Autoreferential semantics for many-valued modal logics fileAutoreferential semantics for many-valued modal logics Zoran Majkic´ ETF, Applied Mathematics Department University of Belgrade

Autoreferential semantics for many-valuedmodal logics

Zoran Majki c

ETF, Applied Mathematics DepartmentUniversity of Belgrade (Serbia)

[email protected]

ABSTRACT.In this paper we consider the class of truth-functional modal many-valued logics withthe complete lattice of truth-values. The conjunction and disjunction logic operators correspondto the meet and join operators of the lattices, while the negation is independently introduced asa hierarchy of antitonic operators which invert bottom and top elements. The non-constructivelogic implication will be defined for a subclass of modular lattices, while the constructive im-plication for distributive lattices (Heyting algebras) isbased on relative pseudo-complementsas in intuitionistic logic. We show that the complete lattices are intrinsically modal, with banalidentity modal operator. We define the autoreferential set-based representation for the classof modal algebras, and show that the autoreferential Kripke-style semantics for this class ofmodal algebras is based on the set of possible worlds equal tothe complete lattice of algebraictruth-values. The philosophical assumption is based on theconsideration that each possibleworld represents a level ofcredibility, so that only propositions with the right logic value (i.e.,level of credibility) can be accepted by this world, then we connect it with paraconsistent prop-erties and LFI logics. The bottom truth value in this complete lattice corresponds to thetrivialworld in which each formula is satisfied, that is, to the world with explosive inconsistency. Thetop truth value corresponds to the world with classical logics, while all intermediate possibleworlds represent the different levels of paraconsistent logics.

KEYWORDS:many-valued logics, modal logics, Kripke-style semantics, paraconsistency.

DOI:10.3166/JANCL.18.79–125c© 2008 Lavoisier, Paris

1. Introduction

A significant number of real-world applications in Artificial Intelligence have todeal with partial, imprecise and uncertain information, and that is the principal reason

Journal of Applied Non-Classical Logics. Volume 18 – No. 1/2008, page 79 to 125

80 Journal of Applied Non-Classical Logics. Volume 18 – No. 1/2008

for introducing non-classic truth-functional many-valued logics, as for example, fuzzy,bilattice-based and paraconsistent logics, etc.

All these logics use the conjunction and disjunction logic operators as meet andjoin lattice operators with truth partial ordering betweenthe set of truth values. Theclass of complete lattices of truth values is the most significant for logics, just becauseeach finite set of truth values, or totally ordering in infinite set of values (as in fuzzylogic) is acompletelattice.

Thus, the many-valued modal logics we consider have a natural algebraic seman-tics based on a complete lattice of truth values, extended byother algebraic operators:negation, implication for a subclass of modular lattices, relative pseudo-complement(for distributive lattices) (Jipsenet al., 2002) for intuitionistic many-valued implica-tion, and unary normal modal operators (the first formal semantics for modal logic wasbased on many-valuedness, proposed by Łukasiewicz in 1918–1920, and consolidatedin 1953 in a 4-valued system of modal logic (Łukasiewicz, 1953)).

There are strong links between the structural rules the logics satisfy and the prop-erties of theiralgebraicmodels (which are, in the case of predicate based logic, themany-valued Herbrand models). The distinctive feature of algebraic semantics is thatit is truth-functional. Besides algebraic models also Kripke-style orrelationalmodelsexist. The distinctive feature of Kripke-style semantics is that accessibility relationsover possible worlds are used in the definition of satisfiability, which is not just amechanical truth-functional translation of the logic formula into the model.

There is a lot of work dedicated to this duality for lattices (Urquhart, 1978) withoperators. A good recent overview can be found in (Brinket al., 1993; Clarket al.,1998; Haim, 2000; Sofronie-Stokkermans, 2003).

Differently from their general approach, we introduce theautoreferential dualityfor complete lattices: we use (autoreferentially) as a set of possible worlds for a Kripkeframe the underlying complete lattice of truth values. Thisis a particular case wherethe frame is a partial order, as for example in (Celaniet al., 1997). Consequently, themain contribution of this paper is a general way of establishing links between algebraicand Kripke-style semantics for the class of many-valued modal logics, based on thisautoreferential duality, and the consequent paraconsistent properties.

The plan of this paper is the following: After a brief introduction into the modalpredicate logics, and a short introduction into complete lattices and modal truth-functional algebraic logics based on Galois connections for modal operators, in Sec-tion 2 we present theautoreferential semanticsfor complete lattices and their modalextensions. The other original contributions to this modalalgebraic theory are givenin the following three Sections: In Section 3 we develop a general RepresentationTheorem for the class of logics based on complete lattices with a hierarchy of nega-tion operators, by using the Dedekind-McNeile Galois connection for the partial orderrelation of the lattice and on the derived closure operator.The obtained set-basedisomorphic algebra has the usual set intersection as meet operator, but the join op-erator is a generalization of the set union: we also introduce the subset of complete

Autoreferential semantics 81

lattices, where the join corresponds exactly to the set union. In Section 4 we enrichthis compact lattice representation by introducing intuitionistic implication (when acomplete lattice is distributive), and a set of modal operators and modal negations. InSection 5 we introduce also the autoreferential world-based Kripke style definition fora many-valued modal logics, with the Kripke frame based on a set of possible worldsequal to the complete lattice of truth values, and show that it is equivalent to standardalgebraic semantics based on many-valued Herbrand models.Finally, in Section 6we discuss the relationships between many-valuedness, autoreference and paraconsis-tency for this class of many-valued logics.

In what follows we will use the algebraic meet and join symbols of a completelattice (X,≤), whereX is a set of algebraic truth-values and≤ a partial order inX , also as symbols for correspondinglogic connectives, conjunction and disjunction,while for other connectives we will maintain the formal distinction.

1.1. Introduction to multi-modal predicate logic

Given two setsA andB, we denote byAB, the set of all functions fromB to A,and byAn, n ≥ 1 the n-th cartesian productA× ...×A.

We denote byM = (W , {Ri | i ≥ 1}, S, V ) the multi-modal Kripke model with:the set of possible worldsW , accessibility relationsRi ⊆ W ×W, non empty set ofindividualsS, and V is a function defined in the following two cases:

1) V : W × F →⋃n<ωS

Sn

, with F a set of functional symbols of the lan-guage, such that for any worldw ∈ W , f ∈ F , V (w, f) : Sn → S is a function(interpretation off in w).

2) V : W × P →⋃n<ω2S

n

, with P a set of predicate symbols of the language,2 = {0, 1} set of logic values (0:false, 1:true), such that for any world w ∈ W ,r ∈ P , the functionV (w, r) : Sarity(r) → 2 defines the extension ofr in a worldw,{d | d ∈ Sarity(r), V (w, r)(d) = 1}, whered denotes a tuple of constants inS.

We define the set of terms of this modal logic as follows: all variablesx ∈ V ar,and constantsd ∈ S are terms; iff ∈ F is a functional symbol of arityn andt1, .., tnare terms, thenf(t1, .., tn) is a term.

An atom is the formular(t1, .., tn) wherer ∈ P is a predicate symbol with arityn, andti, 1 ≤ i ≤ n are the terms. Any atom is a logic formulae. The combinationof logic formulae by logic connectives (conjunction∧, disjunction∨, negation¬,implication⇒, existential modal operators♦i, etc..) is another logic formula.

The extension of a logic formulaϕ, w.r.t a modelM, a worldw ∈ W , and as-signmentg : V ar →

⋃n<ωS

n is denoted by ‖ϕ‖M,w,g. Thus, if r ∈ F⋃P

then ‖r‖M,w,g = V (w, r), that is, for a set of termst1, .., tn wheren is arity of r,‖r(t1, .., tn)‖M,w,g = V (w, r)(‖t1‖M,w,g, .., ‖t1‖M,w,g), and, ift is a variable then‖t‖M,w,g = g(t).

For any formulaϕ we define M |=w,g ϕ iff ‖ϕ‖M,w,g = 1.


The semantics for the universal modal operator♦i, applied to a formulaφ, isdefined by: M |=w,g ♦iϕ iff ∃w′((w,w′) ∈ Ri andM |=w′,g ϕ ) . Theuniversal modal operator�i is equal to¬♦i¬.

A formulaϕ is said to betrue in a modelM if M |=w,g ϕ for each assignmentfunction g and possible worldw. A formula is said to bevalid if it is true in eachmodel. Withφ/g we denote a formula (with variables)φ for a given assignmentg,while ‖φ/g‖ = {w | M |=w,g φ} is the set of worlds where the ground formulaφ/gis satisfied.

Generally normal 2-valued modal logics are not truth-valued (there is no homo-morphism between the syntactical structure of the logic language and the Booleanalgebra (i.e., distributive lattices with complements) of truth functions(2,≤), becausewe have only four unary functions in22: identity, negation, tautology and contra-diction. Thus the existential unary modal operator♦i, and its dual universal modaloperator�i, can not be represented as the function from2 into 2, and it was the rea-son why Jan Łukasiewicz introduced around 1918 more truth-values than the ordinarytwo.

In fact, there is always the equivalenttruth-functionalmany-valued logic, with theset of higher-order truth-values (Majkic, 2006a) in the Boolean-lattice functional space(2W ,≤,∧,∨,¬), isomorphic to the powerset Boolean algebra(P(W),⊆,

⋂,⋃,−),

i.e., there is the isomorphism of Boolean algebrasim : (2W ,≤,∧,∨,¬) ≃ (P(W),⊆,⋂,⋃,−), such that for any two functional algebraic valuesf, g ∈ 2W , f ≤ g iff

im(f) ⊆ im(g), whereim(f) = {w ∈ W | f(w) = 1} is the image off .

The meet, join and negation operations in the Boolean algebra (2W ,≤,∧,∨,¬)are defined by this isomorphism as :

h = f ∧ g :W → 2, such thatim(h) = im(f)⋂im(g),

h = f ∨ g :W → 2, such thatim(h) = im(f)⋃im(g),

h = ¬f : W → 2, such thatim(h) = −im(f), where− is the set complement inW .

The logic implication in this Boolean algebra is defined classically byf ⇒ g =def

¬f∨g. The truth-functional universal (multiplicative) modal operator�i : 2W → 2W

for normal Kripke logic with the binary accessibility relationRi is defined by:h =�if :W → 2, such thatim(h) = {w | ∀w′((w,w′) ∈ Ri implies w′ ∈ im(f)}.

So that, anon truth-functional 2-valued modal logic is equivalently representedby amany-valued truth-functionalalgebraic logic based on Boolean algebra extendedby normal modal algebraic operators(X, 0, 1,≤,∧,∨,¬,�i), whereX = 2W and0, 1 ∈ X are the bottom and top value inX , with also an infinite number of algebraictruth-values (Dugundji, 1940). The same approach can be applied to any many-valuednontruth-functional modal logic, thus it is enough to analyze only thealgebraic truth-functionalmany-valued modal logics, as is done in the rest of this paper. Vice versa,


any algebraic many-valued logic logic program, based on thecomplete lattice of truth-values, can be equivalently represented by a non truth-functional 2-valued multi-modallogic (Majkic, 2006e) with ternary accessibility relations.

Given a modal algebra(X, 0, 1,≤,∧,∨,¬,�i) the standard relational sematicsbased on Stone’s representation theorem, is representd by adescriptive general frameintroduced by Goldblat (Goldblatt, 1976a; Goldblatt, 1976b; Bull et al., 1984), where apossible world is an ultrafilterF defined as a subset ofX which satisfies the followingconditions:

– 1 ∈ F and not0 ∈ F– if x, y ∈ F thenx ∧ y ∈ F– if x ∈ F andx ≤ y theny ∈ F– for eachx ∈ X , eitherx ∈ F or ¬x ∈ F .

Instead of this standard relational semantics for modal logic, where a possibleworld is asubsetof elements inX (an ultrafilter), in this paper we will use an moresimpleautoreferentialsemantics where a possible world is anelementin X .

1.2. Introduction into lattice algebras and their extensions

Posets and lattices (posets such that for all its elements x and y, the set {x, y} hasboth a join (lub—least upper bound) and a meet (glb—greatestlower bound)) with apartial order≤ play an important role in what follows. Aboundedlattice has a greatest(top) and least (bottom) element, denoted by convention by 1and 0.

Finite meets in a poset will be written as1,∧, and finite joins as0,∨. A lattice(poset)X is completeif each (also infinite) subsetS ⊆ X (or, S ∈ P(X) wherePis the symbol for powerset, and∅ ∈ P(X) denotes the empty set) has the least upperbound (supremum) denoted by

∨S ∈ X (whenS has only two elements the supre-

mum corresponds to the join operator∨). Each finite bounded lattice is a completelattice. Each subsetS has the greatest lower bound (infimum) denoted by

∧S ∈ X ,

given as∨{x ∈ X | ∀y ∈ S.x ≤ y}. The complete lattice is bounded and has the

bottom element,0 =∧X ∈ X , and the top element1 =

∨X ∈ X .

A function l : X → Y between posetsX,Y is monotoneif x ≤ x′ impliesl(x) ≤ l(x′) for all x, x′ ∈ X . Such a functionl : X → Y is said to have a right(or upper) adjoint if there is a functionr : Y → X in the reverse direction such thatl(x) ≤ y iff x ≤ r(y) for all x ∈ X, y ∈ Y . Such a situation forms a Galoisconnection and will often be denoted byl ⊣ r. Then l is called a left (or lover)adjoint ofr. If X,Y are complete lattices (posets) thenl : X → Y has a right adjointiff l preserves all joins (it isadditive, i.e., l(x ∨ y) = l(x) ∨ l(y) and l(0X) = 0Ywhere0X , 0Y are bottom elements in complete latticesX andY respectively). Theright adjoint is thenr(y) =

∨{z ∈ X | l(z) ≤ y}. Similarly, a monotone function

r : Y → X is a right adjoint (it ismultiplicative, i.e., has a left adjoint) iffr preservesall meets; the left adjoint is thenl(x) =

∧{z ∈ Y | x ≤ r(z)}.


Each monotone functionl : X → Y on a complete lattice (poset)X has both aleastfixed pointµl ∈ X andgreatestfixed pointνl ∈ X . these can be describedexplicitly as: µl =

∧{x ∈ X | l(x) ≤ x} and νl =

∨{x ∈ X | x ≤ l(x)}.

In what follows we denote byy < x iff ( y ≤ x and notx ≤ y), and we denote byx ⊲⊳ y two unrelated elements inX (so that not(x ≤ y or y ≤ x)). An element in alatticex 6= 0 is ajoin-irreducibleelement iffx = a∨b impliesx = a orx = b for anya, b ∈ X . An element in a latticex ∈ X is anatomiff x > 0 and∄y.(x > y > 0).

A lower set (down closed) is any subsetY of a given poset(X,≤) such that, forall elementsx andy, if x ≤ y andy ∈ Y thenx ∈ Y .

A Heyting algebrais a bounded latticeX with finite meets and joins such that foreach elementx ∈ X , the function(_) ∧ x : X → X has a right adjointx ⇀ (_), alsocalled an algebraic implication. An equivalent definition can be given by considering abonded lattice such that for allx andy inX there is a greatest elementz inX , denotedbyx ⇀ y, such thatz∧x ≤ y, i.e., x ⇀ y =

∨{z ∈ X | z∧x ≤ y} (relative pseudo-

complement). In Heyting algebra we can define negation¬x as a pseudo-complementx ⇀ 0. Thenx ≤ ¬¬x. A complete Heyting algebra is a Heyting algebra whichis complete as a poset. A complete lattice is thus a complete Heyting algebra iff thefollowing distributivityx ∧ (

∨S) =

∨y∈S(x ∧ y) holds.

The negation and implication operators can be represented as the following mono-tone functions:¬ : X → XOP and⇒: X×XOP → XOP , whereXOP is the latticewith inverse partial ordering and∧OP = ∨, ∨OP = ∧.

The smallest complete distributive lattice is denoted by2 = {0, 1} with classictwo values, false and true respectively. It is also complemented Heyting algebra, con-sequently it is Boolean. AGalois algebrais a complete Heyting algebra both with a"nexttime" monotone function FromX to X that preserves all meets (i.e., right ad-joint). Such Galois algebras are often called Heyting algebras with (unary) modaloperators.

1.3. Introduction to sequents and bivaluations

Sequent calculus was developed by Gentzen (Gentzen, 1932),inspired by someideas of Paul Herz (Hertz, 1929). Given a propositional logic languageL (set of logicformulae) a sequent is a consequence pair of formulaes = (φ;ψ) ∈ L × L, denotedalso byφ ⊢ ψ.

A Genzen system, denoted by a pairG = 〈L, 〉 where is a finitary consequencerelation on a set of sequents inL ⊆ L × L, is said to benormal if it satisfies thefollowing conditions: for any sequents = φ ⊢ ψ ∈ L and the set of sequentsΓ ={si = φi ⊢ ψi ∈ L | i ∈ I},

1) (reflexivity) if s ∈ Γ thenΓ s2) (transitivity) if Γ s and for everys′ ∈ Γ, Θ s′, thenΘ s


3) (finiteness) ifΓ s then there is finiteΘ ⊆ Γ such thatΘ s.4) for any homomorphismσ from L into itself (i.e., substitution), ifΓ s then

σ[Γ] σ(s), i.e., {σ(φi) ⊢ σ(ψi) | i ∈ I} (σ(φ) ⊢ σ(ψ)).

Notice that from (1) and (2) we obtain this monotonic property:

5) if Γ s andΓ ⊆ Θ, thenΘ s.

We denote byC : P(L) → P(L) the closure operator such thatC(Γ) =def {s ∈L | Γ s}, with the properties:Γ ⊆ C(Γ) (from reflexivity (1)); it is monotonicΓ ⊆ Γ1 impliesC(Γ) ⊆ C(Γ1) (from (5)), and an involution,C(C(Γ)) = C(Γ) also.Thus we obtain that

6) Γ s iff s ∈ C(Γ).

Any sequent theoryΓ ⊆ L is said to be aclosedtheory iff Γ = C(Γ). This closureproperty corresponds to the fact thatΓ s iff s ∈ Γ.

