Stone algebras: 3-valued logic and rough sets

In this article, we propose 3-valued semantics of the logics compatible with Stone and dual Stone algebras. We show that these logics can be considered as 3-valued by establishing soundness and completeness results. We also establish rough set semantics of these logics where the third value can be interpreted as not certain but possible.

-Can we provide 3-valued (n-valued) logics compatible with reduct algebras of 3-valued Łukasiewicz algebras?  answered this question affirmatively in the case of Kleene algebras. They introduced a logic L K for Kleene algebras, which is sound and complete with respect to a 3-valued consequence relation. In this paper, we B Arun Kumar show that the logic of Stone (dual Stone) algebras are sound and complete with respect to a 3-valued semantics defined via a 3-valued consequence relation.
In other aspects of this paper, we make explicit connections between logic of Stone (dual Stone) algebras and rough sets. Rough set theory, introduced by Pawlak (1982Pawlak ( , 1991 as a tool to deal with uncertainty in an information system. This deals with a domain U and an equivalence relation R on U . In Pawlakian rough sets theory, the equivalence relation R is interpreted as the indiscernibility relation on the domain U . x Ry if and only if x is indiscernible to y with respect to attributes present in the information system. The pair (U , R) is called a (Pawlak) approximation space. For any A ⊆ U , one defines the lower and upper approximations of A in the approximation space (U , R), denoted L A and UA, respectively, as follows. For x ∈ U , let [x] denote the equivalence class of x modulo R, Definition 2 Let (U , R) be an approximation space. For each A ⊆ U , the ordered pair (L A, UA) is called a rough set in (U , R).
Notation 1 RS denotes the collection of all rough sets for an approximation space (U , R).

Notation 2 Let U be a set. Then
-P(U ) denote the power set of U .
-for any A ⊆ U , A c denote the set theoretic complement of A in U .
In rough set theory, the definition ( * ) has been interpreted in the following manner.
1. x certainly belongs to A, if x ∈ L A, i.e., all objects which are indiscernible to x are in A.

x certainly does not belong to
UA \L A. This is the case when some objects indiscernible to x are in A, while some others, also indiscernible to x, are in A c . In rough set terminology, sets of the form UA \ L A are referred as boundary of A.
-  represented a given Kleene algebra in terms of Kleene algebra formed by rough sets for some appropriate approximation space. This imparted the 3-valued and rough set semantics of the logic L K (of Kleene algebras). The interpretations 1, 2 and 3 have been explicitly captured in Kumar (2020). -Panicker and Banerjee (2019) adopted yet other definition of rough sets [first discussed by Pagliani (1998)] to explore the C-algebraic structures of rough sets. As in Pagliani (1998), for an approximation space (U , R) and A ⊆ U , the pair (L A, (UA) c ) is called a rough set. The collection of all the rough sets for an approximation space (U , R) forms a C-algebras. Further they have proved that a C-algebra is embeddable into C-algebra formed by rough sets for some appropriate approximation space. It is worth mention here that the C-algebras are the algebraic counterpart of McCarthy's three-valued logic [cf. Panicker and Banerjee (2019)] and unlike our case where the set of truth values of proposed logic L S is a Stone algebra, a C-algebra may not form even a semilattice.
In the last part of this article, we capture interpretations 1, 2 and 3 via logics compatible with Stone and dual Stone algebras. The rest of this paper is organized as follows. In Sect. 2, we present some basic results of Stone algebras that will be used in the sequel. In Sect. 3, we extend the distributive lattice logic (Dunn 1995) to obtain the logic L S (L DS ) of Stone (dual Stone) algebras. We further propose a 3-valued consequence relation S 1 ( DS 0 ) and show that the logic L S (L DS ) is sound and complete with respect to the S 1 ( DS 0 ).
In Sect. 4, we provide the rough set semantics of the logic L S (L DS ) and capture the the interpretations 1, 2 and 3.
The dual notion of a given Stone algebra is known as dual Stone algebra. To make this article self-contained, we explicitly define the dual Stone algebra. It is well known that B [2] = (B [2] , ∨, ∧, (0, 0), (1, 1)) is a bounded distributive lattice, where ∨ and ∧ are componentwise join and meet inherited from B. Moreover, we have the following results.

