation induces set-theoretic operators from sets of objects to sets of properties, and from sets of properties to sets of objects, respectively. The fundamental notion of a formal concept is defined as a pair of a set of objects and a set of properties connected by the two set-theoretic operators.

For a formal context in [6], it is a known fact that an object o either have an attribute a or not, such a formal context is also called a complete formal context. However, in real situations, we may only have incomplete information. Concretely, for an object, we know a set of attributes that the object has, we also know another set of attributes that the object does not have, for the rest of attributes, we do not know their states. Such a formal context is called an incomplete formal context in the existing literature ([1 , 18]). A difficulty in such a formal context is that the formal concepts cannot be produced exactly by using Wille’s method. We may only have partially-known concepts. Several proposals on partially-known concepts in an incomplete formal context have been made and investigated. For instance, Burmeister and Holzer ([1]) introduced a pair of certain and possible extents and a pair of certain and possible intents of a concept, which are useful in characterizing partially-known concepts. Obiedkov ([14]) used a completion of an incomplete formal context for modeling incomplete formal context. Djouadi et al. ([5]) studied the least and the greatest completions of an incomplete formal context to characterize the family of all possible completions. Li et al. ([11]) introduced the notion of an approximate concept with a pair of sets of attributes as the intent and a set of objects as the extent. In a straightforward way, Li and Wang ([12]) adopted the notion of three-way formal concepts from Qi et al. to an incomplete context. In [10], the authors put forward a method to compress the approximate concept lattice using K-medoids clustering. To describe logical formulas between columns of incomplete contexts, the Kripke-semantics are used for propositional formulas over the set of attributes ([8, 9]). Recently, in [18], Yao introduced a common conceptual (i.e., interpretational) framework of partially-known concepts in incomplete formal contexts by using the notion of interval sets. In [13], Ma et al. introduced the theory of interval sets into the object-oriented concept lattice, and proposed an object-oriented interval-set concept lattice. In [17], the basic ideas underlying fuzzy logic were introduced into the study of threeway formal concept analysis.

Three-valued logics ([4]) are straightforward generalization of Boolean logic based on the most simple bipolar scale ${0, \frac{1}{2}, 1}$ where 1 has positive flavor, 0 has a negative flavor, and $\frac{1}{2}$ is neutral. For an incomplete formal context, since the collection of objects has altogether three different types of states with respect to the set of attributes, an incomplete formal context can be naturally regarded as a three-valued formal context. Moreover, since ${0, \frac{1}{2}, 1}$ forms a simple lattice with respect to the truth order, the existing lattice-valued formal concept approach ([2, 3]) can be employed to generate the collection of formal concepts. The relationship between these two possibly different types of formal concepts in situations of incomplete formal contexts is thus worthy of investigation, which also helps us to gain additional insights from the viewpoint of multiple-valued logics. Furthermore, since the partially-known concepts are represented as interval sets in the proposed common framework, and the collection of three-valued mappings is in one-to-one correspondence with that of interval sets, the role of three-valued logic in generating the collection of partially-known concepts can be thus expected. Such considerations motivate us to present a characterization of partially-known concepts by using three-valued logics.

The rest of the paper proceeds as follows: in order to make the paper as self-contained as possible, we recapitulate in Section 2 the definition of concepts, formal contexts and implications in three-valued logics. In Section 3, we present a novel characterization of partially-known concepts by means of three-valued logics. In Section 4, we evaluate the role of three-valued logic in characterizing attribute implications. A sufficient and necessary condition for computing the true value of an implication correctly in the sense of Kriple semantics is provided. And in Section 5, we complete this paper with some concluding remarks.

2 Preliminaries

2.1 Formal context and concept

Let K = (OB, AT, I) be a complete formal context ([6]), for a set of objects O ⊆ OB and a set of attributes A ⊆ AT, by considering the maximal set of attributes shared by all objects in O and the maximal set of objects sharing all attributes in A, we arrive at a pair of derivation operators and their properties, as listed below.

Definition 2.1. [6] In a complete formal context K = (OB, AT, I), a pair of derivation operators, σ : 2^OB → 2^AT and τ : 2^AT → 2^OB, is defined by: for a set of objects O ⊆ OB and a set of attributes A ⊆ AT, $σ (O) = {a \in AT ∣ \forall o \in O, oIa},$ $τ (A) = {o \in OB ∣ \forall a \in A, oIa} .$

Definition 2.2. [6] A pair (O, A) of a set of objects and a set of attributes is called a formal concept if σ (O) = A and τ (A) = O. The set of objects O and the set of attributes A are, respectively, called the extent and the intent of the formal concept (O, A).

As stated in the introduction part, in real situations, we may only have incomplete information. That is, for an object, we know a set of attributes that the object has, we also know another set of attributes that the object does not have, however, for the rest of attributes, we do not know their states. Such a formal context is called an incomplete formal context in the literature ([1 , 18]). In what follows, we use K = (OB, AT, {+ , ? , -} , J) to denote an incomplete context, where the symbol ? means an unknown state.

In an incomplete context K = (OB, AT, {+, ?, -}, J), for a set of objects we can no longer derive a set of attributes shared by these objects by using the derivation operator σ, and for a set of attributes nor can we derive a set of objects that share all these attributes by using the derivation operator τ. Burmeister and Hozer and Obiedkow generalized derivation operators σ and τ into modal-style derivation operators as given in the next definition.

Definition 2.3. [1, 14] In an incomplete formal context K = (OB, AT, {+ , ? , -} , J), a pair of derivation operators, $\underline{σ}, \bar{σ} : 2^{OB} \to 2^{AT}$ that maps a set of objects O ⊆ OB to a pair of sets of attributes is defined as follows: $\underline{σ} (O) = {a \in AT ∣ \forall o \in O ((o, a, +) \in J)},$ $\begin{matrix} \bar{σ} (O) = {a \in AT ∣ \forall o \in O ((o, a, +) \in \\ J \lor (o, a, ?) \in J)} . \end{matrix}$

Similarly, corresponding to the derivation operator τ, a pair of lower and upper derivation operators that maps a set of attributes A ⊆ AT to a pair of the sets of objects as follows: $\underline{τ} (A) = {o \in OB ∣ \forall a \in A ((o, a, +) \in J)},$ $\begin{matrix} \bar{τ} (A) = {o \in OB ∣ \forall a \in A ((o, a, +) \in \\ J \lor (o, a, ?) \in J)} . \end{matrix}$

2.2 Three-valued logic

Three-valued logics are straightforward generalizations of Boolean logic based on the most simple bipolar scale $3 = {0, \frac{1}{2}, 1}$ , where 1 has a positive flavor, 0 has a negative flavor and $\frac{1}{2}$ is neutral. To date, three-valued logics have been widely used in several applied contexts such as logic programming, electronic circuits, databases and so on.