Each sequent theoryΓ can be considered as a bivaluation (characteristic function)β : L→ 2 such that for any sequents ∈ L, β(s) = 1 iff s ∈ Γ.

2. Many-valued model-theoretic autoreferential semantics

Let L be a propositional logic language obtained as free algebra,from the con-nectives inΣ of an algebra based on a complete lattice(X,≤) of algebraic truth-values (for example meet and join{∧,∨} ⊆ Σ are binary operators, negation¬ ∈ Σand other modal operators are unary operators, while eachx ∈ X ⊆ Σ is a con-stant (nullary operator)) and a setV ar of propositional variables (letters) denoted byp, r, q, ... We will use lettersφ, ψ, .. for the formulae ofL. We define a (many-valued)valuationv as a mappingv : L → X (notice thatX ⊆ L are the constants of thislanguage and we will use the same symbols as for elements of the latticeX), whichis an homomorphism (for example, for anyp, q ∈ V ar, v(p⊙ q) = v(p)⊙ v(q),⊙ ∈{∧,∨,⇒} andv(¬p) = ¬v(p), where∧,∨,⇒,¬ are conjunction, disjunction, im-plication and negation respectively) and is an identity forelements inX , that is, foranyx ∈ X , v(x) = x. We denote byVm the strict subset inXL of all homomorphicmany-valued valuations.

Then we define the following lattice-based consequence binary relation⊢⊆ L×Lbetween the formulae (it is analog to the binary consequencesystem from (Dunn,1995) for the distributive lattice logicDLL), where each consequence pairφ ⊢ ψ is asequentalso.

EXAMPLE 1. — Let us consider the Distributive modal logic (Dunn, 1995) (with� universal modal operator, and its left adjoint existentialmodal operator♦, with♦ ⊣ �) and with a negative modal operator¬. The Gentzen-like systemG of thislogic languageL contains the following axioms (sequents) and rules:

(AXIOMS) G contains the following sequents:


1) φ ⊢ φ (reflexive)2) φ ⊢ 1, 0 ⊢ φ (top/bottom axioms)3) φ ∧ ψ ⊢ φ, φ ∧ ψ ⊢ ψ (product projections: axioms for meet)4) φ ⊢ φ ∨ ψ, ψ ⊢ φ ∨ ψ (co-product injections: axioms for join)5) φ ∧ (ψ ∨ ϕ) ⊢ (φ ∧ ψ) ∨ (φ ∧ ϕ) (distributivity axiom)6) �(φ ∧ ψ) ⊢ �φ ∧�ψ, 1 ⊢ �1 (multiplicative modal property axioms)7) ♦(φ ∨ ψ) ⊢ ♦φ ∨ ♦ψ, ♦0 ⊢ 0 (additive modal property axioms)8) ¬φ ∧ ¬ψ ⊢ ¬(φ ∨ ψ), 1 ⊢ ¬0 (additive modal negation axiom)9) The set of sequents which define the poset of the lattice of truth values(X,≤):

for any twox, y ∈ X , if x ≤ y thenx ⊢ y ∈ G.

(INFERENCE RULES)G is closed under the following inference rules:

1) φ ⊢ψ, ψ ⊢ϕφ ⊢ϕ

(cut/ transitivity rule)

2) φ ⊢ψ, φ ⊢ϕφ ⊢ψ∧ϕ

, φ ⊢ψ, ϕ ⊢ψφ∨ϕ ⊢ψ

(lower/upper lattice bound rules)

3) φ ⊢ψ�φ ⊢�ψ

, φ ⊢ψ♦φ ⊢♦ψ

(monotonicity of modal operators rules)

4) φ ⊢ψ¬ψ ⊢¬φ

(antitonicity of modal negation rule)

5) φ ⊢ψσ(φ) ⊢σ(ψ) (substitution rule:σ is substitution(γ/p)).

Notice that the rules in point 2 are the consequences of diagonal mapping△ : X →Y , whereY = X × X and△x = (x, x), (which is both additive and multiplicativemodal operator), and its Galois adjunctions with the meet (multiplicative) and join(additive) operators∧,∨ : Y → X , i.e., with△ ⊣ ∧ and∨ ⊣ △; that is△x ≤Y (y, z)(i.e., x ≤ y andx ≤ z) iff x ≤ ∧(y, z) = y ∧ z, andx ∨ y = ∨(x, y) ≤ z iff(x, y) ≤Y △z (i.e., x ≤ z andy ≤ z).

The axioms from 1 to 5 and the rules 1 and 2 are taken from (Dunn,1995) fortheDLL and it was shown that this sequent based Genzen-like system is sound andcomplete. If we omit the distributivity axiom 5, we obtain the systemG for completelattice logics. Thus, for a lattice based modal many-valuedlogics we obtain anormalmodal Gentzen-like deductive system, where each sequent isa valid truth-preservingconsequence pair defined by the poset of the complete lattice(X,≤) of truth values(which are also the constants of this modal propositional languageL), so that eachoccurrence of the symbol⊢ can be substituted by the partial order≤ of this completelattice. �

DEFINITION 2 (TRUTH-PRESERVING ENTAILMENT). — For any two formulaeφ, ψ ∈ L, the truth-preserving consequence pair (sequent), denoted by φ ⊢ ψ issatisfied by a given valuationv : L → X if v(φ) ≤ v(ψ). This sequent is a tautologyif it is satisfied by all valuations,i.e., when ∀v ∈ Vm.(v(φ) ≤ v(ψ)).

For a normal Gentzen-like sequent systemG of the many-valued logicL, with theset of sequentsSeqG ⊆ L×L and a set of inference rules inRulG , we tell that a many-valued valuationv is itsmodel if it satisfies all sequents inG. The set of all models ofa given set of sequents (theory)Γ is denoted byModΓ =def {v ∈ Vm | ∀(φ ⊢ ψ) ∈Γ(v(φ) ≤ v(ψ))} ⊆ Vm ⊂ XL.


PROPOSITION3 (SOUNDNESS). — All axioms of the Gentzen like sequent systemG,of a many-valued logicL based on complete lattice(X,≤) of algebraic truth values,are the tautologies, and all its rules are sound for model satisfiability and preserve thetautologies.

PROOF. — It is straightforward to check (see the Example 1) that allaxioms are tau-tologies (all constant sequents specify the poset of the complete lattice(X,≤), thusare tautologies). It is straightforward to check that all rules preserve the tautologies.Moreover, if all premises of any rule inG are satisfied by given many-valued valua-tion v : L → X , then also the deduced sequent of this rule is satisfied by thesamevaluation,i.e., the rules are sound for the model satisfiability. �

It is easy to verify that for any twox, y ∈ X we have thatx ≤ y iff x ⊢ y, thatis the truth-preserving entailment coincides with the partial truth-ordering in a lattice(X,≤). Notice that it is compatible with lattice operators, that is, for any two formulaeφ, ψ ∈ L, φ∧ψ ⊢ ψ andφ ⊢ ψ∨φ. This entailment imposes the following restrictionsto the logic implication: in order to satisfy the Deduction Theorem "z ⊢ x ⇒ y iffz∧x ⊢ y" (i.e., inference rules for elimination and introduction of the logic connective⇒, z ⊢x⇒y

z∧x ⊢yand z∧x ⊢y

z ⊢x⇒y) by this entailment, the logic implication must satisfy

(the case whenz = 1) the requirement that for anyx, y ∈ X , x⇒ y = 1 iff x ≤ y,while it must satisfyx ∧ (x⇒ y) ≤ y in order to satisfy the Modus Ponens inferencerule.

The particularity of this entailment is that any consequence pair (sequent)φ ⊢ ψis algebraically an equationφ ∧ ψ = φ (or,φ ∨ ψ = ψ).

It is easy to verify, that in the case of the classic 2-valued propositional logic thisentailment is equal to the classic propositional entailment, so that the truth-preservingentailment is only a generalization of the classic entailment for a many-valued propo-sitional logics.

REMARK . — It is easy to observe that each sequent is, from the logic point of view,a 2-valued objectso that all inference rules are embedded into the classic 2-valuedframework,i.e., given a bivaluationβ : L × L → 2, we have that a sequents =φ ⊢ ψ is satisfied whenβ(s) = 1, so that we have the relationship between sequentbivaluations and many-valued valuationev used in Definition 2. �

From my point of view, this sequent feature, which is only an alternative formu-lation for the 2-valued classic logic, isfundamentalin the framework of many-valuedlogics, where often the semantics for the entailment, basedon algebraic matrices(X,D) is arbitrary: consider, for example fuzzy logic, where the subset of desig-nated elementsD ⊆ X is an closed interval[a, 1], where0 < a ≤ 1 is an arbitraryvalue, so difficult to fix.

Thus, this correct definition of the 2-valued entailment in the sequent systemG,based only on the lattice ordering, can replace the current entailment based on thealgebraic matrices(X,D), whereD ⊆ X is the subset of designated elements, whichis upward closed, that is, ifx ∈ D andx ≤ y theny ∈ D (thus1 ∈ D), and where


the matrix-entailment, defined byφ ⊢D ψ, is valid iff ∀v ∈ Vm.(v(φ) ∈ D impliesv(ψ) ∈ D). It is easy to verify also thatφ ⊢ ψ impliesφ ⊢D ψ.

This opinion is based also on the consideration that all many-valued,implicationbased(IF), logic program languages, are based on clausesA← B1 ∧ ... ∧Bn whereA is a ground atom andBi are ground literals (ground atoms or negation of groundatoms), we tell that it issatisfiedby a valuationv iff v(A) ≤ v(B1) ∧ ... ∧ v(Bn);Thus, this clause is the sequent(B1 ∧ ... ∧ Bn) ⊢ A, so that a logic program is infact a set of sequents; the valuationv which satisfies all clauses (sequents) is amodelfor such a logic program, and demonstrates how we are able to define the models ofmany-valued logics without the necessity to define the algebraic matrices(X,D).

The other class of "many valued" logic programs, Signed logic programs (Beckertet al., 1999; Imazet al., 1994) and its subclass ofannotation based(AB) programs(Blair et al., 1989; Kifer et al., 1992) a clause is of the formA : f(δ1, .., δn) ←B1 : δ1, ..., Bn : δn , which asserts "the certainty of the atomA is least (or is in)f(δ1, .., δn), whenever the certainty of the atomBi is at least (or is in)δi, 1 ≤ i ≤n", wheref is an n-ary computable function andδi is either a constant or a variableranging over many-valued logic values. But they are not really many-valued logics,but a kind of "meta" many-valued logics, because all annotated literalsBn : δn areclassic 2-valued objects (a valuationv is a model of the literalBn : δn if v(Bn) ≥ δn(or v(Bn) ∈ δn)), and all connectives in the clauses are also classic 2-valued. Infact, in (Majkic, 2006e) it is shown that the annotated elements are 2-valued modalformulae, so that an annotated logic is a kind of 2-valued multi modal logics; the sameresult can be proven also in the more general case of Signed logic programming.

Thus we are able now to introduce the many-valued valuation-based (i.e., model-theoretic) semantics for many-valued logics:

DEFINITION 4. — A many-valued model-theoretic semantics of a given many-valued logicL, with a Gentzen systemG = 〈L, 〉, is the semantic deducibility re-lation |=m , defined for anyΓ = {si = (φi ⊢ ψi) | i ∈ I} and sequents = (φ ⊢ ψ) ∈ L ⊆ L × L by : “ Γ |=m s iff all many-valued models ofΓare the models ofs”. That is: Γ |=m s

iff ∀v ∈ Vm( ∀(φi ⊢ ψi) ∈ Γ(v(φi) ≤ v(ψi)) implies v(φ) ≤ v(ψ))

iff ∀v ∈ModΓ( ∀(φi ⊢ ψi) ∈ Γ(v(φi) ≤ v(ψi)) implies v(φ) ≤ v(ψ))

iff ∀v ∈ModΓ( v(φ) ≤ v(ψ)).

It is easy to verify that any complete-lattice based many-valued logic has theGentzen-like systemG = 〈L, 〉 (see the Example above) which isnormal logic.

THEOREM 5. — The many-valued model theoretic semantics is an adequate seman-tics for a many-valued logicL specified by a Gentzen-like logic systemG = 〈L, 〉,that is, it is sound and complete. Consequently,Γ |=m s iff Γ s.

PROOF. — Let us prove that for any many valued modelv ∈ ModΓ, the obtainedsequent bivaluationβ = eq◦ < π1,∧ > ◦(v × v) : L × L → 2 is the characteristic


function of the closed theoryΓv = C(T ) with T = {φ ⊢ x, x ⊢ φ | φ ∈ L, x =v(φ)}.

From the definition ofβ we have thatβ(φ ⊢ ψ) = β(φ;ψ) = eq◦ < π1,∧ > ◦(v×v)(φ;ψ) = eq◦ < π1,∧ > (v(φ), v(ψ)) = eq < π1(v(φ), v(ψ)),∧(v(φ), v(ψ)) >=eq(v(φ),∧(v(φ), v(ψ))) = eq(v(φ), v(φ) ∧ v(ψ)), whereπ1 : X × X → X is thefirst projection,eq : X ×X → 2⊆ X is the equality characteristic function such thateq(x, y) = 1 if x = y.

Thusβ(φ ⊢ ψ) = 1 iff v(φ) ≤ v(ψ), i.e., when this sequent is satisfied byv.

1) Let us show that for any sequents, s ∈ Γv implies β(s) = 1:

First of all, any sequents ∈ T is of the formϕ ⊢ x or x ⊢ ϕ, wherex = v(ϕ),so that it is satisfied byv (holds thatv(ϕ) ≤ v(ϕ) in both cases). Consequently,all sequents inT are satisfied byv.

By the Proposition 3 we have that all inference rules inG are sound w.r.t themodel satisfiability, thus for any deductionT s (i.e., s ∈ Γv) where allsequents in premises are satisfied by the many-valued valuation (model)v, alsothe deduced sequents = (φ ⊢ ψ) must be satisfied, that is must holdv(φ) ≤v(ψ), i.e., β(s) = 1.

2) Let us show that for any sequents, β(s) = 1 implies s ∈ Γv:

Foranysequents = (φ ⊢ ψ) ∈ L× L if β(s) = 1 thenx = v(φ) ≤ v(ψ) = y(i.e., s is satisfied byv). From the definition ofT , we have thatφ ⊢ x, y ⊢ ψ ∈T , and fromx ≤ y we havex ⊢ y ∈ AxG (whereAxG are axioms (sequents) inG, with {x ⊢ y | x, y ∈ X,x ≤ y} ⊆ AxG , thus satisfied by every valuation) bythe transitivity rule, fromφ ⊢ x, x ⊢ y, y ⊢ ψ, we obtain thatT (φ ⊢ ψ), i.e.,s = (φ ⊢ ψ) ∈ C(T ) = Γv.

So, from 1) and 2) we obtain that β(s) = 1 iff s ∈ Γv, i.e., the sequentbivaluationβ is the characteristic function of a closed set. Consequently, any many-valuedmodelv of this many-valued logicL corresponds to theclosedbivaluationβwhich is a characteristic function of a closed theory of sequents: we define the setof all closed bivaluations obtained from the set of many-valued modelsv ∈ ModΓ:BivΓ = {Γv | v ∈ ModΓ}. From the fact thatΓ is satisfied by everyv ∈ ModΓ

we have that for everyΓv ∈ BivΓ, Γ ⊆ Γv, so thatC(Γ) =⋂BivΓ (intersection of

closed sets is a closed set also). Thus, fors = (φ ⊢ ψ), Γ |=m s

iff ∀v ∈ModΓ( ∀(φi ⊢ ψi) ∈ Γ(v(φi) ≤ v(ψi)) implies v(φ) ≤ v(ψ))

iff ∀v ∈ModΓ( ∀(φi ⊢ ψi) ∈ Γ(β(φi ⊢ ψi) = 1) implies β(φ ⊢ ψ) = 1)

iff ∀v ∈ModΓ( ∀(φi ⊢ ψi) ∈ Γ((φi ⊢ ψi) ∈ Γv) implies s ∈ Γv)

iff ∀Γv ∈ BivΓ( Γ ⊆ Γv implies s ∈ Γv)

iff ∀Γv ∈ BivΓ( s ∈ Γv) , becauseΓ ⊆ Γv for eachΓv ∈ BivΓ


iff s ∈⋂BivΓ = C(Γ), that is,

iff Γ s. �

Thus, in order to define the model-theoretic semantics for a many-valued logicswe do not need to define the problematic matrices: we are able to use only the many-valued valuations, andmany-valued models(i.e., valuations which satisfy all sequentsin Γ of a given many-valued logicL. This point of view is used also for definition ofa new Representation Theorem for many-valued logics in (Majkic, 2006d).

Differently from the classic logic where a formula is a theorem if it is true in allmodels of the logic, here, in a many-valued logicL, but specified by a set of sequents inΓ, for a formulaφ ∈ L which has the same valuex ∈ X (for anyalgebraic truth-valuex) for all many-valued modelsI ∈ ModΓ, we have that its sequent-based versionφ ⊢ x andx ⊢ φ are theorems; that is∀I ∈ ModΓ(I(φ) = x) iff (Γ (φ ⊢ x)andΓ (x ⊢ φ)).

But such a valuex ∈ A does not need to be a designated elementx ∈ D, as inmatrix semantics for a many-valued logic, and it explains why we do not need therigid semantic specification by matrix designated elements. Thus, by a translationof a many-valued logicL into its "meta" sequent-based 2-valued logic, we obtain anunambiguous theory of inference without the introduction of problematic matrices.

REMARK . — There are also two other ways, alternative to 2-valued sequent systems,to reduce the many-valued logics into “meta” 2-valued logics: The first is based on theOntological encapsulation (Majkic, 2004; Majkic, 2006e), where each many-valuedproposition (or many-valued ground atomp(a1, .., an)) is ontologically encapsulatedinto the “flattened” 2-valued atompF (a1, ..., an, x) (by enlarging original atoms withnew logic variable whose domain of values is the set of algebraic truth-values of acomplete latticex ∈ X : ruffly, "p(a1, ..., an) has a valuex" iff pF (a1, ..., an, x)is true). In fact such an atom is equivalent to the following formula of sequents(p(a1, .., an) ⊢ x) ∧ (x ⊢ p(a1, ..., an)).

