Tones of truth. Andrei Popescu. Technical University Munich and Institute of Mathematics Simion Stoilow of the Romanian Academy

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1 Tones of truth Andrei Popescu Technical University Munich and Institute of Mathematics Simion Stoilow of the Romanian Academy Abstract This paper develops a general algebraic setting for the notion of shaded truth value, providing a Lukasiewicz-Moisil construction over an arbitrary truth functional logical system. Among very old, old and new instances of this construction we find Boolean algebras, Lukasiewicz-Moisil [algebras and relation algebras], and shaded residuated lattices. Keywords: shaded truth value, Lukasiewicz-Moisil algebra, truth-functional logics, fuzzy vs. shaded, many-valued logic MSC: 03B50, 03G20, 06D30, 03B52, 03G10. 1 Introduction The tones, or shades of truth are very present in our conceptions and discourse. The simplest forms of shades are the points of view. Consider, for instance, the statement What you are is what you do. This might be true for the Activist, but certainly not for the Contemplationist. By evaluating this statement, one gets (true for Activist,false for Contemplationist), or (1, 0). Thus, by gathering points of view and not jumping to conclusions about the right one, we create complex (or composed) truth values, each having several simple, atomic shades. There are four composed truth values: (0, 0), (0, 1), (1, 0), (1, 1). the Activist the Contemplationist 1

2 Notice that we still have a classical truth structure (a Boolean algebra), but more shaded than the very classical {0, 1}. The evolution was more of a horizontal than a vertical one, the structure gaining node ramification - it looks as if the Activist and the Contemplationist have both pulled the truth structure {0, 1} and deformed it into the rhombus. If we were to have more opinions, more pulling forces would appear, each along its direction, and we would need more nodes and more options in each node, thus a greater Boolean algebra. This constitutes a very simple example of toning, where the shades are independent from each other and a complex truth value is just a collection of simple ones. (This however, together with the Stone representation theorem, shows that evaluating sentences in general Boolean algebras rather than the particular {0, 1}, and, in fact, the very appearance of Boolean algebras, could be seen as a particular instance of the process of toning the reasoning.) Perhaps the most famous truth toning comes from the scholastic modalities - possible and necessary - which make that a sentence has three shades, if we ask, consecutively: is it possible?, is it true?, is it necessary?. Unlike in the points of view situation, the answers to these three questions are not independent, since the shades are hierarchirized. A complex truth value is represented by any 3-place combination of true-false that respects the hierarchy - we obtain a four elements chain: necessarily false < contingently false < contingently true < necessarily true. Of course, one could consider more complex and mixed situations by adding more hidden quantifiers, for instance letting the modalities of truth be necessarily, simply true, normally, possibly, and, since there is no subordination between normally true and simply true, all the complex truth values are necessarily true true normally true possibly true These six complex truth values could be called, from up to down: necessarily false, contingently false, contingently true, surprisingly false, true as expected, and necessarily true. They are ordered as follows: 2