In [4], the authors defined a maximal family of sensible conjunctions, in addition to the existing ones.

Definition 2.4. [4] A conjunction on 3 is a binary mapping ∗ : 3 × 3 ⟶ 3, that is monotonically increasing in the wide sense, and extends the connective AND in Boolean logic:

If x ≤ y then x∗ z ≤ y ∗ z ;

If x ≤ y then z∗ x ≤ z ∗ y ;

0 ∗ 0 =0 ∗ 1 =1 ∗ 0 =0 and 1 ∗ 1 =1 .

In the case of implication, the authors give a general definition, which extend Boolean logic and supposes monotonicity.

Definition 2.5. [4] An implication on 3 is a binary mapping ∗ : 3 × 3 ⟶ 3 such that

If x ≤ y then y→ z ≤ x ∗ z ;

If x ≤ y then z→ x ≤ z ∗ y ;

0 → 0 =1 → 1 =1 and 1 ∗ 0 =0 .

We can easily derive from Definition 2.5 that $0 \to 1 = \frac{1}{2} \to 1 = 1, \frac{1}{2} \to \frac{1}{2} \geq {1 \to \frac{1}{2}, \frac{1}{2} \to 0}$ . There are 14 implications satisfying this definition, as listed in Table 1. Nine of them are known and have been studied.

Table 1
All implications on ${0, \frac{1}{2}, 1}$ in the sense of Definition 2.5

3 Three-valued logic approach to partially-known concepts

In [18], four types of partially-known concepts in an incomplete context were proposed. They are SE-SI concepts (i.e., set extent and set intent), SE-ISI concepts (i.e., set extent and interval set intent), ISE-SI concepts (i.e., interval set extent and set intent) and ISE-ISI concepts (i.e., interval set extent and interval set intent). In what follows, we aim to present a novel characterization of these concepts by means of three-valued logics.

3.1 Three-valued logics and SE-ISI concepts and ISE-SI concepts

For an incomplete formal context, the collection of objects has altogether three different states with respect to the set of attributes, the truth values are {0, ? , 1}, or equivalently, ${0, \frac{1}{2}, 1}$ . Usually, two natural orderings on ${0, \frac{1}{2}, 1}$ are considered, one is the truth ordering that characterizes the nature of objects and the other is a knowledge ordering that reflects our knowledge about objects. For the truth ordering ⪯_t, we have $0 ⪯_{t} \frac{1}{2} ⪯_{t} 1,$ and for the knowledge ordering ⪯_k, we have $\frac{1}{2} ⪯_{k} 0, \frac{1}{2} ⪯_{t} 1 .$

In what follows, we mainly consider the truth ordering, then ${0, \frac{1}{2}, 1}$ forms a simple lattice ([7]). By treating an incomplete context K = (OB, AT, {+ , ? , -} , J) as a lattice-valued context. We can thus use the proposed fuzzy concept lattices to analyse SER-ISI concepts.

Definition 3.1. A tuple (L, ∧ , ∨ , ⊗ , → , 0, 1) is called a complete residuated lattice, if

(L, ∧ , ∨ , 0, 1) is a complete lattice with the least element 0 and the greatest element 1;

(L, ⊗ , 1) is a commutative monoid;

(⊗ , →) is an adjoint pair in L, i.e. a ⊗ b ≤ c ⇔ a ≤ b → c, a, b, c ∈ L .

Definition 3.2. [2, 3] Let L = (L, ∧ , ∨ , ⊗ , → , 0, 1) be a complete residuated lattice, for an L-fuzzy formal context (U, A, I), for all X ∈ L^OB, Y ∈ L^AT, define the operators as follows: $X^{*} (a) = ⋀_{o \in OB} (X (o) \to I (o, a)),$ (1) $Y^{*} (o) = ⋀_{a \in AT} (Y (a) \to I (o, a)) .$ (2) The pair (X, Y) is called a formal concept if and only if X^* = Y and Y^* = X .

In a residuate lattice L = (L, ∧ , ∨ , ⊗ , → , 0, 1), → is called a residual implication. It is a known fact that there exists altogether 2 residual implications on ${0, \frac{1}{2}, 1}$ , that is, the Łukasiewicz implication and G $\ddot{o}$ del implication, as shown in Table 1, respectively. The corresponding conjunction operators on ${0, \frac{1}{2}, 1}$ are listed in Table 2 and Table 3.

Table 2

Łukasiewicz conjunction on ${0, \frac{1}{2}, 1}$

⊗_G	$\frac{1}{2}$	1
0	0	0
$\frac{1}{2}$	0	$\frac{1}{2}$
1	$\frac{1}{2}$	1

Table 3

G $\ddot{o}$ del conjunction on ${0, \frac{1}{2}, 1}$

⊗_G	$\frac{1}{2}$	1
0	0	0
$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$
1	$\frac{1}{2}$	1

Definition 3.3. [18] In an incomplete formal context, a pair of a set of objects and an interval set of attributes $(O, [\underline{A}, \bar{A}])$ , is called a partially-known formal concept with a set extent and an interval-set intent, or simply a partially-known SE-ISI formal concept, if the following conditions hold: $[\underline{σ}, \bar{σ}] (O) = [\underline{A}, \bar{A}],$ $< \underline{τ}, \bar{τ} > ([\underline{A}, \bar{A}]) = O,$ where $[\underline{σ}$ , $\bar{σ}] (O)$ = $[\underline{σ} (O)$ , $\bar{σ} (O)]$ , < $\underline{τ}$ , $\bar{τ}$ > $([\underline{A}$ , $\bar{A}])$ = $\underline{τ}$ $(\underline{A})$ ∩ $\bar{τ}$ $(\bar{A})$ .