The other way, which can be used in the case when a complete lattice (X,≤) ofalgebraic truth-values isfinite, is to represent this sequent formula by a modal 2-valuedatom�xp(a1, ..., an) (where�x is an universal modal operator for a given algebraictruth-value inX) with autoreferential semantics (Majkic, 2006e), where the set ofpossible worlds of this multi modal logic is the set of algebraic truth values in thelattice(X,≤). �

Based on this Genzen-like sequent deductive systemG we are able to define theequivalence relation≈L between the formulae of any modal propositional logic basedon a complete lattice in order to define the Lindenbaum algebra for this logic,(L/ ≈L,⊑), where for any two formulaeφ, ψ ∈ L,

a) φ ≈L ψ iff φ ⊢ ψ andψ ⊢ φ, i.e.,iff ∀v.(v(φ) ≤ v(ψ)) and∀v.(v(φ) ≥ v(ψ)).

Thus the quotient algebraL/ ≈L has as elements equivalence classes, denoted by[φ], and the partial ordering⊑, defined by


b) [φ] ⊑ [ψ] iff φ ⊢ ψ (i.e., if φ ≤ ψ).

It is easy to verify that each equivalence class (set of all equivalent formulae w.r.t≈L) [φ] has exactly one constantx ∈ X which is element of this equivalence class,and we can use it as the representation element for this equivalence class: so that everyformula in this equivalence class has the same truth-value as this constant.

Thus, we have the bijectionis : L/ ≈L→ X between elements in the completelattice(X,≤) and elements in the Lindenbaum algebra, such that for any equivalenceclass[φ] ∈ L/ ≈L, the constantis([φ]) ∈ X is the representation element for thisequivalence class. It is easy to extend this bijection into an isomorphism between theoriginal algebra and this Lindenbaum algebra, by definitionof correspondent connec-tives in the Lindenbaum algebra, for example:

[φ] ∧L [ψ] =def is−1(is([φ]) ∧ is([φ])),

[φ] ∨L [ψ] =def is−1(is([φ]) ∨ is([φ])),

¬L[φ] =def is−1(¬(is([φ]))), etc.

In the autoreferential semantics we will assume that each equivalence class of for-mulae[φ] in this Lindenbaum algebra corresponds to one “state-description”, that is toone possible world in the Kripke-style semantics for the original many-valued modallogic. But, from the isomorphismiswe can take, instead of equivalence class[φ], onlyits representation elementx = is([φ]) ∈ X .

Consequently the set of possible worlds in this autoreferential semantics corre-sponds to the set of truth values in the complete lattice(X,≤).

3. Autoreferential representation theorem for complete lattices

Generally lattices arise concretely as the substructures of closure systems (inter-section systems) where a closure system is a familyF(X) of subsets of a setX suchthatX ∈ F(X) and ifAi ∈ F(X), i ∈ I, then

⋂i∈I Ai ∈ F(X). Then the repre-

sentation problem for general lattices is to establish thatevery lattice can be viewed,up to an isomorphism, as a collection of subsets in a closure system (on some setX),closed under the operations of the system.

Closure operatorsΓ are canonically obtained by the composition of two maps ofGalois connections. The Galois connections can be obtainedfrom any binary relationon a setX (Birkhoff, 1940) (Birkhoffpolarity) in a canonical way: If(X,R) is a setwith a particular relation on a setX , R ⊆ X × X , with mappingsλ : P(X) →P(X)OP , : P(X)OP → P(X), such that for anyU, V ∈ P(X), λU = {x ∈X | ∀u ∈ U.((u, x) ∈ R)} , ρV = {x ∈ X | ∀v ∈ V.((x, v) ∈ R)}, whereP(X) isthepowerset posetwith bottom element empty set∅ and top elementX , andP(X)OP

its dual (with⊆OP inverse of⊆), then we obtain the induced Galois connectionλ ⊣ ρ,i.e., λU ⊆OP V iff U ⊆ ρV .


The following lemma is useful for the relationship of these set-based operatorswith the operation of negation in the complete lattices.

LEMMA 6 (INCOMPATIBILITY RELATION ). — Let (X,≤) be a complete lattice.Then we can use a binary relationR ⊆ X ×X as anincompatibility relation forset-based negation operatorsλ andρ, with the following properties: for anyU, V ⊆X ,

1) λ(U⋃V ) = λU

⋃OPλV = λU

⋂λV with λ∅ = ∅OP = X (additivity)

2) ρ(U⋂OP

V )= ρ(U⋃V )=ρU

⋂ρV with ρXOP =ρ∅ = X (multiplicativity)

3) whileλ(U⋂V ) ≥ λU

⋃λV , ρ(U

⋂V )≥ρU

⋃ρV andλρV ≥ V, ρλU≥U .

We will consider the case when(X,R) is a complete lattice with the binary relationR equal to the partial order≤ in the latticeX . The resulting Galois connection onthis partial order is the familiar Dedekind-McNeile Galoisconnection of antitonicmappingsλ andρ (We denote by↓ x the ideal{y ∈ X | y ≤ x}):

λU = {x ∈ X | ∀u ∈ U.(u ≤ x)},

ρV = {x ∈ X | ∀v ∈ V.(x ≤ v)}.

SettingΓ = ρλ : P(X) → P(X), the operatorΓ is a monotone mapping and aclosure operator, such that for anyU, V ∈ P(X)

1) U ⊆ V implies Γ(U) ⊆ Γ(V ),2) U ⊆ Γ(U), for anyU ∈ P(X),3) ΓΓ(U) = Γ(U)

The set{U ∈ P(X) | U = Γ(U)} is called the set ofstable(closed) sets.

It is easy to verify that for every subsetS ⊆ X , we obtainΓ(S) =↓∨

(S), which isa multiplicative monotone operator: the closure operatorΓ satisfies the multiplicativepropertyΓ(U

⋂V ) = Γ(U)

⋂Γ(V ), andΓ(X) = X which is top element inP(X).

Thus,Γ =↓∨

is auniversal modal operatorfor the complete lattice(P(X),⊆)(which is a Boolean algebra, that is, acomplemented distributivelattice), so that weobtain the modal algebra(P(X),⊆,

⋂,⋃,Γ) for the complete lattice(X,≤).

Notice thatΓ(∅) = Γ({0}) = {0}. Thus, from the monotone property ofΓ andfrom ∅ ⊆ U (for anyU ∈ P(X)) we obtain that{0} = Γ(∅) ⊆ Γ(U). That is, anystable (closed) set contains the bottom element0 ∈ X , and that the minimal stable setis {0} and not an empty set∅, while, naturally, the maximal stable set isX .

Moreover, each stable set is a downclosed ideal and there is the bijection betweenthe set of stable sets and the set of algebraic truth-values inX , so that we are able todefine the representation theorem for a complete lattice(X,≤) as follows:

PROPOSITION 7 (AUTOREFERENTIAL REPRESENTATION THEOREM FOR COM-PLETE LATTICES). — Let (X,≤) be a complete lattice. We define the Dedekind-McNeile coalgebra↓: X → P(X), where↓= Γin, Γ = ρλ is a Dedekind-McNeileclosure operator andin : X → P(X) the inclusion mapx 7→ {x}, such that for anytwox, y ∈ X holds (we denote by− the set substraction)


1)⋃{S ∈ P(X) | S

⋂↓ x ⊆↓ y} = (X− ↓ x)

⋃↓ y.

We denote byF(X) = {↓ x | x ∈ X} the closure system because for eachx ∈ X ,↓ x is a stable set. Then, the(F(X),⊆) is a complete lattice with meet operator

⋂

(set intersection), and join operator⊎

= Γ⋃

=↓∨⋃

differentfrom the set union⋃, such that theAUTOREFERENTIAL REPRESENTATION ISOMORPHISMholds:

2) ↓: (X,≤,∧,∨, 0, 1) ≃ (F(X),⊆,⋂,⊎, {0}, X).

PROOF. — We will show that each stable setU ∈ F(X) is an ideal in(X,≤). In factit holds that for anyx ∈ X , ↓ x = Γin(x) = ρλ({x}) = {x′ ∈ X | x′ ≤ x} is anideal. Thus, for any twox, y ∈ X we have that

1) If x ≤ y then↓ x ⊆↓ y,2) ↓ x

⋂↓ y = {x′ ∈ X | x′ ≤ x}

⋂{y′ ∈ X | y′ ≤ y} = {z ∈ X | z ≤

x ∧ y} =↓ (x ∧ y) ∈ F(X),3) ↓ x

⊎↓ y =↓

∨⋃(↓ x, ↓ y) =↓

∨(↓ x

⋃↓ y) =↓

∨({x′ ∈ X | x′ ≤

x}⋃{y′ ∈ X | y′ ≤ y}) =↓ (x ∨ y) ∈ F(X).

The operator⊎

is the join operator inF(X), that is holds↓ x ⊆↓ y iff ↓ x⊎↓

y =↓ y.

Thus the original lattice(X,≤) is isomorphic to the lattice(F(X),⊆) via mapx 7→↓ x. It is easy to verify that the inverse of↓, i.e., ↓−1= (Γin)−1 = in−1Γ−1 :P(X)→ X is equal to supremum

∨, that is↓−1 (U) =

∨{x ∈ U}, with

∨↓ x = x,

that is,∨↓= idX and↓

∨= idF(X) are the identity functions forx andF(X)

respectively. It is easy to verify thatF(X) is a complete lattice with bottom element↓ 0 = {0} and top element↓ 1 = X , and for anyU =↓ x, V =↓ y ∈ F(X),U

⋂V =↓ (x ∧ y) ∈ F(X) andU

⊎V =↓ (x ∨ y) ∈ F(X). �

REMARK . — What is important to notice is that the autoreferential representation,based only on the lattice ordering of truth values, naturally introduce the modality inthis logic. In fact, the universal modal operatorΓ : P(X) → F(X) ⊂ P(X), basedon the partial order of the complete lattice(X,≤), plays the fundamental role for themodal disjunction

⊎in the canonical autoreferential algebraF(X). We need it be-

cause generally for any twox, y ∈ X we have that↓ x⋃↓ y /∈ F(X). Its restriction

on F(X) is an identity function idF(X), which is a selfadjoint (universal and exis-tential) modal operator. If we denote by� the logic modal operator correspondentto the algebraic modal operatorΓ, we can define the join operator∨ (many-valueddisjunction) from theclassicdisjunction ∨c (for which holds↓ ∨c =

⋃↓, that

is, ↓ (x ∨c y) =↓ x⋃↓ y) and this modal operator, as follows: for any two logic

formulaeφ andψ: φ ∨ ψ = �(φ ∨c ψ). This property will be used when we willdefine the Kripke style semantics for this many-valued logic.

Notice also that in Lewis’s approach he distinguishes the standard orextensionaldisjunction∨c, from the intensionaldisjunction∨ "in such that at least one of thedisjoined propositions is "necessarily" true" (Lewis, 1912) (1912, p.523). In the sameway he defined also the "strict" logic implicationφ⇒ ψ = �(φ⇒c ψ) where⇒c isthe classic extensional implication such thatφ ⇒c ψ = ¬cφ ∨c ψ (here¬c denotes


the classic logic negation). In what follows we will see thatthis property holds alsofor a many-valued implication defined as relative pseudo-complement in a completedistributive latticeX , that is for theintuitionistic logic implication.

Thus, the autoreferential semantics for many-valued logic, based on the partialorder of the lattice, defines the intensional disjunction and implication as a many-valued generalization of Lewis’s 2-valued logic.

It remains to explain what analgebraicuniversal modal operator for a latticeXis, corresponding to the universal algebraic modal operator Γ in P(X). The answer issimple: it is an identity operatorid : X → X . In fact we have that the homomorphism↓ between the many-valued algebra overX and the powerset algebra overP(X) isextended by↓ id = Γ ↓. Then we obtain that↓ x =↓ id(x) = Γ ↓ x =↓

∨↓ x =↓ x,

and that↓ (x ∨ y) =↓ (id ∨c (x, y)) = (from ↓ id = Γ ↓) = Γ ↓ ∨c(x, y) = (from↓ ∨c =

⋃↓) = Γ

⋃↓ (x, y) = Γ

⋃(↓ x, ↓ y) = Γ(↓ x

⋃↓ y).

While, ↓ (x ∧ y) =↓ (id ∧c (x, y)) = (from ↓ id = Γ ↓) = Γ ↓ ∧c(x, y) = (from↓ ∧c =

⋂↓) = Γ

⋂↓ (x, y) = Γ

⋂(↓ x, ↓ y) = Γ(↓ x

⋂↓ y) = (the operatorΓ is

an identity for stable sets)=↓ x⋂↓ y =↓ (x ∧ y).

That is, the modal operatorid has no effect on the conjunction, because↓ x⋂↓

y ∈ F(X) is a stable set, different from↓ x⋃↓ y /∈ F(X) which is not.

Thus we obtain the following modal version of the autoreferential representationfor many-valued algebras based on a complete latticeX :

↓: (X,≤,∧,∨, id, 0, 1) ≃ (F(X),⊆,⋂, Γ

⋃, idF(X), {0}, X)

* (P(X),⊆,⋂,⋃,Γ, ∅, X),

i.e.,F(X) is not a subalgebraof the powerset Boolean distributive algebraP(X) witha modal operatorΓ, as in the standard representation theorems for algebras. In fact,here we do not impose thatX (or F(X)) has to be distributive, as it holds for anyBoolean sublattice. �

EXAMPLE 8. — The smallestnontrivial bilattice is Belnap’s 4-valued bilattice(Belnap, 1977; Fitting, 1991; Majkic, 2007)B = {t, f,⊥,⊤}wheret is true, f isfalse,⊤ is inconsistent (both true and false) orpossible, and⊥ is unknown. As Belnapobserved, these values can be given two natural orders:truth order,≤t, andknowl-edgeorder,≤k, such thatf ≤t ⊤ ≤t t, f ≤t ⊥ ≤t t, ⊥ ⊲⊳t ⊤ and⊥ ≤k f ≤k ⊤,⊥ ≤k t ≤k ⊤, f ⊲⊳k t. That is, the bottom element0 for ≤t ordering isf , and for≤kordering is⊥, and the top element1 for≤t ordering ist, and for≤k ordering is⊤.

Meet and join operators under≤t are denoted∧ and∨; they are natural general-izations of the usual conjunction and disjunction notions.Meet and join under≤k aredenoted⊗ and⊕, such that hold:f ⊗ t = ⊥, f ⊕ t = ⊤,⊤∧⊥ = f and⊤∨⊥ = t.

Thebilattice negation(Ginsberg, 1988) is given by¬f = t, ¬t = f , ¬⊥ = ⊥ and¬⊤ = ⊤, In what follows we will use therelative pseudo-complementsfor implication(Belnap’s 4-valued lattice is distributive), defined byx ⇀ y =

∨{z | z ∧ x ≤t

y}, and thepseudo-complementsfor negation, defined by¬tx = x ⇀ f (which isdifferent from the bilattice negation¬). It is easy to see that holds the De Morgan


law ¬t(x ∧ y) = ¬tx ∨ ¬ty. We have that, w.r.t the truth ordering≤t, ↓ f = {f},↓ ⊥ = {f,⊥}, ↓ ⊤ = {f,⊤} and↓ t = B. Thus,↓ ⊥

⊎↓ ⊤ =↓

∨(↓ ⊥

⋃↓ ⊤) =↓∨

{f,⊥,⊤} =↓ t = B 6= ↓ ⊥⋃↓ ⊤, that is

⊎6=

⋃. �

It is easy to verify that the join operator⊎

in the compact set-based representationF(X) reduces to the standard set union

⋃when the complete lattice(X,≤) is a total

ordering (as, for example, the fuzzy logic with the closed interval of realsX = [0, 1]for the set of truth values). It can be extended to alldistributivelattices, such that foranyx ∈ X andY ⊆ X , x ∧ (

∨Y ) =

∨{x ∧ y | y ∈ Y }.

From Birkhoff’s representation theorem (Birkhoff, 1940) for distributive lattices,every finite (thus complete) distributive lattice is isomorphic to the lattice of lower setsof the poset of join-irreducible elements.

PROPOSITION9 (0-LIFTED BIRKHOFF ISOMORPHISM, (BIRKHOFF, 1940)). —LetX be a complete distributive lattice, then we define the following mapping ↓+: X →P(X): for anyx ∈ X , ↓+ x = ↓ x

⋂X , whereX = {y | y ∈ X and y is

join-irreducible}⋃{0}.

We define the setX+ = {↓+ a |a ∈ X } ⊆ P(X), so that↓+∨

= idX+ : X+ →X+ and

∨↓+= idX : X → X . Thus, the operator↓+ is the inverse of the supremum

operation∨

: X+ → X . The set(X+,⊆) is a complete lattice, such that there is thefollowing 0-lifted Birkhoff isomorphism ↓+: (X,≤,∧,∨) ≃ (X+,⊆,

⋂,⋃

).

PROOF. — Let us show the homomorphic property of↓+:↓+ (x ∧ y) =↓ (x ∧ y)

⋂X = (↓ x

⋂↓ y)

⋂X = (↓ x

⋂X)

⋂(↓ y)

⋂X) =

=↓+ x⋂↓+ y, and

↓+ (x ∨ y) =↓ (x ∨ y)⋂X = (↓ x

⋃↓ y)

⋂X = (↓ x

⋂X)

⋃(↓ y)

⋂X) =

=↓+ x⋃↓+ y. The isomorphic property holds from Bikhoff’s theorem. �

The name “lifted” is used to denote the difference from the original Birkhoff’sisomorphism: that is, we have that for anyx ∈ X , 0 ∈↓+ x , so that↓+ x is never anempty set (it is lifted by the bottom element0).