3 The above is an interesting example, which shows the appearance, via toning, of a natural truth structure that is not totally ordered. 1 This approach to modalities, by enlarging the truth structure, is due to Lukasiewicz. Starting from Lukasiewicz multi-valued logic, and often appealing to examples with modalities for expository reasons [17], Moisil introduced some algebraic structures, the Lukasiewicz algebras, later called Lukasiewicz- Moisil algebras, meant to model the shaded reasoning. These structures are complemented distributive bounded lattices satisfying the de Morgan properties, together with a family of lattice endomorphisms (ϕ i ) i {1,n}, called Chrysippian morphisms, subject to some natural conditions 2 (among which the determination principle, stating that an element is reducible to its Chrysippian, or Boolean, shades) that qualify each such algebra to be a shaded sructure of a Boolean algebra dwelling the center of the lattice. Lukasiewicz-Moisil algebras [2] are the main source of inspiration for this paper, which provides a general and simple (universal-) algebraic approach to shades. Our approach has several similarities to the universal-algebraic treatment of K-Post algebras from [4]. 3 Studying shades more generally than in connection to Boolean algebras is motivated by the fact, as we shall try to argue, that virtually any kind of truth structure (including the whole variety provided by fuzzy logic) is suitable for the process of toning, which is as natural as for Booleans and only requires minimal properties that respectable truth-functional logical connectors (classical or not) should have. On the other hand, looking downwards for examples of toning, one finds Boolean algebras as shaded structures. It is worth pointing out that toning is not the same process as the one (very encountered in building fuzzy systems) of grading truth, that is filling (or 1 In the study of fuzzy logic, there are also used not necessarily totally ordered structures, like the residuated lattices or the BL-algebras [9]; but this is done for purely algebraic reasons - in order that these structures form a variety; the bare intuition of graded truth has little to do with partial order. 2 We shall revisit these in Subsection We shall discuss these similarities in the concluding section. 3

4 completing) in some way a truth structure with intermediate, more sensible values, like taking the continuum [0, 1] instead of {0, 1}. Although the result of toning is still some kind of completion, the interpretation and use of the resulted structure is fundamentally different. As mentioned before, attaching shades brings more of a horizontal growth, following the resulted ramifications, while grading increases the vertical (and perhaps also the topological) dimension of the structure. Here is this paper s itinerary. Section 2 prepares the algebraic apparatus (consisting mainly of basic elements of universal algebra). In Section 3, we define the process of toning on top of a generic algebraic theory (not necessarily equational) for an arbitrary signature of logical connectors, that is meant to axiomatize their behavior. Some properties, connecting shaded with simple truth structures, are provided - among them, a generalization of the adjunction situation from the case of Lukasiewicz-Moisil algebras versus Boolean algebras. Section 4 briefly discusses some logical interpretations of the algebraic results. Section 5 gives examples of shaded structures: Boolean algebras, Lukasiewicz- Moisil algebras, fuzzy shaded algebras, Lukasiewicz-Moisil relation algebras. In Sectiuon 5, some categorical properties are proved - among them, a generalization of the adjunction situation from the case of Lukasiewicz-Moisil algebras versus Boolean algebras [7]. In Section 6, shaded structures of variable formal shade hierarchy are put together, by means of indexed categories. The paper ends with some conclusions. 2 Algebraic preliminaries Throughout this paper, we shall deal with (unsorted) first-order signatures Σ having arbitrary operation symbols (Σ n denotes the set of symbols of n-ary operations - in particular, 0-ary operations are constants) and only one binary relation symbol,. Next, we give a collection of immediate, but sometimes ad hoc adaptations to this almost algebraic framework of some universal algebraic concepts. 4 So the view to congruences and factorization is more algebraic than model-theoretical, allowing, in particular, the canonical projections of direct products to provide congruences. 