In what follows, we will focus on the following question: if we adopt the residual implication on ${0, \frac{1}{2}, 1}$ , what types of interval sets can we obtain? The obtained results are summarized in the following propositions. It is important to note that we don’t distinguish interval set and three-valued mappings in the following discussion, because they are in an one-to-one correspondence. Precisely, given an interval set $[\underline{O}, \bar{O}]$ , we can define a three-valued mapping f as follows: f (o) =1 for $o \in \underline{O}$ , f (o) =0 for $o \in (\bar{O})^{c}$ and $\frac{1}{2}$ else. Conversely, given a three-valued mapping f, an interval set $[\underline{O}, \bar{O}]$ can be constructed as follows: $\underline{O} = f^{- 1} (1), \bar{O} = f^{- 1} (1) \cup f^{- 1} (\frac{1}{2})$ . Such a correspondence is surjective and injective.

Proposition 3.1. Let $L = ({0, \frac{1}{2}, 1}, \land, \lor, \otimes_{L}, \to_{L}, 0, 1)$ be a residuated lattice, (OB, AT, {0, ? , 1} , J) be an L-fuzzy context, then for O ⊆ OB, A ⊆ AT, $O^{*} = [\underline{σ} (O), \bar{σ} (O)],$ $A^{*} = [\underline{τ} (A), \bar{τ} (A)] .$

Example 3.1. Table 1 is an example of an incomplete formal context. It can be naturally treated as an L-fuzzy formal context, where $L = {0, \frac{1}{2}, 1} .$ Let O = {1, 2, 3} , then according to Definition 2.3, we have $\underline{σ} (O) = {a}$ , $\bar{σ} (O) = {a, c},$ and thus, $[\underline{σ}, \bar{σ}] (O) = [\underline{A}, \bar{A}] = [{a}, {a, c}] .$ This result can be reinterpreted from the viewpoint of three-valued logics. Indeed, the three-valued mapping corresponding to O is as follows: $\begin{matrix} f : AT \to {0, \frac{1}{2}, 1}, f (1) = f (2) = f (3) = 1, \\ f (4) = f (5) = f (6) = 0 . \end{matrix}$

According to Definition 3.2, if we employ Łukasiewicz implication operator →, we have

f^∗ (a) = ⋀_o∈OB (f (o) → I (o, a)) = (f (1) → I (1, a)) ∧ (f (2) → I (2, a)) ∧ (f (3) → I (3, a)) ∧ (f (4) → I (4, a)) ∧ (f (5) → I (5, a)) ∧ (f (6) → I (6, a)) = (1 → 1) ∧ (1 → 1) ∧ (1 → 1) ∧ (0 → $\frac{1}{2})$ ∧ (0 → 1) ∧ (0 → 1) = 1,

f^∗ (b) = ⋀_o∈OB (f (o) → I (o, b)) = (f (1) → I (1, b)) ∧ (f (2) → I (2, b)) ∧ (f (3) → I (3, b)) ∧ (f (4) → I (4, b)) ∧ (f (5) → I (5, b)) ∧ (f (6) → I (6, b)) = (1 → 1) ∧ (1 → 0) ∧ (1 → 1) ∧ (0 → 1) ∧ (0 → 1) ∧ (0 → 1) = 0,

f^∗ (c) = ⋀_o∈OB (f (o) → I (o, c)) = (f (1) → I (1, c)) ∧ (f (2) → I (2, c)) ∧ (f (3) → I (3, c)) ∧ (f (4) → I (4, c)) ∧ (f (5) → I (5, c)) ∧ (f (6) → I (6, c)) = (1 → $\frac{1}{2})$ ∧ (1 → $\frac{1}{2})$ ∧ (1 → 1) ∧ (0 → 1) ∧ (0 → 0) ∧ (0 → 1) = $\frac{1}{2}$ ,

similarly, f^∗ (d) = f^∗ (e) =0 . That is, f^∗ is defined as follows:

$\begin{matrix} f^{*} : AT \to {0, \frac{1}{2}, 1}, f (a) = 1, f (c) = \frac{1}{2}, \\ f (b) = f (d) = f (e) = 0 . \end{matrix}$

It can be checked easily that the corresponding interval set is [{a} , {a, c}], as desired.

Proposition 3.2. Let L = (L, ∧ , ∨ , ⊗ _G, → _G, 0, 1) be a residual lattice, (OB, AT, {0, ? , 1} , J) be an L-fuzzy context, then for O ⊆ OB, A ⊆ AT, $O^{*} = [\underline{σ} (O), \bar{σ} (O)],$ $A^{*} = [\underline{τ} (A), \bar{τ} (A)] .$

Proof. Note that the fundamental properties that are used in the proof of Proposition 3.1 are still valid for G $\ddot{o}$ del implication, and hence, the desired result also holds.

Since Łukasiewicz implication and G $\ddot{o}$ del implication are all residual implications on ${0, \frac{1}{2}, 1}$ , we can conclude that the proposed SE-ISI concepts can be equivalently described by using residual implications on ${0, \frac{1}{2}, 1}$ . A natural question then arises: if we employ other types of implications on ${0, \frac{1}{2}, 1}$ (not restricted to residual implication), what types of interval sets can we obtain? Before doing this, we should bear in mind that if we employ a non-residual implication on ${0, \frac{1}{2}, 1}$ , $({0, \frac{1}{2}, 1}, \land, \lor, \to, 0, 1)$ does not form a residual lattice. The following discussion is conducted without such a premise.

Proposition 3.3. In (1)-(2), if we employ Ja $\overset{´}{s}$ kowski implication, Kleene implication, Nelson implication, then for O ⊆ OB, A ⊆ AT, we have $O^{*} = [\underline{σ} (O), \bar{σ} (O)],$ $A^{*} = [\underline{τ} (A), \bar{τ} (A)] .$

Proof. The fundamental properties that used in the proof of Proposition 3.2 are 0 → a = 1, 1 → b = 1 ⇔ b = 1 and $1 \to \frac{1}{2} = \frac{1}{2}$ , which are valid for Ja $\overset{´}{s}$ kowski implication, Kleene implication, Nelson implication, and hence, the desired result holds.

The following proposition shows that if we use Gaines-Rescher implication, then $< \underline{τ}, \bar{τ} >$ in Definition 3.3 coincides with the mapping ∗ in Definition 3.2.

Proposition 3.4. If we use the Gaines-Rescher implication on ${0, \frac{1}{2}, 1}$ , then for $[\underline{A}, \bar{A}] \in 2^{AT}$ , $< \underline{τ}, \bar{τ} > ([\underline{A}, \bar{A}]) = \underline{τ} (\underline{A}) \cap \bar{τ} (\bar{A}) = g^{*}$ , where g denotes the three-valued mapping on AT corresponding to $[\underline{A}, \bar{A}]$ .