Notice that whenX is a distributive lattice then(X+,⊆,⋂,⋃

) is asubalgebraofthe powerset Boolean algebra(P(X),⊆,

⋂,⋃

), differently from the case whenX isnot distributive. Thus, we have↓+: (X,≤,∧,∨, idX) ≃ (X+,⊆,

⋂,⋃, idX+) ⊆ (P(X),⊆,

⋂,⋃,Γ+),

whereΓ+ =↓+∨

: P(X) → P(X) is a modal algebraic monotone multiplicativeoperator. Its reduction to the subset of join-irreducible elementsX ⊆ X is equal to

Γ+ = Γ : P(X)→ P(X).

EXAMPLE 10. — Belnap’s bilattice in Example 1, is a distributive lattice w.r.t the≤t ordering with two join-irreducible elements⊥ and⊤. In that case we have that↓+ t =↓+ (⊥ ∨⊤) =↓+ ⊥

⋃↓+ ⊤ =↓ ⊥

⋃↓ ⊤ = {f,⊥,⊤} 6= ↓ t = B. �

Now we will introduce a hierarchy of negation operators for complete lattices,based on their homomorphic properties: the negation with the lowest requirements(such that it inverts the truth ordering of the lattice of truth values and is able to pro-


duce falsity and truth,i.e., the bottom and top elements of the lattice), denominated"general" negation, can be defined in any complete lattice (see the example below).

DEFINITION 11 (HIERARCHY OF NEGATION OPERATORS). — Let (X,≤,∧,∨) bea complete lattice. Then we define the following hierarchy ofnegation operators onit:

1) A general negation is a monotone mapping between posets (≤OP is inverseof≤), ¬ : (X,≤)→ (X,≤)OP , such that{0, 1} ⊆ {y = ¬x | x ∈ X}.

2) A split negation is a general negation extended into the join-semilattice ho-momorphism,¬ : (X,≤,∨) → (X,≤,∨)OP , with (X,≤,∨)OP = (X,≤OP ,∨OP ),∨OP = ∧.

3) A constructive negation is a general negation extended into full latticehomomorphism, ¬ : (X,≤,∧,∨) → (X,≤,∧,∨)OP , with (X,≤,∧,∨)OP =(X,≤OP ,∧OP∨OP ), and∧OP = ∨.

4) A De Morgan negation is a constructive negation when the lattice homomor-phism is an involution (¬¬x = x).

The names given to these different kinds of negations followfrom the fact thata split negation introduces the second right adjoint negation, a constructive negationsatisfies the constructive requirement (as in Heyting algebras)¬¬x ≥ x, while a DeMorgan negation satisfies well known De Morgan laws:

LEMMA 12 (NEGATION PROPERTIES). — Let (X,≤) be a complete lattice. Thenthe following properties for negation operators hold: for anyx, y ∈ X ,

1) for general negation: ¬(x ∨ y) ≤ ¬x ∧ ¬y, ¬(x ∧ y) ≥ ¬x ∨ ¬y, with¬0 = 1, ¬1 = 0.

2) for split negation: ¬(x∨y) = ¬x∧¬y, ¬(x∧y) ≥ ¬x∨¬y. It is an additivemodal operator with right adjoint (multiplicative) negation∼: (X,≤)OP → (X,≤),and the Galois connection¬x ≤OP y iff x ≤∼ y, such that ∼ ¬x ≥ x and¬ ∼ x ≥ x.

3) for constructive negation: ¬(x ∨ y) = ¬x ∧ ¬y, ¬(x ∧ y) = ¬x ∨ ¬y.It is a selfadjoint operator,¬ =∼, with ¬¬x ≥ x satisfying theproto De Morganinequalities ¬(¬x ∨ ¬y) ≥ x ∧ y and ¬(¬x ∧ ¬y) ≥ x ∨ y.

4) for De Morgan negation (¬¬x = x): it satisfies also De Morgan laws¬(¬x∨¬y) = x ∧ y and ¬(¬x ∧ ¬y) = x ∨ y, and is contrapositive,i.e., x ≤ y iff¬x ≥ ¬y.

PROOF. —

1) From the definition of ageneral negation asmonotonicmapping betweenposets,¬ : (X,≤) → (X,≤)OP , we have thatx ≤ y implies¬x ≤OP ¬y, i.e. itinverts the ordering,¬x ≥ ¬y. Thus,x∧y ≤ x andx∧y ≤ y implies¬(x∧y) ≥ ¬xand¬(x ∧ y) ≥ ¬y, consequently¬(x ∧ y) ≥ ¬x ∨ ¬y. Analogously,x∨ y ≥ x andx∨y ≥ y implies¬(x∨y) ≤ ¬x and¬(x∨y) ≤ ¬y, consequently¬(x∨y) ≤ ¬x∧¬y.From the property∀y(0 ≤ y implies¬0 ≥ ¬y and the fact that1 ∈ {z = ¬y | y ∈ X}must hold¬0 ≥ 1, i.e., ¬0 = 1. Analogously we can show also that¬1 = 0.


2) From the homomorphic definition of thesplit negation (which is also general,thus with¬(x∧ y) ≥ ¬x∨¬y) we have that¬(x∨ y) = ¬x∨OP ¬y = ¬x∧¬y, andfrom ¬0 = 1 = 0OP we conclude that it is a monotone additive mapping, thus it hasthe right multiplicative adjoint∼: (X,≤)OP → (X,≤) with the Galois connection¬ ⊣∼, i.e.,¬x ≤OP y iff x ≤∼ y, that is,¬x ≥ y iff x ≤∼ y. The operators¬ ∼ and∼ ¬ are the closure operators inP(X), it thus holds that¬ ∼ x ≥ x and∼ ¬x ≥ x.

3) Theconstructivenegation is also a split negation, thus it is an additive modaloperator. Let us show that it is also multiplicative, consequently self adjoint,i.e.,∼= ¬: it holds from the lattice homomorphism for meet operators,¬(x ∧ y) =¬x ∧OP ¬y (i.e., = ¬x ∨ ¬y), and from¬1 = 0 = 1OP . Thus¬¬x = ¬ ∼ x ≥x is its constructive property. From this constructive property and the additive andmultiplicative properties we obtain that¬(¬x∧ ¬y) = ¬¬x∨ ¬¬y ≥ x∨ y (because∨ is monotone for both arguments). Analogously,¬(¬x ∨ ¬y) = ¬¬x ∧ ¬¬y ≥x∧ y (because also∧ is monotone for both arguments),i.e., we obtained the proto DeMorgan inequalities.

4) The De Morgan negation is the split negation which is an evolution, thus wehave that¬(¬x∧¬y) = ¬¬x∨¬¬y = x∨y and¬(¬x∨¬y) = ¬¬x∧¬¬y = x∧y.We have that¬x ≥ ¬y implies¬¬x ≤ ¬¬y, so from¬¬x = x we obtain also that¬x ≥ ¬y impliesx ≤ y, thus the contrapositionx ≤ y iff ¬x ≥ ¬y. �

Notice the naturality of this hierarchy, where from the general negation (the mostweak negation) we reach from the antitonicityx ≤ y implies ¬x ≥ ¬y thestronger requirement of the contraposition (x ≤ y iff ¬x ≥ ¬y) for De Morgan (themost strong) negation. Notice also how we pass from inequalities to stronger equalityrequirements. This is valid for any complete lattice where these negation exist (generalnegations always exist). If we impose the stronger requirements on complete lattices,as for example distributivity, then De Morgan negations coincide with the strongestclassic Boolean negation (where for allx ∈ X , ¬x ∧ x = 0 and¬x ∨ x = 1).

The set-based semantics for the split negations (with Galois connections) can begiven by the Bikhoff polarity operatorλ : (P(X),⊆)→ (P(X),⊆)OP , such that foranyU ∈ P(X), λU = {x ∈ X | ∀u ∈ U.((u, x) ∈ R)} = {x ∈ X | ∀u.(u ∈ Uimplies (u, x) ∈ R)}, which in a modal logic can be represented by "M |=x ¬φ iff∀u.(M |=u φ implies (u, x) ∈ R)" (the relationR is an incompatibility relation inLemma 6). The set-based semantics for the constructive negation is obtained in thecase where the incompatibility relationR is symmetric.

EXAMPLE 13. — The example for any complete lattice(X,≤) is ageneralnegationgiven by¬x =

∨Sx whereS = {z ∈ X | z ∧ x = 0}. It is well defined for

complete lattices also whenSx is aninfinite set, as for example in fuzzy logic whereX = [0, 1] is the closed interval of reals between 0 and 1, where for¬0 we havethatS0 = {z ∈ X | z ∧ 0 = 0} = [0, 1]. Notice that differently from distributivelattices, generally we have that in non distributive lattices¬x /∈ Sx, and¬x ∧ x 6= 0(we have that¬x ∧ x ∈ {0, x}, i.e., ¬x ∧ x = x if x ≤ ¬x (it cannot bex > ¬x),and¬x ∧ x = 0 wherex and¬x are not comparable; thus¬x ∧ x ≤ x). Thisnegation indistributivelattices corresponds to the pseudocomplement negation where


for anyx ∈ X , x ∧ ¬x = 0 (thus it cannot be used as paraconsistent negation) and is"constructive"x ≤ ¬¬x but¬¬x � x. This negation in distributive algebras, wherex ∨ ¬x = 1 for anyx ∈ X is also valid, corresponds to the classic Boolean negation.

We are able to define also "non-constructive" general negation in any completelattice by¬ =

∨− ↓: X → X where− is the set complement inX , such that¬x =

0 if x = 1; 1 otherwise. It is easy to verify that for anyx ∈ X , ¬x ≤ ¬x,¬x∨x =1 and¬¬x ≤ x. One example for distributive complete lattice withparaconsistentDe Morgan negation is the fuzzy negation¬x = 1 − x for x ∈ X = [0, 1] (wherex∧¬x = min(x, 1−x) 6= 0). In fact, it is additive and multiplicative (¬(x∧y) = 1−min(x, y) = max(1−x, 1−y) = ¬x∨¬y and¬(x∨y) = 1−max(x, y) = min(1−x, 1 − y) = ¬x ∨ ¬y), thus it is self adjoint with¬¬x = x. Another paraconsistentDe Morgan negation is thebilattice negation(Ginsberg, 1988): in Belnap’s 4-valuedbilattice it corresponds to the epistemic negation, such that¬f = t, ¬t = f , ¬⊥ = ⊥and¬⊤ = ⊤, with x ∧ ¬x 6= 0 (while the pseudocomplement negation¬t given inExample 8 (which is not a bilattice negation) is not paraconsistent). �

The most simple definition for implication for bounded lattices which satisfies theModus Ponens (y ⇒ x, y ⊢ x) and Deduction Theorem (z, y ⊢ x iff z ⊢ y ⇒ x)inference rules where⊢ relation is equal to the≤ partial order in the lattice (as inDefinition 2, and where the logic entailment “preserves the truth”, i.e., x ⊢ y iffx ≤ y), is given by,y ⇒ x = 1 if y ≤ x; x otherwise. But as we can see in thisdefinition the value ofy does not appear in the resulting value of the implication.

Both valuesx andy appear in the resulting value of the implication in the casewhen the lattice is modular (Dedekind’s lattices) and satisfies¬x ∧ x = 0 for everyx ∈ X , when the Deduction Theorem from left to right does not hold.

The lattice is calledmodularwhen “x ≤ z implies x ∨ (y ∧ z) = (x ∨ y) ∧ z”.

PROPOSITION14. — LetX be a completemodular lattice where¬x =∨{z | z ∧

x = 0} and¬x∧x = 0 for everyx ∈ X , then we define the implication⇒: X×X →X for any two elementsx, y ∈ X as follows:

y ⇒ x =def

1 if y ≤ x

¬y ∨ x if y > x

¬y ∨ (x ∧ y) otherwise.

Then,y ⇒ x = 1 iff y ≤ x, and hold the Modus Ponens “rule”,y ∧ (y ⇒ x) ≤ xand half of the Galois connection, that is, “z ≤ y ⇒ x implies z ∧ y ≤ x”.

If we assume that the consequence pair relation⊢ of this logic coincides with thepartial order≤ of this complete modular lattice (in Definition 2), then in this logichold the Modus Ponens ruley ∧ (y ⇒ x) ⊢ x and the partially Deduction Theorem,that is, “z ⊢ y ⇒ x implies z, y ⊢ x”.

PROOF. — The definition of the implication is correct, that is, it ismonotonic w.r.tthe first argument and antitonic w.r.t the second argument. Moreover it is similar tothe definition of the classic material implication.


Let us show that holds the Modus Ponens “rule”, that is,(y ⇒ x) ∧ y ≤ x.

1.1 case wheny ≤ x: then,(y ⇒ x) ∧ y = 1 ∧ y = y ≤ x.

1.2 case wheny > x: then,(y ⇒ x) ∧ y = (¬y ∨ x) ∧ y =(by modular property)= (¬y ∧ y) ∨ x = 0 ∨ x = x.

1.3 case whenx andy are not comparable: then,(y ⇒ x) ∧ y = (¬y ∨ (x ∧ y)) ∧y =(by the modular property andx∧y ≤ y) = (¬y∧y)∨(x∧y) = 0∨(x∧y) =x ∧ y ≤ x.

For the Deduction Theorem we have that ifz ≤ y ⇒ x , then:

2.1 case wheny ≤ x: then from,z ≤ y ⇒ x = 1, we havez = 1, so thatz ∧ y = 1 ∧ y = y ≤ x .

2.2 case wheny > x: then from,z ≤ y ⇒ x = ¬y∨x, we have (by the monotonicityof ∧), z ∧ y ≤ y ∧ (¬y ∨ x) = (by the modular property)= (¬y ∧ y) ∨ x =0 ∨ x = x.

2.3 case whenx andy are not comparable: then from,z ≤ y ⇒ x = ¬y ∨ (x ∧ y),we have (by the monotonicity of∧) that, z ∧ y ≤ y ∧ (¬y ∨ (x∧ y)) = (by themodular property)= (¬y ∧ y) ∨ (x ∧ y) = 0 ∨ (x ∧ y) = x ∧ y ≤ x.

Now for y ⇒ x = 1 andz = 1, and (z ≤ y ⇒ x implies z ∧ y ≤ x), we obtainthat (y ⇒ x = 1 implies z ∧ y = 1 ∧ y = y ≤ x), and from the definition of⇒ wehave thaty ≤ x impliesy ⇒ x = 1. Consequently, in this modular lattice we havey ⇒ x = 1 (or, alternatively, ⊢ y ⇒ x) iff y ≤ x.

Thus, if we assume that⊢ is equal to≤, then the MP “rule”y ∧ (y ⇒ x) ≤ xcoincides with the Modus Ponens rule of inference and we obtain ⊢ (y ∧ (y ⇒ x))⇒x, i.e., that (y ∧ (y ⇒ x)) ⇒ x is an axiom. Analogously, the Galois implicationbecomes half of the Deduction Theorem. �

This class of modular lattices is strictly more general thanthe class of orthomodu-lar lattices (Kalmbach, 1983), where it is also required that ¬¬x = x andx∨ ¬x = 1for everyx.

In any distributive lattice, where¬x ∧ x = 0 is always satisfied for everyx ∈ X ,which is also modular, the definition of the implication given in Proposition 14 isalways possible. But as we will see in what follows, in any distributive lattice theModus Ponens and Deduction Theorem inference rules can be completely satisfied,and the logic implication can be defined precisely by this requirement.

4. Heyting’s and multi-modal extensions of distributive lattices

The meet and join operators correspond to the logic conjunction and disjunction. Inorder to have a full logic language we need also logic implication and logic negation.

100 Journal of Applied Non-Classical Logics. Volume 18 – No.1/2008

For the class of distributive lattices we can introduce the negation operator as thepseudo-complement¬x =

∨{z | z ∧ x = 0}, with ¬x = 1 iff x = 0, used in

intuitionistic logics. In this section we will extend this distributive lattice with otherunary modal operators and, based on them, two dual logic operators: theimplication⇀: X ×X → X , and its dualcoimplicationoperator : X ×X → X .

As we will see, the natural choice for the implication in complete lattices is the in-tuitionistic implication, based on the relative pseudo-complement, which always existsin complete distributive lattices. It is well known that therelative pseudo-complementx ⇀ y =

∨{z | z ∧ x ≤ y} and classic implication satisfy the following left expo-

nential commutative diagram,

(x ⇀ y) ∧ x

≤ =

z ∧ x

y

=

y

≤

≤

(x y) ∨ x

≥ =

z ∨ x

y

=

y

≥

≥

where the arrow(x ⇀ y)∧x ≤ y is the Modus Ponens inference rule(x ⇀ y), x ⊢ y,while for anyx ∈ X the additive modal operatorlx = _∧ x : X → X and its rightadjoint multiplicative operatorrx = x ⇀ _ : X → X define the Galois connectionlx ⊣ rx, that is, "lx(z) = z ∧ x ≤ y iff z ≤ x ⇀ y = rx(y)" corresponding to theDeduction Theorem "z , x ⊢ y iff z ⊢ x ⇀ y" (where the consequence relation⊢is equal to≤, based on the "truth preservation" principle for valid logic derivations, inDefinition2).

The family of operators{lx = _∧ x | x ∈ X}may be considered as the family ofexistential modal operators, derived from the basic lattice meet operator∧, while rxacts as its dual universal modal operator, but withrx 6= ¬lx¬ andlx 6= ¬rx¬.

The fundamental property (from the Galois connection whenz = 1) of relativepseudo-complement is that “x ≤ y iff x ⇀ y = 1” ( i.e., “x ⊢ y iff ⊢ x ⇀ y” ).

The right coexponential commutative diagram is dual (the arrows, that is, the par-tial ordering, are inverted), where the conjunction is replaced by disjunction (cocon-junction) and the implication by its dual coimplication. The dual modal operators, forany x ∈ X are the multiplicative modal operatorrcx = _ ∨ x : X → X and itsleft adjoint additive operatorlcx = x _ : X → X define the Galois connectionlcx ⊣ r

cx, that is, “lcx(y) = x y ≤ z iff y ≤ z ∨ x = rcx(z)” corresponding to the

coDeduction Theorem “y ⊢ z ∨ x iff x y ⊢ z”. The arrow(x y) ∨ x ≥ ycorresponds to the coModus Ponens inference ruley ⊢ (x y) ∨ x.