4 All these would not have been necessarily, if we had chosen to assume, in the next section, that the truth degree order is definable from the logical connectors; but we wanted to cover, for instance, the case of a logic with non-idempotent and perhaps non-residuated conjunction, which neither is reducible to, nor determines the relation of being more true. However, the most relevant cases have definable by operations - it is worth noticing that when this is the case, all the below concepts coincide with the usual universal algebraic ones. 4

5 A Σ-algebra (or, simply, algebra) consists, as usual, of a non-empty set together with interpretations of the operation and relation symbols. A trivial algebra is one consisting of only one element, with operations defined in the unique way and the full relation. A Σ-morphism (or morphism) between two Σ-algebras is a function h that commutes with the operations and preserves, that is x y implies h(x) h(y). An embedding is an injective morphism h such that x y iff h(x) h(y). An isomorphism is a bijective embedding. A is called a subalgebra of B is there exists an embedding of A into B which is a set-theoretic inclusion. A congruence on an algebra A is a pair (, ), where is a usual universal algebraic congruence on A (i.e. an equivalence compatible with the operations), and a relation on A/ such that x y implies x/ y/ for all x, y A; the quotient algebra A = A/(, ) is the one having the support and operational part of the universal algebraic quotient algebra A/ and as relation, ; the function x x/ is a surjective morphism, called the canonical projection; conversely, each surjective morphism is, up to an isomorphism, the canonical projection of a congruence. On each algebra A, there exist (and are equal iff A is trivial) two extreme congruences (, ): the trivial one, with being the diagonal of A and =, 5 and the full one, with being A A and consisting of the only possible pair. Direct products i I B i of arbitrary families of algebras are defined in the usual way (with operations and taken componentwise). The i-th projection function, p i : j I B j B i, is a surjective morphism. Let K be a class of algebras. An algebra A from K is called: - simple (in K), if the only two congruences (, ) on A such that A/(, ) is in K, are the trivial and full congruences; - sudirectly irreducible (in K) if, for each family (B i ) i I of algebras from K, and each embedding h : A i I B i with any π i h surjective, there exist i I such that π i h is an isomorphism; - directly irreducible if there do not exist two non-trivial algebras from K such that A is isomorphic to their direct product. A slight adaptation of universal algebraic arguments shows that, in each class K, simple implies subdirectly irreducible implies directly indecomposable. The (formal) sentences that we consider to describe Σ-algebras are first-order sentences, allowing infinitary conjunctions, but such that each sentence uses a finite number of variables, which are taken from a fixed countably infinite set X; formulas with free variables are read as universally quantified on those variables. T (X) denotes the usual term Σ\{ }-algebra, whose underlying set, formed by words on the alphabet of X Σ {,, (, ),, }, is defined recursively by: - each constant or variable is a term; - if t 1,..., t n are terms and σ Σ n, then σ(t 1,..., t n ) is a term. A basic formula is one of the form t p t, where t, t are terms and p {=, }; a Horn sentence is one of the form [ I e i] e, where e, ei are basic sentences. 5 We assumed, for this case, the natural idntification between A and A/. 5