Proof. According to Definition 3.3, $< \underline{τ}, \bar{τ} > ([\underline{A}, \bar{A}]) = \underline{τ} (\underline{A}) \cap \bar{τ} (\bar{A})$ , and therefore, it is equivalent to show that for any o ∈ OB, $o \in \underline{τ} (\underline{A}) \cap \bar{τ} (\bar{A})$ if and only if g^∗ (o) =1 . Taking into Definition 3.2 into account, g^∗ (o) =1 if and only if for any a ∈ AT, g (a) =1 implies I (o, a) =1 and $g (a) = \frac{1}{2}$ implies I (o, a) =1 or $I (o, a) = \frac{1}{2}$ , or equivalently, for any $a \in \underline{A}$ , I (o, a) =1 and for any $a \in \bar{A}$ , I (o, a) =1 or $I (o, a) = \frac{1}{2}$ , that is, $o \in \underline{τ} (\underline{A}) \cap \bar{τ} (\bar{A})$ .

We thus conclude from Proposition 3.1-Proposition 3.4 that the collection SE-ISI and ISE-SI concepts can be obtained by employing Łukasiewicz implication (or equivalently, G $\ddot{o}$ del implication, Ja $\overset{´}{s}$ kowski implication, Kleene implication, Nelson implication) in E-I direction (i.e., from extension to intension) and Gaines-Rescher implication in I-E direction (i.e., from intension to extension).

However, if we employ Sette and Gaines-Rescher implication in E-I direction, we may obtain a different result.

Proposition 3.5. If we employ Soboci $\overset{´}{n}$ ski implication, Bochvar external implication and Gaines-Rescher implication in (1), then for O ⊆ OB, A ⊆ AT, we have $O^{*} = [\underline{σ} (O), \underline{σ} (O)],$ $A^{*} = [\underline{τ} (A), \underline{τ} (A)] .$

Proof. According to the definition of Soboci $\overset{´}{n}$ ski implication (see Table 1), a → b ∈ {0, 1} for $a, b \in {0, \frac{1}{2}, 1}$ , and thus, O^* reduces to a ordinary set. Moreover, since O^* (a) =1 if and only if for any o ∈ OB, O (o) =1 implies I (o, a) =1, if and only if for any o ∈ O, I (o, a) =1, we conclude that $O^{*} = [\underline{σ} (O), \underline{σ} (O)] .$ $A^{*} = [\underline{τ} (A), \underline{τ} (A)]$ can be shown in a similar manner.

The above proposition shows that if we employ Soboci $\overset{´}{n}$ ski implication, Bochvar external implication and Gaines-Rescher implication, then O^* is equal to the derivation operator in the least completion of an incomplete formal context.

Example 3.2. (Continued from Example 3.1) Let O = {3, 4} , then according to Definition 2.3, we have $\underline{σ} (O) = {b, c}$ , $\bar{σ} (O) = {a, b, c, d},$ and thus, $[\underline{σ}, \bar{σ}] (O) = [\underline{O}, \bar{O}] = [{b, c}, {a, b, c, d}] .$ This result can be reinterpreted from the viewpoint of three-valued logics. Indeed, the three-valued mapping corresponding to O is as follows: $\begin{matrix} f : AT \to {0, \frac{1}{2}, 1}, f (3) = f (4) = 1, \\ f (1) = f (2) = f (5) = f (6) = 0 . \end{matrix}$

According to Definition 3.2, if we employ G $\ddot{O}$ del implication operator →, we have

f^∗ (a) = ⋀_o∈OB (f (o) → I (o, a)) = (f (1) → I (1, a)) ∧ (f (2) → I (2, a)) ∧ (f (3) → I (3, a)) ∧ (f (4) → I (4, a)) ∧ (f (5) → I (5, a)) ∧ (f (6) → I (6, a)) = (0 → 1) ∧ (0 → 1) ∧ (1 → 1) ∧ (1 → $\frac{1}{2})$ ∧ (0 → 1) ∧ (0 → 1) = $\frac{1}{2}$ ,

f^∗ (b) = ⋀_o∈OB (f (o) → I (o, b)) = (f (1) → I (1, b)) ∧ (f (2) → I (2, b)) ∧ (f (3) → I (3, b)) ∧ (f (4) → I (4, b)) ∧ (f (5) → I (5, b)) ∧ (f (6) → I (6, b)) = (0 → 1) ∧ (0 → 0) ∧ (1 → 1) ∧ (1 → 1) ∧ (0 → 1) ∧ (0 → 1) = 1,

f^∗ (c) = ⋀_o∈OB (f (o) → I (o, c)) = (f (1) → I (1, c)) ∧ (f (2) → I (2, c)) ∧ (f (3) → I (3, c)) ∧ (f (4) → I (4, c)) ∧ (f (5) → I (5, c)) ∧ (f (6) → I (6, c)) = (0 → $\frac{1}{2})$ ∧ (0 → $\frac{1}{2})$ ∧ (1 → 1) ∧ (0 → 1) ∧ (0 → 0) ∧ (0 → 1) = 1,

similarly, f^∗ (d) =1, f^∗ (e) =0 . That is, f^∗ is defined as follows: $\begin{matrix} f^{*} : AT \to {0, \frac{1}{2}, 1}, f (b) = f (c) = 1, \\ f (a) = f (d) = \frac{1}{2}, f (e) = 0 . \end{matrix}$

It can be checked easily that the corresponding interval set is [{b, c} , {a, b, c, d}], as desired.

Proposition 3.6. If we employ Sette implication in (1), then for O ⊆ OB, A ⊆ AT, we have $O^{*} = [\bar{σ} (O), \bar{σ} (O)],$ $A^{*} = [\bar{τ} (A), \bar{τ} (A)] .$

Proof. According to the definition of Sette implication (see Table 1), a → b ∈ {0, 1} for $a, b \in {0, \frac{1}{2}, 1}$ , and thus, O^* reduces to a ordinary set, moreover, since O^* (a) =1 if and only if for any o ∈ OB, O (o) =1 implies $I (o, a) \in {\frac{1}{2}, 1}$ , if and only if $a \in \bar{σ} (O)$ . We can thus conclude that $O^{*} = [\bar{σ} (O), \bar{σ} (O)] .$

$A^{*} = [\bar{τ} (A), \bar{τ} (A)]$ can be shown in a similar manner.