Thus, the coimplication is defined byx y = lcx(y) =∧{z | y ≤ z∨x = rcx(z)},

such that “x ≤ y iff ¬(y x) = 1” ( i.e., “x ⊢ y iff ⊢ ¬(y x)” ).

Thusx ⇀ y is an axiom iff¬(y x) is an axiom, that is,⊢ x ⇀ y iff⊢ ¬(y x).

Notice that in Boolean algebras (distributive lattices with ¬x ∨ x = 1 for anyx ∈ X) we have thatx ⇀ y = ¬x ∨ y andx y = ¬x ∧ y.

PROPOSITION15. — Each complete distributive lattice(X,≤) with bottom andtop element0, 1 respectively, can be extended into Heyting algebraB = (X,≤,∧,∨,¬,⇀, 0, 1) with implication⇀ defined byx ⇀ y =

∨{z ∈ X | z ∧ x ≤ y}

and negation¬ : X → X defined by¬x = x ⇀ 0.

The corresponding complete distributive lattice(F(X),⊆,⋂,⊎, ¬,, {0}, X)

is theCANONICAL Heyting algebra with implication defined, for anyU, V ∈ F(X),by U V =

⊎{Z ∈ F(X) | Z

⋂U ⊆ V } and the negation operator (pseudo-

complement)¬ , ¬U = U {0}.

Then for anyx, y ∈ X , ¬ ↓ x =↓ (¬x) and↓ x ↓ y =↓ (x ⇀ y), that is, thefollowing representation for any complete Heyting algebrais valid:

1) ↓: (X,≤,∧,∨,¬,⇀, 0, 1) ≃ (F(X),⊆,⋂,⊎, ¬,, {0}, X), and,

2) ↓+: (X,≤,∧,∨,¬,⇀, 0, 1) ≃ (X+,⊆,⋂,⋃, ¬,, {0}, X),

where¬, are obtained in the same way as above by substitutingF(X) byX+,⊎

by⋃

, and↓ by↓+.

PROOF. — We obtain↓ x↓ y =⊎{↓ z ∈ F(X) | ↓ z

⋂↓ x ⊆↓ y} =

⊎{↓ z ∈

F(X) | ↓ (z∧x) ⊆↓ y} =⊎{↓ z ∈ F(X) | z∧x ≤ y} =↓ ∨{z ∈ X | z∧x ≤ y} =↓

(x ⇀ y) ∈ F(X), that is, it is a stable set. Analogously,¬ ↓ x =↓ (¬x) ∈ F(X) isa stable set. �

Now we are able to introduce also the unary modal operators for these Heytingalgebras, based on complete distributive lattices.

PROPOSITION16. — Each Heyting algebraA = (X,≤,∧,∨,¬,⇀, 0, 1) hasan invariant modal extension denominated banal Galois algebra G = (X,≤,∧,∨, id,¬,⇀, 0, 1), isomorphic to (F(X),⊆,

⋂,

⊎, idF(X), ¬,, {0}, X),

where⊎

= Γ⋃

and the identity functionid : X → X is a self adjoint id ⊣ idmodal operator correspondent to the universal modal operator Γ : P(X) → P(X),whose reduction to the set of stable elements inF(X) ⊂ P(X) is the identity operatoridF(X) =↓

∨: F(X)→ F(X).

We can extend a Heyting algebraA by any other unary additive modal operatorf : X → X , or f : X → XOP (for modal negations in Lemma 12). So that we obtainthe extended representation isomorphism for obtained Galois algebras:

1) ↓: (X,≤,∧,∨, id,¬,⇀, f, 0, 1) ≃ (F(X),⊆,⋂,

⊎, idF(X), ¬,

, f , {0}, X) where f =↓ f ↓−1: F(X) → F(X), and ↓−1=∨

is inverse ho-momorphism of ↓ .


2) ↓+: (X,≤,∧,∨, id,¬,⇀, f, 0, 1) ≃ (X+,⊆,⋂,

⋃, idX+ , ¬,

, f , {0}, X), obtained in the same way as in point 1, by substitutingF(X) byX+,⊎by

⋃, and↓ by↓+.

PROOF. — LetA = (X,≤,∧,∨,¬,⇀, 0, 1) be a Heyting algebra. Then the identityfunctionid : X → X is monotone,i.e., x ≤ x′ impliesid(x) = x ≤ y = id(y), andpreserves the meets: for anyx, y ∈ X , id(x ∧ y) = x ∧ y = id(x) ∧ id(y). Thus it isequivalent to the banal Galois algebraG = (X,≤,∧,∨, id,¬,⇀, 0, 1).

The left adjoint to id is the operatorf : X → X , such thatf(x) =∧{z ∈

Y | id(z) ≥ x} = x, i.e., f = id, so that id is self adjoint.

By introducing this identity modal operator we obtain the Galois banal algebraidentical to the original Heyting algebra. So this is aninvariant modal extensionfor Heyting algebras, and can be considered as minimal modalextension for Heytingalgebras. This fact explains that modal logic is intrinsic in any complete lattice(X,≤),and is based on the partial ordering≤ of the posetX .

Let f : X → X be an additive modal operator (monotone which preserves allmeets), then it has left adjointg : X → X , such thatg(x) ≤ y iff x ≤ f(y). Thenx ≤ y impliesf(x) ≤ f(y), and↓ f(x) ≤↓ f(y), i.e., ↓ f ↓−1 (↓ x) ≤↓ f ↓−1 (↓ y)with ↓ x ⊆↓ y ∈ F(X), that is, the monotone operator↓ f ↓−1: F(X) → F(X)exists.

Thus, for anyU, V ∈ F(X), ↓ f ↓−1 (U⋂V ) =↓ f ↓−1 (↓ (x ∧ y)) =↓

f(x∧y) =↓ f(x)⋂↓ f(y) =↓ f ↓−1 (↓ x)

⋂↓ f ↓−1 (↓ x). Consequently↓ f ↓−1

preserves the meets, and both with a monotone property it means that it is an additiveunary modal operator in a canonical Heyting algebra.

Moreover, fromg(x) ≤ y iff x ≤ f(y) we obtain↓ g ↓−1 (↓ x) ≤↓ y iff ↓ x ≤↓f ↓−1, that is↓ g ↓−1 is the left adjoint of the↓ f ↓−1, with ↓ g ↓−1= ¬ ↓ g ↓−1 ¬.Consequently↓ f ↓−1 is the universal while↓ g ↓−1 is the existential modal operatorin the canonical Heyting (i.e., Galois) algebraF(X). �

EXAMPLE 17. — In Belnap’s bilattice the conflation− is a monotone functionwhich preserves all finite meets (and joins) w.r.t the lattice (B,≤t), thus it is an uni-versal (and existential,i.e., f = g = −, because− = ¬t − ¬t) modal many-valuedoperator: " it is believed that", which extends the 2-valuedbelief of the autoepistemiclogic as follows:

1) if A is true then "it is believed that A",i.e.,−A, is true;2) if A is false then "it is believed that A" is false;3) if A is unknown then "it is believed that A" is inconsistent: it isreally inconsis-

tent to believe in something that is unknown;4) if A is inconsistent (that isboth true and false) then “it is believed that A” is

unknown: really, we can not tell anything about believing insomething that is incon-sistent.


This belief modal operator can be used to define theepistemic(bilattice) negation¬,as a composition of strong negation¬t and this belief operator,i.e., ¬ = ¬t−. Thatis, the epistemic negation is negation as a modal operator (Došen, 1986) and as aparaconsistent negation (Béziau, 2002). We can introduce also Moore’sautoepistemicoperator (Ginsberg, 1988),µ : B → B, for a Belnap’s bilattice, defined byµ(x) = tif x ∈ {⊤, t}; f otherwise.

It is easy to verify that it is monotone w.r.t the≤t, that is multiplicative (µ(x∧y) =µ(x) ∧ µ(u) andµ(t) = t) and additive (µ(x ∨ y) = µ(x) ∨ µ(u) andµ(f) = f ),consequently also it is a selfadjoint (contemporary universal and existential) modal op-erator,µ = ¬tµ¬t. Notice that differently from the belief modal operator−, Moore’smodal operatorµ is not surjective.

Such an autoepistemic intuitionistic logic, based on the 4-valued Belnap’s bilat-tice, (B,≤t,∧,∨,¬t,⇀,−, µ, f, t) can be usefully used for logic programming withincomplete and partially inconsistent information (Majkic, 2005a) (in such logic weuse only the epistemic negation¬ = ¬t−). �

We have seen that the for distributive lattices we can define the "strict" implication(relative pseudo-complement), such that for any two logic formulaeφ andψ holdsthatφ ⇀ ψ = �(φ ⇒c ψ) where⇒c is the classical extensional implication suchthat φ ⇒c ψ = ¬cφ ∨c ψ (here¬c and∨c denote the classic logic negation anddisjunction respectively) and� is not necessarily truth-functional in this distributivelattice. This “strict” implication is theintuitionistic logic implication, where all axiomsof the classic propositional logic are satisfied, except theaxiom for disjunction¬x ∨x = (x ⇀ 0) ∨ x.

Now we will investigate the subclass of distributive complete lattices, where thereexists thealgebraic truth-functionalcounter-partr� : X → X , of the logic modalnecessary operator�, and is an identity, so that¬x ∨ y (i.e., (x ⇀ 0) ∨ y) can beused as implicationx ⇒ y. In what follows, we will show that this “or-implication”satisfies the Modus Ponens, but generally does not satisfy the Deduction Theorem,

PROPOSITION18. — The or-implicationx ⇒ y =def ¬x ∨ y satisfies the ModusPonens but not the Deduction Theorem w.r.t the entailment inDefinition 2, that is, foranyx ∈ X , r = ¬x ∨ _ : X → X is a monotone multiplicative operator, thus rightadjoint of the operatorl : X → X , with l 6= _∧ x.

PROOF. — For the or-implication we have that, from the distributivity, (x⇒ y)∧x =(¬x ∨ y) ∧ x = (¬x ∧ x) ∨ (y ∧ x) = 0 ∨ (y ∧ x) = y ∧ x ≤ y, that is the ModusPonens rule holds.

The operatorr = ¬x∨_ is monotone because∨ is monotone for both arguments.And, from the distributivity,r(y∧z) = ¬x∨(y∧z) = (¬x∨y)∧(¬x∨z) = r(y)∧r(z),andr(1) = ¬x ∨ 1 = 1, thusr is multiplicative, so that its left adjoint operatorl isdefined for anyy ∈ X , by l(y) =

∧{z | y ≤ r(z) = ¬x ∨ z}, with l(y) 6= y ∧ x. �

The Deduction Theorem can not be satisfied because there exist x, y ∈ X suchthat ¬x ∨ y < x ⇀ y =

∨{z | z ∧ x ≤ y}, and if we substitute the relative


pseudo-complementx ⇀ y by "or-implication"¬x ∨ y we are not able to obtainthe commutative exponential diagram as in the case of a relative pseudo-complement.

Thus, in order to satisfy the Deduction Theorem, the or-implicationr(y) = ¬x∨ymust be a right adjoint to l = _ ∧ x, that is, must be equal to the relative pseudo-complementx ⇀ y. In the next proposition we will define the subclass of completedistributive lattices where such a condition is satisfied.

PROPOSITION19. — Each complete distributive lattice(X,≤) with the set of join-irreducible elementsX − {0} equal to the set of atomsX, is a complete distributivelattice, where for everyx, y ∈ X , x ⇀ y = ¬x ∨ y and viceversa.

PROOF. — First of all we will show thatX = X − {0} is a sufficient condition forx ⇀ y = ¬x ∨ y. Each atom is join-irreducible, thusX ⊆ X − {0}. Suppose

thatX ⊂ X − {0}. Then, from the Birkhoff theorem for distributive lattices,∨X <∨

(X − {0}) = 1, and for any atoma ∈ X, ¬a <∨X (becausea <

∨X), so that

¬a∨a ≤∨X∨a =

∨X < 1, whilea ⇀ a = 1, i.e., we obtain that¬a∨a 6= a ⇀ a,

which is a contradiction. ThusX = X − {0}must be correct.

Let us assume thatX = X − {0}, and show that for everyx, y ∈ X , x ⇀ y =¬x ∨ y must hold. We consider the following cases:

1. case whenx ≤ y: in that casex ⇀ y = 1, and, from the Birkhoff iso-morphism (Proposition 9), we have that for unique subsetsSx, Sy ⊆ X suchthat x =

∨Sx, y =

∨Sy, must holdSx ⊆ Sy, i.e., Sy = Sx

⋃△, and

−Sx⋃Sy = −Sx

⋃(Sx

⋃△) = X

⋃△ = X. Thusx ⇀ y = 1 =

∨X =∨

(−Sx⋃Sy) =

∨(−Sx) ∨ (

∨Sy) = ¬x ∨ y.

2. case whenx � y and¬x = 0: Let us show that in this casex = 1 must becorrect. Suppose that∃x 6= 1.¬x = 0, then∄y ⊲⊳ x.(y 6= 0 andy = ¬x), thatis x =

∨X but1 =

∨X, and from the Birkhoff bijection (isomorphism) must

bex = 1, so thatx ⇀ y = 1 ⇀ y =∨{z | z ∧ 1 = z ≤ y} = y = ¬x ∨ y.

This case holds whenx > y. In fact,¬x = 0 must be correct, otherwise thesublattice0 < ¬x < ¬x∨ x, 0 < y < x < ¬x ∨ x is a N5 diagram which cannot be a sublattice of modular (thus distributive) lattice(X,≤).

3. case whenx ⊲⊳ y and¬x 6= 0: it is enough to consider onlyx, y /∈ {0, 1}which are particular subcases of the cases above. It is easy to verify that for anydistributive lattice it holds that(**) x ⇀ y ≥ ¬x ∨ y (because x ⇀ y ≥ ¬x and x ⇀ y ≥ y, fromy ∈ {z | z ∧ x ≤ y}.Suppose thatx ⇀ y > ¬x ∨ y, then there are the following subcases:

3.1 wheny > ¬x =∨{z | z ∧ x = 0}: then¬x ∨ y = y, so thaty ≥ x

(otherwise,¬x = y) which is in contradiction withx ⊲⊳ y.3.2 wheny < ¬x: then we obtain the following N5 sublattice,0 < x <

x ∨ ¬x, 0 < y < ¬x < x ∨ ¬x which can not hold for modular (thusdistributive) lattices.


3.3 wheny = ¬x: thenx ⇀ y < 1 (becausex y). Cannot bex ≤ x ⇀ y.If so, thenx ∧ (x ⇀ y) = x (Modus Ponens in distributive lattices),i.e., y =¬x ≥ xwhich is in contradiction withx ⊲⊳ y. Cannot bex > x ⇀ y (if so, then¬x∨ y = y ≥ x∧ (x ⇀ y) = x ⇀ y which is in contradiction with hypothesisx ⇀ y > ¬x ∨ y). Thus we obtain the N5 sublattice0 < x < x ∨ (x ⇀ y),0 < ¬x ∨ y < x ⇀ y < x ∨ (x ⇀ y) which cannot hold for modular (thus

distributive) lattices.3.4 wheny ⊲⊳ ¬x: then, cannot bex ≤ x ⇀ y (if so, thenx = x ∧ (x ⇀

y) ≤ y (from definition ofx ⇀ y), which is in contradiction withx ⊲⊳ y.Suppose thatx ≥ x ⇀ y, thenx ⇀ y = x ∧ (x ⇀ y) ≤ y (from definitionof x ⇀ y), and fromx ⇀ y ≥ y we obtain thatx ⇀ y = y. Thus,¬x ∨ y ≥y = x ⇀ y, which is in contrast with (**) x ⇀ y ≥ ¬x ∨ y. Consequently,x ⊲⊳ (x ⇀ y), andx ∨ (x ⇀ y) > x ⇀ y andx ∨ (x ⇀ y) > x (noticethat fromx ⊲⊳ y we have thatx ∧ y < x, and¬x ∨ y > x ∧ (¬x ∨ y) =(x ∧ ¬x) ∨ (x ∧ y) = 0 ∨ (x ∧ y) = x ∧ y).

Thus we obtain the N5 sublatticex ∧ y < x < x ∨ (x ⇀ y), x ∧ y < ¬x ∨ y <x ⇀ y < x ∨ (x ⇀ y) which cannot hold for modular (thus distributive) lattices. �

REMARK . — If for everyx, y ∈ X , x ⇀ y = ¬x ∨ y then ¬x ∨ x = x ⇀ x = 1,and from the general property for distributive lattices,¬x ∧ x = 0, we obtain that thedistributive lattices with the property defined in Proposition 19 are Boolean algebras.

�

EXAMPLE 20. — The classic logic with 2-valued lattice(2 = {0, 1},≤) is aBoolean algebra: the set of atoms is equal to the set of join-irreducible elements{1}. The cartesian productX = 22 = 2 × 2 = {(0, 0), (0, 1), (1, 0), (1, 1)},with (x, y) ≤ (v, w) iff x ≤ v and y ≤ w, is a Boolean algebra with the setof atoms (i.e., join-irreducible elements)X = {(0, 1), (1, 0)} and negation definedby ¬(x, y) = (¬x,¬y). It is isomorphic to the Belnap’s 4-valued latticeB4 wheref 7→ (0, 0), t 7→ (1, 1),⊥ 7→ (0, 1) and⊤ 7→ (1, 0).