for some index sets I and J .
Now, as 2 is embedded into algebras 3 ∼ and 3 ¬ , hence the above theorem can be restated in terms of 3 ∼ and 3 ¬ . So, in particular if B is a Boolean algebra, then the Stone algebra B [2] ∼ and dual Stone algebra B [2] ¬ can be embedded into 3 I ∼ and 3 J ¬ , respectively, for appropriate index sets I and J .
(i) An element a ∈ L is said to be completely join irreducible, if a = S implies that a ∈ S, for every subset S of L.
Notation 3 Let J L denote the set of all completely join irreducible elements of L, and J (x) := {a ∈ J L : a ≤ x}, for any x ∈ L.
(ii) A set S is said to be join dense in L, provided for every element a ∈ L, there is a subset S of S such that a = S .
The illustration of importance of completely join irreducible elements can be seen by a result of Birkhoff.
Lemma 1 (Birkhoff 1995) Let L and K be two completely distributive lattices. Further, let J L and J K be join dense in L and K , respectively. Let φ : J L → J K be an order isomorphism. Then the extension map Φ : L → K given by Kumar and Banerjee (2017) characterized the completely join irreducible elements of lattices 3 I and B [2] , where B is a complete atomic Boolean algebra.
Proposition 2  1. The set of completely join irreducible elements of 3 I is given by: 2. Let B be a complete atomic Boolean algebra. The set of completely join irreducible elements of B [2] is given by . Figure 1 shows the Hasse diagrams of J 3 I and J B [2] . We also established the following isomorphisms.

3-Valued semantics of logics for Stone and dual Stone algebras
In this section, we focus on the study of the logics corresponding to the classes of Stone and dual Stone algebras and the structures B [2] ∼ and B [2] ¬ . Our approach to the study is motivated by Dunn's (1999) 4-valued semantics of the De Morgan consequence system: 0,1 (or 0 or 1 ), wherein valuations are defined in the 4-element De Morgan algebra. The 4-valued semantics arises from the fact that each element of a De Morgan algebra can be looked upon as a pair of sets.
In a similar way, we exploit Theorem 3 to provide a 3valued semantics of the logic for Stone algebras. However, by an easy consequence of Stone's representation theorem and Theorem 3, we have: Then there is a set U such that DS can be embedded into dual Stone algebra formed by (P(U )) [2] ¬ .

Bounded distributive lattice logic with negation
Bounded distributive lattices are algebraic models of the bounded distributive lattice logic (B DL L), an extension of distributive lattice logic introduced by Dunn (1995). The study of logics in this section is based on B DL L. Let us present the logic. The language consists of -the set P of propositional variables, whose elements are denoted by p, q, r , . . .. -propositional constants and ⊥, -logical connectives ∨ and ∧.
The set F of well-formed formulas of the logic is then given by the scheme: where p is a propositional variable.
Definition 6 (Dunn 1999) The bounded distributive lattice logic (B DL L) is a binary consequence system ⊆ F × F with the following postulates and rules: The postulates and rules from 1 to 7 precisely define the distributive lattice logic. The term α β in the above representation of logic is called a consequent. Intuitively, α β reflects that β is a consequence of α.
Let us add a unary connective − to the language of B DL L. Let F − be the set of formulas defined using the following rule: By an extension L of B DL L, we mean a binary consequence system ⊆ F − × F − which contains all the postulates and rules of the logic B DL L. By α L β, we shall mean that the consequent α β is derivable in the logical system L (where the notion of derivability is defined in the classical manner).
In this paper, the various semantics of a logic L are defined using valuations.

Definition 7 Let
The notion of local (global) validity is defined in the following manner: Definition 8 Let (A, ∨, ∧, −, 0, 1) be a lattice-based algebra.
If the consequent is valid under all valuations on A, then it is valid in A, and denote it as α A β.
-If the consequent α β is valid in each algebra of A, then we say α β is valid in A, and denote it as α A β.

The logics L S , L DS and their 3-valued semantics
Let U be a set and A ⊆ U . Then for any x ∈ U , either x ∈ A or x ∈ A c . This distinguished property of '∈' leads to the True-False semantics of classical propositional logic. Now, if v is a valuation from classical propositional sentences to P(U ), then v determines a family of 2-valued valuations {v x : Utilizing this fact along with Stone's representation theorem, one establishes the equivalency between True-False semantics, set theoretic semantics and algebraic semantics of classical propositional logic.
In this section, we follow the same approach to establish the completeness results for L S and L DS (defined below). Definition 9 Let ∼ be a unary connective added to the language of B DL L. Then, for α, β ∈ F ∼ , L S denotes the logic B DL L along with following rules and postulates.
∼ α∨ ∼∼ α, Definition 10 Let ¬ be a unary connective added to the language of B DL L. Then, for α, β ∈ F ¬ , L DS denotes the logic B DL L along with following rules and postulates.
Similar to the previous case, using Proposition 3, we can easily establish that v * is indeed a valuation in 3 ¬ . This arises a contradiction to α DS 0 β.
Note that converse of the above statements are not true, for example ∼∼ α S 0 α but ∼∼ α S 1 α and β DS 0 ¬¬β but β DS 1 ¬¬β. This is contrary to the Dunn's "De Morgan consequence relations 0 , 1 and 0,1 " where all these three turn out to be equivalent.