The above proposition shows that if we employ Sette implication, then O^* is equal to the derivation operator in the greatest completion of an incomplete formal context.

We summarize the obtained results in Table 4.

Table 4

An incomplete formal context (U, AT, {+ , ? , -})

	a	b	c	d	e
1	+	+	?	-	-
2	+	-	?	+	+
3	+	+	+	?	-
4	?	+	+	+	-
5	+	+	-	-	-
6	+	+	+	+	?

Table 5

The relationship between 3-valued logic and SE-ISI concepts

	O	A	Residual implication?	Corresponding formal context
G $\ddot{o}$ del	$[\underline{σ} (O), \bar{σ} (O)]$	$[\underline{τ} (A), \bar{τ} (A)]$	yes	The original context
Łukasiewicz	$[\underline{σ} (O), \bar{σ} (O)]$	$[\underline{τ} (A), \bar{τ} (A)]$	yes	The original context
Nelson	$[\underline{σ} (O), \bar{σ} (O)]$	$[\underline{τ} (A), \bar{τ} (A)]$	no	The original context
Kleene	$[\underline{σ} (O), \bar{σ} (O)]$	$[\underline{τ} (A), \bar{τ} (A)]$	no	The original context
Ja $\overset{´}{s}$ kowski	$[\underline{σ} (O), \bar{σ} (O)]$	$[\underline{τ} (A), \bar{τ} (A)]$	no	The original context
Sette	$[\bar{σ} (O), \bar{σ} (O)]$	$[\bar{τ} (A), \bar{τ} (A)]$	no	Greatest completion
Bochvar external	$[\underline{σ} (O), \underline{σ} (O)]$	$[\underline{τ} (A), \underline{τ} (A)]$	no	Least completion
Gaines-Rescher	$[\underline{σ} (O), \underline{σ} (O)]$	$[\underline{τ} (A), \underline{τ} (A)]$	no	Least completion
Soboci $\overset{´}{n}$ ski	$[\underline{σ} (O), \underline{σ} (O)]$	$[\underline{τ} (A), \underline{τ} (A)]$	no	Least completion

3.2 Three-valued logics and ISE-ISI concepts

Definition 3.4. [17] A pair of interval-set based operators $[\underline{σ}, \bar{σ}] : I (2^{OB}) \to I (2^{AT}) \cup \emptyset$ and $[\underline{τ}, \bar{τ}] : I (2^{AT}) \to I (2^{OB}) \cup \emptyset$ is defined by: for an interval set of objects $[\underline{O}, \bar{O}]$ and an interval set of attributes $[\underline{A}, \bar{A}]$ , $[\underline{σ}, \bar{σ}] ([\underline{O}, \bar{O}]) = [\underline{σ} (\underline{O}), \bar{σ} (\bar{O})],$ $[\underline{τ}, \bar{τ}] ([\underline{A}, \bar{A}]) = [\underline{τ} (\underline{A}), \bar{τ} (\bar{A})],$ where $[\underline{σ} (\underline{O}), \bar{σ} (\bar{O})] = \emptyset$ when $\bar{σ} (\bar{O}) \subset \underline{σ} (\underline{O})$ and $[\underline{τ} (\underline{A}), \bar{τ} (\bar{A})] = \emptyset$ when $\bar{τ} (\bar{A}) \subset \underline{τ} (\underline{A}) .$

For ISE-ISI concepts, both extents and intents are interval sets, they can be equivalently represented by three-valued mappings. In what follows, we will examine the following issue: Whether ISE-ISI concepts can be generated by using the existing implications on ${0, \frac{1}{2}, 1}$ ?

The following proposition provides different ISE-ISI concepts by using all possible implications on ${0, \frac{1}{2}, 1} .$

Proposition 3.7. (i) If we use Łukasiewicz implication, then for $[\underline{O}, \bar{O}] \in 2^{OB},$ we have $[\underline{O}, \bar{O}]^{*} = [\bar{σ} (\bar{O} \ \underline{O}) \cap \underline{σ} (\underline{O}), \bar{σ} (\underline{O})]$ ,

(ii) If we use G $\ddot{o}$ del implication, then for $[\underline{O}, \bar{O}] \in 2^{OB},$ we have $[\underline{O}, \bar{O}]^{*} = [\bar{σ} (\bar{O} \ \underline{O}) \cap \underline{σ} (\underline{O}), \bar{σ} (\bar{O})]$ ,

(iii) If we use Sette implication, then for $[\underline{O}, \bar{O}] \in 2^{OB},$ we have $[\underline{O}, \bar{O}]^{*} = [\bar{σ} (\bar{O}), \bar{σ} (\bar{O})]$ ,

(iv) If we use Nelson implication, then for $[\underline{O}, \bar{O}] \in 2^{OB},$ we have $[\underline{O}, \bar{O}]^{*} = [\underline{σ} (\underline{O}), \bar{σ} (\underline{O})]$ ,

(v) If we use Kleene implication, then for $[\underline{O}, \bar{O}] \in 2^{OB},$ we have $[\underline{O}, \bar{O}]^{*} = [\underline{σ} (\bar{O}), \bar{σ} (\underline{O})]$ ,

(vi) If we use Ja $\overset{´}{s}$ kowski implication, then for $[\underline{O}, \bar{O}] \in 2^{OB},$ we have $[\underline{O}, \bar{O}]^{*} = [\underline{σ} (\bar{O}), \bar{σ} (\bar{O})]$ ,

(vii) If we use Soboci $\overset{´}{n}$ ski implication, then for $[\underline{O}, \bar{O}] \in 2^{OB},$ we have $[\underline{O}, \bar{O}]^{*} = [\underline{σ} (\bar{O}), \bar{σ} (\bar{O} \ \underline{O}) \cap \underline{σ} (\underline{O})]$ ,

(viii) If we use Gaines-Rescher implication, then for $[\underline{O}, \bar{O}] \in 2^{OB},$ we have $[\underline{O}, \bar{O}]^{*} = [\bar{σ} (\bar{O} \ \underline{O}) \cap \underline{σ} (\underline{O}), \bar{σ} (\bar{O} \ \underline{O}) \cap \underline{σ} (\underline{O})]$ ,

(ix) If we use Bochvar external implication, then for $[\underline{O}, \bar{O}] \in 2^{OB},$ we have $[\underline{O}, \bar{O}]^{*} = [\underline{σ} (\underline{O}), \underline{σ} (\underline{O})]$ .