It is easy to verify that every latticeXn = (2K ,≤), whereK ≥ 2 and(x1, ..., xk) ≤ (y1, ..., yk) if for all 1 ≤ i ≤ K, xi ≤ yi, is a Boolean alge-bra where negation is defined by¬(x1, ..., xk) = (¬x1, ...,¬xk). But we havealso that in each Boolean algebra(P(S),⊆,

⋂,⋃,−) where− is the set comple-

ment, the setS is exactly the set of atoms). Thus we have the following isomor-phisms of latices:(2,≤) ≃ (P({a1}),⊆), (B4,≤) ≃ (22,≤) ≃ (P({a1, a2}),⊆),..., (2K ,≤) ≃ (P({a1, ..., aK}),⊆),K ≥ 2. �

5. Autoreferential Kripke-style semantics for many-valued logics

We will use the invariant and intrinsic modal properties of complete lattices andHeyting algebras in order to obtain a basic (or canonical) Kripke structure for any suchalgebra in a general way.


By 2 = {0, 1} ⊆ X we denote the set of classic logic values (false and truerespectively), and byH the Herbrand base,i.e., the set of all ground atoms in a many-valued predicate logicL, withH ⊆ L.

Given a many-valued interpretationI : H → X , we denote byI : L → Xits unique extension to all formulae, obtained inductivelyas follows: for any groundatom r(c1, ..., cn) ∈ H , I(r(c1, ..., cn)) = I(r(c1, ..., cn)); for any two formulaeφ, ψ ∈ L, I(φ ∧ ψ) = I(φ) ∧ I(ψ); I(φ ∨ ψ) = I(φ) ∨ I(ψ); I(φ ⇒ ψ) =I(φ) ⇀ I(ψ); I(¬φ) = ¬I(φ), and for any algebraic additive modal operatoroi,I(♦iφ) = oiI(φ), where♦i is a logic symbol for an existential modal operator (and�i is its correspondent universal logic modal operator).

Based on the general Representation Theorem for many-valued logic in (Majkic,2006d), and theautoreferential assumption, we assume that the set of possible worldsof relational Kripke frames is the set of algebraic truth-values in a complete latticeX of this many-valued logic , or, alternatively, the set of join-irreducible elementsX(Majkic, 2006d) for distributive complete lattices with normal additive modal opera-tors only. In what follows we will present the first approach,which is valid also fornon-distributivelattices (for logics where the implication cannot be a relative pseudo-complement).

From the formal mathematical point of view, we can consider the stable sets inF(X) as ’stable characteristic functions’ in2X (2X denotes the set of all functionsfromX to 2), that is, the functions whose image is a stable set (elementin F(X). LetFX ≃ F(X) denote such a set of stable functions, bijective toF(X), whose elementsare functions (i.e., higher-order algebraic truth-values (Majkic, 2006c)). Than anymany-valued valuationv, of a propositional logicL with a set of propositional lettersin P , based on this set of higher-order truth-values is a mappingv : P → FX ⊂ 2X .So that we are able to define the valuationV : P×X → 2, such that for anyp ∈ P, x ∈X , V (p, x) = v(p)(x) ∈ 2, which is identical to Kripke valuations used for standard2-valued Kripke models of propositional modal logics, if weaccept (by autoreferentialassumption) for the set ofpossible worldsthe set of algebraic truth-valuesX .

The philosophical autoreferential assumption is based on the consideration thateach possible world represents a level ofcredibility, so that only the propositions withthe right logic value (i.e., level of credibility) can be accepted by this world.

Now we will define the accessibility relation for any given modal operatoroi.

DEFINITION 21. — Let oi : X → X be a monotonic operator withS ={(oi(x), x) | x ∈ X}, and the setSm ⊆ S obtained by elimination of each element(x, y) ∈ S if there exists another element(x, z) ∈ S such thatz ≤ y.

Then we define the accessibility relation foroi by

Ri = Sm⋃{(x, y) | ∄(x, v) ∈ Sm} where

{y = 0 if x = 0; otherwise

y = z if ∃(v, z) ∈ Sm.(x ≤ v).


Notice that if x 6= 0 and ∄y ∈ X.(x ≤ oi(y)) thenx /∈ π1(Ri), whereπ1 is thefirst projection.

EXAMPLE 22. — In the case of the believe (conflation) modal operator− in Bel-nap’s bilattice, we obtain thatR− = Sm = S = {(f, f), (⊤,⊥), (⊥,⊤), (t, t)},while for the autoepistemic Moore’s operatorµ, S = {(f, f), (f,⊥), (t,⊤), (t, t)},Sm = {(f, f), (t,⊤)} ⊂ S andRµ = {(f, f), (t,⊤), (⊤,⊤), (⊤, t), (⊥,⊤), (⊥, t)}.

�

DEFINITION 23. — Let oi : (X,≤) → (X,≤)OP be a split negation operator. Wedefine the binary relationS as follows:

(a) S = {(x, oi(x)) | x ∈ X},

(b) For everyy ∈ X substituteS by the relation(S − Sy)⋃{(

∨π1Sy, y), where

Sy = {(x, y) | (x, y) ∈ S} ⊂ S andπ1 is the first projection.

Then we define the relationRi for oi by: S ⊆ Ri and if (x, y) ∈ Ri andx′ ≤ x,y′ ≤ y then(x′, y′) ∈ Ri.

The split negation operatorsoi correspond to the monotone additive operatorsoi : X → XOP , and the correspondent relationsRi are "perp" (Perpendicular orincompatibility) relations, introduced by Goldblatt in (Goldblatt, 1974).

Notice that the point (b) for eachy ∈ X substitutes the equivalence classSy bythe single pair(x, y) wherex =

∨π1Sy =

∨{z | (z, y) ∈ Sy}, so that the following

lemma holds (byπ2 we denote the second projection):

LEMMA 24. — Let S be a binary relation in Definition 23 for a split negation op-erator oi = ¬ : (X,≤) → (X,≤)OP . Then¬r : (S1,≤) → (S2,≤)OP , whereS1 = π1S ⊆ X , S2 = π2S ⊆ X is the bijective domain reduction of the split opera-tor ¬ (i.e., for eachx ∈ S1, ¬rx = ¬x), with its inverse¬−1

r : (S2,≤)OP → (S1,≤),such that

1) for any twox, x′ ∈ S1, x < x′ iff ¬x > ¬x′,2) for anyx ∈ X we have thatx′ = ¬−1

r ¬x ∈ S1, such that¬x = ¬rx′ withx ≤ x′.

PROOF. — Let us show that¬r is a domain reduction of¬. In fact, after the actionsdefined in point (b) of Definition 23, we have that the cardinalities of S1 andS2 areequal, and for anyy ∈ S2 we have the unique pair(x, y) ∈ S wherex =

∨π1Sy =∨

{z | (z, y) ∈ Sy}. Let us show that¬rx = y: in fact, by definition¬rx =def ¬x =¬

∨{z | (z, y) ∈ Sy} =

∧{¬z | (z, y) ∈ Sy} (from the act that¬ is a split negation,

thus an additive operator)= y (because from the definition of the equivalence classSyin Definition 23 we have that for each(z, y) ∈ Sy,¬z = y). Thus, the definition of¬ras the domain reduction of¬ is correct. We have also that for any two(x, y), (x′, y′) ∈S we have thatx 6= x′ iff y 6= y′, thus¬r is a bijection, with inverse mapping¬−1r : S2 → S1 defined by, for anyy ∈ S2, ¬−1

r (y) = x, where(x, y) ∈ S. Thus,¬−1r ¬r = idS1

and¬r¬−1r = idS2

. Consequently, also¬−1r is monotone,i.e., for any


two ¬rx,¬rx′ ∈ S2, if ¬rx >OP ¬rx′ (i.e., ¬rx < ¬rx′, then¬−1r (¬rx) = x >

¬−1r (¬rx′) = x′, and from the monotonicity of¬ (and its domain reduction¬r) we

obtain that the point (1) of this lemma holds.

Point (2): from the definition ofS, x′ is the join of all elements inπ1Sy in anequivalence classSy, wherey = ¬x, with x ∈ π1Sy, so thatx ≤ x′. �

EXAMPLE 25. — Finite cases: In the case of the bilattice (epistemic) negation op-erator¬ in Belnap’s bilattice, which is paraconsistent De Morgan negation (thus splitnegation also), we obtain:S = {(f, t), (⊤,⊤), (⊥,⊥), (t, f)}, with S1 = S2 = X = B4, ¬r = ¬−1

r = ¬, andR¬ = {(f, f), (f,⊥), (f,⊤), (f, t)(⊥,⊥), (⊥, f), (⊤,⊤), (⊤, f), (t, f)}.

Infinite cases: In the case of the fuzzy logic,¬x = 1− x, it is a paraconsistent DeMorgan negation thus also split negation, withS = {(x, 1 − x) | x ∈ X = [0, 1]}.Thus, from Definition 23, we obtain a symmetricinfinite incompatibility relationR¬

as in the example above: it must be so because both negations are selfadjoint, so thatthis incompatibility relation must be symmetric. Other cases for paraconsistent splitnegations are given in belief and confidence level logics andtheir fuzzy extensions(Majkic, 2005b). �

Now we are able to define themany-valued relationalKripke-style semantics (the2-valued Kripke-style semantics is presented in (Majkic, 2006e)) for a predicate modallogicL based on the modal Heyting algebras in Proposition 16:

DEFINITION 26. — Given a predicate modal logicL, based on the modal Heytingalgebra in Proposition 16, we define the Kripke model for thislogicM = (K,S, V )

with the frameK = 〈(X,≤), {Ri, Ri}〉, where eachRi, Ri, i = 1, 2, ..., is the ac-cessibility relation for existential (additive) modal operators oi and negation modaloperator oi respectively, given in Definition 21, with a decreasing valuation V :X×P →

⋃n<ω2S

n

, such that for any predicate symbolr ∈ P with arityn, and tupleof constants(c1, ..., cn) ∈ Sn (i.e., ground atomr(c1, ..., cn) ∈ H), there exists aparticular valuey ∈ X , such that ∀x ∈ X(V (x, r)(c1, ..., cn) = 1 iff x ≤ y).

Then, for any worldx ∈ X and assignmentg, we define the many-valued satisfac-tion relation for a formulaφ ∈ L, denoted byM |=x,g φ, as follows:

1) M |=x,g r(x1, ..., xn) iff V (x, r)(g(x1), ..., g(xn)) = 1 ,2) M |=x,g φ ∧ ψ iff M |=x,g φ and M |=x,g ψ,3) M |=x,g φ∨ψ iff ∀y(( y ≤ x and notM |=y,g φ) implies M |=y,g ψ),4) M |=x,g φ⇒ ψ iff ∀y(( y ≤ x andM |=y,g φ) implies M |=y,g ψ),5) M |=x,g ¬φ iff M |=x,g φ⇒ ∅ , where∅ is a contradiction formula,6) M |=x ♦iφ iff ∃y((x, y) ∈ Ri and M |=y φ), for any monotone modal

operator♦i.7) M |=x ∼i φ iff ∀y( M |=y φ implies (y, x) ∈ Ri), for any modal

negation operator∼i, so that‖ ∼i φ‖ = λ‖φ‖ = {x | ∀y ∈ ‖φ‖. (y, x) ∈ Ri)}.

By definitionM is a Kripke-style model of a logicL if it satisfies all formulae inL.


The reason that we use both negations, the pseudocomplement¬ and modal (weak)negations∼i is justified by necessity to consider also theparaconsistentmany-valuedlogics in which¬ can not be used as paraconsistent negation (in Proposition 30).

Notice that in the worldx = 0 (the bottom element inX) every formulaφ ∈F (L) is satisfied: because of that we will denominate this world bytrivial world.The semantics for implication is similar (but different) tothe Kripke style definitionfor intuitionistic implication (where the time sequence for the accessibility relation isused, upward (in the future) closed and based on the principle "persistence of truthin time" (Kripke, 1965)). Otherwise, the definition for disjunction is not standard. Infact, it holds that(M |=x,g φ or M |=x,g ψ) implies M |=x,g φ ∨ ψ , butnotvice versa. It results from the fact that the disjunction in the canonical representationis the set operator

⊎, (generally) different from the standard set union

⋃.

The satisfaction for disjunction can be given alternatively by3.a M |=x,g φ ∨ ψ iff ∃y, z( x ≤ y ∨ z and M |=y,g φ and M |=z,g ψ).

Consequently the conjunction disjunction and implicationare defined indepen-dently as in intuitionistic logic (McKinseyet al., 1948). In fact, the Definition 26for a Kripke model of many-valued modal logics is similar butnot equal(down-ward closed instead of upward closed hereditary subsets, and the worlds are not thetime-points as in Kripke interpretation but the autoreferential truth-levels of credibil-ity) to that of Kripke-style intuitionistic logic because not only the implication (asin (McKinsey et al., 1948)) but also thedisjunction is intensional: it is based onLewis’s approach, as discussed in Remark after Proposition7, where he distinguishthe standard orextensionallogic connectives∧c,∨c,⇒c, from theintensionallogicconnectives∧,∨,⇒, such thatx ⊙ y = �(x ⊙c y), for ⊙ ∈ {∧,∨,⇒} where� is“necessarily” a universal logic modal operator corresponding to the algebraic modaloperatorΓ : P(X)→ P(X).

The accessibility relation for the multiplicative modal operator “necessary”� isthe partial order≥=≤−1 in Definition 26. For its left adjoint (additive) dual modaloperator⋄, ⋄ ⊣ �, such that the Galois connection⋄x ≤ y iff x ≤ �y holds, the acces-sibility relation can be obtained by the "gaggle" theory of Dunn (Dunn, 1991; Dunn,1993), for this simple Galois connection with the followingtraces and tonicities:

function tonicity trace� + + 7→ +⋄ + − 7→ −

From (Dunn, 1991) it holds that the binary accessibility relations of these two unaryadjoint operators are mutually inverse,i.e., R⋄ = (R�)−1 = (X × X − R�)−1 =X ×X− ≤.

It is intuitive also because these two operators are usuallygiven as two mappings⋄ : (X,≤) → (X,≤) and� : (X,≤)OP → (X,≤)OP , where(X,≤)OP is a posetwith inverted arrows (ordering).


We will consider that, the downward closure for the satisfaction holds for eachground formulaφ/g, that is ifM |=x,g φ than for everyy ≤ x holdsM |=y,g φ:it is satisfied for all ground atoms, directly from the definition of the mappingV inDefinition 26, and will be proven in Theorem 28 bellow.

Thus, we can use the standard Kripke definition for the satisfaction relation forthese intensional connectives, that is:

1. for the principal modal operator�:M |=x,g �r(x1, ..., xn) iff ∀y( y ≤ x impliesM |=y,g r(x1, ..., xn)) iffM |=x,g r(x1, ..., xn). Thus, it is invariant for ground atoms. Moreover, for

any ground formulaφ/g such that‖φ/g‖ = {x | M |=x,g φ} ∈ F(X), i.e.,‖φ/g‖ =↓ α for someα ∈ X , we have thatM |=x,g �φ iff ∀y( y ≤ ximpliesM |=y,g φ) iff ∀y( y ≤ x implies y ∈↓ α) iff x ≤ α iffM |=x,g φ. That is,� is an identity.

But if ‖φ/g‖ /∈ F(X) then� is not an identity, but‖�φ/g‖ = Γ‖φ/g‖ ∈F(X).

Also in the case of the dual, existential, modal operator⋄ we have for that anyground formulaφ/g such that‖φ/g‖ = {x | M |=x,g φ} ∈ F(X), i.e.,‖φ/g‖ =↓ α for someα ∈ X , holdsM |=x,g ⋄ φ iff ∃y( x ≤ y andM |=y,g φ) iff ∃y( x ≤ y andy ∈↓ α) iff x ≤ α iff M |=x,g φ. Thatis, ⋄ is an identity.

2. for the intensional conjunction:M |=x,g φ ∧ ψ iff M |=x,g �(φ ∧c ψ) iff ∀y( y ≤ x impliesM |=y,g φ∧c ψ) iff ∀y( y ≤ x implies(M |=y,g φ andM |=y,g ψ)) iffM |=x,g φ andM |=x,g ψ.

Thus, intensional conjunction has the same semantics as extensional conjunc-tion.

3. for the intensional implication:M |=x,g φ ⇒ ψ iff M |=x,g �(φ ⇒c ψ) iff ∀y( y ≤ x im-pliesM |=y,g φ ⇒c ψ) iff ∀y( y ≤ x implies (M |=y,g φ impliesM |=y,g ψ)) iff ∀y(( y ≤ x andM |=y,g φ) implies M |=y,g ψ)).

Here‖(φ ⇒c ψ)/g‖ = {y | M |=y,g φ implies M |=y,g ψ} /∈ F(X), butit holds that‖�(φ ⇒c ψ)/g‖ = Γ‖(φ ⇒c ψ)/g‖ = Γ(

⋃{Z | Z

⋂‖φ/g‖ ⊆

‖φ/g‖}) ∈ F(X).

If ‖φ/g‖, ‖ψ/g‖ ∈ F(X) then‖�(φ⇒c ψ)/g‖ = ‖φ/g‖ ‖φ/g‖ ∈ F(X).

4. for the intensional disjunction:M |=x,g φ ∨ ψ iff M |=x,g �(φ ∨c ψ) iff ∀y( y ≤ x impliesM |=y,g φ ∨c ψ) iff ∀y( y ≤ x implies(M |=y,g φ or M |=y,g ψ)) iff∀y( y ≤ x implies( not M |=y,g φ impliesM |=y,g ψ)) iff ∀y(( y ≤ xand notM |=y,g φ) implies M |=y,g ψ)).


Here‖(φ ∨c ψ)/g‖ = {y | M |=y,g φ or M |=y,g ψ} = ‖φ/g‖⋃‖ψ/g‖ /∈

F(X), thus� is not an identity, but holds that‖�(φ ∨c ψ)/g‖ = Γ‖(φ ∨cψ)/g‖ = Γ(‖φ/g‖

⋃‖ψ/g‖) = ‖φ/g‖

⊎‖ψ/g‖ ∈ F(X).

Notice that in the case when(X,≤) is a distributive lattice (when we are usingalso logic implication and negation defined as in Heyting algebras) and we areusing for the set of worlds the setX of join-irreducible elements inX (thesecond alternative for autorefrential assumption, based on the representationisomorphisms in point 2 of Propositions 15 and 16), we obtainthe standardsemantics for extensional disjunction, that is,M |=x,g φ∨ψ iff M |=x,g φor M |=x,g ψ.