α DSP(U )
By Theorem 4, 3 ∼ is embedded to a Stone algebra of P(U ) [2] for some set U . If this embedding is denoted by φ, φ • v is a valuation in P(U ) [2] .
Let U be a set, and P(U ) [2] be the corresponding Stone algebra. Let v be a valuation on Consider any γ, δ ∈ F ∼ , with v(γ ) := (A, B) and v(δ) := (C, D), A, B, C, D ⊆ U . It is easy to show that (for a complete proof, we refer to Kumar and Banerjee (2017)), Then v x (α) = 1, and as α S 1 β, by definition, v x (β) = 1. This implies x ∈ C , whence A ⊆ C . On the other hand, if x / ∈ D , v x (β) = 0. Then using Lemma 2, we have v x (α) = 0, so that x / ∈ B , giving B ⊆ D . 2. We prove second part only. For this let α DS 0 β. Let U be a set, and P(U ) [2] be the corresponding dual Stone algebra. Let v be a valuation on P(U ) [2] , we show that v(α) ≤ v(β). Very similar to the previous case, for any γ ∈ F ¬ with v(γ ) := (A, B), A, B ⊆ U , and for each Consider any γ, δ ∈ F ¬ , with v ( = (A , B ), v(β) := (C , D ), and x ∈ A . Then v x (α) = 1, and as Finally, we have the following 3-valued semantics of the logics L S and L DS .

Rough set models for 3-valued logics
For an approximation space (U , R), RS ⊆ P(U ) × P(U ). So RS has a natural ordering ≤ (inherited from P(U ) × P(U )). Pomykała and Pomykała (1988) showed that (RS, ≤) is a Stone algebra. Gehrke and Walker (1992) characterized the lattice structure of rough sets. They showed that (RS, ≤) ∼ = 2 I × 3 J for some appropriate index sets I and J . Comer (1995) proved that for any index sets I and J , there is an approximation space (U , R) such that the lattices 2 I ×3 J and RS are isomorphic. Hence, any Stone (dual Stone) algebra is embeddable into Stone (dual Stone) algebra formed by rough sets. Alternatively, we can also prove this assertion by using Theorem 4.
An easy consequence we get the following rough set semantic for the logic L S (L DS ). Kumar (2020) captured the interpretations 1, 2 and 3 through the logic compatible with Kleene algebras. Now, we follow the same approach to capture the interpretations 1, 2 and 3 through the logics L S and L DS . Let us define the following semantic consequence relations.
Definition 12 1. Let α be a formula in F ∼ and v be a valuation in RS ∼ for some approximation space 2. Let α, β ∈ F ¬ and α β be a consequent.
α β is valid in an approximation space (U , R), if and only if α RS ¬ 0 β.
α β is valid in a class F of approximation spaces if and only if α β is valid in all approximation spaces (U , R) ∈ F. β. Now, suppose α β is valid in the class of all approximation spaces. We want to show that α A SRS β. Let v be a valuation in RS ∼ as taken above. We have to show that L A ⊆ LB and UA ⊆ UB. Let x ∈ L A, i.e., v, x RS ∼ 1 α. Hence, by our assumption, v, x RS 1 β, i.e., x ∈ LB. So L A ⊆ LB. Now, let y / ∈ UB, using Lemma 2, we have v, y RS ∼ 0 β. By our assumption, v, y RS ∼ 0 α, i.e., y / ∈ UA. 2. The proof of this part is very similar to that of part 1 which uses lemma 2.

Conclusions
This paper presents a relationship between Stone algebras, rough sets and 3-valued logics. We have drawn a line parallel to the line of Boolean algebra-2-valued Boolean algebra-Stone's representation theorem-classical propositional logic. We have shown that the logic L S (L DS ) is truly a 3-valued logic via a 3-valued semantics. Further this 3-valued semantics of the logic L S can be interpreted in rough set theory, where the third value can be treated as not certain but possible.  analyzed the Stone and dual Stone negations in perp frames (Dunn 1999(Dunn , 1994(Dunn , 1996, where negations are viewed as modal operators. We introduced Stone and dual Stone frames and showed that the logics L S and L DS are sound and complete, respectively, in these classes of frames. Thus, the perp semantics of the logics L S and L DS are established. Hence, in view of Theorems 5-9 we can conclude that algebraic, 3-valued, rough set and perp semantics of the logic L S (L DS ) are all equivalent.
In future, we would like to discuss the following.
1. Düntsch and Orłowska (2011), discrete duality for Stone algebras have been obtained. So naturally it would be interesting to investigate the relationship between the frame defined there, the logic L S , 3 − valued consequence relation S 1 and the Stone frames defined in . 2. There has been a lot of study on Topological Boolean algebras (TBAs). Similarly, can we define Topological Stone algebras? Can we obtain representation results of these Topological Stone algebras in terms of B [2] and RS? 3. Hilbert style axiomatization of the logic of Stone algebras. 4. Zhou and Zhao (2011) studied the Stone-like representation theorems of 3-valued Łukasiewicz algebras. Naturally, it would be interesting to investigate the Stone-like representation theorems for the class of Stone algebras determined by rough sets. 5. Applications of the logic L S in approximate reasoning.

Conflict of interest
The authors declare that they have no conflict of interest.
Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors.