Connecting Systems of Mathematical Fuzzy Logic with Fuzzy Concept Lattices

Abstract

In this paper our aim is to explore a new look at formal systems of fuzzy logics using the framework of (fuzzy) formal concept analysis (FCA). Let L be an extension of MTL complete with respect to a given L-chain. We investigate two possible approaches. The first one is to consider fuzzy formal contexts arising from L where attributes are identified with L-formulas and objects with L-evaluations: every L-evaluation (object) satisfies a formula (attribute) to a given degree, and vice-versa. The corresponding fuzzy concept lattices are shown to be isomorphic to quotients of the Lindenbaum algebra of L. The second one, following an idea in a previous paper by two of the authors for the particular case of Gödel fuzzy logic, is to use a result by Ganter and Wille in order to interpret the (lattice reduct of the) Lindenbaum algebra of L-formulas as a (classical) concept lattice of a given context.

1 Introduction

In this paper our aim is to explore a new look at formal systems of fuzzy logics using the framework of (fuzzy) formal concept analysis (FCA).

The possibility of connecting descriptions of real-world contexts with powerful formal instruments is what makes (fuzzy) FCA a promising framework, merging the intuitions of intended semantics with the advantages of formal semantics. In the case of classical logic, a first attempt has been done in [8].

To build a bridge between systems of fuzzy logic and FCA, we explore two possible approaches. In the first one, given a fuzzy logic L we consider fuzzy FCA tables where attributes are described by formulas of the logic L, while L-evaluations play the role of objects: every object (L-evaluation) satisfies attributes (formulas) to a given degree, and vice-versa, every attribute (formula) is satisfied to a given extent by objects (evaluations).

The second one is, following the idea in [5] for the particular case of Gödel fuzzy logic [12], is to use Ganter and Wille’s result [10, Theorem 3] in order to interpret the lattice reduct of the Lindenbaum algebra of L-formulas as a lattice of the set of formal concepts of a given context. Then, in order to endow the lattice of concepts with a structure of L-algebra, suitable operations on formal concepts have to be defined.

The paper is structured as follows. After this brief introduction, we recall some background notions in Sect. 2, in Sect. 3 we introduce concept lattices of formulas and evaluations, and in Sect. 4 we recall the construction of [5]. Both approaches will be used to obtain formal concepts for formulas of the 3-valued Łukasiewicz logic.

2 Preliminaries

2.1 Basic Notions on Formal Concept Analysis

We recollect the basic definitions and facts about formal concept analysis needed in this work. For further details on this topics we refer the reader to [10].

Recall that an element j of a distributive lattice H is called a join-irreducible if j is not the bottom of H and if whenever $j = a{\sqcup } b$, then $j= a$ or $j= b$, for $a,b \in L$. Meet-irreducible elements are defined dually. Given a lattice $\mathbf{H} =(H,\sqcap ,\sqcup ,1)$, we denote by $\mathfrak {J}(H)$ the set of its join-irreducible elements, and by $\mathfrak {M}(H)$ the set of its meet-irreducible elements.

Let G and M be arbitrary sets of objects and attributes, respectively, and let $I\subseteq G\times M$ be an arbitrary binary relation. Then, the triple $\mathbb {K} = (G,M,I)$ is called a formal context. For $g\in G$ and $m\in M$, we interpret $(g,m)\in I$ as “the object g has attribute m”. For $A\subseteq G$ and $B\subseteq M$, a Galois connection between the powersets of G and M is defined through the following operators:

$$ A^* =\{m\in M\mid \forall g\in A : gIm\} \qquad B^\circ =\{g\in G\mid \forall m\in B : gIm\}$$

Every pair (A, B) such that $A^*=B$ and $B^\circ =A$ is called a formal concept. A and B are the extent and the intent of the concept, respectively. Given a context $\mathbb {K}$, the set $\mathfrak {B}(\mathbb {K})$ of all formal concepts of $\mathbb {K}$ is partially ordered by $(A_1, B_1)\le (A_2, B_2)$ if and only if $A_1 \subseteq A_2$ (or, equivalently, $B_2\subseteq B_1$). The basic theorem on concept lattices [10, Theorem 3] states that the set of formal concepts of the context $\mathbb {K}$ is a complete lattice $(\mathfrak {B}(\mathbb {K}), \sqcap , \sqcup )$, called concept lattice, where meet and join are defined by:

(1)

for a set J of indexes. The following proposition is fundamental for our purposes.

Proposition 1

([10, Proposition 12]). For every finite lattice H there is (up to isomorphisms) a unique context $\mathbb {K}_H$, with $L\cong \mathfrak {B}(\mathbb {K}_H)$:

$$ \mathbb {K}_H := (\mathfrak {J}(H), \mathfrak {M}(H),\le ).$$

The context $\mathbb {K}_H$ is called the standard context of the lattice H.

Since H is finite, $\mathfrak {J}(H)$ is finite as well. Hence, the concept $(\mathfrak {J}(H),\emptyset )$ is the top element of $\mathfrak {B}(\mathbb {K}_H)$. We denote it by $\top _G$, emphasizing the fact that the join-irreducible elements of L are the objects of our context. Analogously, the concept $(\emptyset ,\mathfrak {M}(H))$ is the bottom element of $\mathfrak {B}(\mathbb {K}_H)$, and we denote it by $\bot _M$.