In this particular case when the set of worlds is the setX ⊂ X , for the logicconnectives∧,∨,¬,⇒ we obtain Kripke-stile semantics of intuitionistic logic,but with inverted accessibility relation, of complete distributive lattice, for theoperator�. This version is presented in (Majkic, 2006d) for the complete dis-tributive lattices with additive normal modal operators.

Based on all these considerations we can see how important a role the closureoperator and universal "necessity" operatorΓ have, based on the Dedekind-McNeileGalois connection over partial order of a complete lattice(X,≤) of algebraic truth-values.

Thus, the autoreferential semantics based on the lattice preorder, determinate boththe Kripke frame for many-valued modal logic and the Kripke-style semantics forlogic connectives. The next proposition shows that there isa bijective correspondence(see also in (Majkic, 2006d)) between thealgebraicmany-valued Herbrand modelsof a modal logicL, based on the complete lattice of truth values(X,≤), and therelational(or Kripke-stile) models (based on the set of possible worlds equal to the setof truth valuesX) of the same logicL.

PROPOSITION27. — (DUALITY INVARIANCE) For any given Kripke model of amodal logicL, based on a complete lattice(X,≤) of truth values, which satisfiesDefinition 26, we are able to obtain the algebraic many-valued Herbrand modelI :H → X of the same logicL, such that for any ground atomr(c1, .., cn) ∈ H holds(a) I(r(c1, .., cn)) =

∨{x ∈ X | M |=x r(c1, .., cn)}.

Vice versa, for any given algebraic many-valued Herbrand model I : H → Xof a logicL, we are able to obtain the Kripke model ofL, which satisfies Definition26, such that for any worldx ∈ X , a predicate symbolr ∈ P with arity n, and atuple of constants(c1, ..., cn) ∈ Sn holds(b) V (x, r)(c1, ..., cn) = 1 iff x ≤I(r(c1, ..., cn)).

PROOF. — Easy to verify, directly from Definition 26. The next theorem shows theclear relationships between the canonicalrepresentation algebrabased on the com-plete lattice(F(X),⊆) and the Kripke style model of a many-valued modal logicL.

�


THEOREM 28. — Let K = 〈(X,≤), {Ri}〉 be a frame andI : H → X be aHerbrand valuation of a many-valued logicL, given by Definition 26. Then, for agiven assignmentg and for any formulaφ, the set of worlds whereφ/g (a formulaφ where all variables are substituted by the assignmentg) holds is ‖φ/g‖ = ↓(I(φ/g)) ∈ F(X).

PROOF. — By structural induction: We have thatx ∈ ‖φ/g‖ iff M |=x φ/g iffM |=x,g φ.

1. For any ground atomr(c1, ..., cn) ∈ H , x ∈ X , M |=x,g r(c1, ..., cn) iffx ≤ I(r(c1, ..., cn)), thus‖r(c1, ..., cn)‖ =↓ I(r(c1, ..., cn)). Notice that it canbe obtained directly from the equation (a) of Proposition 27, i.e., by applyingthe operator↓ to both sides of this equation (a) we obtain↓ I(r(c1, .., cn)) =↓∨{x ∈ X | M |=x r(c1, .., cn)} = {x ∈ X | M |=x r(c1, .., cn)} =‖r(c1, ..., cn)‖.

Suppose by hypothesis that‖φ/g‖ = ↓ (I(φ/g)) and ‖ψ/g‖ = ↓ (I(ψ/g)),then:

2. FromM |=x φ/g ∧ ψ/g iff M |=x φ/g andM |=x ψ/g, holds thatx ∈‖φ/g ∧ ψ/g‖ iff x ∈ ‖φ/g‖

⋂‖ψ/g‖, (and by structural induction hypothesis)

iff x ∈↓ I(φ/g)⋂↓ I(ψ/g) =↓ I(φ/g ∧ ψ/g). Thus,‖φ/g ∧ ψ/g‖ =↓

I(φ/g ∧ ψ/g).

3. FromM |=x,g φ ∨ ψ iff ∀y(( y ≤ x and not M |=y,g φ) impliesM |=y,g ψ) iff ∀y(((x ≤ y and notM |=y φ/g) implies M |=y ψ/g)iff ∀y((y ≤ x and not y ≤ I(φ/g) implies y ≤ I(ψ/g) iff ∀y((y ∈↓x

⋂(X − ‖φ/g‖) impliesy ∈ ‖ψ/g‖) iff ↓ x

⋂(X − ‖φ/g‖) ⊆ ‖ψ/g‖.

So thatS = ‖φ/g ∨ ψ/g‖ = {x | ↓ x⋂

(X − ‖φ/g‖) ⊆ ‖ψ/g‖}.

Then,S = idF(X)(S) =↓∨S =

⊎↓ S (from the homomorphism↓) =⊎

x∈S ↓ x =⊎{↓ x | ↓ x

⋂(X − ‖φ/g‖) ⊆ ‖ψ/g‖} =↓

∨⋃{↓ x | ↓

x⋂

(X − ‖φ/g‖) ⊆ ‖ψ/g‖} = (from (1) in Proposition 7 )=↓∨

((X − (X −‖φ/g‖))

⋃‖ψ/g‖) =↓

∨(‖φ/g‖

⋃‖ψ/g‖) = (from inductive hypothesis)=↓∨

(↓ I(φ/g)⋃↓ I(ψ/g)) = (from the homomorphism↓) =↓

∨↓ (I(φ/g) ∨

I(ψ/g)) =↓ (I(φ/g) ∨ I(ψ/g)).

That is,‖φ/g ∨ ψ/g‖ =↓ I(φ/g ∨ ψ/g).

The proof for the alternative definition 3.a is as follows: FromM |=x,g φ∨ψiff ∃y, z( x ≤ y ∨ z and M |=y,g φ and M |=z,g ψ), holds thatx ∈ ‖φ/g ∨ ψ/g‖ iff (by structural induction hypothesis)∃y, z( x ≤ y ∨ zand y ≤ I(φ/g) and z ≤ I(ψ/g) iff x ≤

∨{y ∨ z | y ≤ I(φ/g) and

z ≤ I(ψ/g) } = I(φ/g) ∨ I(ψ/g) = I(φ/g ∨ ψ/g). Thus,‖φ/g ∨ ψ/g‖ =↓I(φ/g ∨ ψ/g).

4. Fromx ∈ ‖φ/g ⇒ ψ/g‖ iff M |=x φ/g ⇒ ψ/g iff ∀y(((x ≤ y andM |=y φ/g) implies M |=y ψ/g) iff ∀y((y ≤ x and y ≤ I(φ/g)


implies y ≤ I(ψ/g) iff ∀y((y ∈↓ x⋂‖φ/g‖ implies y ∈ ‖ψ/g‖) iff

↓ x⋂‖φ/g‖ ⊆ ‖ψ/g‖. So thatS = ‖φ/g ⇒ ψ/g‖ = {x | ↓ x

⋂‖φ/g‖ ⊆

‖ψ/g‖}.

Then,S = idF(X)(S) =↓∨S =

⋃↓ S (from the homomorphism↓) =⋃

x∈S ↓ x =⋃{↓ x | ↓ x

⋂‖φ/g‖ ⊆ ‖ψ/g‖} =

⋃{S′ ∈ F(X) | S′

⋂↓

I(φ/g) ⊆↓ I(ψ/g)} =↓ I(φ/g) ↓ I(ψ/g) =↓ (I(φ/g) ⇀ I(ψ/g)) =↓I(φ/g ⇒ ψ/g) (from the homomorphism of the valuationI). Consequently,‖φ/g ⇒ ψ/g‖ =↓ I(φ/g ⇒ ψ/g).

5. For any additive algebraic modal operatoroi we obtain the existential logic modaloperator♦i, so thatM |=x ♦iφ/g iff ∃y ∈ X((x, y) ∈ Ri andM |=y φ/g),iff ∃y((x, y) ∈ Ri andy ≤ I(φ/g)) .

Let us denote it byα = I(φ/g), then the condition above,(∗) ∃y((x, y) ∈ Riandy ≤ α), is satisfied in one of the following possible cases:

5.1.1 Case whenx ∈ π1(Sm). Then (∗) corresponds tox = oi(y) andoi(y) ≤ oi(α), (becauseoi is monotone), that is,x ≤ oi(α), i.e., x ∈↓ oi(α);

5.1.2 Case whenx /∈ π1(Sm), but (x, y) ∈ Ri. Then,∃(v, y) ∈ Sm.(x ≤ v)i.e., x ≤ v = oi(y), and from(∗), y ≤ α, and from the monotonicity ofoi,oi(y) ≤ oi(α), we obtainx ≤ oi(α), i.e., x ∈↓ oi(α).

Vice versa, let us show that for anyx ∈↓ oi(α), the condition(∗) is satisfied:

5.2.1 Case whenx = oi(α). We have (oi(α), α) ∈ Sm ⊆ Ri, thus, y = αand (∗) is satisfied;

5.2.2 Case whenx < oi(α) and ∃y.x = oi(y). Then (x, y) ∈ Sm ⊆ Ri.Suppose thaty > α, then from the monotonicity ofoi it will hold that oi(y) >oi(α) > x, which is in contrast withx = oi(y). So, it must hold thaty ≤ α,and (∗) is satisfied;

5.2.3 Case whenx < oi(α) and ∄y.x = oi(y). Then (z, y) ∈ Sm ⊆ Ri suchthat z = oi(y) ≥ x, and consequentlyoi(α) ∈ {oi(y1) | (x, y1) ∈ Ri\Sm},and (x, y) ∈ Ri\Sm for y = α (from the fact thatx ≤ oi(y) = oi(α)), so that(∗) is satisfied.

Thus, we have‖♦iφ/g‖ =↓ oi(α) =↓ oi(I(φ/g)) =↓ I(♦iφ/g).

6. For any negation operatoroi we obtain the modal operator∼i, we haveM |=x ∼iφ iff ∀y( M |=y φ implies (y, x) ∈ Ri) iff ∀y( y ∈ ‖φ‖ implies(y, x) ∈ Ri) iff (by inductive hypothesis‖φ‖ =↓ α whereα = I(φ/g)

) ∀y( y ≤ α implies (y, x) ∈ Ri) , i.e., ‖ ∼i φ‖ = {x | ∀y( y ≤ α

implies (y, x) ∈ Ri)} .

It is easy to verify that for eachx ≤ ¬α andy ≤ α, (y, x) ∈ Ri (from Definition23 and the point 2 of Lemma 24), we have thatx ∈ ‖ ∼i φ‖ . Thus ‖ ∼iφ‖ ⊇ ↓ α.


Suppose that there existsx > ¬α such thatx ∈ ‖ ∼i φ‖. In that case there aretwo possible cases (hereS is the binary relation in Lemma 24 of this negationoperator:

6.1 There existsβ ∈ π1S ⊆ π1Ri such thatx = ¬β > ¬α: then (from thepoint 1 of Lemma 24)β < α, and(β,¬β) ∈ S ⊆ Ri, but(α,¬β) /∈ Ri (fromDefinition 23) so that∀y( y ≤ α implies (y,¬β) ∈ Ri)} is false (the casewheny = α cannot be satisfied), in contradiction with the hypothesys.

6.2 Does not existβ such thatx = ¬β > ¬α: then (α, x) ∈ Ri only if∃α1.¬α1 > x with (α1,¬α1) ∈ S ⊆ Ri (from Definition 23), but from¬α1 >x > ¬α we have (from the point 1 of Lemma 24) thatα1 < α, so that(α, α1) /∈

Ri, and, consequently,(α, x) /∈ Ri. Thus ∀y( y ≤ α implies (y, x) ∈ Ri)}is false, in contradiction with hypothesys. Consequently,‖ ∼i φ‖ = ↓ α. �

REMARK . — Notice that in the proof for existential modal operators,we used onlythemonotonicproperty for these operators and not its additive property.That meansthat the definition of the satisfaction for existential modal operators (point 6 in Defi-nition 26) can be extended toany monotonic operator, and, particularly touniversalmodal operators also. In this case, we are able to havethe sameexistential defini-tion for both joined modal operators, but with each one basedon its proper specificaccessibility relation.

This is important if we consider that the satisfaction for universal modal operatorsM |=x �iφ/g is notgiven by the classic definition, that is by∀y ∈ X((x, y) ∈ R♦i

impliesM |=y φ/g), whereR♦iis the accessibility relation given by Definition 21 for

the existential modal operator. It can not be defined byM |=x ¬♦i¬φ/g because, indistributive lattices, for two adjoint modal operators♦i ⊣ �i generally does not hold♦i = ¬�i¬ and�i = ¬♦i¬.

Thus, we can define the semantics for the satisfaction of universal modal operatorby this new way, that is

7.a M |=x �iφ/g iff ∃y ∈ X((x, y) ∈ R�iandM |=y φ/g), whereR�i

is an accessibility relation obtained by Definition 21 applied to the universal modaloperator�i, such thatR�i

6= R♦i, or by dual result, when♦i is an additive modal

operator, obtained for accessibility relations of adjointunary modal operators and forthe "gaggle" theory of Dunn (hereA denotes the set complement ofA)

7.b M |=x �iφ/g iff ∀y ∈ X((x, y) ∈ (R♦i)−1 impliesM |=y φ/g).

This is really a new result which generalizes the relationalsemantics fornormalKripke modal logic: for any couple of adjoint modal operators l ⊢ r (Galois con-nection), which results in a normal Kripke modal logic, we are able to define thesatisfaction relation of these two (universal versus existential) operators based on twoaccessibility relations, but in the same standard way used only for existential modaloperators.


This result is particularly important when we are dealing with positive modal logic(Celaniet al., 1997; Dunn, 1995) (without negation) for which Definition 21 holdsalso if we substitute point 7, with this new definition for universal modal operator,based on its proper accessibility relation which is different from the relation used fora existential modal operator. �

EXAMPLE 29. — In the case of Belnaps’ 4-valued logic, where the logic values canbe considered as 0-arity atoms, we can verify thatM |=⊥,g ¬t⊤ iff M |=⊥,g (⊤ ⇒∅) iff ∀y(( y ≤ ⊥ and M |=y,g ⊤) implies M |=y,g ∅) which is true(consider thatM |=y,g ∅ is true only ify = f , and false otherwise), thus‖¬t⊤‖ =↓I(¬t⊤) =↓ I(⊥) =↓ ⊥ = {f,⊥}.

WhileM |=⊤,g ¬t⊤ iff M |=⊤,g ⊤ ⇒ ∅ iff ∀y(( y ≤ ⊤ andM |=y,g ⊤)implies M |=y,g ∅) is false (consider he case wheny = ⊤, then y ≤ ⊤ andM |=y,g ⊤) is true, while M |=⊤,g ∅ is false), consequently⊤ /∈ ‖¬t⊤‖.

The Kripke frame for autoepistemic intuitionistic logic based on Belnap’s4-valued logic is very simple, with two accessibility relations: R− ={(f, f), (⊤,⊥), (⊥,⊤), (t, t)}, for the belief (conflation) modal operator, andRµ = {(f, f), (t,⊤), (⊤,⊤), (⊤, t), (⊥,⊤), (⊥, t)}, for the Moore’s autoepistemicmodal operatorµ.

Both of them can be obtained from Definition 21 by applying it independently tothe existential or universal operator, just because in these two cases they are selfadjoint(equal).

Let us consider a more general case when the existential and universal operator aredifferent, given by the following truth table:

M Lt t ⊤⊤ t ⊤⊥ ⊥ ff ⊥ f

It is easy to verify that the operatorM is the multiplicative monotone operator (i.e.,M(α ∧ β) = M(α) ∧M(β) andM(t) = t), used for universal modal logic operator�. The operatorL is its correspondent (by Galois connectionL ⊣ M ) additiveoperator (i.e.,L(α∨ β) = L(α) ∨L(β) andL(f) = f ), used for an existential modallogic operator♦, such thatM = ¬tL¬t (because the Belnap’s latticeB4 is a Booleanalgebra) with¬t¬t = id an identity.

The accessibility relations for these two monotone operators, obtainedfrom Definition 21 are, R♦ = {(⊤,⊤), (f, f)} and R� ={(t,⊤), (⊤,⊤), (⊤, t), (⊥, f), (f, f)}, and we will denominate them asjoint acces-sibility relations.

Thus, for the existential modal operator the satisfaction is as usualM |=x ♦φ/giff ∃y ∈ X((x, y) ∈ R♦ andM |=y φ/g), while for the universal modal operator


we have two alternative definitionsM |=x �φ/g iff M |=x ¬♦¬φ/g iff ∀y ∈X((x, y) ∈ R♦ impliesM |=y φ/g), based on the relationR♦, or, independently,based on accessibility relationR�, M |=x �φ/g iff ∃y ∈ X((x, y) ∈ R� andM |=y φ/g). �

Finally, from the canonical representation for a complete distributive lattice basedmodal intuitionistic logics in Proposition 16, we obtainedthat the isomorphism, be-tween the original Galois algebraG = (X,≤,∧,∨,¬,⇀, {oi}i≤ω, 0, 1) with unarymodal operators in{oi}i≤ω and itscanonicalrepresentation algebraGK = (F(X),⊆,⋂,⊎, ¬,, {oi}i≤ω, 0, 1), whereoi =↓ oi ↓−1, i = 1, 2, .., corresponds to the rep-

resentation of any ground formulaφ/g by the set of worlds‖φ/g‖ in the canonicalKripke model for the modal algebraG.

So, for example, a ground termφ/g ∧ ψ/g in a predicate modal logicL (based onthe Galois algebraG) with a many-valued Herbrand modelI : H → X , correspondsto the set‖φ/g‖

⋂‖ψ/g‖ in the canonical algebraGK ≃ G, where‖φ/g‖ =↓ y ( y is

an algebraic truth-value in a Herbrand modelI for the ground formulaφ/g) is equaltothe set of worlds in the canonical Kripke modelM = (K,S, V ) (given by Definition26) whereφ/g holds.

