Epistemic Łukasiewicz logic of partial knowledge

Abstract

We offer a new logic, called Epistemic Łukasiewicz logic of partial knowledge that is represented as multimodal epistemic Łukasiewicz logic $K\text{\L} _\text {P}(n)$ with n knowledge operators $\Box _i$ $(1 \le i \le n)$ interpreted in a non-archimedean monadic MV-algebra. We choose knowledge operators, which can be estimated by some grading (different kinds of knowledge): absolute knowledge or partial knowledge. We consider a very special type of partial knowledge. Actually, we take infinitesimal elements (the radical) of perfect MV-algebras as a range of this estimation. The choice of infinitesimal elements seems suitable for actual situations like a measure of partial information.

1 Introduction

In many studies of distributed systems, a multiagent model is used. An agent can be a processor, sensor or finite state machine, interconnected by a communication network with other ‘agents’. Typically each agent has a local state that is a function of its initial state, the messages received from other agents, observations of the external environment and possible internal actions. It has become customary when using formal models of distributed systems to use modal epistemic logic as one of the tools for studying the knowledge of such systems. We recall that a similar link between formal systems and distributed system can be encountered, for example, also between Łukasiewicz logic and artificial neural networks, see (Di Nola et al. 2016; Di Nola and Vitale 2020) or Łukasiewicz logic and decision theory, see (Vitale 2020).

The basic logic for handling a system with n-agents is known as $S5_n$, introduced in Porter (2003). The logic $S5_n$ is obtained from ordinary classical propositional logic by adding ‘knowledge operators’ - $\Box _i$. It models a community of ideal knowledge agents who have the properties of veridical knowledge (everything they know is true), positive introspection (they know what they know) and negative introspection (they know what they do not know).

Very often the agent is encountered with the uncertainty of information about some problem that needs a solution, and this solution depends on the degree of uncertainty. In other words, we need an algorithm based on partial (incomplete, not exact) information. But, the difference between incompletely and partially is that: incompletely in an incomplete manner; partially is a partial degree, i.e., related to only a part, not general or complete. In order to model a distributed system that is sensitive with respect to infinitesimal variations of information in a communication network, we use infinitesimal elements of a special type of MV-algebras—perfect MV-algebras that are non-archimedean MV-algebras. Summarizing saying above, we choice MV-algebra giving a possibility to interpret the valuation of Lukasiewicz sentences into some degree of uncertainty belonging to some MV-algebras.

Taking as a motivation mentioned above facts, in distinct from the classical case we suggest a new logics—multimodal epistemic Łukasiewicz logic $K\text{\L} _\text {P}(n)$ with n knowledge operators $\Box _i$ $(1 \le i \le n)$ that are interpreted in a non-archimedean monadic MV-algebra. Epistemic logic is a modal logic extended by some additional axioms, where the modal operator is interpreted as “knowledge”, which can be estimated by some grading (different kinds of knowledge): absolute knowledge or partial knowledge. We consider a very special type of partial knowledge. Actually, we take infinitesimal elements, the radical, of perfect MV-algebras as a range of this estimation. The choice of infinitesimal elements seems suitable for actual situations like a measure of partial information.

The logic $K\text{\L} _\text {P}(n)$ is obtained extending the language of the logic $\text{\L} _\text {P}$, the algebraic models of which are perfect MV-algebras, by adding n ‘knowledge operators’ with corresponding axioms. The knowledge operators model a community of ideal knowledge agents who have the properties of veridical knowledge (everything they know is true), fuzzy knowledge (everything they know is quasi-true), positive introspection (they know what they know) and negative introspection (they know what they do not know) and so on. The knowledge operators permit the following interpretation:

• $\Box _i\alpha$ - “i knows proposition $\alpha$”;

• $\diamondsuit _i\alpha$ - “i does not know that proposition $\alpha$ is false”.

where i belongs to s set A of agents.

There are MV-algebras which are not semisimple, i.e., the intersection of their maximal ideals (the radical of A) is different from $\{0\}$. Nonzero elements in the radical of A are called infinitesimals. It is worth stressing that the existence of infinitesimals in some MV-algebras is due to the remarkable difference of behavior between Boolean algebras and MV-algebras.

Perfect MV-algebras, that were introduced by B. Belluce, A. Di Nola, and A. Lettieri in Belluse et al. (1993), are those MV-algebras generated by their infinitesimal elements or, equivalently, generated by their radical (Belluce et al. 2007). They generate the smallest non-locally finite subvariety of the variety MV of all MV-algebras. An important example of a perfect MV-algebra is the subalgebra S of the Lindenbaum algebra $\text{\L}$ of first-order Łukasiewicz logic generated by the classes of formulas which are valid when interpreted in [0, 1] but non-provable. Hence perfect MV-algebras are directly connected with the very important phenomenon of incompleteness in Łukasiewicz first-order logic (see (Belluce and Chang 1963; Scarpellini 1962). Infinitesimal elements of perfect MV-algebra spring to mind the idea of quasi-false and quasi-truth. Following this idea, A. Di Nola, R. Grigolia, and E. Turunen have been published the monograph Fuzzy Logic of Quasi-Truth: An Algebraic Treatment (Di Nola et al. 2016).

For our main aim, as an algebraic model of multi-monadic Łukasiewicz logic $K{\text{\L} }_\text {P}(n)$ we use multi-monadic MV-algebras, that is a generalization of monadic MV-algebras, which are algebraic models of monadic Łukasiewicz logic that is a special kind of modal logic. Monadic MV-algebras (monadic Chang algebras by Rutledge’s terminology) were introduced and studied by Rutledge in Rutledge (1959), using a functional approach, as an algebraic model for the predicate calculus of Łukasiewicz infinite-valued logic, in which only a single individual variable occurs. Rutledge followed the study of monadic Boolean algebras investigated by Halmos (1962). Extending the signature of MV-algebra by unary monadic (modal) operation, A. Di Nola and R. Grigolia in Di Nola and Grigolia (2004) define and study monadic MV-algebras as pairs of MV-algebras one of which there is a special case of relatively complete subalgebra named m-relatively complete. An m-relatively complete subalgebra determines a unique monadic operator. A necessary and sufficient condition is given for a subalgebra to be m-relatively complete.

We also mention the papers similar to this paper—(Hansoul and Teheux 2006) concerning the modal Łukasiewicz logic, (Di Nola et al. 2020; Harel et al. 2000; Parikh 1978; Pratt 1980, 1991; Segerberg 1977) concerning the multimodal case.

The paper is organized in the following way. In Sect. 1, an introduction and some preliminaries and motivation are given. Let us note that Sects. 2, 3, 4 represent preliminaries for the concepts that are necessary for the main results. Section 2 represents a definition of MV-algebras and perfect MV-algebras. Section 3 represents a definition of a variety of monadic MV-algebras and its subvariety of monadic perfect MV-algebras. Section 4 represents multi-monadic perfect $K{\text{\L} }_\text {P}(n)$-algebra that are algebraic models of multi-monadic Łukasiewicz logic $K{\text{\L} }_\text {P}(n)$. In Sect. 5, we introduce multi-monadic Łukasiewicz logic and its algebraic counterpart. It is proved deduction theorem and completeness theorem. Conclusion is given in Sect. 6.

2 MV-algebras and perfect MV-algebras

In this section, we give a definition of MV-algebras and perfect MV-algebras, which are algebraic models of Łukasiewicz logic ${\text{\L} }$ and the logic ${\text{\L} }_\text {P}$, respectively.

Infinite-valued logic has been introduced by Łukasiewicz (1920); Łukasiewicz and Tarski (1930). Taking into account Łukasiewicz’s idea on infinite-valued logic, afterward C. C. Chang has developed its algebraic counterpart, i.e., the variety of MV-algebras (Chang 1958), and proved the completeness theorem for Łukasiewicz logic with respect to the variety MV of MV-algebras.

An MV-algebra $A=(A, \oplus , \odot , \lnot ,0,1)$ where $(A,\oplus , 0)$ is an abelian monoid, and the following identities hold: $x \oplus 1 = 1$, $\lnot \lnot x=x$, $\lnot 0 = 1$, $x \odot y = \lnot ( \lnot x \oplus \lnot y)$, $\lnot (\lnot x\oplus y) \oplus y = \lnot ( \lnot y \oplus x) \oplus x$ (see (Chang 1958)). We shall write ab for $a \odot b$ and $a^{n}$ for $\underbrace{a \odot \dots \odot a}_{n\;\text {times}}$, for given $a,b \in A$. Every MV-algebra has an underlying ordered structure defined by

$$\begin{aligned} x \le y \,\, \text{ iff }\,\, \lnot x \oplus y=1. \end{aligned}$$

$(A,\le ,0,1)$ is a bounded distributive lattice. Moreover, the following property holds in any MV-algebra:

$$\begin{aligned} x\odot y \le x\wedge y \le x\vee y\le x\oplus y. \end{aligned}$$

The unit interval of real numbers [0, 1] endowed with the following operations: $x \oplus y = \min (1,x+y), x \odot y = \max (0, x + y-1), \lnot x = 1 - x$, becomes an MV-algebra that is named standard MV-algebra. Note that standard MV-algebra is an archimedean chain. It is well known that the MV-algebra $S=([0,1], \oplus , \odot , \lnot , 0,1)$ generate the variety ${\mathbf {MV}}$ of all MV-algebras, i.e., ${\mathcal {V}}(S) = {\mathbf {MV}}$.

MV-algebras are algebraic models of Łukasiewicz logic Ł (Chang 1958). Perfect MV-algebras are remarkable examples of MV-algebras because they are non-archimedean in a very strong way. The class of perfect MV-algebras is a full subcategory of the category of MV-algebras. In general, there are MV-algebras which are not semisimple. Roughly speaking, we can say that a non-semisimple MV-algebra A has a nonzero radical. We call a nonzero element from the radical of A an infinitesimal. The first example of non-simple MV-chain was given by Chang in Chang (1958), where the MV-algebra C is described.

Chang’s MV-algebra C (Chang 1958), which is our main interest, is defined on the set

$$\begin{aligned} C = \{0, c, ... , nc, ... , 1 - nc, ... , 1 - c, 1\}, \end{aligned}$$

by the following operations (consider $0 = 0c$): $x \oplus y =$

$$\begin{aligned} x \oplus y = {\left\{ \begin{array}{ll} (\textit{m} + \textit{n})\textit{c} \quad \text {if}\ \textit{x} = \textit{nc} \ \text {and} \ \textit{y} = \textit{mc}\\ 1- (\textit{m} - \textit{n})\textit{c} \quad \text {if}\ \textit{x} = 1 - \textit{nc} \ \text {and} \\ \textit{y} = \textit{mc}\ \text {and} \ 0< \textit{n}< \textit{m}\\ 1- (\textit{n} - \textit{m})\textit{c} \quad \text {if}\ \textit{x} = \textit{nc}\ \text {and}\\ \textit{y} = 1 - \textit{mc}\ \text {and}\ 0< \textit{m} < \textit{n}\\ 1\quad \text {otherwise} \end{array}\right. } \end{aligned}$$

$$\begin{aligned} \lnot x = {\left\{ \begin{array}{ll} 1 - nc\quad \text {if}\ x = nc\\ nc\quad \text {if}\ x = 1 -nc \end{array}\right. } \end{aligned}$$

The algebra C has remarkable properties:

• (1) C is generated by its radical,

• (2) $C=\text {Rad}(C)\cup \lnot \text {Rad}(C)$,

• (3) $C/\text {Rad}(C)\cong \{0,1\}$,

• (4) $2x = 1$ for every $x\in \lnot \text {Rad}(C)$,

• (5) $x^2 = 0$ for every $x\in \text {Rad}(C)$.

Hence C is just made by infinitesimal elements and co-infinitesimal elements. Let us take note that if we take C as truth values of logical formulas, then the values from $\lnot \text {Rad}(C) - \{1\}$ are considered as quasi-truth values. In other words, if the valuation $v(\alpha ) \in \lnot \text {Rad}(C) - \{1\}$, then $v(\alpha \underline{\vee } \alpha ) = 1$.

We say that an MV-algebra A is perfect if for each element $x\in A$, $\text {ord}(x)<\infty$ iff $\text {ord}(\lnot x)=\infty$, where the order of an element x, in symbols $\text {ord}(x)$, is the least integer m such that $mx = 1$; if no such integer m exists, then $\text {ord}(x) = \infty$.

The unit interval of real numbers [0, 1] endowed with the following operations: $x \oplus y = \min (1,x+y), x \odot y = \max (0, x + y-1), \lnot x = 1 - x$, becomes an MV-algebra. It is well known that the MV-algebra $S=([0,1], \oplus , \odot , \lnot , 0,1)$ generate the variety ${\mathbf {MV}}$ of all MV-algebras, i. e. ${\mathcal {V}}(S) = {\mathbf {MV}}$.

3 Monadic perfect MV-algebras

In this section, we define a variety of monadic MV-algebras and its subvariety of monadic perfect MV-algebras which are algebraic models of the logic ${\text{\L} }_\text {P}$.

The propositional calculi, which have been described by Łukasiewicz and Tarski in Łukasiewicz and Tarski (1930), are extended to the corresponding predicate calculi. The predicate Łukasiewicz (infinitely valued) logic $Q\text{\L}$ is defined in the following standard way. The existential (universal) quantifier is interpreted as supremum (infimum) in a complete MV-algebra. Then the valid formulas of predicate calculus are defined as all formulas having value 1 for any assignment. The functional description of the predicate calculus is given by Rutledge in Rutledge (1959). Scarpellini in Scarpellini (1962) has proved that the set of valid formulas is not recursively enumerable. Monadic MV-algebras were introduced and studied by Rutledge in Rutledge (1959) as an algebraic model for the predicate calculus $Q\text{\L}$ of Łukasiewicz infinite-valued logic, in which only a single individual variable occurs. Rutledge followed the study of monadic Boolean algebras made by P.R. Halmos. In view of the incompleteness of the predicate calculus, the result of Rutledge in Rutledge (1959), showing the completeness of the monadic predicate calculus, has been of great interest.

Let $\text{\L}$ denote a first-order language based on $\cdot , +, \rightarrow , \lnot , \exists$ and let $\text{\L} _\text {m}$ denote a propositional language based on $\cdot , +, \rightarrow , \lnot , \exists$. Let $Form(\text{\L} )$ and $Form(\text{\L} _\text {m})$ be the set of all formulas of $\text{\L}$ and $\text{\L} _\text {m}$, respectively. We fix a variable x in $\text{\L}$, associate with each propositional letter p in $\text{\L} _\text {m}$ a unique monadic predicate $p^{*}(x)$ in $\text{\L}$ and define by induction a translation $\Psi : Form(\text{\L} _\text {m})\rightarrow Form(\text{\L} )$ by putting:

• $\Psi (p) = p^{*}(x)$ if p is propositional variable,

• $\Psi (\alpha \circ \beta ) = \Psi (\alpha ) \circ \Psi (\beta )$, where $\circ = \cdot , +, \rightarrow$,

• $\Psi (\exists \alpha ) = \exists x \Psi (\alpha )$.

Through this translation $\Psi$, we can identify the formulas of $\text{\L} _\text {m}$ with monadic formulas of $\text{\L}$ containing the variable x.

An algebra $A=(A, \oplus , \odot , \lnot , \exists , 0, 1)$ is said to be a monadic MV-algebra (Di Nola and Grigolia 2004) (MMV-algebra for short) if $A=(A, \oplus , \odot , \lnot , 0, 1)$ is an MV-algebra and in addition $\exists$ satisfies the following identities:

• E1. $x \le \exists x$,

• E2. $\exists (x \vee y) = \exists x \vee \exists y$,

• E3. $\exists (\lnot (\exists x)) = \lnot (\exists x)$,

• E4. $\exists (\exists x \oplus \exists y) = \exists x \oplus \exists y$,

• E5. $\exists (x \odot x) = \exists x \odot \exists x$,

• E6. $\exists (x \oplus x) = \exists x \oplus \exists x$.

• E7. $\exists (x \odot \exists y) = \exists x \odot \exists y$.

Sometimes we shall denote a monadic MV-algebra $A=(A, \oplus , \odot , \lnot , \exists , 0, 1)$ by $(A, \exists )$, for brevity. We can define a unary operation $\forall x = \lnot (\exists \lnot x)$ corresponding to the universal quantifier.

Let $A_1$ and $A_2$ be any MMV-algebras. A mapping $h : A_1 \rightarrow A_2$ is an MMV-homomorphism if h is an MV-homomorphism and for every $x \in A_1$ $h(\exists x)=\exists h(x)$. Denote by ${\mathbf {MMV}}$ the variety and the category of MMV-algebras and MMV-homomorphisms.

As it is well known, MV-algebras form a category that is equivalent to the category of abelian lattice ordered groups ($\ell$-groups, for short) with strong unit (Mundici 1986). Let us denote by $\Gamma$ the functor implementing this equivalence. If G is an $\ell$-group, then for any element $u\in G$, $u > 0$ we let $[0,u] = \{x\in G: 0 \le x \le u\}$ and for each $x,y \in [0,u]$$x\oplus y = u \wedge (x + y)$ and $\lnot x = u - x$.

Let us introduce some notations: let $C_0 = \Gamma (Z,1)$, $C_1 = C \cong \Gamma (Z\times _{lex} Z, (1,0))$ with generator $(0,1) = c_1 (= c)$, where $\times _{lex}$ is the lexicographic product. Let us denote $\text {Rad}(A) \cup \lnot \text {Rad} (A)$ through $R^*(A)$, where $\lnot \text {Rad} (A) = \{\lnot x: x\in \text {Rad} (A)\}$.

Let $(A, \oplus , \odot , \lnot , \exists , 0, 1)$ be a monadic MV-algebra. Let $\exists A= \{x \in A : x= \exists x\}$. By Di Nola and Grigolia (2004), $(\exists A,\oplus , \odot , \lnot , 0, 1)$ is an MV-subalgebra of the MV-algebra $(A, \oplus , \odot , \lnot , 0, 1)$.

A subalgebra $A_0$ of an MV-algebra A is said to be relatively complete if for every $a\in A$ the set $\{b\in A_0 : a\le b\}$ has a least element.

Let $(A, \oplus , \odot , \lnot , \exists , 0, 1)$ be a monadic MV-algebra. By Rutledge (1959), the MV-algebra $\exists A$ is a relatively complete subalgebra of the MV-algebra $(A, \oplus , \odot , \lnot , 0, 1)$, and $\exists a =inf\{b\in \exists A : a\le b\}$.

A subalgebra $A_0$ of an MV-algebra A is said to be m-relatively complete (Di Nola and Grigolia 2004), if $A_0$ is relatively complete and two additional conditions hold:

• $(\#)$ $(\forall a\in A)(\forall x \in A_0)(\exists v \in A_0)(x\ge a \odot a \Rightarrow v\ge a \& v \odot v\le x)$,

• $(\# \#)$ $(\forall a\in A)(\forall x \in A_0)(\exists v \in A_0)(x\ge a \oplus a \Rightarrow v\ge a \& v \oplus v\le x)$.

Notice that two-elements Boolean subalgebra of the standard MV-algebra $S=([0,1], \oplus , \odot , \lnot , 0,1)$ is relatively complete, but not m-relatively complete.

Proposition 1

(Di Nola and Grigolia 2004). Let A be monadic MV-algebra. Then $\exists A$ is m-relatively complete subalgebra of the monadic MV-algebra A.

Proposition 2

Let A be an MV-algebra.

• (1) Di Nola and Grigolia (2004); Rutledge (1959). If $A_0$ is m-relatively complete totally ordered MV-subalgebra of the MV-algebra A, then $A_0$ is a maximal totally ordered subalgebra of A.

• (2) Di Nola and Grigolia (2004); Rutledge (1959). If $(A, \exists )$ is a totally ordered monadic MV-algebra, then $A=\exists A(=A_0)$.

• (3) Di Nola and Grigolia (2004); Rutledge (1959). $(A, \exists )$ is a subdirectly irreducible monadic MV-algebra if and only if $\exists A(=A_0)$ is totally ordered.

• (4) Rutledge (1959). Any monadic MV-algebra $(A, \exists )$ is isomorphic to a subdirect product of monadic MV-algebras $(A_i, \exists )$ such that $\exists A_i$ is totally ordered.

A monadic MV-algebra $A=(A, \oplus , \odot , \lnot , \exists , 0, 1)$ is said to be MMV(C)-algebra if A in addition satisfies the identity (Di Nola and Lettieri 1999):

$$\begin{aligned} (Perf) \ \ 2(x^2) = (2x)^{2}, \end{aligned}$$

that is ${\mathbf {MMV(C)}} = {\textbf {MMV}}+(Perf)$.

Denote the variety of perfect MV-algebras by ${\mathbf {MMV(C)}}$, which is a subvariety of the variety ${\mathbf {MMV}}$.

Let

$$\begin{aligned} \text {Alt}_2^C =\forall (2x_1^2) \vee \forall (2x_1^2 \rightarrow 2x_2^2) \vee \forall (2x_1^2 \wedge 2x_2^2 \rightarrow 2x_{3}^2). \end{aligned}$$

Let $\mathbf {MMV(C)}^1$ (Di Nola et al. 2018) be the subvariety of $\mathbf {MMV(C)}$ defined by the identity $\text {Alt}_2^C = 1$.

Theorem 3

(Di Nola et al. 2018). The identity $\text {Alt}_2^C = 1$ is true in finitely generated subdirectly irreducible algebra $A\in \mathbf {MMV(C)}$ if and only if A contains as a maximal homomorphic image the monadic Boolean algebra $(2^2, \exists )$.

Let us consider the identity

$$\begin{aligned} (K\text{\L} _\text {P}) \ \ (\exists x)^2 \wedge (\exists \lnot x)^2 = 0. \end{aligned}$$

It holds

Theorem 4

(Di Nola et al. 2018). The identity $(\exists x)^2 \wedge (\exists \lnot x)^2 = 0$ is satisfied in the subdirectly irreducible MMV(C) algebra $(A, \exists )$ if and only if the MV-algebra reduct of that is perfect MV-algebra.

$\text {MMV}(C)^1$-algebra A is said to be $\text {MMV}(C)_K^1$-algebra if it satisfies the identity $(K\text{\L} _\text {P})$. Denote by $\mathbf {MMV(C)_K^1}$ the variety of all $\text {MMV}(C)_K^1$-algebras.

The main interest for us is the $\text {MMV}(C)_K^1$-algebra $(K, \oplus , \odot , \lnot , \exists , 0, 1) = R^{*} (C^2) = \text {Rad} (C^2) \cup \lnot \text {Rad} (C^2)$, where for any $(x_1,x_2) \in K$ we have $\exists (x_1,x_2) = (max (x_1,x_2), max (x_1,x_2))$ and $\forall (x_1,x_2) = (min (x_1,x_2), min (x_1,x_2))$. In other words, $\exists K = \{(x,x): x\in C\}$, i. e. $\exists K$ is a diagonal of $C^2$. Notice, that $\exists K$ is a subalgebra of $K \ (= R^{*}(C^2))$, where $\exists x =x$ (and $\forall x =x$ as well) for every $x\in \exists K$. So, the MV-reduct $\exists K$ is isomorphic to C.

4 Multi-monadic perfect $K\text{\L} _\text {P}(n)$-algebra

In this section, we define a variety of multi-monadic perfect algebras that are algebraic models of multi-monadic epistemic Łukasiewicz logic $K{\text{\L} }_\text {P}(n)$.

An algebra $A=(A, \oplus , \odot , \lnot , \exists _1, ... , \exists _n, 0, 1)$ is said to be $K\text{\L} _\text {P}(n)$-algebra if it satisfies the following identities:

• KE1. $x \le \exists _i x$, $0 < i \le n$,

• KE2. $\exists _i (x \vee y) = \exists _i x \vee \exists _i y$, $0 < i \le n$,

• KE3. $\exists _i (\lnot (\exists _i x)) = \lnot (\exists _i x)$, $0 < i \le n$,

• KE4. $\exists _i (\exists _i x \oplus \exists _i y) = \exists _i x \oplus \exists _i y$, $0 < i \le n$,

• KE5. $\exists _i (x \odot x) = \exists _i x \odot \exists _i x$, $0 < i \le n$,

• KE6. $\exists _i (x \oplus x) = \exists _i x \oplus \exists _i x$, $0 < i \le n$,

• KE7. $\exists _i (x \odot \exists _i y) = \exists _i x \odot \exists _i y$, $0 < i \le n$,

• KE8. $\exists _i x \le \forall _i \exists _i x$, $0 < i \le n$,

• KE9. $2(x^2) = (2x)^{2}$,

• KE10. $\forall _i (2x_1^2) \vee \forall _i (2x_1^2 \rightarrow 2x_2^2) \vee \forall _i (2x_1^2 \wedge 2x_2^2 \rightarrow 2x_{3}^2) = 1$, $0 < i \le n$,

• KE11. $(\exists x)^2 \wedge (\exists \lnot x)^2 = 0$.

The algebra $A=(A, \oplus , \odot , \lnot , \exists _1, ... , \exists _n, 0, 1)$ is said to be K(n)-algebra if $(A, \oplus , \odot , \lnot , \exists _i, 0, 1)$ is algebra K for $0 < i \le n$, and denote the algebra by K(n).

5 Multi-monadic epistemic Łukasiewicz logic $K\text{\L} _\text {P}(n)$

In this section we introduce and investigate multi-monadic epistemic Łukasievicz $K{\text{\L} }_\text {P}$, and prove deduction theorem and completeness theorem with respect to the multi-monadic perfect MV-algebras.

The formulas of ${\text{\L} }$ukasiewicz logics are built from a countable set of propositional variables $\text {Var} = \{p, q, . . .\}$ using the connectives & (conjunction), $\rightarrow$ (implication) and $\bot$ (falsity truth constant). We introduce the connectives $\wedge , \vee , \leftrightarrow , \lnot , \underline{\vee }$ and $\top$ (the semantics counterpart will be denoted, respectively, by $\wedge , \vee , \text{\L}eftrightarrow , \oplus$ and 1, and $\odot$ for &) as the following abbreviations: $\varphi \wedge \psi = \varphi \& (\varphi \rightarrow \psi ), \ \varphi \vee \psi = (\varphi \rightarrow \psi ) \rightarrow \psi , \ \varphi \leftrightarrow \psi = (\varphi \rightarrow \psi ) \& (\psi \rightarrow \varphi ), \ \lnot \varphi = \varphi \rightarrow \bot , \ \varphi \underline{\vee } \psi = \lnot (\lnot \varphi \& \lnot \psi ), \top = \lnot \bot .$

Infinite-valued ${\text{\L} }$ukasiewicz logic ${\text{\L} }$ is axiomatized by the following axioms schemata:

• ${\text{\L} }1.$ $\varphi \rightarrow (\psi \rightarrow \varphi )$,

• ${\text{\L} }2.$ $(\varphi \rightarrow \psi ) \rightarrow ((\psi \rightarrow \chi ) \rightarrow (\varphi \rightarrow \chi ))$,

• ${\text{\L} }3.$ $((\varphi \rightarrow \psi ) \rightarrow \psi )\rightarrow ((\psi \rightarrow \varphi ) \rightarrow \varphi )$,

• ${\text{\L} }4.$ $(\lnot \varphi \rightarrow \lnot \psi ) \rightarrow (\psi \rightarrow \varphi )$.

The inference rule is Modus Ponens: $\varphi , \varphi \rightarrow \psi / \psi$.

${\text{\L} }$ukasiewicz logic ${\text{\L} }_\text {P}$ is axiomatized by the axioms of ${\text{\L} }$ plus the schema:

$$ \begin{aligned} (\text{\L} _\text {P}) \ \ (\varphi \& \varphi ))\underline{\vee } (\varphi \& \varphi )) \leftrightarrow (\varphi \underline{\vee } \varphi )) \& (\varphi \underline{\vee } \varphi )). \end{aligned}$$

We extend ${\text{\L} }$ukasiewicz logic ${\text{\L} }$ to the multimodal ${\text{\L} }$ukasiewicz logic $K\text{\L} _\text {P}(n)$ by adding n unary knowledge (modal) operators $\Box _i$ and $\diamondsuit _i$ ($i=1, ... , n$) to the language of ${\text{\L} }$.

We suggest the following schemata of axioms for multimodal epistemic logic $K\text{\L} _\text {P}(n)$: to the schemata of axioms of ${\text{\L} }_\text {P} \ ( = \text{\L} +(\text{\L} _\text {P}))$ we add

• 1) $\Box _i\varphi \rightarrow \varphi , \ \ i=1,...,n$,

• 2) $\Box _i \varphi \rightarrow \Box _i \Box _i \varphi , \ \ i=1,...,n$,

• 3) $\Box _i (\varphi \wedge \psi ) \leftrightarrow (\Box _i \varphi \wedge \Box _i \psi ), \ \ i=1,...,n$,

• 4) $\Box _i (\varphi \& \varphi ) \leftrightarrow (\Box _i \varphi \& \Box _i \varphi ), \ \ i=1,...,n$,

• 5) $\Box _i (\varphi \underline{\vee } \varphi ) \leftrightarrow (\Box _i \varphi \underline{\vee } \Box _i \varphi ), \ \ i=1,...,n$,

• 6) $\diamondsuit _i \varphi \rightarrow \Box _i \diamondsuit _i \varphi , \ i=1,...,n$,

• 7) $\Box _i( (\varphi _1 \& \varphi _1))\underline{\vee } (\varphi _1 \& \varphi _1)) \vee \Box _i((\varphi _1 \& \varphi _1))\underline{\vee } (\varphi _1 \& \varphi _1)) \rightarrow (\varphi _2 \& \varphi _2))\underline{\vee } (\varphi _2 \& \varphi _2)) \vee \Box _i ((\varphi _1 \& \varphi _1))\underline{\vee } (\varphi _1 \& \varphi _1)) \wedge (\varphi _2 \& \varphi _2))\underline{\vee } (\varphi _2 \& \varphi _2)) \rightarrow (\varphi _3 \& \varphi _3))\underline{\vee } (\varphi _3 \& \varphi _3))), \ \ i=1,...,n$;

• 8) $\Box _i (\varphi \underline{\vee } \varphi ) \vee \Box _i (\lnot \varphi \underline{\vee } \lnot \varphi ), \ \ i=1,...,n$.

Inference rules: $\varphi , \varphi \rightarrow \psi / \psi$, $\varphi / \Box _i \varphi , \ i=1,...,n$.

Notice that Axiom 7) is a logical analog of algebraic identity KE11.

Recall, that the logic $K\text{\L} _\text {P}(n)$ is obtained from Łukasiewicz propositional logic $\text{\L} _\text {P}$, corresponding to perfect MV-algebras, by adding n ‘knowledge operators’, with corresponding axioms. The knowledge operators model a community of ideal knowledge agents who have the properties of veridical knowledge (everything they know is true), fuzzy knowledge (everything they know is quasi-true), positive introspection (they know what they know) and negative introspection (they know what they do not know) and so on. The knowledge operators permit the following interpretation:

• $\Box _i\alpha$ - “i knows proposition $\alpha$”;

• $\diamondsuit _i\alpha$ - “i does not know that proposition $\alpha$ is false”.

An evaluation is a mapping $e_i: \text {Var} \rightarrow K(n)$ that is extended to the set of all formulas in the following way:

• $e_i(\varphi \& \psi ) = e_i(\varphi ) \odot e_i(\psi )$,

• $e_i(\varphi \rightarrow \psi ) = e_i(\varphi ) \Rightarrow e_i(\psi )$,

• $e_i(\bot ) = 0$,

• $e_i(\Box _i \varphi ) = (max (e_i(\varphi )), max (e_i(\varphi ))) \in K(n), \ \ i=1,...,n$.

Let $e(\varphi ) = \bigwedge _{i=1}^n e_i(\varphi )$. A formula $\varphi$ is said to be valid when it is evaluated to 1 in all evaluations $e_i, \ \ i=1,...,n$. In other words, a formula $\varphi$ is said to be valid when it is evaluated to 1 for any evaluation e.

Semantically we can define the notion of “partial knowledge”: $\Box _i \alpha$ - “agent i has partial knowledge about proposition $\alpha$” if $e_i(\alpha ) \in \lnot \text {Rad}(K(n) - \{1\}$. Taking into account the completeness theorem (that we will prove below) we can say that “agent i has partial knowledge about proposition $\alpha$” if $\nvdash \Box _i \alpha$ and $\vdash \Box _i \alpha \ \underline{\vee } \ \Box _i \alpha$.

Lemma 5

(Hajek 1998). The following formulas are theorems in ${\text{\L} }$:

• (1) $\vdash (\varphi \rightarrow (\psi \rightarrow \chi )) \leftrightarrow ((\varphi \& \psi ) \rightarrow \chi )$;

• (2) $\vdash ((\varphi _1 \rightarrow \psi _1) \& (\varphi _2 \rightarrow \psi _2)) \rightarrow ((\varphi _1 \& \varphi _2) \rightarrow (\psi _1 \& \psi _2))$;

• (3) $\vdash (\varphi \& (\varphi \rightarrow \psi )) \rightarrow \psi$;

• (4) $\vdash \varphi \leftrightarrow \lnot \lnot \varphi$;

• (5) $\vdash \varphi \rightarrow (\psi \rightarrow \varphi )$,

• (6) $\vdash (\varphi \rightarrow \psi ) \rightarrow (\lnot \psi \rightarrow \lnot \varphi )$.

Lemma 6

The following formulas are theorems in ${\text{\L} }$:

• (1) $(\psi ^m \rightarrow \varphi ) \rightarrow (((\lnot \psi )^m \rightarrow \varphi ) \rightarrow ((\psi ^m \wedge (\lnot \psi )^m) \rightarrow \varphi ) \rightarrow \alpha )$, $0< m \in \omega$,

• (2) $\lnot \psi \rightarrow (\psi \rightarrow \chi )$,

• (3) $(\psi \wedge \lnot \psi ) \rightarrow ((\chi \wedge \lnot \chi ) \rightarrow ((\psi \rightarrow \chi ) \wedge \lnot (\psi \rightarrow \chi ))$,

• (4) $\chi \rightarrow ((\psi \wedge \lnot \psi ) \rightarrow ((\psi \rightarrow \chi ) \wedge \lnot (\psi \rightarrow \chi ))$,

• (5) $\lnot \psi \rightarrow ((\chi \wedge \lnot \chi ) \rightarrow ((\psi \rightarrow \chi ) \wedge \lnot (\psi \rightarrow \chi ))$.

Proof

The formulas $(\psi ^m \rightarrow \varphi ) \rightarrow (((\lnot \psi )^m \rightarrow \varphi ) \rightarrow ((\psi ^m \wedge (\lnot \psi )^m) \rightarrow \varphi ) \rightarrow \alpha )$, $0< m \in \omega$, $\lnot \psi \rightarrow (\psi \rightarrow \chi )$, $\lnot \chi \rightarrow ((\psi \wedge \lnot \psi ) \rightarrow ((\psi \rightarrow \chi ) \wedge \lnot (\psi \rightarrow \chi ))$, $\lnot \psi \rightarrow ((\chi \wedge \lnot \chi ) \rightarrow ((\psi \rightarrow \chi ) \wedge \lnot (\psi \rightarrow \chi ))$ and $(\psi \wedge \lnot \psi ) \rightarrow ((\chi \wedge \lnot \chi ) \rightarrow ((\psi \rightarrow \chi ) \wedge \lnot (\psi \rightarrow \chi ))$ are logically true in ${\text{\L} }$. So they are theorems. $\square$

Theorem 7

(Deduction theorem). Let $\Gamma$ be a set of formulas and let $\varphi$ and $\psi$ be formulas. $\Gamma \cup \{\varphi \}\vdash \psi$ iff there is an m such that $\Gamma \vdash (\Box _i \varphi )^m \rightarrow \psi$ (where $(\Box _i\varphi )^m$ is $\Box _i \varphi \& ... \& \Box _i \varphi$, m factors), $i=1, ... , n$.

Proof

If $m > 1$ and $\Gamma \vdash (\Box _i \varphi )^m \rightarrow \psi$, then $\Gamma \vdash (\Box _i\varphi \& (\Box _i \varphi )^{m-1}) \rightarrow \psi$ , $\Gamma \vdash \Box _i \varphi \rightarrow ((\Box _i \varphi )^{m-1} \rightarrow \psi )$ (Lemma 5 (1)), and since we have $\varphi /\Box _i \varphi$, hence $\Gamma \cup \{\varphi \} \vdash ((\Box _i \varphi )^{m-1} \rightarrow \psi )$. Replacing this we get $\Gamma \cup \{\varphi \}\vdash \Box _i \varphi \rightarrow \psi$ and $\Gamma \cup \{\varphi \}\vdash \psi$.

Conversely, let us assume that $\Gamma \cup \{\varphi \}\vdash \psi$ and let $\gamma _1, ... , \gamma _k$ be a derivation of $\psi$ from $\Gamma \cup \{\varphi \}$. We prove by induction that, for each $j=1,...,k$, there is $n_j$ such that $\Gamma \vdash (\Box _i \varphi )^{n_j} \rightarrow \gamma _j$. This is clear for $\gamma _j$ being an axiom or a member of $\Gamma \cup \{\varphi \}$. Let us assume that $\gamma _j$ results by modus ponens from previous members $\gamma _t, \gamma _t \rightarrow \gamma _j$. Then by induction hypothesis we have $\Gamma \vdash (\Box _i \varphi )^n \rightarrow \gamma _t$, $\Gamma \vdash (\Box _i \varphi )^m \rightarrow (\gamma _t \rightarrow \gamma _j)$, thus by Lemma 5 (2), $\Gamma \vdash (\varphi ^n \& \varphi ^m) \rightarrow (\gamma _t \& (\gamma _t \rightarrow \gamma _j))$, thus, according to Lemma 5 (3), $\Gamma \vdash \varphi ^{n+m} \rightarrow \gamma _j$.

Now let $\gamma _j$ a result by necessitation rule from $\gamma _t$, i. e. $\gamma _j = \Box _i \gamma _t$. Then by induction hypothesis we have $\Gamma \vdash (\Box _i \varphi )^m \rightarrow \gamma _t$. But $(\Box _i \varphi )^m \rightarrow \gamma _t \vdash \Box _i((\Box _i \varphi )^m \rightarrow \gamma _t)$. But $\vdash _{{\text{\L} }} \Box _i((\Box _i \varphi )^{m} \rightarrow \gamma _t) \rightarrow (\Box _i(\Box _i \varphi )^{m}) \rightarrow \Box _i \gamma _t)$. By modus ponens we have $\Gamma \cup \{\varphi \}\vdash (\Box _i \varphi )^{m} \rightarrow \gamma _j)$. This completes the proof. $\square$

Theorem 8

(Soundness of $K\text{\L} _\text {P}(n)$) Any theorem of $K\text{\L} _\text {P}(n)$ is valid.

Proof

It is easy to check that any axiom of $K\text{\L} _\text {P}(n)$ is valid and the inference rules preserve validity. Consequently, any theorem of $K\text{\L} _\text {P}(n)$ is valid. $\square$

To show that any valid formula is a theorem of $K\text{\L} _\text {P}(n)$ we need an auxiliary lemma.

Lemma 9

Let $\varphi$ be a formula containing propositional variables $p_1,...,p_k$ and let $e \ (= \bigwedge _{i=1}^n e_i(\varphi ))$ be some evaluation for $p_1,...,p_k$. Then let $p_i'$ be $p_i$ if $e(p_i)= 1$, $p_i'$ be $\lnot p_i$ if $e(p_i)= 0$ and $p_i'$ be $p_i \wedge \lnot p_i$ if $e(p_i)\in K(n) - \{0,1\}$, and let $\varphi '$ be $\varphi$ if $e(\varphi )=1$, $\varphi '$ be $\lnot \varphi$ if $e(\varphi )=0$, $\varphi '$ be $\varphi \wedge \lnot \varphi$ if $e(\varphi )\in K(n) - \{0,1\}$. Then $p_1',...,p_n' \vdash \varphi '$.

Proof

We prove this assertion by induction on the number n of connectives of the formula $\varphi$. If $n=0$, then the formula $\varphi$ is a propositional variable $p_1$ and the assertion of the lemma is came to $p_1 \vdash p_1$, $\lnot p_1 \vdash \lnot p_1$ and $p_1 \wedge \lnot p_1 \vdash p_1 \wedge \lnot p_1$. Let us suppose that the lemma is true for any $j < n$.

Case 1. $\varphi = \lnot \psi$. The number of connectives in $\psi$ less than n.

• (a) Let $e(\psi ) =1$. Then $e(\varphi ) =0$. Thus $\psi ' = \psi$ and $\varphi ' = \lnot \varphi$. By the induction hypotheses we have $p_1',...,p_k' \vdash \psi$. Hence, by Lemma 5 (4) and modus ponens, $p_1',...,p_k' \vdash \lnot \lnot \psi$. But $\lnot \lnot \psi$ is $\varphi '$.

• (b) Let $e(\psi ) =0$. Thus $\psi ' = \lnot \psi$ and $\varphi ' = \varphi$. By the induction hypotheses we have $p_1',...,p_k' \vdash \lnot \psi$. But $\varphi = \lnot \psi$.

• (c) Let $e(\psi ) \in K(n) - \{0,1\}$. Thus $\psi ' = \psi \wedge \lnot \psi$. Then $p_1',...,p_k' \vdash \psi \wedge \lnot \psi$. But $\psi \wedge \lnot \psi$ is equivalent to $\varphi \wedge \lnot \varphi$.

Case 2. Let $\varphi = \psi \rightarrow \chi$. Then the number of connectives in $\varphi$ and $\psi \rightarrow \chi$ less than n. So, by the induction hypotheses $p_1',...,p_k' \vdash \psi '$ and $p_1',...,p_k' \vdash \chi '$.

• (a) Let $e(\psi ) = 0$. Then $e(\varphi ) = 1$, and $\psi ' = \lnot \psi$ and $\varphi ' = \varphi$. Thus, $p_1',...,p_k' \vdash \lnot \psi$ and, by the Lemma 6 (2), and modus ponens, $p_1',...,p_k' \vdash \psi \rightarrow \chi$. But $\varphi = \psi \rightarrow \chi$.

• (b) Let $e(\chi ) = 1$. Then $e(\varphi ) = 1$, and $\chi ' = \chi$, and $\varphi ' = \varphi$. So, we have $p_1',...,p_k' \vdash \chi$, and then, by Lemma 5 (5), $p_1',...,p_k' \vdash \psi \rightarrow \chi$, where $\varphi = \psi \rightarrow \chi$.

• (c) Let $e(\psi ), e(\chi ) \in \lnot \text {Rad} (K(n)) - \{1\}$ ($e(\psi ), e(\chi ) \in \text {Rad} (K(n)) - \{0\}$) and $e(\psi )\le e(\chi )$. Then $e(\varphi ) = 1$. Thus, $p_1',...,p_k' \vdash \psi \wedge \lnot \psi$, $p_1',...,p_k' \vdash \chi \wedge \lnot \chi$ and by Lemma 6 (3), and modus ponens, $p_1',...,p_k' \vdash (\psi \rightarrow \chi ) \wedge \lnot (\psi \rightarrow \chi )$. But $((\psi \rightarrow \chi ) \wedge \lnot (\psi \rightarrow \chi )) \rightarrow (\psi \rightarrow \chi )$. So, by modus ponens, $p_1',...,p_k' \vdash \varphi$.

• (d) Let Let $e(\psi )\in \text {Rad} (K(n)) - \{0\}, e(\chi ) \in \lnot \text {Rad} (K(n)) - \{1\}$. Then $e(\varphi ) = 1$. Thus, $p_1',...,p_k' \vdash \psi \wedge \lnot \psi$, $p_1',...,p_k' \vdash \chi \wedge \lnot \chi$ and by Lemma 6 (5), and modus ponens, $p_1',...,p_k' \vdash (\psi \rightarrow \chi ) \wedge \lnot (\psi \rightarrow \chi )$. But $\varphi ' = (\psi \rightarrow \chi ) \wedge \lnot (\psi \rightarrow \chi )$ and, hence, $((\psi \rightarrow \chi ) \wedge \lnot (\psi \rightarrow \chi )) \rightarrow ((\psi \rightarrow \chi )$. So, $p_1',...,p_k' \vdash \varphi$.

• (e) Let $e(\psi )\in \lnot \text {Rad} (K(n)) - \{1\}, e(\chi ) \in \text {Rad} (K(n)) - \{0\}$. Then $e(\varphi ) \in \text {Rad} (K(n)) - \{0\}$. Thus, $p_1',...,p_k' \vdash \psi \wedge \lnot \psi$, $p_1',...,p_k' \vdash \chi \wedge \lnot \chi$ and by Lemma 6 (5), and modus ponens, $p_1',...,p_k' \vdash (\psi \rightarrow \chi ) \wedge \lnot (\psi \rightarrow \chi )$. But $\varphi ' = (\psi \rightarrow \chi ) \wedge \lnot (\psi \rightarrow \chi )$. So, $p_1',...,p_k' \vdash \varphi$.

Case 3. Let $\varphi = \Box _i \psi$. Then the number of connectives in $\psi$ less than n. So, by the induction hypotheses $p_1',...,p_k' \vdash \psi '$.

• (a) Let $e(\psi ) = 1$. Then $e(\Box _i \psi ) = e(\varphi ) = 1$. Thus, $p_1',...,p_k' \vdash \psi$. But $\psi /\Box _i \psi$, and hence $p_1',...,p_k' \vdash \Box _i \psi$. But $\Box _i \psi = \varphi$.

• (b) Let $e(\psi ) = 0$. Then $e(\Box _i \psi ) = e(\varphi ) = 0$. Thus, $p_1',...,p_k' \vdash \lnot \psi$ and, according to 1) $\vdash \Box _i \psi \rightarrow \psi$ and by Lemma 5 (6) $(\Box _i \psi \rightarrow \psi ) \rightarrow (\lnot \psi \rightarrow \lnot \Box _i \psi )$, and modus ponens, we have $p_1',...,p_k' \vdash \lnot \Box _i \psi$. But $\varphi ' = \lnot \Box _i \psi$.

• (c) Let $e(\psi )\in K(n)) - \{0,1\}$. Then $e(\Box _i \psi ) \in K(n)) - \{0,1\}$. Thus, $p_1',...,p_k' \vdash \psi \wedge \lnot \psi$. But, $\psi \wedge \lnot \psi / \Box _i (\psi \wedge \lnot \psi )$ $\Rightarrow p_1',...,p_k' \vdash \Box _i (\psi \wedge \lnot \psi )$ $\Rightarrow p_1',...,p_k' \vdash \Box _i (\psi ), p_1',...,p_k' \vdash \Box _i (\lnot \psi )$. But, $\vdash \Box _i \lnot \psi \rightarrow \lnot \Box _i \psi$. So, $p_1',...,p_k' \vdash \Box _i \psi \wedge \lnot \Box _i \psi$. $\square$

Lemma 10

$\vdash _{L_P} (\alpha \wedge \lnot \alpha )^n \leftrightarrow \alpha ^n \wedge (\lnot \alpha )^n$.

Proof

$\lnot (\alpha \wedge \lnot \alpha )^n$ and $\lnot (\alpha ^n \wedge (\lnot \alpha )^n)$ are tautologies in $\text{\L} _\text {P}$. So, $\vdash _{L_\text {P}} (\alpha \wedge \lnot \alpha )^n \leftrightarrow \alpha ^n \wedge (\lnot \alpha )^n$. $\square$

Theorem 11

(Completeness of $K\text{\L} _\text {P}(n)$) Any valid formula of $K\text{\L} _\text {P}(n)$ is a theorem of $K\text{\L} _\text {P}(n)$.

Proof

For simplicity we will also write $e(\varphi )$ instead of $\bigwedge _{i=1}^n e_i(\varphi )$. Let us suppose that $\varphi$ is valid and it contains propositional variables $p_1,...,p_k$. So, for every evaluation e, according to Lemma 9, $p_1',...,p_k' \vdash \varphi$, since $e(\varphi ) =1$ and, hence, $\varphi ' = \varphi$. So, when $e(p_k) =1$, according to Lemma 9, $p_1',..., p_{k-1}', p_k \vdash \varphi$, and when $e(p_k) = 0$, according to the same lemma, $p_1',...,p_{k-1}', \lnot p_k \vdash \varphi$, and when $e(p_k) \in K(n)) - \{0,1\}$, we have $p_1', ... , p_{k-1}', p_k \wedge \lnot p_k \vdash \varphi$. From here, according to the deduction theorem, we obtain (1) $p_1', ... , p_{k-1}' \vdash (\Box _i p_k)^{m_1} \rightarrow \varphi$, (2) $p_1', ... , p_{k-1}' \vdash (\Box _i (\lnot p_k))^{m_2} \rightarrow \varphi$ and (3) $p_1', ... , p_{k-1}' \vdash (\Box _i (p_k \wedge \lnot p_k)^{m_3} \rightarrow \varphi$.

Taking into account the axiom of $K\text{\L} _\text {P}(n)$,3), from (3) we have $p_1', ... , p_{k-1}' \vdash (\Box _i p_k \wedge \Box _i \lnot p_k)^{m_3} \rightarrow \varphi$.

Since $\vdash \lnot \Box _i p_k \rightarrow \Box _i \lnot p_k$, from (2) we have $p_1', ... , p_{k-1}' \vdash (\lnot \Box _i p_k)^{m_2} \rightarrow \varphi$. From (3) we have (4) $p_1', ... , p_{k-1}' \vdash (\Box _i p_k \wedge \lnot \Box _i p_k)^{m_3} \rightarrow \varphi$. According to Lemma 10, from (4) we have

$$\begin{aligned} p_1', ... , p_{k-1}' \vdash (\Box _i p_k)^{m_3} \wedge (\lnot \Box _i p_k)^{m_3} \rightarrow \varphi \ \ \ (\star ). \end{aligned}$$

Applying Lemma 6 (1), (1), (2), ($\star$) and modus ponens we obtain $p_1', ... , p_{k-1}' \vdash \varphi$. Applying the same procedures for other variables finally after k steps we come to $\vdash \varphi$. $\square$

6 Conclusion

We have proposed new epistemic logics ($K\text{\L} _\text {P}(n)$) aimed to model the partial knowledge of a distributed finite system of agents acting independently. This situation is represented by the axiomatization of the proposed logic. The semantics of $K\text{\L} _\text {P}(n)$ is given by multimodal perfect MV-algebras. Indeed, an agent partially knows about a proposition (a formula of $K\text{\L} _\text {P}(n)$) with some degree of information. This degree is expressed, via the evaluation map, by an element of its algebraic model.

Among the remarkable algebraic properties of the model is that it is sensitive to infinitesimal variations of the degree of information. By this, we can consider distributed finite systems of agents whose partial knowledge can infinitesimally vary, but it always is infinitesimally close to a crisp (complete) knowledge.