Epistemic Łukasiewicz logic of partial knowledge

We offer a new logic, called Epistemic Łukasiewicz logic of partial knowledge that is represented as multimodal epistemic Łukasiewicz logic KŁP(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\text{\L} _\text {P}(n)$$\end{document} with n knowledge operators □i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Box _i$$\end{document}(1≤i≤n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1 \le i \le n)$$\end{document} 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.


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  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'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-algebrasperfect 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 Ł P (n) with n knowledge operators i (1 ≤ i ≤ 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 Ł P (n) is obtained extending the language of the logic Ł 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: i α -"i knows proposition α"; ♦ i α -"i does not know that proposition α 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 MValgebras.
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 Ł 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 .
For our main aim, as an algebraic model of multi-monadic Łukasiewicz logic K Ł P (n) we use multi-monadic MValgebras, 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.
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 Ł P (n)-algebra that are algebraic models of multi-monadic Łukasiewicz logic K Ł P (n). In Sect. 5, we introduce multimonadic Łukasiewicz logic and its algebraic counterpart. It is proved deduction theorem and completeness theorem. Conclusion is given in Sect. 6.

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 Ł and the logic Ł 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, ⊕, , ¬, 0, 1) where (A, ⊕, 0) is an abelian monoid, and the following identities hold: (Chang 1958)). We shall write ab for a b and a n for a · · · a n times , for given a, b ∈ A. Every MV-algebra has an underlying ordered structure defined by x ≤ y iff ¬x ⊕ y = 1.
(A, ≤, 0, 1) is a bounded distributive lattice. Moreover, the following property holds in any MV-algebra: The unit interval of real numbers [0, 1] endowed with the following operations: becomes an MV-algebra that is named standard MV-algebra. Note that standard MValgebra is an archimedean chain. It is well known that the MV-algebra S = ([0, 1], ⊕, , ¬, 0, 1) generate the variety MV of all MV-algebras, i.e., V(S) = 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 The algebra C has remarkable properties: (1) C is generated by its radical, 2x = 1 for every x ∈ ¬Rad(C), (5) x 2 = 0 for every x ∈ Rad(C).
Hence C is just made by infinitesimal elements and coinfinitesimal elements. Let us take note that if we take C as truth values of logical formulas, then the values from where the order of an element x, in symbols ord(x), is the least integer m such that mx = 1; if no such integer m exists, then ord(x) = ∞.
The unit interval of real numbers [0, 1] endowed with the following operations:

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 Ł 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 QŁ 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 QŁ 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 Ł denote a first-order language based on ·, +, →, ¬, ∃ and let Ł m denote a propositional language based on ·, +, → , ¬, ∃. Let Form(Ł) and Form(Ł m ) be the set of all formulas of Ł and Ł m , respectively. We fix a variable x in Ł, associate with each propositional letter p in Ł m a unique monadic predicate p * (x) in Ł and define by induction a translation : Form(Ł m ) → Form(Ł) by putting: Through this translation , we can identify the formulas of Ł m with monadic formulas of Ł containing the variable x.
Let A 1 and A 2 be any MMV-algebras. A mapping h : A 1 → A 2 is an MMV-homomorphism if h is an MVhomomorphism and for every x ∈ A 1 h(∃x) = ∃h(x). Denote by MMV the variety and the category of MMValgebras and MMV-homomorphisms.
As it is well known, MV-algebras form a category that is equivalent to the category of abelian lattice ordered groups ( -groups, for short) with strong unit (Mundici 1986). Let us denote by the functor implementing this equivalence. If G is an -group, then for any element u ∈ G, u > 0 we let [0, u] = {x ∈ G : 0 ≤ x ≤ u} and for each x, y ∈ [0, u] x ⊕ y = u ∧ (x + y) and ¬x = u − x.
A subalgebra A 0 of an MV-algebra A is said to be relatively complete if for every a ∈ A the set {b ∈ A 0 : a ≤ b} has a least element.
A subalgebra A 0 of an MV-algebra A is said to be mrelatively complete (Di Nola and Grigolia 2004), if A 0 is relatively complete and two additional conditions hold: Notice that two-elements Boolean subalgebra of the standard MV-algebra S = ([0, 1], ⊕, , ¬, 0, 1) is relatively complete, but not m-relatively complete.