Proposition 3.7 tells that if we use different types if implications on ${0, \frac{1}{2}, 1}$ , we can get different interval sets. In particularly, if we use Sette, Bochvar external or Gaines-Rescher implication, then the obtained interval set reduces to an ordinary set.

According to Definition 3.4, if $\underline{σ} (\underline{O}) \subseteq \bar{σ} (\bar{O})$ , then the interval-set based operator $[\underline{σ}, \bar{σ}]$ satisfies $[\underline{σ}, \bar{σ}] ([\underline{O}, \bar{O}]) = [\underline{σ} (\underline{O}), \bar{σ} (\bar{O})] .$ The following proposition states that there doesn’t exist an implication → on ${0, \frac{1}{2}, 1}$ such that $[\underline{σ}, \bar{σ}]$ can be realized by using →.

Proposition 3.8. Let → be one of implication operators on ${0, \frac{1}{2}, 1}$ in Table 1 and $[\underline{O}, \bar{O}] \subseteq 2^{OB}$ , then $[\underline{σ}, \bar{σ}]$ cannot be obtained by using →, under the premise of $\underline{σ} (\underline{O}) \subseteq \bar{σ} (\bar{O})$ .

4 The role of three-valued logic in characterizing attribute implication

4.1 Attribute implications

Attribute implications are important tools of conceptual data analysis. Let K = (OB, AT, I) be a complete context. For sets A, B ⊆ AT of attributes, we designate by A → B an attribute implication, where A is called the premise, B is called the conclusion of the implication. We say that this implication holds (is valid) in K, if and only if, for every object o ∈ OB to which all attributes from A apply, all attributes from B apply, too, or equivalently, A ⊆ σ ({o}) implies B ⊆ σ ({o}) .

By treating attributes in M as propositional variables, we can also interpret A → B as a propositional formula ∧_a∈A → ∧ _b∈B. Such attribute implications contain a lot of information about the data, and the concept lattice of a given complete context K is determined up to isomorphism by the knowledge of a set of attribute implications valid in K and generating the set of all attribute implications valid in K with respect to the ARMSTRONG rules ^[1].

For incomplete contexts, attribute implications can be generated as follows. Let K = (OB, AT, {+, ?, -}, J) be an incomplete formal context. An attribute implication A → B is certainly valid in K if it holds in any completion of K; an attribute implication A → B is possibly valid in K, if it holds in some completion of K; otherwise, the attribute implication is invalid in K. Usually, we assign the value 1 to those certainly valid attribute implications, the value $\frac{1}{2}$ to those possibly but not certainly valid attribute implications and 0 to those invalid attribute implications.

4.2 Evaluating attribute implications with Kleene logic

In [1], the three-valued Kleene logic is used to evaluate attribute implications. Every pair (o, a) (o ∈ OB, a ∈ AT) can be associated with a propositional variable, whose value is given by J (o, a), that is, $J (o, a) \in {0, \frac{1}{2}, 1}$ . The value of an attribute implication A → B in the context K is actually the conjunction of the values of corresponding implications on single objects $&_{o \in OB} (&_{a \in A} J (o, a) \to &_{a \in B} J (o, a)) .$ (3) (3) is also called a Kripke semantics. However, as stated in [1], the Kleene logic is not guaranteed to compute the truth-value of an implication, as well as of any other propositional formula, which can be seen from the fact that the implication {a} → {a} is valid in every completion of any incomplete context, yet one also easily realizes that it will get the truth value $\frac{1}{2}$ whenever a gets the value. However, one has the following result, which is helpful to check the validity and possibility of attribute implications with disjoints sets for premise and conclusion.

Proposition 4.1. Let K = (OB, AT, {+ , - , ?} , J) be an incomplete context, and let Φ be a propositional formula with propositional variables in AT. Then the three-valued Kleene-logic computes the truth value of Φ with respect to K correctly in the sense of our Kriple-semantics, whenever every propositional variable occurs within Φ at most once.

Proposition 4.2. [1] Let A → B be an attribute implication with respect to some incomplete context K, then the three-valued Kleene logic computes its truth value correctly when it is applied to A → (B \ A) instead of A → B.

Proposition 4.2 shows a way to evaluate an attribute implication syntactically with respect to any incomplete context, because the implications A → B and A → (B \ A) always contain the same information.

Note that Proposition 4.1 and 4.2 hold for attribute implications with disjoint sets for premise and conclusion. By considering a more general case, we can present the following sufficient and necessary condition for computing the true value of an attribute implication correctly by means of three-valued Kleene logic.

Let A → B be an attribute implication and C = A ∩ B. For o ∈ OB, c ∈ C, a ∈ A \ C, b ∈ B \ C, we call (J (o, a), J (o, b), J (o, c)) a truth vector of o with respect to A → B if J (o, c) & J (o, a) → J (o, c) & J (o, b) = &_o ∈OB (& _a∈A J (o, a) → &_a∈B J (o, a)). Denote

$\begin{matrix} H_{0} = {(J (o, c), J (o, a), J (o, b)) ∣ J (o, c) & J (o, a) \\ \to J (o, c) & J (o, b) = 0}, \\ H_{\frac{1}{2}} = {(J (o, a), J (o, b), J (o, c)) ∣ J (o, c) & J (o, a) \\ \to J (o, c) & J (o, b) = \frac{1}{2}}, \\ H_{1} = {(J (o, a), J (o, b), J (o, c)) ∣ J (o, c) & J (o, a) \\ \to J (o, c) & J (o, b) = 1} . \end{matrix}$

Owing to Proposition 4.1, we will consider the case C≠ ∅ in the following discussion.

Proposition 4.3. Let K = (OB, AT, {+ , - , ?} , J) be an incomplete context, and let A → B be an attribute implication with A\ C ≠ ∅ and B\ C ≠ ∅. Then the three-valued Kleene-logic computes the truth value of A → B with respect to K correctly in the sense of our Kriple-semantics if and only if $&_{o \in OB} (&_{a \in A} J (o, a) \to &_{b \in B} J (o, b)) = \frac{1}{2}$ implies $H_{\frac{1}{2}} \ {(+, +, ?), (?, +, ?)} \neq \emptyset .$

Proposition 4.3 holds under the preliminary condition A\ C ≠ ∅ and B\ C ≠ ∅. For the left cases, we have the following results.

Proposition 4.4. Let K = (OB, AT, {+ , - , ?} , J) be an incomplete context, and let A → B be an attribute implication with A\ C = ∅ and B\ C ≠ ∅. Then the three-valued Kleene-logic computes the truth value of A → B with respect to K correctly in the sense of our Kriple-semantics.

Proof. The preliminary condition A\ C = ∅ and B\ C ≠ ∅ means A ⊂ B, which leads immediately to &_a∈AJ (o, a) ≥ & _b∈BJ (o, b) for any o ∈ OB. Then the desired result follows from the following table (i.e., Table 6).

Table 6

A comparative analysis between the computed value and the true value w.r.t. different truth vectors

J(o,a)	J(o,b)	J(o,c)	computed value	true value true value	correct or wrong
+	+	?	?→?=?	+	wrong
+	?	?	?→?=?	?	correct
?	+	?	?→?=?	+	wrong
?	?	?	?→?=?	?	correct
?	-	?	?→-=?	?	correct
-	?	?	-→?=+	+	correct
-	-	?	-→-=+	+	correct
+	-	?	?→-=?	?	correct
-	+	?	-→?=+	+	correct
+	+	+	+→+=+	+	correct
-	-	+	-→-=+	+	correct
?	+	+	?→+=+	+	correct
+	?	+	+→?=?	?	correct
?	?	+	?→?=?	?	correct
?	-	+	?→-=?	?	correct
-	?	+	-→?=+	+	correct
+	-	+	+→-=-	-	correct
-	+	+	-→+=+	+	correct
+	+	-	-→-=+	+	correct
-	-	-	-→-=+	+	correct
?	+	-	-→-=+	+	correct
+	?	-	-→-=+	+	correct
?	?	-	-→-=+	+	correct
?	-	-	-→-=+	+	correct
-	?	-	-→-=+	+	correct
+	-	-	-→-=+	+	correct
-	+	-	-→-=+	+	correct

Proposition 4.5. Let K = (OB, AT, {+ , - , ?} , J) be an incomplete context, and let A → B be an attribute implication with A\ C ≠ ∅ and B\ C = ∅. Then the three-valued Kleene-logic computes the truth value of A → B with respect to K correctly in the sense of our Kriple-semantics.

Proof. The preliminary condition A\ C ≠ ∅ and B\ C = ∅ means A ⊃ B, which leads immediately to &_a∈AJ (o, a) ≤ & _b∈BJ (o, b) for any o ∈ OB. Then the desired result follows from the following table (i.e., Table 7).

Table 7

Truth table corresponding to Proposition 4.4

&_a∈AJ (o, a)	&_a∈AJ (o, b)	computed value	truth value	correct or wrong
+	+	+→+=+	+	correct
+	?	+→?=?	?	correct
+	-	+→-=-	-	correct
?	?	?→?=?	?	correct
?	-	?→-=?	?	correct
-	-	-→-=+	+	correct

Proposition 4.6. Let K = (OB, AT, {+ , - , ?} , J) be an incomplete context, and let A → B be an attribute implication with A\ C = ∅ and B\ C = ∅. Then the three-valued Kleene-logic computes the truth value of A → B with respect to K correctly in the sense of our Kriple-semantics if and only if ∀o ∈ OB, & _a∈AJ (o, a) = & _b∈BJ (o, b) ∈ {0, 1}.

Proof. The preliminary condition A\ C = ∅ and B\ C = ∅ means A = B, which is a certainly valid attribute implication. Then the desired result follows immediately from the fact that ? → ? = ? holds in three-valued Kleene logic (see Table 8).

Table 8

Truth table corresponding to Proposition 4.5

&_a∈AJ (o, a)	&_b∈BJ (o, b)	computed value	truth value	correct or wrong
+	+	+→+=+	+	correct
?	+	?→+=+	+	correct
?	?	?→?=?	?	correct
-	+	-→+=+	+	correct
-	?	-→?=+	+	correct
-	-	-→-=+	+	correct

5 Concluding remarks

In this paper, we present a novel characterization of partially-known concepts by means of three-valued logics. Concretely, we examine in detail what type of implications on ${0, \frac{1}{2}, 1}$ can generate those proposed four types of partially-known concepts by using the existing lattice-valued concept lattice. We also evaluate the role of three-valued logic in characterizing attribute implications. A sufficient and necessary condition for computing the true value of an implication correctly in the sense of Kriple semantics is provided, which improve the existing results.

Our results are based on the preliminary assumption that the possible attribute values of each object are independent, that is, all completions of the incomplete formal context are possible. In the future study, we will focus on the situation where objects in incomplete contexts are not independent to each other, and present the logical characterization of partially known concepts. Furthermore, it is an interesting topic to extend the present study to incomplete fuzzy formal context and provide the corresponding characterizations by using fuzzy logics.

Footnotes

Appendix

1. Proof of Proposition 3.1

Proof. According to Definition 3.2, O^* (a) = ⋀ _o∈OB (O (o) → I (o, a)) , due to the one-to-one correspondence between the collection of three-valued mappings and that of interval sets, it is equivalent to show that for any $a \in AT, a \in \underline{σ} (O)$ if and only if $O^{*} (a) = 1, a \in \bar{σ} (O)$ if and only if $O^{*} (a) \in {\frac{1}{2}, 1}$ .

Considering the fact that O is a crisp set (that is, O (o) is either 0 or 1) and the fundamental property a → _Lb = 1 ⇔ a ≤ b of Łukasiewicz implication, we then obtain that O^* (a) =1 if and only if for any o ∈ OB, O (o) =1 implies I (o, a) =1, if and only if for any o ∈ O, I (o, a) =1. Combining the definition of $\underline{σ}$ , we thus conclude that $a \in \underline{σ} (O)$ if and only if O^* (a) =1.

Similarly, $O^{*} (a) \in {\frac{1}{2}, 1}$ if and only if for any o ∈ OB, O (o) =1 implies $I (o, a) = \frac{1}{2}$ or I (o, a) =1, if and only if for any o ∈ O, $I (o, a) = \frac{1}{2}$ or I (o, a) =1 Combining the definition of $\bar{σ}$ , we thus conclude that $a \in \bar{σ} (O)$ if and only if $O^{*} (a) \in {\frac{1}{2}}$ .