2.2 On t-Norm Based Fuzzy Logics

In this paper we investigate logical systems based on left continuous t-norms, that are binary, commutative, associative and monotonically non-decreasing operations over [0, 1] that have 1 as unit element. A t-norm operator $\odot$ is used to interpret a conjunction connective, while its corresponding implication connective $\rightarrow$ is modelled by the residuum of $\odot$, that is, defined by $x \rightarrow y = max\{z\mid x\odot z\le y\}$ for all $x,y,z\in [0,1]$. It has been shown that the necessary and sufficient condition for a t-norm $\odot$ to have a residuum (i.e. satisfying the condition $x \odot y \le z$ iff $x \le y \rightarrow z$ for all $x,y,z\in [0,1]$) is the left-continuity $\odot$.

In [7] the authors introduce MTL, the logic of all left-continuos t-norms and their residua [13]. MTL encompasses the Basic fuzzy Logic BL of Hájek [12], which is the logic of continuous t-norms and their residua. For axiomatisations of MTL and BL, we refer the reader to [7] and [12] respectively.

Other relevant t-norm based fuzzy logics can be obtained as schematic extensions of MTL or BL. Gödel logic G is the schematic extension of BL obtained by adding the idempotency axiom, $\varphi \rightarrow (\varphi \odot \varphi )$. Łukasiewicz logic $\L$ is the schematic extension of BL obtained by adding the double negation axiom $\lnot \lnot \varphi \rightarrow \varphi$. Adding $\varphi \odot \varphi \leftrightarrow \varphi \odot \varphi \odot \varphi$ to $\L$ we obtain the 3-valued Łukasiewicz logic $\L _3$.

Our interest in $\L _3$ is given by the recent paper [6], where authors characterize this logic as the logic of prototypes and counterexamples. Gödel logic will be used as a stepping stone for developing the methodology to be applied to the case of $\L _3$.

Each schematic extension L of MTL determines a subvariety $\mathbb {V}(\mathrm {L})$ of the variety of MTL algebras $\mathbb {MTL}$, that is the class of algebras $\mathbf {A}=(A,\wedge ,\vee ,\odot ,\rightarrow ,\bot ,\top )$ of type (2, 2, 2, 2, 0, 0) such that $(A,\wedge ,\vee ,\bot ,\top )$ is a bounded lattice, with top $\top$ and bottom $\bot$, $(A,\odot ,\top )$ is a commutative monoid, satisfying the residuation equivalence, $x \odot y \le z$ if and only if $x \le y \rightarrow z$, and the prelinearity equation $(x \rightarrow y ) \vee (y \rightarrow x)=\top$¹. Negation is usually defined as $\lnot x=x\rightarrow \bot$.

The notion of logical consequence for a logic $\mathrm {L}$ relative to a class $\mathcal A \subseteq \mathbb {V}(\mathrm {L})$ is defined as follows: for any set of formulas $T\cup \{\varphi \}$, $\varphi$ is a logical consequence of T, written $T \models _\mathcal{A} \varphi$, whenever for all algebra $\mathbf{A} \in \mathcal{A}$ and each evaluation e of formulas on $\mathbf A$, if $e(\psi ) = 1$ for all $\psi \in T$, then $e(\varphi ) = 1$ as well.

Given a logic $\mathrm {L}$, two formulas $\varphi$ and $\psi$ are logically equivalent, in symbols $\varphi \equiv _{\mathrm {L}}\psi$, if and only if $\varphi \leftrightarrow \psi$ is a ${\mathrm L}$-tautology, that is, if $\models _{ \mathbb {V}(\mathrm {L})} \varphi \leftrightarrow \psi$. The Lindenbaum Algebra $\mathbf {Lind}(\mathrm {L})$ of $\mathrm {L}$ is the algebra whose elements are the equivalence classes of formulas of $\mathrm {L}$, with respect to $\equiv _{\mathrm {L}}$. The free k-generated algebra $\mathbf {F}_k(\mathbb {V}(\mathrm {L}))$ in $\mathbb {V}(\mathrm {L})$ is the subalgebra of the Lindenbaum algebra $\mathbf {Lind}(\mathrm {L})$ of the formulas over the first k variables. Combinatorial representations of $\mathbf {F}_k(\mathbb {G})$ and $\mathbf {F}_k(\mathbb {MV}_3)$, where $\mathbb {G} = \mathbb {V}(G)$ and $\mathbb {MV}_3 = \mathbb {V}(\L _3)$, can be found in [1].

3 The Concept Lattice of Formulas and Evaluations

Suppose L is an axiomatic extension of MTL that is complete with respect to a given L-chain M, that is, $\models _{\mathbb {V}(\mathrm {L})}\; = \; \models _M$. In what follow, we will denote by $\mathcal L$ the set of propositional L-formulas built from a finite set of propositional variables V, and by $\varOmega$ the set of truth-evaluations of propositinal variables into the L-chain M, that is, $\varOmega = \{e: V \rightarrow M\}$. Of course, every evaluation of variables uniquely extends to an evaluation of any propositional formula using the truth-functions interpreting the connectives.

In our FCA-based analysis of the notion of consequence in the logic L, we will consider attributes described as propositional formulas from $\mathcal L$, and objects as evaluations from $\varOmega$. In this setting, a formal context will be specified by a triple

$$K = (\varOmega _0, \mathcal{L}_0, R),$$

where $\varOmega _0 \subseteq \varOmega$ and $\mathcal{L}_0 \subseteq \mathcal L$ are finite sets, and $R: \varOmega _0 \times \mathcal{L}_0 \rightarrow M$ is a M-valued fuzzy relation defined as $R(e, \varphi ) = e(\varphi )$.