6 The operations on T (X) are defined in the obvious fashion. The reader is assumed to be familiar with basic categorical notions, like category, functor, natural transformation, or adjunction. 3 General framework for shade We fix Σ to be a (unsorted) first-order signature which has only one symbol of relation,, binary, and arbitrary operation symbols of any arity. We also fix E to be an arbitrary set of (infinitarily-conjuncted first-order, formed using a countably infinite set of variables, X) Σ-axioms, such that, among its consequences, we encounter the following: each operation is, on each argument, either increasing or decreasing. Formally, for each n-ary σ Σ\{ } and each i {1,..., n}, one of the two sentences is a consequence of E: x i x i σ(x 1,..., x n ) σ(x 1,..., x i,..., x n ) ; x i x i σ(x 1,..., x i,..., x n ) σ(x 1,..., x n ). We also assume that no sentence asserting for one of the operations that is constant on one of its arguments is a consequence of E. (This is true if there exists one (Σ, E)-algebra with operations non-constant - of course, if this is not the case, it means that the presence of one of the arguments is futile, and we can eliminate it.) The signature Σ contains the logical connective symbols from a generic logical system; although usually there exist only unary or binary connectors, there might be the case of an unconventional logic where some complex ways of combining sentences, like ϕ ψ χ, meaning ϕ implies χ in the presence of ψ, are not reducible to binary and unary operations. For instance, define the mentioned ternary operator, on [0, 1] by (z 2 y 3 )/x, if 0 < x, y < 1/2 x y z = (zy)/x, if 0 < x, 1/2 y 9/10 1, if x = 0 or y 9/10. This would correspond to a logic of reactivity, the catalyser y not acting uniformly. The general expected behavior of logical connectors is described by the axioms E - using not only equational, but first-order formulae, we allow a very wide variety of important conditions, like: - true is different from false ; - there are only n truth values ; - the connector σ is continuous (say, w.r.t. the order topology) in its i-th variable ; 6

7 - the binary connector σ coincides with suprema of, which therefore exist ; 6 - any other algebraic definition of (that places the discussion inside a pure universal algebraic framework). The (Σ, E)-algebras are truth structures for the logical system, that: - provide a set of truth values (degrees), wich is partially ordered by the relation of being less true ; - give interpretations for the logical connectors. In addition, the connectors are assumed to be monotone w.r.t., the increase of one argument systematically inducing the increase or decrease of the result - this is a very natural assumption and holds for all known logical connectors for truth-functional logics. 7 In what follows, if not otherwise mentioned, by sentence we mean infinitarilyconjuncted first-order sentence using variables from X. For each n 0 and w = w 1... w n { 1, 1} n, Σ w shall denote the set of operation from Σ n that, for each i {1,..., n}, are E-consequently increasing on the i-th argument if and only if w i = 1 (and, of course, are decreasing if w i = 1). For the whole section, we fix a triple (I,, d), consisting of a non-empty partial ordered set (I, ) together with a decreasing involution d : I I. This triple represents a hierarchy of formal shades, d providing, for each shade i, its dual, or complement. For each term t T (X), denote by t i the term obtained from t by replacing each occurance of each variable x by ϕ i (x), where ϕ i is a new symbol of unary operator. Extend this definition to sentences: for each Σ-sentence e, let e i be the sentence obtained by replacing each occurrence of each variable x by ϕ i (x). (For instance, ( x)x = x becomes ( x)ϕ i (x) = ϕ i (x).) Define the signature Σ I by adding to Σ a family (ϕ i ) i I of new unary operators. Define E I (a set of Σ I -sentences) as consisting of the following axioms: (A1) ϕ i ϕ j (x) = ϕ j (x), i, j I; (A2) ϕ i (x) ϕ j (x), i, j I, i j; (A3) x y ϕ i (x) ϕ i (y), i I; (A4)(the determination principle) [ i I ϕ i(x) ϕ i (y) ] x y; (A5) ϕ i (σ(x 1,..., x n )) = σ(y 1,..., y n ), σ Σ w, w = w 1,... w n { 1, 1} n, where, for each k {1,..., n}, y k = ϕ i (x k ) if w k = 1 and y k = ϕ d(i) (x k ) if w k = 1; 8 (A6) e i, i I, e is a sentence from E. Let us discuss the above axioms and their suitability for the notion of shaded truth. Let L be a (Σ I, E I )-algebra. An element from L is regarded as a complex 6 As we shall see, some of these conditions, provided they are regular, are inherited by the shaded structures as well. 7 An exception would be the equivalence,, but it is always defined in terms of (some) conjunction and implication. 8 Notice that the y k -s are not variables, but denotations that we use for certain terms. 7

8 truth values with shades that behave according to the axioms E (E describes the behavior of the simple, or atomic truth values, bricks to construct more complex values, that do not rely upon other logical principles, but only have to take into consideration more hierarchirized points of view). Each of the operators ϕ i provides, for any complex truth x, its actual value of the formal shade i, or, in other words, the status of the modality i w.r.t. x. The axioms (A2) and (A3) are very natural, stating that the instanciation of formal shades should respect both the formal hierarchy and the truth degree order. (A1) states that the process of taking a shade is stationary, since it resides in obtaining a truth of an atomic nature, untoned and hence insensible to shades. (A4) says that all complex truths are totally reducible to their shades, while (A6) forces simple truth values to respect the initial logical requirements. The axiom (A5) is the most interesting: when the operator σ is decreasing, a shade has to appeal to its complement - increasing or decreasing the acual i-shade of x results in increasing or decreasing the actual d(i)-shade of σ(..., x,...). In the case of n-valued Lukasiewicz-Moisil algebras for example (see Subsection 5.3), we have ϕ i (x) = ϕ n i ( x). As for modalities, a natural principle is that not necessarily means possibly not; also, a certain kind of strict implication, ϕ necessarily implies χ, could be translated as: possibly ϕ implies necessarily χ, because necessity means that all precautions (for the unlucky case of ϕ being true) should be taken by χ - this shall be further discussed in Subsection 5.2. In the situation of pure points of view ( being the diagonal on I and d(i) = i), complex truths just act componentwise w.r.t. connectors. The canonical example of (Σ I, E I )-algebra constructed from a (Σ, E)-algebra S is provided by the structure, Θ = S [I], having as support the set S [I] of increasing functions from I to S (that assign actual values to the formal shades) and: - defined pointwise; 9 - for each n 0, w = w 1... w n { 1, 1} n, σ Σ w, f 1,... f n Θ, i I, Θ σ (f 1,..., f n )(i) = S σ (s 1,..., s n ), where, for each k {1,..., n}, s k = f k (i) if w k = 1 and s k = f k (d(i)), if w k = 1; - for each i I and f Θ, Θ i (f) : I S is the constant function f(i). That S [I] is indeed a (Σ I, E I )-algebra is shown by Lemma 2.(2). Lemma 1 Let L be a Σ I -algebra satisfying (A1)-(A4). Then: (1) For each x L, [( i I)L ϕi (x) = L ϕi (y)] iff x = y; (2) If I has a first element 0 (a last element 1), then ϕ 0 (x) x (x ϕ 1 (x)). 9 For a model A, we denote A σ the operation corresponding to σ; but the relation corresponding to is denoted too - remember that also denotes the relation on I. 8

9 (3) The following are equivalent for an x L: (a) ( i I) ϕ i (x) = x; (b) ( i I) ϕ i (x) = x; (c) ( i I, y L) ϕ i (y) = x. Proof: (1): This is a weaker form of (and follows immediately from) the determination principle. (2): Fix j I and notice that, for each i I, ϕ i ϕ 0 (x) = ϕ 0 (x) ϕ j (x) = ϕ i ϕ j (x), then apply the determination principle. (3): (a) implies (b) is true because I is non-empty. For (b) implies (c), take an i I and let y = ϕ i (x); then ϕ i (y) = x by (A1). (c) implies (a) also follows from (A1). q.e.d. Let L be an Σ I -algebra satisfying (A1)-(A5). Define its center to be the Σ-algebra C(L) with: - C(L) = {x L / ( i I)ϕ i (x) = x} - for each σ Σ n, C(L) σ (a 1,..., a n ) = L σ (a 1,..., a n ). The operations on C(L) are well-defined: if a 1,..., a n C(L), then, for each i I, k {1,..., n}, L ϕi (a k ) = a k and L ϕd(i) (a k ) = a k ; hence, by (A5), ϕ i (L σ (a 1,..., a n )) = L σ (a 1,..., a n ), so L σ (a 1,..., a n )) is also in C(L). The center of L consists of all the truth values from L that are reducible to each one of their shades - they are actually the simple, untoned truth values. Lemma 2 Let S be a (Σ, E)-algebra, L a Σ I -algebra satisfying (A1)-(A5), e a Σ-sentence. Then (1) C(L) = Σ e iff L = ΣI e i ; if L is a (Σ I, E I )-algebra, then C(L) is a (Σ, E)- algebra; (2) S [I] = Σ I ; C(S [I] ) is isomorphic to S; (3) There exists an embedding of L into C(L) [I] ; (4) E = Σ e iff E I = ΣI e i. Proof: (1): Because, for each x L, x is of the form ϕ(y) iff x C(l), it follows that L = ΣI e is equivalent to e holds in L relative to C(L), in the sense that L satisfies a statement 10 that is obtained from e by replacing each ( x) with ( x C(L)) and each ( x) by ( x C(L)); but the last expresses precisely C(l) = Σ e. 10 We use this informal language because we would not want to through much technicalities into a simple ideea. 9

10 (2): We first show that S [I] satisfies (A1)-(A5). (A1) is obvious, (A2) follows from the increasingness of functions, while (A3) and (A4) are true because each function is uniquely determined by its values and is increasing. (A5) follows immediately from the definition of the operations on S [I]. Because S [I] satisfies these axioms, it makes sense to talk about its center, C(S [I] ), which turns up isomorphic to S if we notice that the constant functions behave both orderly and algebraically like their corresponding elements from S (the i-d(i) variations in defining the operations on S [I] does not affect constant functions). Now, applying point (1), for each e E, S [I] = ΣI e i, because C(S [I] ) S = Σ e; so the axiom (A6) is also satisfied. Hence S [I] is a (Σ I, E I )-algebra. (3): Denote Ω = C(L) [I]. Define u : L C(L) [I] as follows: for each x L, u(x) : I C(L) is u(x)(i) = L ϕi (x) for all i I. u is well defined as a function, since each u(x) is, by (A2), increasing. u is an order embedding (and hence also injective), because of (A3) and (A4). u commutes with the ϕ i -s because, for each x L and i I, both u(l ϕi (x)) and Ω ϕi (u(x)) are the constant ϕ i (x) function. Finally, let us check that u commutes with each operation σ Σ w, where w = w 1... w n ; let x 1,... x n L, i I. Then On the other hand, u(l σ (x 1,..., x n ))(i) = L ϕi (L σ (x 1,..., x n )). Ω σ (u(x 1 ),..., u(x n ))(i) = C(L) σ (y 1,..., y n ) = L σ (y 1,..., y n ), where, for each k, y k = u(x k )(i) = L ϕi (x k ) if w k = 1, and y k = u(x k )(d(i)) = L ϕd(i) (x k ) if w k = 1. Now, that u(l σ (x 1,..., x n ))(i) = Ω σ (u(x 1 ),..., u(x n ))(i), follows from the fact that L satisfies (A5). (4) only if : Let L be a (Σ I, E I )-algebra. Then C(L), being a (Σ, E)-algebra, satisfies e, and hence, by point (1), L satisfies e i. if : Let S be a (Σ, E)-algebra. Then S C(S [I] ) by point (2), and C(S [I] ) satisfies e, because of point (1) and S [I] being a (Σ I, E I )-algebra; hence S satisfies e. q.e.d. Remark 1 Lemma 2.(3) says that each (Σ I, E I )-algebra is an algebra of increasing functions (although not all the Σ I -subalgebras of S [I] are necessarily (Σ I, E I )-algebras); just like each Boolean algebra is an algebra of (bare, since we require no order on I - see Subsection 5.1) functions from I to {0, 1}, and every Lukasiewicz-Moisil algebra is an algebra of increasing functions to some Boolean algebra. Lemma 3 Let L be a (Σ I, E I ) algebra. Then L is simple between (Σ I, E I )- algebras whenever C(L) is simple between (Σ, E)-algebras. 10

11 Proof: Assume that C(L) is simple and also, by absurd, that L is not simple. Then there exists a non-full and non-trivial congruence ( 0, 0 ) on L such that L/( 0, 0 ) is a (Σ I, E I )-algebra. Define 1 to be the restriction to C(L) of 0 ; this is obviously an equivalence relation on C(L) compatible with the operations from Σ. Now, define 1 C(L)/ 1 C(L)/ 1 by x/ 1 1 y/ 1 iff x/ 0 0 y/ 0 for all x, y C(L). Because C(L) is a Σ-subalgebra of L, we have that ( 1, 1 ) is a Σ-congruence on C(L). Furthermore, one can easily check that C(L)/( 1, 1 ) is isomorphic to C(L/( 0, 0 )), the isomorphism being x/ 1 x/ 0. But C(L/( 0, 0 )) is a (Σ, E)-algebra, since L/( 0, 0 ) is a (Σ I, E I )-algebra. It remains to show that ( 1, 1 ) is neither full, nor trivial. Consider the cases: - 0 is not the total relation on L - so there exist x, y L with x 0 y; because of the determination principle in the (Σ I, E I )-algebra L/( 0, 0 ), it follows that there exists i I with ϕ i (x) 0 ϕ i (y); hence neither 1 is the total relation on C(L); - 0 is the total relation on L, but 0 is not the total relation on L/ 0 - impossible, since L/ 0 has only one element and 0 must be reflexive; - 0 is not the diagonal on L; then x 0 y for some x, y L, x y; by the determination principle, ϕ i (x) ϕ i (y) for some i I; also, 0 being compatible with the Σ I operations, ϕ i (x) 0 ϕ i (y); hence 1 is not the diagonal on C(L); - 0 is the diagonal on L, but there exist x, y L with x/ 0 0 y/ 0, but x y; then, for some i, ϕ i (x) ϕ i (y) etc.; we get that 1 is the diagonal on C(L), but 1. In none of the above cases, ( 1, 1 ) is full or trivial. q.e.d. Lemma 4 If Alg (Σ,E) is closed to arbitrary direct products (to arbitrary direct products and subalgebras) then so is Alg (ΣI,E I ). Proof: Being closed to arbitrary direct products and subalgebras means being axiomatized by some set of Horn sentences; but one can easily see that the definition of E I from E is deduction-independent, the class Alg (ΣI,E I ) being invariant to different axiomatizations of Alg (Σ,E) ; this means that we can assume E to be formed only by Horn sentences; so E I consists also of Horn sentences. Thus half of the proposition is proved. Consider now that any direct product of (Σ, E)-algebras is a (Σ, E)-algebra. Let (L i ) i I be a family of (Σ I, E I )-algebras. Then their direct product L = j J L i satisfies all the Horn clauses satisfied by the L i -s, including (A1)-(A5). It remains to show L = ΣI (A6). According to Lemma 2.(1), it suffices to prove that C(L) = Σ E. But C(L) j J C(L j) (as one can immediately see), the last being, according to our hypothesis, a (Σ, E)-algebra. q.e.d. 11

12 We are now going to describe a certain set of properties that are inherited by the models of (Σ I, E I ) from those of (Σ, E). Recall that X denotes a fixed countable set of variables. For each x X and t T (x), define recursively Is(x, t) the isotonicity of x in t by: - if t is a constant or a variable, then Is(x, t) = 1; - if t = σ(t 1,..., t n ), - Is(x, t) = p if, for each k {1,..., n}, w k Is(x, t k ) = p; - 0, otherwise. Notice that the isotonicity of a variable in a term can only be 1, 1, or 0, meanig, respectively (and this can be proved): in each (Σ, E)-algebra L, L t is increasing, decreasing, or none of these, in x, where L t : L X L is the term function associated to t. We extend the definition of isotonicity Horn sentences, as follows: for each variable x and Horn sentence e of the form j J t j r j t j t r t, where r, r j {=, }, let Is(x, e) = p if Is(x, t) = Is(x, t ) = Is(x, t j ) = Is(x, t j) = p { 1, 0, 1} for all j J and I(x, e) = 0 otherwise. Call a Horn sentence e regular if Is(x, e) 0 for each variable x (or, equivalently, for each variable x appearing in e). Proposition 1 1.(a) Any (Σ I, E I )-algebra satisfies all the regular Σ-Horn sentences that are consequences of E. 1.(b) E I can be equivalently express by: - (A1)-(A5); - all the regular Σ-Horn sentences from E; - e i, where e E is not a regular Horn sentence. 2. Suppose that is the diagonal on I and d : I I is the identity function. Then (a) Any (Σ I, E I )-algebra satisfies all the Σ-Horn sentences that are consequences of E. (b) E I can be equivalently express by: - (A1)-(A5); - all the Horn sentences from E; - e i, where e E is not a Horn sentence. Proof: 1.(a): Let L be a (Σ I, E I )-algebra and e a Horn sentence of the form j J t j r j t j ) t r t, where t, t j {=, }. Using the determination principle, in order to show 12