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.
A monadic MV-algebra A = (A, ⊕, , ¬, ∃, 0, 1) is said to be MMV(C)-algebra if A in addition satisfies the identity (Di Nola and Lettieri 1999): Denote the variety of perfect MV-algebras by MMV(C), which is a subvariety of the variety MMV. Let Let MMV(C) 1 (Di Nola et al. 2018) be the subvariety of MMV(C) defined by the identity Alt C 2 = 1.
Theorem 3 (Di Nola et al. 2018). The identity Alt C 2 = 1 is true in finitely generated subdirectly irreducible algebra A ∈ MMV(C) if and only if A contains as a maximal homomorphic image the monadic Boolean algebra (2 2 , ∃).
Let us consider the identity

MMV(C) 1 -algebra A is said to be MMV(C) 1
K -algebra if it satisfies the identity (K Ł P ). Denote by MMV(C) 1 K the variety of all MMV(C) 1 K -algebras. The main interest for us is the MMV( where ∃x = x (and ∀x = x as well) for every x ∈ ∃K . So, the MV-reduct ∃K is isomorphic to C.

Multi-monadic perfect K Ł 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 Ł P (n).

Multi-monadic epistemic Łukasiewicz logic K Ł P (n)
In this section we introduce and investigate multi-monadic epistemic Łukasievicz K Ł P , and prove deduction theorem and completeness theorem with respect to the multi-monadic perfect MV-algebras. The formulas of Łukasiewicz logics are built from a countable set of propositional variables Var = {p, q, ...} using the connectives & (conjunction), → (implication) and ⊥ (falsity truth constant). We introduce the connectives ∧, ∨, ↔, ¬, ∨ and (the semantics counterpart will be denoted, respectively, by ∧, ∨, ⇔, ⊕ and 1, and for &) as the following abbreviations: Infinite-valued Łukasiewicz logic Ł is axiomatized by the following axioms schemata: The inference rule is Modus Ponens: ϕ, ϕ → ψ/ψ. Łukasiewicz logic Ł P is axiomatized by the axioms of Ł plus the schema:
Notice that Axiom 7) is a logical analog of algebraic identity KE11.
Recall, that the logic K Ł P (n) is obtained from Łukasiewicz propositional logic Ł 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: i α -"i knows proposition α"; ♦ i α -"i does not know that proposition α is false".
An evaluation is a mapping e i : Var → K (n) that is extended to the set of all formulas in the following way: Let e(ϕ) = n i=1 e i (ϕ). A formula ϕ is said to be valid when it is evaluated to 1 in all evaluations e i , i = 1, ..., n. In other words, a formula ϕ is said to be valid when it is evaluated to 1 for any evaluation e.
Semantically we can define the notion of "partial knowledge": i α -"agent i has partial knowledge about proposition α" if e i (α) ∈ ¬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 α" if i α and i α ∨ i α.

Lemma 6
The following formulas are theorems in Ł: )) are logically true in Ł. So they are theorems.
Theorem 7 (Deduction theorem). Let be a set of formulas and let ϕ and ψ be formulas. ∪{ϕ} ψ iff there is an m such that Conversely, let us assume that ∪ {ϕ} ψ and let γ 1 , ..., γ k be a derivation of ψ from ∪ {ϕ}. We prove by induction that, for each j = 1, ..., k, there is n j such that ( i ϕ) n j → γ j . This is clear for γ j being an axiom or a member of ∪ {ϕ}. Let us assume that γ j results by modus ponens from previous members γ t , γ t → γ j . Then by induction hypothesis we have , thus, according to Lemma 5 (3), ϕ n+m → γ j . Now let γ j a result by necessitation rule from γ t , i. e. γ j = i γ t . Then by induction hypothesis we have . This completes the proof.
Theorem 8 (Soundness of K Ł P (n)) Any theorem of K Ł P (n) is valid.
Proof It is easy to check that any axiom of K Ł P (n) is valid and the inference rules preserve validity. Consequently, any theorem of K Ł P (n) is valid.
To show that any valid formula is a theorem of K Ł P (n) we need an auxiliary lemma.
Proof We prove this assertion by induction on the number n of connectives of the formula ϕ. If n = 0, then the formula ϕ 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.

Author Contributions
All authors contributed to the study conception and design. Material preparation and results were performed by Antonio Di Nola, Revaz Grigolia and Gaetano Vitale. All authors read and approved the final manuscript.
Funding Open access funding provided by Università degli Studi di Salerno within the CRUI-CARE Agreement. Not applicable.
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Conflict of interest
The authors declare that they have no conflict of interest.
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