In this way, each attribute or formula $\varphi \in \mathcal{L}_0$ determines a fuzzy set of objects $\varphi ^*: \varOmega _0 \rightarrow M$, with $\varphi ^*(e) = R(e,\varphi )$, for all $e \in \varOmega _0$, and vice-versa, each object or evaluation $e \in \varOmega _0$ determines a fuzzy set of attributes $e^\circ : \mathcal{L}_0 \rightarrow M$, with $e^\circ (\varphi ) = R(e,\varphi )$, for all $\varphi \in \mathcal{L}_0$. More than that, following Pollandt [14] and Bělohlávek’s [2] models of FCA, this correspondence is extended to a Galois connection between fuzzy sets of formulas and fuzzy sets of evaluations as follows.

Definition 1

Let $F \in \mathcal{F}(\mathcal{L}_0)$ be a fuzzy subset of formulas (fuzzy theory) and let $E \in \mathcal{F}(\varOmega _0)$ be a fuzzy set of evaluations. Define:

• $F^*$ is the fuzzy subset of $\varOmega _0$ defined as $F^*(e) = \inf _{\varphi \in \mathcal{L}_0} F(\varphi ) \rightarrow R(e, \varphi )$, for all $e \in \varOmega _0$,

• $E^\circ$ is the fuzzy subset of $\mathcal{L}_0$ defined as $E^\circ (\varphi ) = \inf _{e \in \varOmega _0} E(e) \rightarrow R(e, \varphi )$, for all $\varphi \in \mathcal{L}_0$.

A pair (E, F) is a logic fuzzy concept if $F^* = E$ and $E^\circ = F$.

In other words, $F^*$ is the fuzzy set of models of the fuzzy theory F, and $E^\circ$ is the fuzzy set of formulas satisfied by the fuzzy set of evaluations E. Moreover, as it is known, $^{*\circ }$ is a closure operation on the set $\mathcal{F}(\mathcal{L}_0)$ of M-valued fuzzy sets of formulas, hence $F \le F^{*\circ }$. Actually the mapping $^{*\circ }: \mathcal{F}(\mathcal{L}_0) \rightarrow \mathcal{F}(\mathcal{L}_0)$, defined by

$$F^{*\circ }(\varphi ) = \inf _{e \in \varOmega _0} [ \inf _{\psi \in \mathcal{L}_0} F(\psi ) \rightarrow e(\psi )] \rightarrow e(\varphi ) $$

can be considered as a graded logical consequence relation, that it is even a bit more general than the one central to the so-called graded approach to fuzzy logic, developed by authors like J. A. Goguen, J. Pavelka, V. Nóvak and G. Gerla, as discussed e.g. in [11].

In what follows, we will denote by $\mathbf{C}(K) = (C(K), \preceq )$ the lattice of fuzzy concepts induced by a context K, where the ordering $\preceq$ is defined as

$$(E, F) \preceq (E', F') \text{ iff } E \le E' \text{ and } F \ge F',$$

and the meet and join operations are defined as:

$$(E, F) \sqcap (E', F') = (E \cap E', (F \cup F')^{*\circ }), \quad (E, F) \sqcup (E', F') = ((E \cup E')^{\circ *}, F \cap F'),$$

where $\cap$ and $\cup$ denote intersection and union of fuzzy sets, defined with the $\min$ and $\max$ operations respectively.

This lattice is bounded and the bottom element is the concept $\bot _K = (\emptyset , \mathcal{L}_0)$, while the top element is $\top _K = (\varOmega _0, T_{\varOmega _0})$, where $T_{\varOmega _0}$ if the fuzzy set of formulas defined by $T_{\varOmega _0}(\psi ) = \inf _{e \in \varOmega _0} e(\psi )$.

Let us see how it looks like the fuzzy concept in $\mathbf{C}(K)$ induced by (the crisp set of) a single formula $\varphi \in \mathcal{L}_0$, i.e. the pair $(\varphi ^*, \varphi ^{*\circ })$, where for the sake of a simpler notation we have used $\varphi ^*$ for $\{\varphi \}^*$ and $\varphi ^{*\circ }$ for $(\{\varphi \}^*)^\circ$. An easy computation shows that:

• $\varphi ^*(e) = R(e,\varphi ) = e(\varphi )$, for all $e \in \varOmega _0$;

• $\varphi ^{*\circ }(\psi ) = \inf _{e \in \varOmega _0} R(e, \varphi ) \rightarrow R(e, \psi ) = \inf _{e \in \varOmega _0} e(\varphi \rightarrow \psi )$, for all $\psi \in \mathcal{L}_0$.

Further, if we consider a finite set of formulas or theory T, using the same notation convention as above, the corresponding concept $(T^*, T^{*\circ })$ is as follows, where $\bigwedge T$ denotes the $\wedge$-conjunction of all the formulas in T, i.e. $\bigwedge T= \bigwedge _{\varphi \in T}\varphi$:

• $T^*(e) = \inf _{\varphi \in T} R(e,\varphi ) = \inf _{\varphi \in T} e(\varphi ) = e(\bigwedge T)$, for all $e \in \varOmega _0$;

• $T^{*\circ }(\psi ) = \inf _{e \in \varOmega _0} T^*(e) \rightarrow R(e, \psi ) = \inf _{e \in \varOmega _0} e(\bigwedge T \rightarrow \psi )$, for all $\psi \in \mathcal{L}_0$.

