Algebraic Completeness of Connexive and Bi-Intuitionistic Multilattice Logics

Abstract

In this paper, we introduce the notions of connexive and bi-intuitionistic multilattices and develop on their base the algebraic semantics for Kamide, Shramko, and Wansing’s connexive and bi-intuitionistic multilattice logics which were previously known in the form of sequent calculi and Kripke semantics. We prove that these logics are sound and complete with respect to the presented algebraic structures.

1 Introduction

Shramko introduced multilattice logic $\textbf{ML}_n$ as a generalization of logics based on lattices. In particular, $\textbf{ML}_n$ is a generalization of a class of many-valued logics based on four-valued Belnap-Dunn’s logic of first degree entailment (Belnap, 1977a, b; Dunn, 1976) and its algebraic framework, De Morgan lattices. This class contains Arieli and Avron’s four-valued bilattice logic (Arieli & Avron, 1996), Shramko and Wansing’s sixteen-valued trilattice logic (Shramko & Wansing, 2005), and Zaitsev’s eight-valued tetralattice logic (Zaitsev, 2009). Multilattice logic is based on the notion of n-lattice (multilattice), i.e., a lattice with n orders and some required relations between them. The algebraic completeness theorem for $\textbf{ML}_n$ was only formulated in Shramko (2016), the proof was found later in Grigoriev and Petrukhin (2019b). The family of multilattice logics contains not only $\textbf{ML}_n$, but its several modifications: bi-intuitionistic multilattice logic $\textbf{BML}_n$ and its connexive version $\textbf{CML}_n$ (Kamide et al., 2017), first-order multilattice logic $\textbf{FML}_n$ (Kamide & Shramko, 2017b), modal multilattice logic $\textbf{MML}_n$ (Kamide & Shramko, 2017a) and its modifications $\textbf{MML}_n^\textbf{MNT4}$, $\textbf{MML}_n^\textbf{S4}$, and $\textbf{MML}_n^\textbf{S5}$ (Grigoriev & Petrukhin, 2021, 2019a) as well as congruent and monotonic modal multilattice logics (Grigoriev & Petrukhin, 2022), a fragment of $\textbf{ML}_n$, called $\textbf{MLL}_n$, determined by logical multilattices ($\textbf{ML}_n$ itself is determined by ultralogical multilattices) (Grigoriev & Petrukhin, 2022), linear multilattice logics $\textbf{EML}_n$ and $\textbf{LML}_n$ (Kamide, 2019), and alternative multilattice logics $\textbf{SM}_n$ and $\textbf{IM}_n$ (Kamide, 2021). The algebraic completeness theorem has been established for $\textbf{ML}_n$ and $\textbf{MML}_n$ in Grigoriev and Petrukhin (2019b), for $\textbf{MML}_n^\textbf{MNT4}$, $\textbf{MML}_n^\textbf{S4}$, and $\textbf{MML}_n^\textbf{S5}$ in Grigoriev and Petrukhin (2021), for congruent and monotonic modal multilattice logics and $\textbf{MLL}_n$ in Grigoriev and Petrukhin (2022), for $\textbf{SM}_n$ and $\textbf{IM}_n$ in Kamide (2021).

However, for a few members of the multilattice family of logics, the algebraic completeness theorem has not been established yet. Among such logics are connexive multilattice logic $\textbf{CML}_n$ and bi-intuitionistic multilattice logic $\textbf{BML}_n$. These two logics are considered a pair since $\textbf{CML}_n$ is a connexive version of $\textbf{BML}_n$. $\textbf{BML}_n$ is a multilattice version of bi-intuitionistic logic BiInt introduced by Rauszer (1974, 1977, 1980); $\textbf{CML}_n$ is a multilattice version of bi-intuitionistic connexive logic BCL introduced by Wansing (2008) as one of sixteen variants of bi-intuitionistic logic and separately studied later by Kamide and Wansing (2016).

Kamide et al. (2017) formulated $\textbf{BML}_n$ and $\textbf{CML}_n$ in the form of sequent calculi (based on sequent calculi for BiInt and BCL offered in Kamide and Wansing (2016)) and Kripke-style semantics (based on Rauszer’s semantics for BiInt (Rauszer, 1977, 1980)), but algebraic semantics has not been developed. Moreover, the notions of bi-intuitionistic and connexive multilattices have not been introduced. This paper fills this gap: we introduce such notions and prove that sequent calculi for $\textbf{BML}_n$ and $\textbf{CML}_n$ are sound and complete with respect to bi-intuitionistic and connexive multilattices. Additionally, we show how the notions of bi-intuitionistic and connexive multilattices can be modified to get an alternative algebraic semantics for $\textbf{ML}_n$ and $\textbf{MLL}_n$.

The structure of the paper is as follows. The next section is devoted to the preliminaries regarding algebraic aspects of our topic. Section 3 describes Kamide, Shramko, and Wansing’s sequent calculi for the logics in question. In Sect. 4, we introduce the notions of connexive and bi-intuitionistic multilattices. Section 5 contains a proof of the algebraic completeness theorem. Section 6 consists of concluding remarks.

2 Preliminaries

We begin with some preliminaries about lattices, following their presentation in Dunn and Restall (2002).

Definition 2.1

(Lattice) A lattice is a structure $\langle L, \cap , \cup \rangle$, where L is a non-empty set and $\cap$ and $\cup$ are binary operations on L, with the relation $a \leqslant b$ defined as $a \cap b = a$. Postulates characterising the operations are as follows, for each $a,b\in L$:

• Idempotence: $a \cap a = a$, $a \cup a = a$

• Commutativity: $a \cap b = b \cap a$, $a \cup b = b \cup a$

• Associativity: $a \cap (b \cap c) = (a \cap b) \cap c$, $a \cup (b \cup c) = (a \cup b) \cup c$

• Absorption: $a \cap (a \cup b) = a$, $a \cup (a \cap b) = a$.

Definition 2.2

(Distributive lattice) $\langle L, \cap , \cup \rangle$ is a distributive lattice iff it is a lattice satisfying the following postulate, for any $a,b,c\in L$: $a \cap (b \cup c) \leqslant (a \cap b) \cup c.$

Multilattices generalize the notion of lattice. We are ready now to present the notion of a multilattice and some other important related notions.

Definition 2.3

(Multilattice) (p. 204, Definition 4.1, Shramko 2016)

• 1. A multilattice (or n-lattice, or n-dimensional multilattice) is a structure $\mathcal {M}_n=\langle S,\leqslant _1,\ldots ,\leqslant _n\rangle$, where $n>1$, $S\not =\emptyset$, $\leqslant _1,\ldots ,\leqslant _n$ are partial orderings such that $\langle S,\leqslant _1\rangle ,\ldots ,\langle S,\leqslant _n\rangle$ are lattices with the corresponding pairs of meet and join operations $\langle \cap _1,\cup _1\rangle ,\ldots ,\langle \cap _n,\cup _n\rangle$.

• 2. A multilattice is called complete iff all meets and joins exist, with respect to all n orderings.

• 3. A multilattice is called interlaced iff each of the operations $\cap _1,\cup _1,\ldots ,\cap _n,\cup _n$ is monotone with respect to all n orderings.

• 4. A multilattice is called distributive iff all $2(2n^2-n)$ distributive laws are satisfied, i.e. $a\otimes (b\oplus c) = (a\otimes b)\oplus (a\otimes c)$, where $a,b,c\in S$, $\otimes ,\oplus \in \{\cup _1,\cap _1,\ldots ,\cup _n,\cap _n\}$, and $\otimes \not =\oplus$.

Remark 2.4

In what follows, we are going to deal with complete, interlaced, and distrubutive multilattices exclusively in our research.

Definition 2.5

(Multilattice with inversions) (p. 204, Definition 4.2, Shramko 2016) Let $\mathcal {M}_n=\langle S,\leqslant _1,\ldots ,\leqslant _n\rangle$ be a multilattice. Then for any $j \leqslant n$ an unary operation $-_j$ on S is said to be a (pure) j-inversion iff for any $k \leqslant n$, $k \not = j$ the following conditions are satisfied, where $a,b\in S$:

$$\begin{aligned} a\leqslant _jb\text {~implies~}{-_j}b\leqslant _j{-_j}a; \end{aligned}$$

(anti)

$$\begin{aligned} a\leqslant _kb\text {~implies~}{-_j}a\leqslant _k{-_j}b; \end{aligned}$$

(iso)

$$\begin{aligned} {-_j-_j}a=a. \end{aligned}$$

(per2)

Definition 2.6

(Multifilter) (p. 207, Definition 5.1, Shramko 2016) Let $\mathcal {M}_n=\langle S,\leqslant _1,\ldots ,\leqslant _n\rangle$ be a multilattice, with pairs of meet and join operations $\langle \cap _1,\cup _1\rangle ,\ldots ,\langle \cap _n,\cup _n\rangle$. $\mathcal {F}_n\subset S$ is a multifilter on $\mathcal {M}_n$ iff the following condition holds, for each $j,k\leqslant n$, $j\not =k$, and $a,b\in S$:

$$\begin{aligned} a\cap _jb\in \mathcal {F}_n \text {~iff~} a\in \mathcal {F}_n \text {~and~} b\in \mathcal {F}_n; \end{aligned}$$

(filter)

A multifilter $\mathcal {F}_n$ is a prime multifilter on $\mathcal {M}_n$ iff the following condition holds, for each $j,k\leqslant n$, $j\not =k$, and $a,b\in S$:

$$\begin{aligned} a\cup _jb\in \mathcal {F}_n \text {~iff~} a\in \mathcal {F}_n \textit{~or~} b\in \mathcal {F}_n. \end{aligned}$$

(prime)

Definition 2.7

(Logical multilattice) (p. 207, Definition 5.1, Shramko 2016) A pair $\langle \mathcal {M}_n,\mathcal {F}_n\rangle$ is a logical multilattice iff $\mathcal {M}_n=\langle S,\leqslant _1,\ldots ,\leqslant _n\rangle$ is a multilattice and $\mathcal {F}_n$ is a prime multifilter.