$A^{*} = [\underline{τ} (O), \bar{τ} (O)]$ can be shown in a similar manner, and hence we omit the detailed proof here.

2. Proof of Proposition 3.7

Proof. We need only show the proof of (i), and the others can be shown in a similar manner.

We use f to denote the three-valued set on the universe corresponding to the interval set $[\underline{O}, \bar{O}]$ , then it is equivalent to show that f^* (a) =1 if and only if $a \in \bar{σ} (\bar{O} \ \underline{O}) \cap \underline{σ} (\underline{O}),$ and $f^{*} (a) \in {\frac{1}{2}, 1}$ if and only if $a \in \bar{σ} (\underline{O}), a \in AT$ .

Considering that for Łukasiewicz implication →, a → b = 1 if and only if $(a, b) \in {(1, 1), (\frac{1}{2}, \frac{1}{2}), (\frac{1}{2}, 1), (0, 0), (0, \frac{1}{2}), (0, 1)}$ , $a, b \in {0, \frac{1}{2}, 1}$ . we then have from Definition 3.2 that f^* (a) =1 if and only if for any object o ∈ OB, $f (o) = \frac{1}{2}$ implies $I (o, a) \in {\frac{1}{2}, 1}$ and f (o) =1 implies I (o, a) =1, if and only if $I (o, a) \in {\frac{1}{2}, 1}$ holds for any $o \in \bar{O} \ \underline{O}$ and f (o) =1 holds for $o \in \underline{O}$ , equivalently, $a \in \bar{σ} (\bar{O} \ \underline{O})$ and $a \in \underline{σ} (\underline{O}),$ that is, $a \in \bar{σ} (\bar{O} \ \underline{O}) \cap \underline{σ} (\underline{O}) .$ Similarly, $f^{*} (a) \in {\frac{1}{2}, 1}$ if and only if for any object o ∈ OB, f (o) =1 implies $I (o, a) \in {\frac{1}{2}, 1}$ , if and only if $I (o, a) \in {\frac{1}{2}, 1}$ holds for any $o \in \underline{O}$ , if and only if $a \in \bar{σ} (\underline{O})$ .

3. Proof of Proposition 3.8

Proof Denote by f the three-valued function corresponding to $[\underline{O}, \bar{O}]$ . Suppose, on the contrary, that $[\underline{σ}, \bar{σ}]$ can be obtained by using →, that is, $f^{*} = [\underline{σ}, \bar{σ}] ([\underline{O}, \bar{O}])$ . We have from $[\underline{σ}, \bar{σ}] ([\underline{O}, \bar{O}]) = [\underline{σ} (\underline{O}), \bar{σ} (\bar{O})]$ that for any object o ∈ OB,a ∈ AT, f^∗ (a) =1 if and only if $a \in \underline{σ} (\underline{O})$ , if and only if for any $o \in \underline{O}, I (o, a) = 1$ , if and only if for any o ∈ OB, f (o) =1 implies that I (o, a) =1, and hence, → satisfies the following condition $\begin{matrix} 0 \to 0 = 0 \to \frac{1}{2} = 0 \to 1 = \frac{1}{2} \to 0 = \frac{1}{2} \to \frac{1}{2} \\ = \frac{1}{2} \to 1 = 1 \to 1 = 1, 1 \to 0 \neq 1, 1 \to \frac{1}{2} \neq 1 . \end{matrix}$ Similarly, we also conclude that $f^{*} (a) \in {\frac{1}{2}, 1}$ if and only if $a \in \bar{σ} (\bar{O})$ , which together with the definition of derivation operator $\bar{σ}$ implies that $1 \to \frac{1}{2} = \frac{1}{2}$ and $\frac{1}{2} \to 0 \notin {\frac{1}{2}, 1}$ , that is a contradiction!

4. Proof of Proposition 4.3

Proof Firstly, we need show that for o ∈ OB, c ∈ C,a ∈ A \ C, b ∈ B \ C, J (o, c)& J (o, a) → J (o, c) &J (o, b) computes the truth value of A → B,if and only if (J (o, a) , J (o, b) , J (o, c)) ∉ {(+ , + , ?),(? , + , ?)}, which can be seen from the following table (i.e., Table 5).

Now, we will show the proof by considering the necessity part and sufficiency part, respectively.

“Necessity”: Suppose, on the contrary, that $&_{o \in OB} (&_{a \in A} J (o, a) \to &_{b \in B} J (o, b)) = \frac{1}{2}$ and $H_{\frac{1}{2}}$ \ {(? , + , +) , (? , ? , +)} = ∅ , that is, $\emptyset \neq H_{\frac{1}{2}} \subseteq {(+, +, ?), (?, +, ?)} .$ However, we observe from Table 5 that when $\emptyset \neq H_{\frac{1}{2}} \subseteq {(+, +, ?), (?, +, ?)},$ the three-valued Kleene-logic computes the truth value of A → B with respect to K wrongly, that is a contradiction!

“Sufficiency”: If &_o∈OB (& _a∈AJ (o, a) → & _b∈B $J (o, b)) = \frac{1}{2}$ implies $H_{\frac{1}{2}} \ {(+, +, ?), (?, +, ?)} \neq \emptyset$ ,there are three cases to be considered below.

Case 1: &_o∈OB (& _a∈AJ (o, a) → & _b∈BJ (o, b)) =1: We then have from Table 5 that the three-valued Kleene-logic computes the truth value of A → B with respect to K correctly in the sense of our Kriple-semantics.

Case 2: &_o∈OB (& _a∈AJ (o, a) → & _b∈BJ (o, b)) =0: We then have from Table 5 that the three-valued Kleene-logic computes the truth value of A → B with respect to K correctly in the sense of our Kriple-semantics,

Case 3: $&_{o \in OB} (&_{a \in A} J (o, a) \to &_{b \in B} J (o, b)) = \frac{1}{2}$ : We then have from the preliminary condition that there exists a truth vector $(m, l, n) \in H_{\frac{1}{2}} \ {(+, +, ?), (?, +, ?)},$ which together with Table 5 shows that the computed value coincide with the truth value of A → B, that is, the three-valued Kleene-logic computes the truth value of A → B with respect to K correctly in the sense of our Kriple-semantics.

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