Note that, as discussed above, $T^{*\circ }$ accounts for a certain notion of graded consequence from T, in the sense that $T^{*\circ }(\psi )$ provides the degree in which $\psi$ is implied by T, relative to the set of interpretations $\varOmega _0$. It is a graded consequence that resembles Pavelka’s notion of truth degree of a formula in a theory (see e.g. [11, 12]), although they do not coincide. It is also related to the so-called degree preserving logic $\models _\mathrm{L}^{\le }$ companion of L, see e.g. [4]. Indeed, it is easy to check the following lemma.

Lemma 1

For any $\psi \in \mathcal{L}_0$, $T^{*\circ }(\psi )= 1$ iff $e(\bigwedge T \rightarrow \psi ) = 1$ for all $e \in \varOmega _0$.

Therefore, when $\varOmega _0 = \varOmega$, $T^{*\circ }(\psi )= 1$ holds if, and only if, $\inf _{\varphi \in T}e(\varphi ) \le e(\psi )$, i.e. iff $T \models _\mathrm{L}^{\le } \psi$. That is, the core of $T^{*\circ }$ is nothing but the set of consequences of T (restricted to $\mathcal{L}_0$) under the degree preserving logic companion of L.

Lemma 2

$T_1^{*\circ } = T_2^{*\circ }$ iff $\bigwedge T_1$ and $\bigwedge T_2$ are logically equivalent relative to $\varOmega _0$, i.e. $e(\bigwedge T_1) = e(\bigwedge T_2)$ for every evaluation $e \in \varOmega _0$.

Proof

The direction right-to-left is trivial. As for the converse, if $T_1^{*\circ } = T_2^{*\circ }$, then in particular, for all $\chi$, $T_1^{*\circ }(\chi ) = 1$ iff $T_2^{*\circ }(\chi ) = 1$. Take $\chi = \bigwedge T_1$. Since $T_1^{*\circ }(\bigwedge T_1) = 1$, then $T_2^{*\circ }(\bigwedge T_1) = 1$ as well, and by Lemma 1 this happens iff for all $e\in \varOmega _0$, $e(\bigwedge T_2) \le e(\bigwedge T_1)$. Analogously, if we take $\chi = \bigwedge T_2$, we would get that, for all $e\in \varOmega _0$, $e(\bigwedge T_1) \le e(\bigwedge T_2)$.    $\square$

Notice again that in case $\varOmega _0 = \varOmega$, then $T_1^{*\circ } = T_2^{*\circ }$ iff $\bigwedge T_1$ and $\bigwedge T_2$ are logically equivalent in the usual sense.

The set $C^{cg}(K)$ of concepts of the form $(T^*, T^{*\circ })$, with $T \subseteq \mathcal{L}_0$ a finite (crisp) set of formulas, is in fact what is known as the set of crisply generated concepts in the fuzzy concept lattice $\mathbf{C}(K)$ [3]. As already mentioned, for the purpose of building concepts, we can always replace a finite theory T by the $\wedge$-conjunction of its formulas $\bigwedge T$. Indeed, for every concept of the form $(T^*, T^{*\circ })$ with T a finite set of formulas, there is always a formula $\varphi$ (e.g. $\bigwedge T$) such that $(T^*, T^{*\circ }) = (\varphi ^*, \varphi ^{*\circ })$. Thus $C^{cg}(K) = \{ (\varphi ^*, \varphi ^{*\circ }) \mid \varphi \in \mathcal{L}_0 \}$ and we can safely restrict ourselves to deal with concepts induced by a single formula.

The lattice operations in $\mathbf{C}(K)$ over concepts from $C^{cg}(K)$ take the following form.

Lemma 3

For any $\varphi , \psi \in \mathcal{L}$,

$$\begin{aligned}&(\varphi ^*, \varphi ^{*\circ }) \sqcap (\psi ^*, \psi ^{*\circ }) = ((\varphi \wedge \psi )^*, (\varphi \wedge \psi )^{*\circ }), \end{aligned}$$

(2)

$$\begin{aligned}&(\varphi ^*, \varphi ^{*\circ }) \sqcup (\psi ^*, \psi ^{*\circ }) = ((\varphi \vee \psi )^*, (\varphi \vee \psi )^{*\circ }). \end{aligned}$$

(3)

Proof

By definition, $(\varphi ^*, \varphi ^{*\circ }) \sqcap (\psi ^*, \psi ^{*\circ }) = (\varphi ^* \cap \psi ^*, (\varphi ^* \cap \psi ^*)^\circ )$, but since $(\varphi ^* \cap \psi ^*)(e) = \min (\varphi ^*(e), \psi ^*(e)) = \min (e(\varphi ), e(\psi )) = e(\varphi \wedge \psi ) = (\varphi \wedge \psi )^*(e)$, we have $(\varphi ^* \cap \psi ^*, (\varphi ^* \cap \psi ^*)^\circ ) = ((\varphi \wedge \psi )^*, (\varphi \wedge \psi )^{*\circ })$.

Analogously, by definition $(\varphi ^*, \varphi ^{*\circ }) \sqcup (\psi ^*, \psi ^{*\circ }) = ((\varphi ^* \cup \psi ^*)^{\circ *}, \varphi ^{*\circ } \cap \psi ^{*\circ })$, but $(\varphi ^{*\circ } \cap \psi ^{*\circ })(\chi ) = \min (\inf _{e} e(\varphi \rightarrow \chi ), \inf _e e(\psi \rightarrow \chi )) = \inf _e \min (e(\varphi \rightarrow \chi ), e(\psi \rightarrow \chi )) = \inf _e e(\varphi \vee \psi \rightarrow \chi ) = (\varphi \vee \psi )^{*\circ }(\chi )$. Therefore $((\varphi ^* \cup \psi ^*)^{\circ *}, \varphi ^{*\circ } \cap \psi ^{*\circ }) = ((\varphi ^* \cup \psi ^*)^{\circ *}, (\varphi \vee \psi )^{*\circ }) = ((\varphi \vee \psi )^*, (\varphi \vee \psi )^{*\circ })$.    $\square$

As it proven in [3], $C^{cg}(K)$ is indeed a $\sqcap$-subsemilattice of $\mathbf{C}(K)$ in the general case. Indeed, notice that the $\sqcap$ operation is closed in $C^{cg}(K)$, since the concept induced by the conjunction $\bigwedge T$ of a set of formulas $T \subseteq \mathcal{L}_0$, even if $\bigwedge T$ does not belong to $\mathcal{L}_0$, is the same concept induced by the crisp set of formulas T, and hence it belongs to $C^{cg}(K)$. However, this is not the case for a disjunction of a set of formulas. However, if we can guarantee that the concept induced by a disjunction also belongs to $C^{cg}(K)$, then $C^{cg}(K)$ is actually a sublattice of C(K).

Lemma 4

If $\mathcal{L}_0$ is closed by $\vee$ (modulo logical equivalence) then $\sqcup$ is closed in $C^{cg}(K)$, and $\mathbf{C}^{cg}(K) = (C^{cg}(K), \sqcap , \sqcup , \top _K, \bot _k)$ is a sublattice of $\mathbf{C}(K)$.

In the following we will assume $\mathcal{L}_0 = \mathcal{L}$ to avoid any problem. In such a case, we can also enrich the lattice $\mathbf{C}^{cg}(K)$ with some further operations in a natural way so to come up with a residuated lattice structure, inherited from the L-algebras.

Definition 2

We define the following two operations on fuzzy concepts from $C^{cg}(K)$. For any $\varphi , \psi \in \mathcal{L}$, let us define:

$$\begin{aligned}&(\varphi ^*, \varphi ^{*\circ }) \boxtimes (\psi ^*, \psi ^{*\circ }) = ((\varphi \odot \psi )^*, (\varphi \odot \psi )^{*\circ }), \end{aligned}$$

(4)

$$\begin{aligned}&(\varphi ^*, \varphi ^{*\circ }) \Rightarrow (\psi ^*, \psi ^{*\circ }) = ((\varphi \rightarrow \psi )^*, (\varphi \rightarrow \psi )^{*\circ }). \end{aligned}$$

(5)

It is easy to check that $\boxtimes$ and $\Rightarrow$ endow the lattice $C^{cg}(K)$ with a structure of residuated lattice, in particular with the structure of a L-algebra.

Proposition 2

$\mathbf{C^{cg}(K)} = (C^{cg}(K), \sqcap , \sqcup , \boxtimes , \Rightarrow , \top _K, \bot _K)$ is an L-algebra that is isomorphic to the quotient algebra $\mathcal{L}/\!\equiv _{\varOmega _0}$, where $\varphi \equiv _{\varOmega _0} \psi$ iff $e(\varphi ) = e(\psi )$ for all $e \in \varOmega _0$.

Proof

Elements of $\mathcal{L}/\!\equiv _{\varOmega _0}$ are equivalence classes of formulas from $\mathcal L$, according to the congruence relation $\equiv _{\varOmega _0}$. Given a formula $\varphi \in \mathcal{L}$, let us denote by $[\varphi ]$ its equivalence class. Since the class of L-algebras is a variety, it is closed under quotients, hence $\mathcal{L}/\!\!\equiv _{\varOmega _0}$ is an L-algebra as well. Now consider the mapping $\lambda : \mathcal{L}/\!\equiv _{\varOmega _0} \rightarrow C^{cg}(K)$ defined by $\lambda ([\varphi ]) = (\varphi ^*, \varphi ^{*\circ })$. It is easy to check that this mapping is one-to-one thanks to Lemma 2, and moreover it is an algebraic homomorphism with respect to the operations involved: $\lambda ([\varphi ] \wedge [\psi ])) = \lambda ([\varphi ]) \sqcap \lambda ([\psi ])$, etc. Therefore, $\mathbf{C^{cg}(K)}$ is an L-algebra as well, isomorphic to $\mathcal{L}/\!\equiv _{\varOmega _0}$.    $\square$

Corollary 1

If $\varOmega _0 = \varOmega$, then $\mathbf{C^{cg}(K)}$ is isomorphic to the Lindenbaum algebra $\mathbf{Lind}(\mathrm{L}) = \mathcal{L} /\!\equiv _\mathrm{L}$.

3.1 An Example: The Case of Ł$_3$

In this section, we provide an example of the construction of the concept lattice of formulas and evaluations for the Łukasiewicz 3-valued logic $\L _3$.

Let $\mathcal{L}_0 =\{\varphi _1,\varphi _2,\dots ,\varphi _{12}\}$ be the set of all Ł$_3$-formulas (up to logical equivalence) on one variable x, where²

$$\begin{aligned} \varphi _1&= x^2 \wedge (\lnot x)^2 = \bot ,&\varphi _2&= (\lnot x)^2,&\varphi _3&= x\wedge \lnot x, \\ \varphi _4&= x^2,&\varphi _5&= \lnot x,&\varphi _6&= (x\vee \lnot x)^2, \\ \varphi _7&= \lnot x^2 \wedge \lnot (\lnot x)^2,&\varphi _8&= x,&\varphi _9&= \lnot x^2, \\ \varphi _{10}&= x \vee \lnot x,&\varphi _{11}&= \lnot (\lnot x)^2,&\varphi _{12}&= \lnot x^2 \vee \lnot (\lnot x)^2=\top . \end{aligned}$$

Further, let us consider all possible 3-valued evaluations on the variable x as the set of objects: $\varOmega _0=\{e_0, e_1, e_2\}$, where $e_0(x)=0, e_1(x)=\frac{1}{2}, e_2(x)=1$. The following table shows the values of each formula of $\mathcal{L}_0$ under each evaluation.

As described before in this section, the triple $K=\{\varOmega _0,\mathcal{L}_0, R\}$, where $R: \varOmega _0 \times \mathcal{L}_0 \rightarrow \{0,\frac{1}{2},1\}$ is a 3-valued fuzzy relation defined as $R(e, \varphi ) = e(\varphi )$, identifies a formal context.

First of all, we aim at obtaining all the concepts induced by a single formula. For instance, consider the formula $\varphi _8 = x$. Then, $\varphi _8^*(e_0)=e_0(\varphi _8)=0$, $\varphi _8^*(e_1)=\frac{1}{2}$, and $\varphi _8^*(e_2)=1$. We denote the fuzzy set of objects (evaluations) $\varphi _8^*$ by the tuple $(0,\frac{1}{2},1)$. Let us compute the fuzzy set of attributes (formulas) $\varphi _8^{*\circ }$:

$$\begin{aligned} \varphi _8^{*\circ }(\varphi _1)&= \inf _{e \in \varOmega _0} e(\varphi _8 \rightarrow \varphi _1)=0,&\varphi _8^{*\circ }(\varphi _2)&= \inf _{e \in \varOmega _0} e(\varphi _8 \rightarrow \varphi _2)=0,\\ \varphi _8^{*\circ }(\varphi _3)&= \inf _{e \in \varOmega _0} e(\varphi _8 \rightarrow \varphi _3)=0,&\varphi _8^{*\circ }(\varphi _4)&= \inf _{e \in \varOmega _0} e(\varphi _8 \rightarrow \varphi _4)=1/2,\\ \varphi _8^{*\circ }(\varphi _5)&= \inf _{e \in \varOmega _0} e(\varphi _8 \rightarrow \varphi _5)=0,&\varphi _8^{*\circ }(\varphi _6)&= \inf _{e \in \varOmega _0} e(\varphi _8 \rightarrow \varphi _6)=1/2,\\ \varphi _8^{*\circ }(\varphi _7)&= \inf _{e \in \varOmega _0} e(\varphi _8 \rightarrow \varphi _7)=0,&\varphi _8^{*\circ }(\varphi _8)&= \inf _{e \in \varOmega _0} e(\varphi _8 \rightarrow \varphi _8)=1,\\ \varphi _8^{*\circ }(\varphi _9)&= \inf _{e \in \varOmega _0} e(\varphi _8 \rightarrow \varphi _9)=0,&\varphi _8^{*\circ }(\varphi _{10})&= \inf _{e \in \varOmega _0} e(\varphi _8 \rightarrow \varphi _{10})=1,\\ \varphi _8^{*\circ }(\varphi _{11})&= \inf _{e \in \varOmega _0} e(\varphi _8 \rightarrow \varphi _{11})=1,&\varphi _8^{*\circ }(\varphi _{12})&= \inf _{e \in \varOmega _0} e(\varphi _8 \rightarrow \varphi _{12})=1. \end{aligned}$$

We indicate the value of $\varphi _8^{*\circ }$ by the tuple $(0,0,0,\frac{1}{2},0,\frac{1}{2},0,1,0,1,1,1)$. The pair $(\varphi _8^*,\varphi _8^{*\circ })$ is the formal concept induced by the furmula $\varphi _8$. In the same way, we can compute all the formal concepts induced by single formulas of $\mathcal{L}_0$, obtaining:

$$\begin{aligned} (\varphi _1^*,\varphi _1^{*\circ })&= \left( (0,0,0),(1,1,1,1,1,1,1,1,1,1,1,1)\right) ,\\ (\varphi _2^*,\varphi _2^{*\circ })&= \left( (1,0,0),(0,1,0,0,1,1,0,0,1,1,0,1)\right) ,\\ (\varphi _3^*,\varphi _3^{*\circ })&= \left( (0,1/2,0),(1/2,1/2,1,1/2,1,1/2,1,1,1,1,1,1)\right) ,\\ (\varphi _4^*,\varphi _4^{*\circ })&= \left( (0,0,1),(0,0,0,1,0,1,0,1,0,1,1,1)\right) ,\\ (\varphi _5^*,\varphi _5^{*\circ })&= \left( (1,1/2,0),(0,1/2,0,0,1,1/2,0,0,1,1,0,1)\right) ,\\ (\varphi _6^*,\varphi _6^{*\circ })&= \left( (1,0,1),(0,0,0,0,0,1,0,0,0,1,0,1)\right) ,\\ (\varphi _7^*,\varphi _7^{*\circ })&= \left( (0,1,0),(0,0,1/2,0,1/2,0,1,1/2,1,1/2,1,1)\right) ,\\ (\varphi _8^*,\varphi _8^{*\circ })&= \left( (0,1/2,1),(0,0,0,1/2,0,1/2,0,1,0,1,1,1)\right) ,\\ (\varphi _9^*,\varphi _9^{*\circ })&= \left( (1,1,0),(0,0,0,0,1/2,0,0,0,1,1/2,0,1)\right) ,\\ (\varphi _{10}^*,\varphi _{10}^{*\circ })&= \left( (1,1/2,1),(0,0,0,0,0,1/2,0,0,0,1,0,1)\right) ,\\ (\varphi _{11}^*,\varphi _{11}^{*\circ })&= \left( (0,1,1),(0,0,0,0,0,0,0,1/2,0,1/2,1,1)\right) ,\\ (\varphi _{12}^*,\varphi _{12}^{*\circ })&= \left( (1,1,1),(0,0,0,0,0,0,0,0,0,1/2,0,1)\right) . \end{aligned}$$

Note that all the formal concepts above are precisely all the crisply generated concepts. Indeed, the concept generated by $\{\psi _1,\dots ,\psi _k\}\subseteq \mathcal{L}_0$ coincides with the concept generated by the single formula $\bigwedge _{i=1,\dots ,k}\psi _i$, which is logically equivalent to a formula of $\mathcal{L}_0$. We also observe that $\varphi _{12}^{*\circ }=(\inf _{e\in \varOmega _0}e(\varphi _1),\dots ,\inf _{e\in {\varOmega _0}}e(\varphi _{12}))=T_{\varOmega _0}\ne \emptyset$.

As described in the previous part of the section, we can endow the set $\mathcal{C}^{cg}(K)$ with the operations defined in (2)–(5). We obtain in this way an algebra $\mathbf{C^{cg}(K)}$ of crisply generated concepts of $\mathcal{L}_0$ which is isomorphic to the free 1-generated Ł$_3$ algebra, depicted in Fig. 1, via the isomorphism $\lambda$ that associates each formula $\varphi \in \mathcal{L}_0$ with the concept $(\varphi ^*,\varphi ^{*\circ })$.

Fig. 1.

The Lindenbaum-Tarski algebra of Ł$_3$ over one generator.

Consider now the set of objects (evaluations) $\varOmega _B=\{e_0,e_2\}\subseteq \varOmega _0$. Again, the triple $K_B=\{\varOmega _B,\mathcal{L}_0, R_B\}$, where $R_B: \varOmega _B \times \mathcal{L}_0 \rightarrow \{0,\frac{1}{2},1\}$ is a 3-valued fuzzy relation defined as $R(e, \varphi ) = e(\varphi )$, identifies a formal context. Actually, the fuzzy relation $R_B$ is in fact a crisp relation, since the evaluation $e_0$ and $e_2$ only evaluate x to either 0 or 1. In this new setting, we can compute all the formal concepts induced by single formulas of $\mathcal{L}_0$, obtaining:

$$\begin{aligned} (\varphi _1^*,\varphi _1^{*\circ })&= (\varphi _3^*,\varphi _3^{*\circ }) = (\varphi _7^*,\varphi _7^{*\circ }) = \left( (0,0),(1,1,1,1,1,1,1,1,1,1,1,1)\right) ,\\ (\varphi _4^*,\varphi _4^{*\circ })&= (\varphi _8^*,\varphi _8^{*\circ }) = (\varphi _{11}^*,\varphi _{11}^{*\circ }) = \left( (0,1),(0,0,0,1,0,1,0,1,0,1,1,1)\right) ,\\ (\varphi _2^*,\varphi _2^{*\circ })&= (\varphi _5^*,\varphi _5^{*\circ }) = (\varphi _9^*,\varphi _9^{*\circ }) = \left( (1,0),(0,1,0,0,1,1,0,0,1,1,0,1)\right) ,\\ (\varphi _6^*,\varphi _6^{*\circ })&= (\varphi _{10}^*,\varphi _{10}^{*\circ }) = (\varphi _{12}^*,\varphi _{12}^{*\circ }) = \left( (1,1),(0,0,0,0,0,1,0,0,0,1,0,1)\right) , \end{aligned}$$

which, in fact, they turn out to be classical concepts. Not surprisingly, endowing this set of concepts $\mathcal{C}^{cg}(K_B)$ with the operations defined in (2)–(5) we obtain an algebra of concepts which is isomorphic to the free 1-generated Boolean algebra. Such algebra is obtained as a quotient of $\mathbf{C^{cg}(K)}$. As it is easily seen using Proposition 2, this holds in general, that is, an algebra of concepts $\mathbf{C^{cg}(K')}$, with $K'=\{\varOmega _0',\mathcal{L}_0, R\}$ and $\varOmega _0'\subseteq \varOmega _0$ is a quotient of the algebra $\mathbf{C^{cg}(K)}$.

4 The Natural Concept Lattice of a Logic

In this section we recall the construction of concept lattices applied in [5] to characterize formal concept lattices associated to Gödel algebras.

Proposition 1 states that for every finite lattice H there is always a canonical way to build the standard context $\mathbb {K}_H$, whose concept lattice $\mathfrak {B}(\mathbb {K}_H)$ is isomorphic to H. Let $\mathbf {A}=(A,\wedge ,\vee ,\rightarrow ,\top ,\bot )$ be a finite algebra in a variety $\mathbb {V}\subseteq \mathbb {MTL}$, and let $C_{\mathbf {A}} = \mathfrak {B}((\mathfrak {J}(\mathbf {A}), \mathfrak {M}(\mathbf {A}), \le ))$ be the concept lattice of its standard context. Then, the lattice $\mathbf {C}_{\mathbf {A}} = (C_{\mathbf {A}}, \sqcap ,\sqcup ,\top _G,\bot _M)$, is isomorphic to the lattice reduct of $\mathbf {A}$.

Pushing further this approach, when $\mathbb {V}$ is a locally finite variety the k-generated free algebras $\mathbf {F}_k(\mathbb {V})$ are finite, and hence we can apply to them the above construction. As the elements of $\mathbf {F}_k(\mathbb {V})$ are equivalence classes of logical formulas in k variables, this amount to associate every logical formula to its natural formal concept.

For some cases it is possible to extend the lattice isomorphism to a full isomorphism of algebras by defining suitable operations between the formal concepts. In [5] the authors use this methodology to obtain formal concepts for every Gödel logic formula. This is possible because in Gödel algebras lattice and monoidal conjunctions coincide, and hence it is natural to define an implication operator between concepts by using the residum of the concepts meet.

Comparing to Sect. 3.1, in the next subsection we apply the above sketched construction to $\mathbf {F}_1(\mathbb {V}({\L }_3))$.

4.1 Constructing the Concept Lattice of the Logic Ł$_3$

Consider the set $\mathcal{L}_0 =\{\varphi _1,\varphi _2,\dots ,\varphi _{12}\}$ of all Ł$_3$-fomulas (up to logical equivalence) on one variable x. The formulas of $\mathcal{L}_0$ are exhibited in Fig. 1. Let $H = (\mathcal{L}_0,\le )$ be the lattice reduct of the free 1-generated Ł$_3$ algebra $\mathbf {F}_1$ depicted in Fig. 1. The sets of join irreducible elements and meet irreducible elements of L are $\mathfrak {J}(H)=\{\varphi _2,\varphi _3,\varphi _4,\varphi _7\}$, and $\mathfrak {M}(H)=\{\varphi _{6},\varphi _{9},\varphi _{10},\varphi _{11}\}$, respectively. By Proposition 1, we can identify $\mathfrak {J}(H)$ and $\mathfrak {M}(H)$ with the set of objects and attributes, respectively, of a standard context $\mathbb {K}_H = (\mathfrak {J}(H), \mathfrak {M}(H),\le )$. The following table shows the relation $\le$

The corresponding standard context lattice is depicted in Fig. 2. Clearly, by Proposition 1, it is isomophic to the lattice reduct of the free 1-generated Ł$_3$ algebra of Fig. 1, via a lattice isomorphim f defined as follows. For each $\varphi \in \mathcal{L}_0$, let $J_\varphi$ be the maximal subset of $\mathfrak {J}(H)$ such that $\varphi =\bigvee J_\varphi$, and $M_\varphi$ be the maximal subset of $\mathfrak {M}(H)$ such that $\varphi =\bigwedge M_\varphi$. Then the map f associates each $\varphi \in \mathcal{L}_0$ with the formal concept $(J_\varphi ,M_\varphi )\in \mathbb {K}_H$.

Fig. 2.

The concept lattice associated with the lattice reduct of $\mathbf {F}_1$

To extend the above defined lattice isomorphism to an algebraic isomorphism between $\mathbf {F}_1(\mathbb {\L }_3)$ and the concept lattice of the standard context $\mathbb {K}_{H}$, it is necessary to define a proper monoidal conjunction between concepts of $\mathbb {K}_H$. Of course, an obvious way to define such an operation is through the isomorphism f, that is, for each pair of concepts $(E, F), (E', F') \in \mathbb {K}_H$, to define $(E, F) \otimes (E', F') = (J_{\varphi \odot \psi },M_{\varphi \odot \psi })$, where $f^{-1}((E, F)) = \varphi$ and $f^{-1}((E', F')) = \psi$. However, this does not shed any light on how the operation works on the elements of the concepts. To have a much better insight in the operation seems not to be an easy task, even in the case of locally finite subvarieties of $\mathbb {MTL}$ (such as $\mathbb {\L }_3$), and it will be faced in some future paper.

5 Conclusions and Further Developments

To obtain a direct relation between a formal concept and a fuzzy logic formula, in this work we have explored two ways to obtain concept lattices isomorphic to Lindenbaum algebras of many-valued logics. The first approach naturally gives the desired isomorphism between the concept lattice and the algebra of formulas, while to complete the second approach additional research has to be done.

To depict the two constructions we have chosen the logic $\L _3$. In [6], $\L _3$ has been characterized as a logic of prototypes and counterexamples. The construction of possible worlds in [6] gives a lattice of functions $\varOmega _0^{\varOmega _0^n}$ very similar to the concept lattice of our first approach in Sect. 3.1. Hence, putting together the characterization of [6] with the constructions presented here, it will be ideally possible to build a formal concept semantics of prototypes and counterxamples for the logic $\L _3$.