Definition 2.8

(Ultramultifilter, ultralogical multilattice) (p. 207–208, Definition 5.2, Shramko 2016) Let $\mathcal {M}_n=\langle S,\leqslant _1,\ldots ,\leqslant _n\rangle$ be a multilattice, with j-inversions defined with respect to every $\leqslant _j$ ($j \leqslant n$). Then $\mathcal {F}_n$ is an n-ultrafilter (ultramultifilter) on $\mathcal {M}_n$ iff it is a prime multifilter, such that for every $j,k\leqslant n$, $j\not =k$, and $a\in S$:

$$\begin{aligned} a\in \mathcal {U}_n \text {~iff~} {-_k-_j}a\not \in \mathcal {U}_n. \end{aligned}$$

(ultra)

A pair $\langle \mathcal {M}_n,\mathcal {U}_n\rangle$ is an ultralogical multilattice iff $\mathcal {M}_n$ is a multilattice and $\mathcal {U}_n$ is an ultramultifilter.

Definition 2.9

(Language) The formulas of $\textbf{ML}_n$, $\textbf{BML}_n$, and $\textbf{CML}_n$ are built from the set $\mathcal {P}=\{p_n\mid n\in {\mathbb {N}}\}$ of propositional variables, negations $\lnot _1,\ldots ,\lnot _n$, conjunctions $\wedge _1,\ldots ,\wedge _n$, disjunctions $\vee _1,\ldots ,\vee _n$, implications $\rightarrow _1,\ldots ,\rightarrow _n$, and co-implications $\leftarrow _1,\ldots ,\leftarrow _n$. The notion of a formula is defined in a standard inductive way.

Definition 2.10

(Valuation in $\textbf{ML}_n$) Let $\mathcal {M}_n=\langle S,\leqslant _1,\ldots ,\leqslant _n\rangle$ be a multilattice. A valuation v is defined as a mapping from $\mathcal {P}$ to S. It is extended into complex formulas as follows: $v(\lnot _jA)=-_jv(A)$, $v(A\wedge _jB)=v(A)\cap _jv(B)$, $v(A\vee _jB)=v(A)\cup _jv(B)$, $v(A\rightarrow _jB)={-_k-_j}v(A)\cup _jv(B)$, and $v(A\leftarrow _jB)=v(A)\cap _j{-_k-_j}v(B)$.

Remark 2.11

This definition of the valuation for implications and co-implications is applicable only to the case of ultralogical multilattices and $\textbf{ML}_n$ (as well as its modal extensions studied in Grigoriev and Petrukhin (2019b, 2021, 2022)). We will need another definition for these connectives for the case of $\textbf{BML}_n$ and $\textbf{CML}_n$.

Definition 2.12

(Entailment in $\textbf{ML}_n$) The entailment relation in $\textbf{ML}_n$ is defined as follows:

$\Gamma \models _{\textbf{ML}_n}\Delta$ iff for each ultralogical multilattice $\langle \mathcal {M}_n,\mathcal {U}_n\rangle$ and each valuation v, it holds that if $v(C)\in \mathcal {U}_n$, for each $C\in \Gamma$, then $v(D)\in \mathcal {U}_n$, for some $D\in \Delta$.

Definition 2.13

(Entailment in $\textbf{MLL}_n$) The entailment relation in $\textbf{MLL}_n$ is defined as follows:

$\Gamma \models _{\textbf{MLL}_n}\Delta$ iff for each logical multilattice $\langle \mathcal {M}_n,\mathcal {U}_n\rangle$ and each valuation v, it holds that if $v(C)\in \mathcal {U}_n$, for each $C\in \Gamma$, then $v(D)\in \mathcal {U}_n$, for some $D\in \Delta$.

Remark 2.14

In $\textbf{ML}_n$ if $j,k\leqslant n$ and $j\not = k$, then $\lnot _k\lnot _jA$ is equivalent to $\lnot _j\lnot _kA$; $\lnot _k\lnot _j$ behaves as Boolean negation. In $\textbf{BML}_n$ and $\textbf{CML}_n$, if $j,k\leqslant n$, $j\not = k$, and $j<k$, then $\lnot _j\lnot _kA$ behaves as intuitionistic negation and $\lnot _k\lnot _jA$ behaves as dual intuitionistic negation. In all the logics in question A is equivalent to $\lnot _j\lnot _jA$ and $\lnot _j$ behaves as De Morgan negation.

3 Sequent Calculi

Let us describe Kamide, Shramko, and Wansing’s sequent calculus for the logic $\textbf{CML}_n$ Kamide et al. (2017). A sequent is understood as an ordered pair written as $\Gamma \Rightarrow \Delta$, where $\Gamma$ and $\Delta$ are finite sets of formulas. The axioms, for any propositional variable P:

$$\begin{aligned}{} & {} (Ax) P\Rightarrow P \qquad (Ax_\lnot ) \lnot _jP \Rightarrow \lnot _jP \end{aligned}$$

The structural rules:

$$\begin{aligned} (Cut) \dfrac{\Gamma \Rightarrow \Delta ,A\quad A,\Theta \Rightarrow \Lambda }{\Gamma ,\Delta \Rightarrow \Theta ,\Lambda } \quad (W \Rightarrow ) \dfrac{\Gamma \Rightarrow \Delta }{A,\Gamma \Rightarrow \Delta } \quad ( \Rightarrow W) \dfrac{\Gamma \Rightarrow \Delta }{\Gamma \Rightarrow \Delta ,A} \quad \end{aligned}$$

The non-negated logical rules:

$$\begin{aligned}{} & {} (\wedge _j\Rightarrow ) \dfrac{A,B,\Gamma \Rightarrow \Delta }{A\wedge _jB,\Gamma \Rightarrow \Delta } (\Rightarrow \wedge _j) \dfrac{\Gamma \Rightarrow \Delta ,A\qquad \Gamma \Rightarrow \Delta ,B}{\Gamma \Rightarrow \Delta ,A\wedge _jB}\\{} & {} (\vee _j\Rightarrow ) \dfrac{A,\Gamma \Rightarrow \Delta \qquad B,\Gamma \Rightarrow \Delta }{A\vee _jB,\Gamma \Rightarrow \Delta } (\Rightarrow \vee _j) \dfrac{\Gamma \Rightarrow \Delta ,A,B}{\Gamma \Rightarrow \Delta ,A\vee _jB}\qquad \\{} & {} (\rightarrow _j\,\Rightarrow ) \dfrac{\Gamma \Rightarrow \Delta ,A\qquad B,\Theta \Rightarrow \Lambda }{A\rightarrow _jB,\Gamma ,\Theta \Rightarrow \Delta ,\Lambda } (\Rightarrow \,\rightarrow _j) \dfrac{A,\Gamma \Rightarrow B}{\Gamma \Rightarrow A\rightarrow _jB}\\{} & {} (\leftarrow _j\,\Rightarrow ) \dfrac{A\Rightarrow \Delta ,B}{A\leftarrow _jB\Rightarrow \Delta } (\Rightarrow \,\leftarrow _j) \dfrac{\Gamma \Rightarrow \Delta ,A\qquad B,\Theta \Rightarrow \Lambda }{\Gamma ,\Theta \Rightarrow \Delta ,\Lambda ,A\leftarrow _jB} \end{aligned}$$

The jj-negated logical rules:

$$\begin{aligned}{} & {} (\lnot _j\wedge _j\Rightarrow ) \dfrac{\lnot _jA,\Gamma \Rightarrow \Delta \qquad \lnot _jB,\Gamma \Rightarrow \Delta }{\lnot _j(A\wedge _jB),\Gamma \Rightarrow \Delta } (\Rightarrow \lnot _j\wedge _j) \dfrac{\Gamma \Rightarrow \Delta ,\lnot _jA,\lnot _jB}{\Gamma \Rightarrow \Delta ,\lnot _j(A\wedge _jB)}\\{} & {} (\lnot _j\vee _j\Rightarrow ) \dfrac{\lnot _jA,\lnot _jB,\Gamma \Rightarrow \Delta }{\lnot _j(A\vee _jB),\Gamma \Rightarrow \Delta } (\Rightarrow \lnot _j\vee _j) \dfrac{\Gamma \Rightarrow \Delta ,\lnot _jA\qquad \Gamma \Rightarrow \Delta ,\lnot _jB}{\Gamma \Rightarrow \Delta ,\lnot _j(A\vee _jB)}\\{} & {} (\lnot _j{\rightarrow _j^c}\Rightarrow ) \dfrac{\Gamma \Rightarrow \Delta ,A\qquad \lnot _jB,\Theta \Rightarrow \Lambda }{\lnot _j(A\rightarrow _jB),\Gamma ,\Theta \Rightarrow \Delta , \Lambda }\qquad (\Rightarrow \lnot _j{\rightarrow _j^c})\dfrac{A,\Gamma \Rightarrow \lnot _jB}{\Gamma \Rightarrow \lnot _j(A\rightarrow _jB)}\\{} & {} (\lnot _j{\leftarrow _j^c}\Rightarrow )\dfrac{\lnot _jA\Rightarrow \Delta ,B}{\lnot _j(A\leftarrow _jB)\Rightarrow \Delta }\qquad (\Rightarrow \lnot _j{\leftarrow _j^c})\dfrac{\Gamma \Rightarrow \Delta ,\lnot _jA\qquad B,\Theta \Rightarrow \Lambda }{\Gamma ,\Theta \Rightarrow \Delta ,\Lambda ,\lnot _j(A\leftarrow _jB)}\\{} & {} (\lnot _j\lnot _j\Rightarrow ) \dfrac{A,\Gamma \Rightarrow \Delta }{\lnot _j\lnot _jA,\Gamma \Rightarrow \Delta }\qquad (\Rightarrow \lnot _j\lnot _j) \dfrac{\Gamma \Rightarrow \Delta ,A}{\Gamma \Rightarrow \Delta ,\lnot _j\lnot _jA} \end{aligned}$$

The kj -negated logical rules (we presuppose that $j<k$ in the case of the rules for $\lnot _j\lnot _k$ and $\lnot _k\lnot _j$ ):

$$\begin{aligned}{} & {} (\lnot _k\wedge _j\Rightarrow ) \dfrac{\lnot _kA,\lnot _kB,\Gamma \Rightarrow \Delta }{\lnot _k(A\wedge _jB),\Gamma \Rightarrow \Delta } (\Rightarrow \lnot _k\wedge _j) \dfrac{\Gamma \Rightarrow \Delta ,\lnot _kA\qquad \Gamma \Rightarrow \Delta ,\lnot _kB}{\Gamma \Rightarrow \Delta ,\lnot _k(A\wedge _jB)}\\{} & {} (\lnot _k\vee _j\Rightarrow ) \dfrac{\lnot _kA,\Gamma \Rightarrow \Delta \qquad \lnot _kB,\Gamma \Rightarrow \Delta }{\lnot _k(A\vee _jB),\Gamma \Rightarrow \Delta }(\Rightarrow \lnot _k\vee _j) \dfrac{\Gamma \Rightarrow \Delta ,\lnot _kA,\lnot _kB}{\Gamma \Rightarrow \Delta ,\lnot _k(A\vee _jB)}\\{} & {} (\lnot _k{\rightarrow _j}\Rightarrow ) \dfrac{\Gamma \Rightarrow \Delta ,\lnot _kA\qquad \lnot _kB,\Theta \Rightarrow \Lambda }{\lnot _k(A\rightarrow _jB), \Gamma ,\Theta \Rightarrow \Delta ,\Lambda } \quad (\Rightarrow \lnot _k{\rightarrow _j}) \dfrac{\lnot _kA,\Gamma \Rightarrow \lnot _kB}{\Gamma \Rightarrow \lnot _k(A\rightarrow _jB)}\\{} & {} (\lnot _k{\leftarrow _j}\Rightarrow ) \dfrac{\lnot _kA\Rightarrow \Delta ,\lnot _kB}{\lnot _k(A\leftarrow _jB),\Rightarrow \Delta } \quad (\lnot _k{\leftarrow _j}\Rightarrow ) \dfrac{\Gamma \Rightarrow \Delta ,\lnot _kA\qquad \lnot _kB,\Theta \Rightarrow \Lambda }{\Gamma ,\Theta \Rightarrow \Delta ,\Lambda ,\lnot _k(A\leftarrow _jB)} \qquad \\{} & {} (\lnot _j\lnot _k\Rightarrow ) \dfrac{\Gamma \Rightarrow A}{\lnot _j\lnot _kA,\Gamma \Rightarrow } \qquad (\Rightarrow \lnot _j\lnot _k) \dfrac{A,\Gamma \Rightarrow }{\Gamma \Rightarrow \lnot _j\lnot _kA}\\{} & {} (\lnot _k\lnot _j\Rightarrow ) \dfrac{\Rightarrow \Delta ,A}{\lnot _k\lnot _jA\Rightarrow \Delta } \qquad (\Rightarrow \lnot _k\lnot _j) \dfrac{A\Rightarrow \Delta }{\Rightarrow \Delta ,\lnot _k\lnot _jA} \end{aligned}$$

Kamide, Shramko, and Wansing’s sequent calculus for $\textbf{BML}_n$Kamide et al. (2017) is obtained from the sequent calculus for $\textbf{CML}_n$ by the replacement of the rules $(\lnot _j{\rightarrow _j^c}\Rightarrow )$, $(\Rightarrow \lnot _j{\rightarrow _j^c})$, $(\lnot _j{\leftarrow _j^c}\Rightarrow )$, and $(\Rightarrow \lnot _j{\leftarrow _j^c})$ with the following ones:

$$\begin{aligned}{} & {} (\lnot _j{\rightarrow _j^b}\Rightarrow ) \dfrac{\lnot _jB\Rightarrow \Delta ,\lnot _jA}{\lnot _j(A\rightarrow _jB)\Rightarrow \Delta } (\Rightarrow \lnot _j{\rightarrow _j^b}) \dfrac{\Gamma \Rightarrow \Delta ,\lnot _jB\qquad \lnot _jA,\Theta \Rightarrow \Lambda }{\Gamma ,\Theta \Rightarrow \Delta ,\Lambda ,\lnot _j(A\rightarrow _jB)}\\{} & {} (\lnot _j{\leftarrow _j^b}\Rightarrow ) \dfrac{\Gamma \Rightarrow \Delta ,\lnot _jB\qquad \lnot _jA,\Theta \Rightarrow \Lambda }{\lnot _j(A\leftarrow _jB),\Gamma ,\Theta \Rightarrow \Delta ,\Lambda }(\Rightarrow \lnot _j{\leftarrow _j^b}) \dfrac{\lnot _jB,\Gamma \Rightarrow \lnot _jA}{\Gamma \Rightarrow \lnot _j(A\leftarrow _jB)} \end{aligned}$$

Let $\textbf{L}\in \{\textbf{CML}_n,\textbf{BML}_n\}$. We write $\vdash _\textbf{L}\Gamma \Rightarrow \Delta$ iff there is a proof of the sequent $\Gamma \Rightarrow \Delta$ in the sequent calculus for the logic L. The notion of the proof is defined in a standard manner for sequent calculi.

These calculi are multilattice versions of a sequent calculus BL for bi-intuitionistic logic and a sequent calculus BCL for bi-intuitionistic connexive logic, respectively, developed by Kamide and Wansing (2016). As mentioned in Kamide et al. (2017), since the cut-elimination theorem does not hold for BCL and BL (Kamide & Wansing, 2016), the cut-elimination theorem also does not hold for $\textbf{CML}_n$ and $\textbf{BML}_n$.

To obtain Kamide and Shramko (2017b) sequent calculus for $\textbf{ML}_n$ from the sequent calculus for $\textbf{BML}_n$ one needs to change the rules of which have only one formula in ancedent or consequent of a sequent: this restriction should be rejected; the rules for $\lnot _j\lnot _k$ and $\lnot _k\lnot _j$ will coincide, the condition that in their formulation $j<k$, should be omitted. The sequent calculus for $\textbf{ML}_n$ is cut-free (Kamide & Shramko, 2017b).

To obtain the sequent calculus for the logic $\textbf{MLL}_n$ from (Grigoriev & Petrukhin, 2022) from the sequent calculus for $\textbf{ML}_n$ one needs to delete the rules for $\lnot _j\lnot _k$ ($\lnot _k\lnot _j$) as well as all the rules for implications, coimplications, and their negations. The sequent calculus for $\textbf{MLL}_n$ is cut-free (Grigoriev & Petrukhin, 2022).

4 Connexive and Bi-Intuitionistic Multilattices

Definition 4.1

(De Morgan multifilter) Let $\mathcal {M}_n=\langle S,\leqslant _1,\ldots ,\leqslant _n\rangle$ be a multilattice (cf. Definition 2.3) and $\mathcal {F}_n$ be a prime multifilter on $\mathcal {M}_n$ (cf. Definition 2.6). Then for any $j \leqslant n$ an unary operation $-_j$ on S is said to be a j-pseudo-inversion and $\mathcal {F}_n$ is called De Morgan multifilter iff for any $k \leqslant n$, $k \not = j$ the following conditions are satisfied, where $a,b\in S$:

$$\begin{aligned}&-_j(a\cap _jb)\in \mathcal {F}&\text {~iff~} -_ja\cup _j-_jb\in \mathcal {F}; \end{aligned}$$

(DM1)

$$\begin{aligned}&-_j(a\cup _jb)\in \mathcal {F}&\text {~iff~} -_ja\cap _j-_jb\in \mathcal {F}; \end{aligned}$$

(DM2)

$$\begin{aligned}&-_k(a\cap _jb)\in \mathcal {F}&\text {~iff~} -_ka\cap _j-_kb\in \mathcal {F}; \end{aligned}$$

(DM3)

$$\begin{aligned}&-_k(a\cup _jb)\in \mathcal {F}&\text {~iff~} -_ka\cup _j-_kb\in \mathcal {F}; \end{aligned}$$

(DM4)

$$\begin{aligned}&{-_j}{-_j}a\in \mathcal {F}&\text {~iff~} a\in \mathcal {F}. \end{aligned}$$

(per2)

A De Morgan multifilter $\mathcal {F}_n$ is called De Morgan ultramultifilter iff it satisifes the condition (ultra). A pair $\langle \mathcal {M}_n,\mathcal {F}_n \rangle$ is called De Morgan logical (resp. ultralogical) multilattice iff $\mathcal {M}_n$ is a multilattice and $\mathcal {F}_n$ is a De Morgan multifilter (resp. ultramultifilter).

Definition 4.2

(Connexive multilattice and connexive multifilter) Let $\mathcal {M}_n=\langle S,\leqslant _1,\ldots ,\leqslant _n\rangle$ be a multilattice with the j-pseudo-inversion operations $-_1,\ldots ,-_n$ and $\mathcal {F}_n$ be a De Morgan multifilter. Then for any $j,k\leqslant n$ such that $j\not =k$ the corresponding pairs $\langle \supset _1,\subset _1\rangle ,\ldots ,\langle \supset _n,\subset _n\rangle$ of binary operations called relative pseudo-complement and relative pseudo-difference operations are defined as follows, where $a,b,c\in S$ and $j<k$ in the conditions ($-_j-_k$) and ($-_k-_j$):

A multilattice $\mathcal {M}_n$ is called connexive iff $\mathcal {M}_n$ is a multilattice with the j-pseudo-inversion operations $-_1,\ldots ,-_n$ and the pairs $\langle \supset _1,\subset _1\rangle ,\ldots ,\langle \supset _n,\subset _n\rangle$ of relative pseudo-complement and relative pseudo-difference operations (where $j,k\leqslant n$ and $j\not =k$). A De Morgan multifilter $\mathcal {F}_n$ is called a connexive multifilter iff it satisfies the above presented conditions. A pair $\langle \mathcal {M}_n,\mathcal {F}_n \rangle$ is called connexive logical multilattice iff $\mathcal {M}_n$ is a connexive multilattice and $\mathcal {F}_n$ is a connexive multifilter.

Definition 4.3

(Bi-intuitionistic multilattice and bi-intuitionistic multifilter) Bi-intuitionistic multilattice, multiliter, logical multilattice satisfy all those conditions which hold for their connexive counterparts, except ($-_j{\supset _j^c}$) and ($-_j{\subset _j^c}$), as well as satisfy the following ones, for each $j,k\leqslant n$, $j\not =k$, and $a,b,c\in S$:

Definition 4.4

(Valuation in $\textbf{CML}_n$ and $\textbf{BML}_n$) Let $\mathcal {M}_n=\langle S,\leqslant _1,\ldots ,\leqslant _n\rangle$ be a connexive (resp. bi-intuitionistic) multilattice. A valuation v is defined as a mapping from $\mathcal {P}$ to S. It is extended into complex formulas as follows: $v(\lnot _jA)=-_jv(A)$, $v(A\wedge _jB)=v(A)\cap _jv(B)$, $v(A\vee _jB)=v(A)\cup _jv(B)$, $v(A\rightarrow _jB)=v(A)\supset _jv(B)$, and $v(A\leftarrow _jB)=v(A)\subset _jv(B)$.

Definition 4.5

(Entailment in $\textbf{CML}_n$ and $\textbf{BML}_n$) The entailment relation in $\textbf{CML}_n$ and $\textbf{BML}_n$ is be defined as follows, for any finite sets of formulas $\Gamma$ and $\Delta$:

• $\Gamma \models _{\textbf{CML}_n}\Delta$ iff for each logical connexive multilattice $\langle \mathcal {M}_n,\mathcal {F}_n\rangle$ and each valuation v, it holds that if $v(C)\in \mathcal {F}_n$, for each $C\in \Gamma$, then $v(D)\in \mathcal {F}_n$, for some $D\in \Delta$.

• $\Gamma \models _{\textbf{BML}_n}\Delta$ iff for each logical bi-intuitionistic multilattice $\langle \mathcal {M}_n,\mathcal {F}_n\rangle$ and each valuation v, it holds that if $v(C)\in \mathcal {F}_n$, for each $C\in \Gamma$, then $v(D)\in \mathcal {F}_n$, for some $D\in \Delta$.

For any finite sets of formulas $\Gamma$ and $\Delta$, we write $\bigwedge _j \Gamma$ for the j-conjunction of all formulas from $\Gamma$ and $\bigvee _j \Delta$ for the j-disjunction of all formulas from $\Delta$. If $\bigwedge _j \Gamma =\emptyset$, then $\bigwedge _j \Gamma =p\subset _jp$. If $\bigvee _j \Delta =\emptyset$, then $\bigvee _j \Delta =p\supset _jp$.

Definition 4.6

(Validity of sequents. The case of $\textbf{CML}_n$ and $\textbf{BML}_n$) A sequent $\Gamma \Rightarrow \Delta$ is valid in the logic $\textbf{L}\in \{\textbf{CML}_n,\textbf{BML}_n\}$ (symbolically, $\models _\textbf{L}\Gamma \Rightarrow \Delta$) iff $\Gamma \models _\textbf{L}\Delta$.

In a similar fashion, modifying the above presented algebraic semantics for $\textbf{CML}_n$ and $\textbf{BML}_n$, we can propose an alternative algebraic semantics for the sequent calculi for $\textbf{ML}_n$ and $\textbf{MLL}_n$.

Definition 4.7

(Classical multilattice and classical multifilter) Let $\mathcal {M}_n=\langle S,\leqslant _1,\ldots ,\leqslant _n\rangle$ be a multilattice with the j-pseudo-inversion operations $-_1,\ldots ,-_n$ and $\mathcal {F}_n$ be a De Morgan ultramultifilter. Then for any $j,k\leqslant n$ such that $j\not =k$ the pairs $\langle \supset _1,\subset _1\rangle ,\ldots ,\langle \supset _n,\subset _n\rangle$ of binary operations called pseudo-complement and pseudo-difference operations are defined as follows, where $a,b,c\in S$:

A multilattice $\mathcal {M}_n$ is called classical iff $\mathcal {M}_n$ is a multilattice with the j-pseudo-inversion operations $-_1,\ldots ,-_n$ and the pairs $\langle \supset _1,\subset _1\rangle ,\ldots ,\langle \supset _n,\subset _n\rangle$ of pseudo-complement and pseudo-difference operations (where $j,k\leqslant n$ and $j\not =k$). A De Morgan ultramultifilter $\mathcal {F}_n$ is called a classical ultramultifilter iff it satisfies the above presented conditions. A pair $\langle \mathcal {M}_n,\mathcal {F}_n \rangle$ is called classical ultralogical multilattice iff $\mathcal {M}_n$ is a classical multilattice and $\mathcal {F}_n$ is a classical ultramultifilter.

Definition 4.8

(Entailment in $\textbf{ML}_n$ and $\textbf{MLL}_n$) The entailment relation in $\textbf{ML}_n$ and $\textbf{MLL}_n$ can be defined as follows, for any finite sets of formulas $\Gamma$ and $\Delta$¹:

• $\Gamma \models _{\textbf{ML}_n}^A\Delta$ iff for each ultralogical classical multilattice $\langle \mathcal {M}_n,\mathcal {F}_n\rangle$ and each valuation v, it holds that if $v(C)\in \mathcal {F}_n$, for each $C\in \Gamma$, then $v(D)\in \mathcal {F}_n$, for some $D\in \Delta$.

• $\Gamma \models _{\textbf{MLL}_n}^A\Delta$ iff for each logical De Morgan multilattice $\langle \mathcal {M}_n,\mathcal {F}_n\rangle$ and each valuation v, it holds that if $v(C)\in \mathcal {F}_n$, for each $C\in \Gamma$, then $v(D)\in \mathcal {F}_n$, for some $D\in \Delta$.

Definition 4.9

(Validity of sequents. The case of $\textbf{ML}_n$ and $\textbf{MLL}_n$) A sequent $\Gamma \Rightarrow \Delta$ is valid in the logic $\textbf{L}\in \{\textbf{ML}_n,\textbf{MLL}_n\}$ (symbolically, $\models _\textbf{L}\Gamma \Rightarrow \Delta$) iff $\Gamma \models _\textbf{L}^A\Delta$.

5 Soundness and Completeness Proofs

Lemma 5.1

All the rules of the sequent calculus for $\textbf{CML}_n$ are sound with respect to logical connexive multilattices.