That is the result of theautoreferentialrepresentation, where the set of possibleworlds is assumed to be the set of algebraic truth values (thecomplete lattice), usedfor the canonical and Kripke style semantics for modal many-valued logics.

6. Autoreference, manyvaluedness and paraconsistency

I will begin this section with some words (Cadoliet al., 1996) from my friend fromRoma, Marco Cadoli (1965-11/21/2006) recently deceased:

“The main motivation for adopting MV logic was the necessityto over-come some semantic limitations of classical logic. One of the draw-backs of classical logic that has been more frequently pointed out is that ageneric unsatisfiable formula —e.g.a∧¬a— impliesanyformula b. Thework of Anderson and Belnap (Andersonet al., 1975) in relevance logichas been one of the first attempts at defining a semantically well-foundedlogical system that does not have this undesiderable feature. As shown byBelnap in (Belnap, 1977) their system can indeed be characterized by amultivalued semantics.”

Theparaconsistentlogic goes beyond consistency but prevents triviality,i.e., theexplosive inconsistency of classic logic described above.In paraconsistent logics wecan have a kind of local inconsistency which has no total explosive feature as in 2-valued classical logic. Thus, in the presence of these localinconsistencies paracon-sistent logic still separates propositions into two non-empty classes, derivable andnon-derivable ones.


The most important criteria for choosing paraconsistent logics, as suggested in(Nelson, 1959; da Costa, 1974; Majkic, 2008), and recently formalized into the Logicof Formal Inconsistencies (LFI) (Carnielliet al., 2006), is their abstract degree of non-triviality, rather than the mere absence of contradiction.

The LFI internalizes the very notions of consistency and inconsistency at theobject-language level, instead of the usual meta-logic level. It designs an expressivelogic language, able of recovering both consistent reasoning and allowing for someinconsistency.

We will denote byL a many-valued predicate logic, with a Herbrand baseH (theset of all ground atoms), based on a complete distributive lattice (X,≤) of algebraictruth-values, obtained as free algebra by a carrier setH of ground atoms and the setof logic connectives in

∑, that is, the set of standard logic connectives{∧,∨,⇒,¬},

corresponding to the Heyting algebra in Proposition 15, andthefinitenumber of unarymodal (normal andnonnormal) operators. We will denote byAX the extended many-valued Heyting algebra by a finite set of modal operators{oi : X → X}i<ω, that is,AX = (X,≤,∧,∨,⇀,¬, {oi : X → X}i<ω}.

Any many-valued valuationv : H → X , i.e., Herbrand interpretation, can beuniquely extended to all formulae inLG (the set of all ground formulae inL) by thehomomorphismv : LG → AX . By (AX , D) we will denote the matrix whereD ⊆ Xis any subset of designated algebraic values.

We will formally define the multi-modal many-valued logic asa pair (LG, )where ⊆ P(LG) × LG is a consequence relationfor a logicLG, generallydefined as follows: for anytheory ∆ ( a subset of ground formulae ofLG), theground formulaφ ∈ LG is a logical consequence of∆, denoted by∆ φ, iff∀v(∀ψ ∈ ∆.v(ψ) ∈ D implies v(φ) ∈ D).

It is said that a theory∆ is closediff {φ ∈ LG |∆ φ} = ∆.

In what follows we will briefly present basic concepts in LFI,and invite the readersto use (Carnielliet al., 2002; Carnielliet al., 2006) for more information. We willomit ’ground’ when we use the word formula; notice that each ground atom can beseen as a propositional variable, so that grounded predicate logic can be equivalentlyconsidered as a propositional logic with the set of propositional variables isomorphicto the Herbrand base of this predicate logic.

The logic(LG, ) with a ’negation’∼, which has to be antitonic and must satisfy∼ 1 = 0, and ∼ 0 = 1 , but different from strongsupplementary negation(Carnielliet al., 2006) based on pseudocomplement,¬α = α ⇀ 0 , where⇀, defined asrelative pseudocomplement, is adeductive implication(Carnielli et al., 2006) whichsatisfies the Modus Ponens and Deduction Theorem (Majkic, 2006b) (notice that w.r.t(Carnielliet al., 2006) here we inverted the symbols for these two negations), is:

1) contradictory, if ∀∆∃φ(∆ φ, and ∆ ∼ φ).2) trivial, if ∀∆∀φ(∆ φ).3) explosive, if ∀∆∀φ∀ψ(∆, φ,∼ φ ψ).


4) gently explosive, if ∀∆∀φ∀ψ(∆, φ,∼ φ,O(φ) ψ),whereO(φ) is a possibly empty set of formulae which depends only onφ, satisfyingthe following: there are formulaeα, β such thatO(α), α 1 β and O(α),∼ α 1 β.

5) consistent, if it is both explosive and non-trivial.6) paraconsistent, if it is inconsistent yet not trivial.

In (Carnielliet al., 2006) it was shown that for tarskian (monotonic) logic hold:

1) if logic is trivial then it is both contradictory and explosive,2) an explosive logic fails non-triviality iff it fails non-contradiction.

Finally, in (Carnielliet al., 2006) a Logic of Formal Inconsistency (LFI) is definedas a logic which is not explosive but isgently explosive. Obviously, LFI represents avery significant part of the class of parconsistent logics where∼ is calledparaconsis-tent negation.

Differently from LFI, we consider all kinds of many-valued logics (also non mono-tonic where the property(∆ φ and∆ ⊆ Λ) impliesΛ φ) based on a completedistributive lattice of algebraic truth values does not hold.

In what follows we will demonstrate that all Many-Valued logics based on a Com-pletedistributiveLattice of truth-values (MVCL logics) are gently explosive, that isthey are LFIs.

Now we will define the paraconsistent negation∼ for MVCL logics. Given anunary operator⊙ : X → X we will denote its image byim⊙ = {y = ⊙(x) | x ∈ X}.For a paraconsistent negation we need the condition∃x ∈ X.(x∧ ∼ x 6= 0) (in thatcase we take thatx ∧ ¬x ∈ D is a designated element), so it must hold2⊂ im ∼.

PROPOSITION30. — Strong negation¬, defined by the pseudocomplement¬α =α ⇀ 0 in a complete distributive lattice(X,≤), can not be used as a paraconsistentnegation.

PROOF. — In fact, for anyx ∈ X we have thatx ∧ ¬x = x ∧ (∨{y | x ∧ y = 0} =∨

{x ∧ y | x ∧ y = 0} = 0. Thus,∄x ∈ X.x ∧ ¬x 6= 0 , and¬ is the supplementarynegation. �

An atom in a lattice(X,≤) is an elementx ∈ X different from 0, such that∄y 6= 0.(y < x). Now we will show how we are able to define the paraconsistentnegation for any complete distributive lattice(X,≤).

PROPOSITION31. — Any bounded lattice(X,≤), with cardinality bigger than 2,can have a paraconsistent negation∼.

PROOF. — For any fixedα ∈ X , such thatα /∈ {0, 1}, we define∼ x = 1 if x = 0; 0 if x = 1; α . It is easy to show that for any twox, y ∈ X ,

if x ≤ y then¬x ≥ ¬y. So that∼ is a general negation given by Definition 11. It iseasy to verify thatα ∧ ¬α = α 6= 0.

But in some cases there are also other kinds of paraconsistent negation:


1) WhenX is an infinite set and there exists the isomorphism (which preservesthe ordering) with the closed interval of reals[a, b], ω : (X,≤,∧,∨) ≃ ([a, b],≤,min,max), then we define for anyx ∈ X , ∼ x = ω−1(b + a− ω(x)), whereω−1 is the inverse ofω.

2) WhenX is finite with the cardinalityn = |X | ≥ 3. We define the followingisomorphism σ : (X,≤,∧,∨) ≃ ({ i

n| 1 ≤ i ≤ n},≤,min,max), and we define

for any x ∈ X , ∼ x = σ−1(1+nn− σ(x)), whereσ−1 is the inverse ofσ. �

EXAMPLE 32. — Let us consider the following cases:

1) In the case of fuzzy logic, where the complete lattice is a closed interval of reals,X = [0, 1], we have the case 1.1 whereω is defined byω(x) = a+(b− a) ∗x for anyx ∈ X , andω−1(y) = (y− a)/(b− a) for anyy ∈ [a, b]. Thus ∼ x = ω−1(b+ a−ω(x)) = (b+a−ω(x)−a)/(b−a) = 1−(ω(x)−a)/(b−a) = 1−ω−1(ω(x)) = 1−xis a standard negation of fuzzy logic.The set of designated elements isD = {y | y ∈ X and a ≤ y} for some0 < a ≤ 1.

2) In the case of the 3-valued logic whereX = {0, 12 , 1}, andσ(x) = 1+2x

3 , weobtain ∼ (0) = 1, ∼ (1

2 ) = 12 and ∼ (1) = 0.

The set of designated elements isD = { 12 , 1}.

3) In the case of Belnap’s 4-valued logic,X = {f,⊤,⊥, t}, where0 correspondsto f and1 to t, we have two atoms,⊤ and⊥, and we obtain that∼= ¬t− is stan-dard epistemic bilattice negation, where¬t is a pseudocomplement (supplementarynegation) and− the belief modal operator (conflation).The set of designated elements isD = {⊤, t}.

4) As one example we can consider the followingexplicit paraconsistentexten-sion of fuzzy logic, whereX = [0, .5)

⋃{.5−, .5+}

⋃(.5, 1], where.5−, .5+ substitute

the "normal" value.5, but only with the property that.5− ⊲⊳ .5+. So thatX is nota total ordering. In this case the number of atoms is zero. We can consider.5+ asan "inconsistent" value (like⊤ in Belnap’s bilattice), and.5− as an "unknown" value(like ⊥ in Belnap’s bilattice). We define the negation by∼ x = x if x ∈ {.5−, .5+};1− x otherwise. The set of designated elements isD = {.5−, .5+}

⋃(.5, 1]. �

Now we will introduce the particular non normal modal operators (they are notmonotone functions), used in (Majkic, 2006e) for the reduction of any MVCL logicinto 2-valued modal logic, and, based on them, the paraconsistent negation∼. Thenwe will define the consistency operator◦ : X → X as follows:

DEFINITION 33. — Given a complete lattice(X,≤) we define the family of unarymodal operators [x] : X → X , for eachx ∈ X , as follows: for any α ∈ X[x]α = 1 if α = x; 0 otherwise.

Based on them we define the following global modalCONSISTENCYoperator◦ :X → X for a lattice(X,≤): for anyα ∈ X ◦α =def

∧{∼ [x]α | x ∈ X\2},

where\ is the set substraction.

Notice that in the case of the classic 2-valued logic we obtain for any formulaφ,◦ φ = 1, because the meet of the empty set is equal to1.


Otherwise, for a n-valued logic,◦ α = 1 for α ∈ 2 and is otherwise equal to0.

THEOREM 34 (MANYVALUEDNESS AND PARACONSISTENCY). — Any many-valued logic based on a complete lattice(X,≤) with |X | ≥ 3 is a LFI with theconsistency operator defined in Definition 33.

PROOF. — It is easy to verify that the following matrix holds:

∼ [x1] [x2] ....... [xi] ....... [xn] ◦1 0 0 0 ....... 0 ....... 0 1x1 y1 1 0 ....... 0 ....... 0 0x2 y2 0 1 ....... 0 ....... 0 0. . . . ....... . ....... . .. . . . ....... . ....... . .xi yi 0 0 ....... 1 ....... 0 0. . . . ....... . ....... . .. . . . ....... . ....... . .xn yn 0 0 ....... 0 ....... 1 00 1 0 0 ....... 0 ....... 0 1

where for all1 ≤ i ≤ n, xi ∈ X\2 and yi ∈ X\2, and n ≥ 1.

It is easy to verify that ∀x ∈ X.(x ∧ ∼ x ∧ ◦ x = 0), consequently∀∆∀φ∀ψ(∆, φ,∼ φ,O(φ) ψ), while ◦ 1, 1 1 0 and ◦ 0,∼ 0 1 0.

Thus, a many-valued logic based on a complete lattice(X,≤) of algebraic truth-values is gently explosive,i.e., it is a LFI. �

Consequentially, for any finite many-valued logic (each finite lattice is a completelattice) we are able to define the paraconsistent negation inProposition 31 in orderto obtain a LFI. It is possible also in the case of infinite set of algebraic truth-values,when this set is a complete lattice (as, for example, in the case of the fuzzy logic inExample 20).

The relationship between the many-valuedness and the autoreferential assumptionis given in precedent sections by representation theorems and dual Kripke-style seman-tics for many-valued logics based on complete lattices of truth-values (Proposition 16and Definitions 21,26).

All that remains is to explain the last relationship in the triangle manyvaluedness-autoreference-paraconsistency: the relationship between theautoreferenceassumptionfor Kripke-style semantics of many-valued logics andparaconsistency.

Let us consider the multi-modal many-valued logic(L, ) based on the extendedmany-valued Heyting algebraAX = (X,≤,∧,∨,⇀,¬, {oi : X → X}i<ω}, and aparticular set of formulae (theory)∆ ⊂ L, with the Herbrand baseH , for which theKripke style semantics is given by Definitions 21 and 26.


Let I : H → X be a many-valued Herbrand model of this theory∆, andI :LG → AX its unique homomorphic extension to all ground formulae. Then from theTheorem 28 we have that the set of ground formulae, which are satisfied in a worldx ∈ X is: S(x) =def {φ ∈ LG | I(φ) ≥ x}.

If we consider a worldx as the matrix(AX , D(x)) whereD(x) = {y ∈ X | y ≥x} is the set of designated elements of this matrix, then the setS(x) is the set offormulae which are satisfied w.r.t this matrix. That is∀φ ∈ S(x)(I(φ) ∈ D(x)), sothatI is a model for all formulae inS(x).

Thus we can introduce the following preorder between matrices : (AX , D(x)) �(AX , D(y)) iff S(x) ⊇ S(y) iff x ≤ y.

Consequently, the bottom matrix is(AX , D(0)) = (AX , X) . It corresponds tothe bottom world0, where the setS(0) = LG is the set of all ground formulae: thisworld is trivial , and corresponds to classic explosive inconsistency. So that this worldcan not be paraconsistent.

The top world1 corresponds to the matrix(AX , D(1)) = (AX , {1}) , withthe smallest set of formulaeS(1). In this world we can not have any true formulaφ ∧ ¬φ, because ifI(φ) ∈ D(1) = {1}, that is,I(φ) = 1, thenI(¬φ) = 0, i.e.,I(¬φ) /∈ D(1) = {1}, and, consequentlyI(φ∧ ¬φ) /∈ D(1) = {1}. That is, also thisworld can not be paraconsistent.

Thus, to be able to permit paraconsistent reasoning we need at least three differentworlds, that is n-valued logic withn ≥ 3.

In that case, for the world0 < x < 1 we are able to have the formulaeφ ∧ ¬φsuch thatI(φ ∧ ¬φ) = x, that is,I(φ ∧ ¬φ) ∈ D(x). Consequently, this world isparaconsistent.

An interesting point is how the hierarchy between the possible worlds (lattice or-dering of truth-values) corresponds to the hierarchy of matrices for the same theory,and, consequently, to the hierarchy of paraconsistent logics. The level of paraconsis-tency is inverse to the truth-level of the possible world: the higher truth level of thepossible world corresponds to a lower paraconsistent logic(with a smaller number ofparaconsistent formulae).

7. Conclusion

The main result of this paper is a general way of establishinglinks between al-gebraic and Kripke-style semantics for the class of many-valued modal logics basedon a complete lattice of truth values, by introducing autoreferential duality: it meansthat we will use as a set of possible worlds in Kripke-style semantics for these modallogics exactly the lattice of its truth values.

This autoreferential approach, where a Kripke frame is based on posets (the com-plete lattices of truth values), is the result of the consideration that each possible world


represents a level ofcredibility, so that only the propositions with the right logic value(i.e., level of credibility) can be accepted by this world. The lowest credible world0 isthe trivial world of explosive inconsistency: so any derivable formula in this world isstrongly doubted. The top world1 has maximal credibility: it corresponds to classiclogic where we have no doubt about true formulae. The intermediate worlds have adegree of credibility, and we can consider true also formulae which in classic logicwould be inconsistent: these worlds permit paraconsistentreasoning.

We developed a general Representation Theorem for the classof logics based oncomplete lattices and on the Dedekind-McNeile Galois connection for the partial orderrelation of the lattice, with the derived closure operator.The obtained set-based iso-morphic algebra has the usual set intersection of a meet operator, but the join operatoris a generalization of the set union: we introduced also the subset of complete lattices,that is, the distributive complete lattices, where join corresponds exactly to set union.

We have shown that each distributive complete lattice naturally supports intuition-istic implication and negation, and a set of modal operators, so that it can be extendedinto an Galois algebra. We defined the Kripke frame and Kripkemodels for this classof many-valued modal algebraic logics, and have shown that each Kripke model, forsuch modal logics, is equivalent to the standard algebraic semantics based on many-valued Herbrand models. Differently from the standard method based on thenaturalduality theorem(Clarket al., 1998), where a class of relational structures would be thefamily of duals of algebras, which is difficult to describe ina simple logic language,our approach offers a very simple and compact description. We demonstrated a gen-eral way to obtain paraconsistent negation and consistencyoperators for this class ofmany-valued logics, so that they can be used as LFI gently explosive logics.

We discovered how the possible world Kripke-style semantics for many-valuedlogics, based on a complete lattice of algebraic truth-values, can be interpreted asa paraconsistent approach to these logics. Finally we established the relationshipsbetween many-valuedness, autoreferential assumption andthe paraconsistency.

Acknowledgements

The author wish to thank Walter Carnielli for his support anduseful commentsabout earlier version of this paper, about paraconsistencyand LFI. The last Sectionwas written by author after this correspondence.

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Autoreferential semantics for many-valued modal logics fileAutoreferential semantics for many-valued modal logics Zoran Majkic´ ETF, Applied Mathematics Department University of Belgrade