Proof

Consider the rule $(\Rightarrow \lnot _j\wedge _j)$. Suppose $\models _{\textbf{CML}_n} \Gamma \Rightarrow \Delta ,\lnot _jA,\lnot _jB$. Thus, for each logical connexive multilattice $\langle \mathcal {M}_n,\mathcal {F}_n\rangle$ and each valuation v, it holds that if $v(C)\in \mathcal {F}_n$, for each $C\in \Gamma$, then $v(D)\in \mathcal {F}_n$, for some $D\in \Delta \cup \{\lnot _jA,\lnot _jB\}$. Assume that $v(C)\in \mathcal {F}_n$, for each $C\in \Gamma$. Thus, $v(D)\in \mathcal {F}_n$, for some $D\in \Delta \cup \{\lnot _jA,\lnot _jB\}$. If $v(D)\in \mathcal {F}_n$, for some $D\in \Delta$, then $\models _{\textbf{CML}_n} \Gamma \Rightarrow \Delta ,\lnot _j(A\wedge _jB)$. If $v(D)\in \mathcal {F}_n$, for some $D\in \{\lnot _jA,\lnot _jB\}$, that is $v(\lnot _jA)\in \mathcal {F}_n$ or $v(\lnot _jB)\in \mathcal {F}_n$, then, since $\mathcal {F}_n$ is prime, $v(\lnot _jA)\cup _jv(\lnot _jB)\in \mathcal {F}_n$. By Definition 4.4, $-_jv(A)\cup _j-_jv(B)\in \mathcal {F}_n$. By (DM1), $-_j(v(A)\cap _jv(B))\in \mathcal {F}_n$. By Definition 4.4, $v(\lnot _j(A\wedge _jB))\in \mathcal {F}_n$. Therefore, $\models _{\textbf{CML}_n} \Gamma \Rightarrow \Delta ,\lnot _j(A\wedge _jB)$.

Consider the rule $(\Rightarrow \lnot _k\wedge _j)$. Suppose that $\models _{\textbf{CML}_n} \Gamma \Rightarrow \Delta ,\lnot _kA$ and $\models _{\textbf{CML}_n} \Gamma \Rightarrow \Delta ,\lnot _kB$. Then for each logical connexive multilattice $\langle \mathcal {M}_n,\mathcal {F}_n\rangle$ and each valuation v, it holds that if $v(C)\in \mathcal {F}_n$, for each $C\in \Gamma$, then $v(D)\in \mathcal {F}_n$, for some $D\in \Delta \cup \{\lnot _kA\}$ as well as for each logical connexive multilattice $\langle \mathcal {M}_n,\mathcal {F}_n\rangle$ and each valuation v, it holds that if $v(C)\in \mathcal {F}_n$, for each $C\in \Gamma$, then $v(D)\in \mathcal {F}_n$, for some $D\in \Delta \cup \{\lnot _kB\}$. Assume that $v(C)\in \mathcal {F}_n$, for each $C\in \Gamma$. Thus, $v(D)\in \mathcal {F}_n$, for some $D\in \Delta \cup \{\lnot _kA\}$ as well as $v(D)\in \mathcal {F}_n$, for some $D\in \Delta \cup \{\lnot _kB\}$. If $v(D)\in \mathcal {F}_n$, for some $D\in \Delta$, then $\models _{\textbf{CML}_n} \Gamma \Rightarrow \Delta ,\lnot _k(A\wedge _jB)$. If $v(\lnot _kA)\in \mathcal {F}_n$ and $v(\lnot _kB)\in \mathcal {F}_n$, then, since $\mathcal {F}_n$ is a filter, $v(\lnot _kA)\cap _jv(\lnot _kB)\in \mathcal {F}_n$. By Definition 4.4 and (DM3), $\lnot _kv(A\wedge _jB)\in \mathcal {F}_n$. Thus, $\models _{\textbf{CML}_n} \Gamma \Rightarrow \Delta ,\lnot _k(A\wedge _jB)$.

The cases of the other rules for $\vee _j$, $\wedge _j$, $\lnot _j\vee _j$, $\lnot _j\wedge _j$, $\lnot _k\vee _j$, and $\lnot _k\wedge _j$ are considered similarly with the help of the fact that $\cup _j$ and $\cap _j$ are lattice operations and with the use of (DM1)–(DM4). The rules for $\lnot _j\lnot _j$ are easily checked due to (per2).

Consider the rule $(\Rightarrow \rightarrow _j)$. Suppose that $\models _{\textbf{CML}_n} A,\Gamma \Rightarrow B$. Then for each logical connexive multilattice $\langle \mathcal {M}_n,\mathcal {F}_n\rangle$ and each valuation v, it holds that if $v(C)\in \mathcal {F}_n$, for each $C\in \Gamma \cup \{A\}$, then $v(B)\in \mathcal {F}_n$. Since $\mathcal {F}_n$ is a multifilter, we conclude that for each logical connexive multilattice $\langle \mathcal {M}_n,\mathcal {F}_n\rangle$ and each valuation v, it holds that if $v(A\wedge _j\bigwedge _j\Gamma )\in \mathcal {F}_n$, then $v(B)\in \mathcal {F}_n$. Using Definition 4.4 and ($\supset _j$), we infer that for each logical connexive multilattice $\langle \mathcal {M}_n,\mathcal {F}_n\rangle$ and each valuation v, it holds that if $v(\bigwedge _j\Gamma )\in \mathcal {F}_n$, then $v(A\rightarrow _jB)\in \mathcal {F}_n$. Hence, $\models _{\textbf{CML}_n}\Gamma \Rightarrow A\rightarrow _jB$.

Consider the rule $(\rightarrow _j\Rightarrow )$. Suppose that $\models _{\textbf{CML}_n} \Gamma \Rightarrow \Delta ,A$ and $\models _{\textbf{CML}_n} B,\Theta \Rightarrow \Lambda$. Then, for each logical connexive multilattice $\langle \mathcal {M}_n,\mathcal {F}_n\rangle$ and each valuation v, it holds that if $v(C)\in \mathcal {F}_n$, for each $C\in \Gamma$, then $v(D)\in \mathcal {F}_n$, for some $D\in \Delta \cup \{A\}$; as well as for each logical connexive multilattice $\langle \mathcal {M}_n,\mathcal {F}_n\rangle$ and each valuation v, it holds that if $v(C)\in \mathcal {F}_n$, for each $C\in \Theta \cup \{B\}$, then $v(D)\in \mathcal {F}_n$, for some $D\in \Lambda$. Since $\mathcal {F}_n$ is a prime multifilter, we have that for any valuation v, if $v(\bigwedge _j\Gamma )\in \mathcal {F}_n$, then $v(\bigvee _j\Delta )\in \mathcal {F}_n$ or $v(A)\in \mathcal {F}_n$; if $v(B)\in \mathcal {F}_n$ and $v(\bigwedge _j\Theta )\in \mathcal {F}_n$, then $v(\bigvee _j\Lambda )\in \mathcal {F}_n$. Assume that $v(A\rightarrow _jB),v(\bigwedge _j\Gamma ),v(\bigwedge _j\Theta )\in \mathcal {F}_n$. Therefore, $v(\bigvee _j\Delta )\in \mathcal {F}_n$ or $v(A)\in \mathcal {F}_n$. If $v(\bigvee _j\Delta )\in \mathcal {F}_n$, then $\models _{\textbf{CML}_n} A\rightarrow _jB,\Gamma ,\Theta \Rightarrow \Delta ,\Lambda$. If $v(A)\in \mathcal {F}_n$, then $v(B)\in \mathcal {F}_n$, since $v(A\rightarrow _jB)\in \mathcal {F}_n$ (this is a consequence of ($\supset _j$)). Since $v(B)\in \mathcal {F}_n$ and $v(\bigwedge _j\Theta )\in \mathcal {F}_n$, then $v(\bigvee _j\Lambda )\in \mathcal {F}_n$. Thus, $\models _{\textbf{CML}_n} A\rightarrow _jB,\Gamma ,\Theta \Rightarrow \Delta ,\Lambda$.

Consider the rule $(\lnot _j{\leftarrow _j^c}\Rightarrow )$. Suppose that $\models _{\textbf{CML}_n}\lnot _jA\Rightarrow \Delta , B$. Then, for each logical connexive multilattice $\langle \mathcal {M}_n,\mathcal {F}_n\rangle$ and each valuation v, it holds that if $v(\lnot _jA)\in \mathcal {F}_n$, then $v(D)\in \mathcal {F}_n$, for some $D\in \Delta \cup \{B\}$. Since $\mathcal {F}_n$ is a prime multifilter, $v(\lnot _jA)\in \mathcal {F}_n$ implies $v(\bigvee _j\Delta )\cup _j v(B)\in \mathcal {F}_n$. By Definition 4.4 and ($-_j{\subset _j^c}$), $v(\lnot _j(A\leftarrow _jB))\in \mathcal {F}_n$ implies $v(\bigvee _j\Delta )\in \mathcal {F}_n$. Hence, $\lnot _j(A\leftarrow _jB)\models _{\textbf{CML}_n}\Delta$.

Consider the rule $(\Rightarrow \lnot _j{\leftarrow _j^c})$. Suppose that $\models _{\textbf{CML}_n}\Gamma \Rightarrow \Delta ,\lnot _jA$ and $\models _{\textbf{CML}_n}B,\Theta \Rightarrow \Lambda$. Then, for each logical connexive multilattice $\langle \mathcal {M}_n,\mathcal {F}_n\rangle$ and each valuation v, it holds that if $v(C)\in \mathcal {F}_n$, for each $C\in \Gamma$, then $v(D)\in \mathcal {F}_n$, for some $D\in \Delta \cup \{\lnot _jA\}$; as well as for each logical connexive multilattice $\langle \mathcal {M}_n,\mathcal {F}_n\rangle$ and each valuation v, it holds that if $v(C)\in \mathcal {F}_n$, for each $C\in \Theta \cup \{B\}$, then $v(D)\in \mathcal {F}_n$, for some $D\in \Lambda$. Suppose that $v(\bigwedge _j\Gamma ),v(\bigwedge _j\Theta )\in \mathcal {F}_n$. Then $v(\bigvee _j\Delta )\in \mathcal {F}_n$ or $v(\lnot _jA)\in \mathcal {F}_n$. If $v(\bigvee _j\Delta )\in \mathcal {F}_n$, then $\models _{\textbf{CML}_n}\Gamma ,\Theta \Rightarrow \Delta ,\Lambda ,\lnot _j(A\leftarrow _jB)$. If $v(\lnot _jA)\in \mathcal {F}_n$, then $v(\lnot _j(A\leftarrow _jB))\in \mathcal {F}_n$ or $v(B)\in \mathcal {F}_n$ (this is a consequence of ($-_j{\supset _j^c}$)). If $v(\lnot _j(A\leftarrow _jB))\in \mathcal {F}_n$, then $\models _{\textbf{CML}_n}\Gamma ,\Theta \Rightarrow \Delta ,\Lambda ,\lnot _j(A\leftarrow _jB)$. If $v(B)\in \mathcal {F}_n$, then $v(\bigvee _j\Lambda )\in \mathcal {F}_n$, since $v(\bigwedge _j\Theta )\in \mathcal {F}_n$; $v(\bigvee _j\Lambda )\in \mathcal {F}_n$ implies $\models _{\textbf{CML}_n}\Gamma ,\Theta \Rightarrow \Delta ,\Lambda ,\lnot _j(A\leftarrow _jB)$.

The rules for $\leftarrow _j$, $\lnot _j{\rightarrow _j}$, $\lnot _k{\rightarrow _k}$, $\lnot _k{\leftarrow _k}$, $\lnot _j\lnot _k$, and $\lnot _k\lnot _j$ are considered similarly, use the conditions ($\subset _j$), ($-_j{\supset _j^c}$), ($-_k{\supset _j}$), ($-_k{\subset _j}$), ($-_j-_k$), and ($-_k-_j$), respectively. $\square$

Lemma 5.2

All the rules of the sequent calculus for $\textbf{BML}_n$ are sound with respect to logical bi-intuitionistic multilattices.

Proof

For most of the rules the proof is the same as in Lemma 5.1, for the rules for $\lnot _j{\rightarrow _j}$ and $\lnot _j{\leftarrow _j}$ use the conditions ($-_j{\supset _j^b}$) and ($-_j{\subset _j^b}$). $\square$

Theorem 5.3

(Soundness) Let $\textbf{L}\in \{\textbf{BML}_n,\textbf{CML}_n\}$. For every pair of finite sets of formulas $\Gamma$ and $\Delta$, it holds that if $\textbf{L}\vdash \Gamma \Rightarrow \Delta$, then $\Gamma \models _\textbf{L} \Delta$.

Proof

By the fact that both axioms of L are valid, and the induction on the length of derivation with the help of Lemmas 5.1 and 5.2. $\square$

Definition 5.4

(Class of equivalence) The class of equivalence [A] of a formula A is the set of formulas $\{B\mid \textbf{L}\vdash A\Rightarrow B\text {~and~}\textbf{L}\vdash B\Rightarrow A\text {~and~}{} \textbf{L}\vdash \lnot _j A\Rightarrow \lnot _jB\text {~and~}{} \textbf{L}\vdash \lnot _j B\Rightarrow \lnot _j A \}$, for any $j\leqslant n$, where $\textbf{L}\in \{\textbf{BML}_n,\textbf{CML}_n\}$. The class of equivalence $[\Gamma ]$ of a set of formulas $\Gamma$ is the set $\{[C]\mid C\in \Gamma \}$.

Definition 5.5

(Lindenbaum-Tarski algebra) A Lindenbaum-Tarski algebra (LT-algebra) is a structure $\mathcal {M}_n^\textbf{L}=\langle [\mathscr {F}],\leqslant _1,\ldots ,\leqslant _n\rangle$, where $\textbf{L}\in \{\textbf{BML}_n,\textbf{CML}_n\}$ and $\mathscr {F}$ is the set of all formulas, which satisfies the following conditions, for any formulas $A,B\in \mathscr {F}$:

$$\begin{aligned}{}[A]\leqslant _j[B]&\text {~iff~}[A]=[A\wedge _jB];\\ -_j[A]&=[\lnot _jA];\\ [A]\cap _j[B]&=[A\wedge _jB];\\ [A]\cup _j[B]&=[A\vee _jB];\\ [A]\supset _j[B]&=[A\rightarrow _jB];\\ [A]\subset _j[B]&=[A\leftarrow _jB]. \end{aligned}$$

Fact 5.6

Let $\textbf{L}\in \{\textbf{BML}_n,\textbf{CML}_n\}$. For any formulas A and B, any $j\leqslant n$, it holds that

• $\textbf{L}\vdash A\Rightarrow B$, $\textbf{L}\vdash B\Rightarrow A$, $\textbf{L}\vdash \lnot _j A\Rightarrow \lnot _j B$, and $\textbf{L}\vdash \lnot _j B\Rightarrow \lnot _j A$ iff $[A]=[B];$

• $\textbf{L}\vdash A\Rightarrow B$ and $\textbf{L}\vdash \lnot _jB\Rightarrow \lnot _jA$ iff $[A]\leqslant _j[B]$.

Lemma 5.7

The following sequents are provable in $\textbf{L}\in \{\textbf{CML}_n,\textbf{BML}_n \}$, where $j,k\leqslant n$ and in (10) and (11) we suppose that $j<k$:

• (1) $A\wedge _jA\Rightarrow A; A\Rightarrow A\wedge _jA; \lnot _k(A\wedge _jA)\Rightarrow \lnot _k A; \lnot _kA\Rightarrow \lnot _k(A\wedge _jA);$

• (2) $A\vee _jA\Rightarrow A; A\Rightarrow A\vee _jA; \lnot _k(A\vee _jA)\Rightarrow \lnot _k A; \lnot _kA\Rightarrow \lnot _k(A\vee _jA);$

• (3) $A\wedge _jB\Rightarrow B\wedge _jA; B\wedge _jA\Rightarrow A\wedge _jB; \lnot _k(A\wedge _jB)\Rightarrow \lnot _k(B\wedge _jA);$ $\lnot _k(B\wedge _jA)\Rightarrow \lnot _k(A\wedge _jB);$

• (4) $A\vee _jB\Rightarrow B\vee _jA; B\vee _jA\Rightarrow A\vee _jB; \lnot _k(A\vee _jB)\Rightarrow \lnot _k(B\vee _jA);$ $\lnot _k(B\vee _jA)\Rightarrow \lnot _k(A\vee _jB);$

• (5) $A\wedge _j(B\wedge _jC)\Rightarrow (A\wedge _jB)\wedge _jC; (A\wedge _jB)\wedge _jC \Rightarrow A\wedge _j(B\wedge _jC); \lnot _k(A\wedge _j(B\wedge _jC))\Rightarrow \lnot _k((A\wedge _jB)\wedge _jC); \lnot _k((A\wedge _jB)\wedge _jC) \Rightarrow \lnot _k(A\wedge _j(B\wedge _jC));$

• (6) $A\vee _j(B\vee _jC)\Rightarrow (A\vee _jB)\vee _jC; (A\vee _jB)\vee _jC \Rightarrow A\vee _j(B\vee _jC); \lnot _k(A\vee _j(B\vee _jC))\Rightarrow \lnot _k((A\vee _jB)\vee _jC); \lnot _k((A\vee _jB)\vee _jC) \Rightarrow \lnot _k(A\vee _j(B\vee _jC));$

• (7) $A\wedge _j(A\vee _jB)\Rightarrow A; A\Rightarrow A\wedge _j(A\vee _jB); \lnot _k(A\wedge _j(A\vee _jB))\Rightarrow \lnot _k A; \lnot _kA\Rightarrow \lnot _k(A\wedge _j(A\vee _jB));$

• (8) $A\vee _j(A\wedge _jB)\Rightarrow A; A\Rightarrow A\vee _j(A\wedge _jB); \lnot _k(A\vee _j(A\wedge _jB))\Rightarrow \lnot _k A; \lnot _kA\Rightarrow \lnot _k(A\vee _j(A\wedge _jB));$

• (9) $A\otimes (B\oplus C) \Rightarrow (A\otimes B)\oplus (A\otimes C); (A\otimes B)\oplus (A\otimes C)\Rightarrow A\otimes (B\oplus C); \lnot _k(A\otimes (B\oplus C)) \Rightarrow \lnot _k((A\otimes B)\oplus (A\otimes C)); \lnot _k((A\otimes B)\oplus (A\otimes C))\Rightarrow \lnot _k(A\otimes (B\oplus C))$, where $\otimes ,\oplus \in \{\wedge _1,\vee _1,\ldots ,\wedge _n,\vee _n\}$ and $\otimes \not =\oplus ;$

• (10) $\lnot _j\lnot _k A\Rightarrow A\rightarrow _k(A\leftarrow _kA); A\rightarrow _k(A\leftarrow _kA)\Rightarrow \lnot _j\lnot _k A;$

• (11) $\lnot _k\lnot _j A\Rightarrow (A\rightarrow _kA)\leftarrow _kA; (A\rightarrow _kA)\leftarrow _kA\Rightarrow \lnot _k\lnot _j A$.

Proof

We prove the case (10).

We prove the case (11).

The other cases are proved similarly. $\square$

Lemma 5.8

Let ${\widetilde{v}}$ be a valuation introduced in Definition 4.4 such that ${\widetilde{v}}(P) = [P]$, for all $P\in \mathcal {P}$ (such a valuation is said to be a canonic one). Then ${\widetilde{v}}(A) = [A]$, for any formula A.

Proof

By a structural induction on a formula A. Use Definition 5.5. $\square$

Lemma 5.9

(Lindenbaum lemma for $\textbf{CML}_n$) Let $\textbf{L}$ be $\textbf{CML}_n$. For every pair of finite sets of formulas $\Gamma$ and $\Delta$, it holds that $\not \vdash _\textbf{L}\Gamma \Rightarrow \Delta$ implies that there is a connexive multifilter $\, \mathcal {F}_n^\textbf{L}$ on the Lindenbaum-Tarski algebra $\mathcal {M}_n^\textbf{L}$ and $[C]\in \mathcal {F}_n^\textbf{L}$, for each $C\in \Gamma$, while $[D]\not \in \mathcal {F}_n^\textbf{L}$, for each $D\in \Delta$.

Proof

We follow the standard strategy of the proof of the Lindenbaum lemma which was adopted for the case of multilattice logic $\textbf{ML}_n$ in (Lemma 4.12, Grigoriev and Petrukhin 2019b).

Suppose that $\not \vdash _{\textbf{L}}\Gamma \Rightarrow \Delta$. Let $F_1,\ldots ,F_m,\ldots$ be an enumeration of the set of all formulas. We postulate the following identities:

By Definition 5.4, we have $[\Sigma ]=\{[B]\mid B\in \Sigma \}$. We need to show that $[\Sigma ]$ is the required connexive multifilter on $\mathcal {M}_n^\textbf{L}$.

By the induction on i, one may easily prove that ($\divideontimes$) for each i, it holds that $\not \vdash _\textbf{L} \Omega _i\Rightarrow \Sigma$. Moreover, $\not \vdash _\textbf{L} \Omega _i\Rightarrow$, otherwise, by ($\Rightarrow$W), $\vdash _\textbf{L} \Omega _i\Rightarrow \Sigma$. It is easy to justify that $[\Gamma ]\subseteq [\Sigma ]$, i.e. $[C]\in [\Sigma ]$ (for each $C\in \Gamma )$, and $[D]\not \in [\Sigma ]$ (for each $D\in \Delta )$.

Let us show that $[\Sigma ]$ satisfies condition (filter). Suppose that $[A],[B]\in [\Sigma ]$. Then $A,B\in \Sigma$ and there are l and m such that $\vdash _{\textbf{L}}\Omega _l\Rightarrow A$ and $\vdash _{\textbf{L}}\Omega _m\Rightarrow B$. Assume that $[A]\cap _j[B]\not \in [\Sigma ]$. Then, by Definition 5.5, $[A\wedge _jB]\not \in [\Sigma ]$ which yields $A\wedge _jB\not \in \Sigma$. Then there is i such that $A\wedge _jB=F_{i+1}$ and $\vdash _{\textbf{L}}\Omega _{i},F_{i+1}\Rightarrow$. We have (double lines indicate multiple applications of a rule):

It contradicts the fact ($\divideontimes$). Thus, $[A]\cap _j[B]\in [\Sigma ]$.

Suppose that $[A]\cap _j[B]\in [\Sigma ]$, but $[A]\not \in [\Sigma ]$ or $[B]\not \in [\Sigma ]$. Then $\vdash _{\textbf{L}}\Omega _l\Rightarrow A\wedge _jB$, for some l. Assume that $[A]\not \in [\Sigma ]$. Then $A=F_{i+1}$ and $\Omega _{i},F_{i+1}\Rightarrow$, for some i. We have:

It contradicts the fact ($\divideontimes$). Hence, $[A]\in [\Sigma ]$. The case $[B]\not \in [\Sigma ]$ is treated similarly. Therefore, $[\Sigma ]$ is a multifilter.

Let us show that $[\Sigma ]$ satisfies condition (prime). Assume that $[ A]\cup _j[ B]\in [\Sigma ]$ while $[ A],[ B]\not \in [\Sigma ]$. We have $\vdash _\textbf{L}\Omega _i\Rightarrow A\vee _j B$ as well as $\vdash _\textbf{L}\Omega _l, A\Rightarrow$ and $\vdash _\textbf{L}\Omega _m, B\Rightarrow$, for some i, l, and m. Thus,

It contradicts the fact ($\divideontimes$). Hence, $[ A]\in [\Sigma ]$ or $[ B]\in [\Sigma ]$.

Suppose that $[ A]\in [\Sigma ]$ while $[ A]\cup _j[ B]\not \in [\Sigma ]$. Then we have:

It contradicts the fact ($\divideontimes$). The case when $[ B]\in [\Sigma ]$ is treated similarly. Therefore, $[\Sigma ]$ is a prime multifilter.

Let us show that $[\Sigma ]$ satisfies condition (DM1). Assume that $-_j([A]\cap _j[B])\in [\Sigma ]$, while $-_j[A]\cup _j-_j[B]\not \in [\Sigma ]$. Since we already know that $[\Sigma ]$ is a prime multifilter, $-_j[A]\not \in [\Sigma ]$ and $-_j[B]\not \in [\Sigma ]$. We have $\vdash _\textbf{L}\Omega _i\Rightarrow \lnot _j(A\wedge _jB)$ as well as $\vdash _\textbf{L}\Omega _l,\lnot _j A\Rightarrow$ and $\vdash _\textbf{L}\Omega _m, \lnot _jB\Rightarrow$, for some i, l, and m. Thus,

It contradicts the fact ($\divideontimes$). Hence, $-_j[ A]\in [\Sigma ]$ or $-_j[ B]\in [\Sigma ]$.

Assume that $-_j[A]\cup _j-_j[B]\in [\Sigma ]$, while $-_j([A]\cap _j[B])\not \in [\Sigma ]$. Since $[\Sigma ]$ is a prime multifilter, $-_j[A]\in [\Sigma ]$ or $-_j[B]\in [\Sigma ]$. Hence, $\vdash _\textbf{L}\lnot _j(A\wedge _jB),\Omega _i\Rightarrow$ as well as $\vdash _\textbf{L}\Omega _l\Rightarrow \lnot _j A$ or $\vdash _\textbf{L}\Omega _m\Rightarrow \lnot _jB$, for some i, l, and m. Suppose that $\vdash _\textbf{L}\Omega _l\Rightarrow \lnot _j A$. Thus,

It contradicts the fact ($\divideontimes$). The case when $-_j[B]\in [\Sigma ]$ is treated similarly. Therefore, $[\Sigma ]$ satisfies condition (DM1).

By a similar reasoning, one can show that $[\Sigma ]$ satisfies conditions (DM2)–(DM4) as well.

Let us show that $[\Sigma ]$ satisfies condition (per2). Assume that $-_j-_j[A]\in [\Sigma ]$, while $[A]\not \in [\Sigma ]$. We have $\vdash _\textbf{L}\Omega _i\Rightarrow \lnot _j\lnot _jA$ as well as $\vdash _\textbf{L}\Omega _l, A\Rightarrow$, for some i and l. Thus,

It contradicts the fact ($\divideontimes$). Hence, $[ A]\in [\Sigma ]$.

Assume that $[A]\in [\Sigma ]$, while $-_j-_j[A]\not \in [\Sigma ]$. Hence, $\vdash _\textbf{L}\lnot _j\lnot _jA,\Omega _i\Rightarrow$ as well as $\vdash _\textbf{L}\Omega _l\Rightarrow A$, for some i and l. Thus,

It contradicts the fact ($\divideontimes$). Hence, ${-_j-_j}[A]\in [\Sigma ]$. Thus, $[\Sigma ]$ satisfies condition (per2).

Let us show that $[\Sigma ]$ satisfies condition ($-_j-_k$). Assume that $-_j-_k[A]\in [\Sigma ]$, while $[A]\supset _k([A]\subset _k[A])\not \in [\Sigma ]$. We have $\vdash _\textbf{L}\Omega _i\Rightarrow \lnot _j\lnot _kA$ as well as $\vdash _\textbf{L}\Omega _l, A\rightarrow _k(A\leftarrow _kA)\Rightarrow$, for some i and l. Recall that, by Lemma 5.7, $\vdash _\textbf{L}\lnot _j\lnot _kA\Rightarrow A\rightarrow _k(A\leftarrow _kA)$. Thus,

It contradicts the fact ($\divideontimes$). Hence, $[A]\supset _k([A]\subset _k[A])\in [\Sigma ]$.

Assume that $[A]\supset _k([A]\subset _k[A])\in [\Sigma ]$, while $-_j-_k[A]\not \in [\Sigma ]$. Hence, $\vdash _\textbf{L}\lnot _j\lnot _kA,\Omega _i\Rightarrow$ as well as $\vdash _\textbf{L}\Omega _l\Rightarrow A\rightarrow _k(A\leftarrow _kA)$, for some i and l. Recall that, by Lemma 5.7, $\vdash _\textbf{L} A\rightarrow _k(A\leftarrow _kA)\Rightarrow \lnot _j\lnot _kA$. Thus,

It contradicts the fact ($\divideontimes$). Hence, ${-_j-_k}[A]\in [\Sigma ]$. Thus, $[\Sigma ]$ satisfies condition ($-_j-_k$).

Similarly, one can show that $[\Sigma ]$ satisfies condition ($-_k-_j$).

Let us show that $[\Sigma ]$ satisfies condition ($\supset _j$). Assume that $C\not \in [\Sigma ]$ or $[A]\supset _j[B]\in [\Sigma ]$. Suppose that $[A]\cap _j[C]\in [\Sigma ]$, while $[B]\not \in [\Sigma ]$. Since we already know that $[\Sigma ]$ is a multifilter, $[A]\in [\Sigma ]$ and $[C]\in [\Sigma ]$. We have $\vdash _\textbf{L}\Omega _i,B\Rightarrow$ as well as $\vdash _\textbf{L}\Omega _l\Rightarrow A$ and $\vdash _\textbf{L}\Omega _m \Rightarrow C$, for some i, l, and m. Suppose that $C\not \in [\Sigma ]$. Then $\vdash _\textbf{L}\Omega _o,C \Rightarrow$, for some o. Thus,

It contradicts the fact ($\divideontimes$). Assume that $[A]\supset _j[B]\in [\Sigma ]$. Then $\vdash _\textbf{L}\Omega _t \Rightarrow A\rightarrow _jB$, for some t. Thus,

It contradicts the fact ($\divideontimes$). Hence, $[B]\in [\Sigma ]$. Consequently, $[A]\cap _j[C]\in [\Sigma ]$ implies $[B]\in [\Sigma ]$. Therefore, if $C\not \in [\Sigma ]$ or $[A]\supset _j[B]\in [\Sigma ]$, then $[A]\cap _j[C]\in [\Sigma ]$ implies $[B]\in [\Sigma ]$.

Assume that $[A]\cap _j[C]\not \in [\Sigma ]$ or $[B]\in [\Sigma ]$. Since $[\Sigma ]$ is a multifilter, $[A]\not \in [\Sigma ]$ or $[C]\not \in [\Sigma ]$. Suppose that $[C]\in [\Sigma ]$, while $[A]\supset _j[B]\not \in [\Sigma ]$. We have $\vdash _\textbf{L}\Omega _l\Rightarrow C$ and $\vdash _\textbf{L}\Omega _m,A\rightarrow _jB\Rightarrow$, for some l, and m. Suppose that $[A]\not \in [\Sigma ]$. Then $\vdash _\textbf{L}\Omega _i,A\Rightarrow$, for some i. Thus,

It contradicts the fact ($\divideontimes$). Suppose that $[C]\not \in [\Sigma ]$. Then $\vdash _\textbf{L}\Omega _t,C\Rightarrow$, for some t. Thus,

Consequently, $[A]\supset _j[B]\in [\Sigma ]$. Hence, if $[A]\cap _j[C]\not \in [\Sigma ]$ or $[B]\in [\Sigma ]$, then $[C]\in [\Sigma ]$ implies $[A]\supset _j[B]\in [\Sigma ]$. Therefore, $[\Sigma ]$ satisfies condition ($\supset _j$).

The cases regarding conditions ($\subset _j$), ($-_k{\supset _j}$), ($-_k{\subset _j}$), ($-_j{\supset _j^c}$), and ($-_j{\subset _j^c}$) are considered similarly. Therefore, $[\Sigma ]$ is a connexive multifilter. $\square$

Lemma 5.10

(Lindenbaum lemma for $\textbf{BML}_n$) Let $\textbf{L}$ be $\textbf{BML}_n$. For every pair of finite sets of formulas $\Gamma$ and $\Delta$, it holds that $\not \vdash _\textbf{L}\Gamma \Rightarrow \Delta$ implies that there is a bi-intuitionistic multifilter $\, \mathcal {F}_n^\textbf{L}$ on the Lindenbaum-Tarski algebra $\mathcal {M}_n^\textbf{L}$ and $[C]\in \mathcal {F}_n^\textbf{L}$, for each $C\in \Gamma$, while $[D]\not \in \mathcal {F}_n^\textbf{L}$, for each $D\in \Delta$.

Proof

Similarly to Lemma 5.9. $\square$

Lemma 5.11

$\langle \mathcal {M}_n^\textbf{CML},\mathcal {F}_n^\textbf{CML}\rangle$, where $\mathcal {F}_n^\textbf{CML}$ is a connexive multifilter constructed in Lemma 5.9, is a connexive logical multilattice.

Proof

Due to Lemmas 5.7 and 5.9 operations $-_j$, $\cap _j$, $\cup _j$, $\supset _j$, and $\subset _j$ on $[\mathscr {F}]$ satisfy the conditions listed in Definition 4.2. To be more exact, the correspondence between the properties required by the definition and the provable sequents from the lemmas is as follows: the condition that $\langle \cap _1,\cup _1\rangle ,\ldots ,\langle \cap _n,\cup _n\rangle$ are pairs of lattice meet and join operations satisfying distributivity is justified by the provability of (1)–(9) (Lemma 5.7). The conditions regarding both the behaviour of the connective and the properties of a multifilter (that is (filter), (prime), (DM1)–(DM4), (per2), ($\supset _j$), ($\subset _j$), ($-_k{\supset _j}$), ($-_k{\subset _j}$), ($-_j-_k$), ($-_k-_j$), ($-_j{\supset _j^c}$), and ($-_j{\subset _j^c}$)) are justified by Lemma 5.9 and in the case of ($-_j-_k$) and ($-_k-_j$) by Lemma 5.7 as well. $\square$

Lemma 5.12

$\langle \mathcal {M}_n^\textbf{BML},\mathcal {F}_n^\textbf{BML}\rangle$, where $\mathcal {F}_n^\textbf{BML}$ is a bi-intuitionistic multifilter constructed in Lemma 5.10, is a bi-intuitionistic logical multilattice.

Proof

Follows from Lemmas 5.7 and 5.10. $\square$

Theorem 5.13

(Soundness and completeness) Let $\textbf{L}\in \{\textbf{BML}_n,\textbf{CML}_n \}$. For every pair of finite sets of formulas $\Gamma$ and $\Delta$, it holds that $\Gamma \models _\textbf{L}\Delta$ iff $\models _\textbf{L}\Gamma \Rightarrow \Delta$ iff $\vdash _\textbf{L}\Gamma \Rightarrow \Delta$.

Proof

The equivalence $\Gamma \models _\textbf{L}\Delta$ iff $\models _\textbf{L}\Gamma \Rightarrow \Delta$ holds due to Definition 4.6. As for the equivalence $\models _\textbf{L}\Gamma \Rightarrow \Delta$ iff $\vdash _\textbf{L}\Gamma \Rightarrow \Delta$, its soundness part is justified by Theorem 5.3. As for the completeness part, assume that $\textbf{L}\not \vdash _\textbf{L}\Gamma \Rightarrow \Delta$. By Lemma 5.9, there is a connexive (resp. bi-intuitionistic) multifilter $\mathcal {F}_n$ on $\mathcal {M}_n^\textbf{L}$ such that $[C]\in \mathcal {F}_n$, for all $C\in \Gamma$, and $[D]\not \in \mathcal {F}_n$, for all $D\in \Delta$. By Lemma 5.11, if $\textbf{L}=\textbf{CML}_n$, then $\langle \mathcal {M}_n^\textbf{L},\mathcal {F}_n^\textbf{L}\rangle$ is a connexive logical multilattice. By Lemma 5.12, if $\textbf{L}=\textbf{BML}_n$, then $\langle \mathcal {M}_n^\textbf{L},\mathcal {F}_n^\textbf{L}\rangle$ is a bi-intuitionistic logical multilattice. By Lemma 5.8, there is a canonic valuation ${\widetilde{v}}$ such that ${\widetilde{v}}(C)\in \mathcal {F}_n$, for all $C\in \Gamma$, and ${\widetilde{v}}(D)\not \in \mathcal {F}_n$, for all $D\in \Delta$, i.e. $\textbf{L}\not \models \Gamma \Rightarrow \Delta$. $\square$

Theorem 5.14

(Soundness and completeness) Let $\textbf{L}\in \{\textbf{ML}_n,\textbf{MLL}_n \}$. For every pair of finite sets of formulas $\Gamma$ and $\Delta$, it holds that $\Gamma \models _\textbf{L}^A\Delta$ iff $\models _\textbf{L}\Gamma \Rightarrow \Delta$ iff $\vdash _\textbf{L}\Gamma \Rightarrow \Delta$.

Proof

Similarly to Theorem 5.13. $\square$

6 Conclusion

We offered the algebraic semantics for connexive and bi-intuitionistic multilattice logics previously being formulated only with the help of sequent calculi and Kripke semantics. As for topics for future research, we leave an investigation of modal extensions of $\textbf{CML}_n$ and $\textbf{BML}_n$ by Tarski, Kuratowski, and Halmos closure and interior operators (see (Grigoriev & Petrukhin, 2021) for a systematic study of the extensions of $\textbf{ML}_n$ by these operators). Yet another topic is the study of congruent and monotonic modal multilattice logics on the basis of $\textbf{CML}_n$ and $\textbf{BML}_n$ (such modal logics on the basis of $\textbf{ML}_n$ and $\textbf{MLL}_n$ were explored in Grigoriev and Petrukhin (2022)).