13 L = ΣI e, it suffices to prove that, for each i I, L = ΣI ϕ i (t j ) r j ϕ i (t j) ϕ(t) r ϕ(t ). (I) j J But, because e is regular and using repeatedly the axiom (A5) until we get with the ϕ i -s to the leaves of the terms, we have L = ΣI ϕ i (t) = t $i, L = ΣI ϕ i (t ) = t $i, L =ΣI ϕ i (t j ) = t $i j, L = ΣI ϕ i (t j) = t $i j, (II) for all j J, where t $i is defined by replacing in t each variable x with ϕ i (x) or ϕ d(i) (x), according to its isotonicity - whether Is(x, t) is 1 or 1; t $i, t $i j, t $i j are defined similarly. Notice that, because of regularity, each variable x has constant isotonicity 1 or 1 in all terms, hence x will be replaced by the same term, either ϕ i (x), or ϕ d(i) (x), in all the terms. Using (II), (I) becomes equivalent to L = ΣI j J t $i j r j t $i j t $i r t $i. A similar argument as the one from Lemma 2.(1) proves the last relation to be equivalent to C(L) = Σ e, which is true because C(L) is a (Σ, E)-algebra. 1.(b): Let L be a (Σ I, E I )-algebra. Then, according to point 1.(a), it also satisfies all the regular Horn sentences from E. Conversely, let L be a Σ I - algebra satisfying the discussed three types of axioms. We want to show that L also satisfies e i, where e E is a regular Horn sentence. But, according to Lemma 2.(1), L = ΣI e i iff C(L) = Σ e; on the other hand, C(L) is a Σ- subalgebra of L, hence C(L) = Σ e. Thus L = ΣI e i. 2.(a): Because of the hypotheses, as far as only Σ-operations are concerned, C(L) [I] is the I-power algebra of S and, according to Lemma 2.(3), L is a subalgebra of C(L) [I]. Hence it satisfies all the Horn clauses from E. 2.(b): The argument is similar to that of 1.(b), only more simple. q.e.d. Remark 2 In particular, Proposition 1.1(a) says that, if the order in (Σ, E) is equationally defined by a sentence e (of the form x y iff t = t, where t, t T ({x, y}), then the order in (Σ I, E I ) is also equationally defined, by the same sentence. Hence, if we start with a purely algebraic theory, it remains so after toning. We now define two adjoint functors between the categories of shaded and simple truth structures. Let C : Alg (ΣI,E I ) Alg (Σ,E) and T : Alg (Σ,E) Alg (ΣI,E I ), where: - the functor C is already defined on objects, L C(L); 13

14 - if L h L is a (Σ I, E I )-morphism, C(h) : C(L) C(L ) is the restriction and corestriction of h - this is consistent, since, h, commuting with each ϕ i, takes elements x with x = L ϕi (x) for some i into elements y with y = L ϕ i (y) for the same i; C(h) is obviously a (Σ, E)-morphism; - if S is a (Σ, E)-algebra, then T (S) = S [I] ; - if S g S is a (Σ, E)-morphism, then T (g) : S [I] S [I] is defined by T (g)(f) = g f for each f S [I] ; T (g) is well defined as a function, g f being increasing, because g and f are so. Moreover, T (g) is a (Σ I, E I )-morphism, because: f 1 f 2 implies g f 1 g f 2, since g is increasing; T (g) commutes with the ϕ i -s, since it takes the constant function f(i) into the constant function g(f(i)); let Θ = S [ I] and Θ = S [I], σ Σ w, w = w 1... w n, f 1,... f n Θ - we need to show T (g)(θ σ (f 1,..., f n )) = Θ σ(t (g)(f 1 ),..., T (g)(f n )), that is that is g Θ σ (f 1,..., f n ) = Θ σ(g f 1,..., g f n ), i I, g Θ σ (f 1,..., f n )(i) = Θ σ(g f 1 (i),..., g f n (i)). Looking at the definition of Θ σ and Θ σ, the last becomes i I, g S σ (a 1,..., a n )(i) = S σ(g(a 1 ),..., g(a n )), where, for each k {1,..., n}, a k = f k (i) if w k = 1 and a k = f k (d(i)) if w k = 1. The last is true, because g commutes with σ, being a Σ-morphism. It is immediate that C and T take identities into identities and commute with morphism composition, thus they are functors. Proposition 2 (1) C is surjective on objects, T is fully faithful and injective on objects. (2) There exists an adjunction (C, T, η, ɛ), 11 with ɛ functorial isomorphism and η consisting of embeddings. Proof: Let L be a (Σ I, E I )-algebra and S an (Σ, E)-algebra. Define the function χ S,L : Alg (ΣI,E I )(L, T (S)) Alg (Σ,E) (C(L), S) by χ S,L (h)(a) = h(a)(i), where i is an arbitrary element from I, 11 C being left adjoint, T right adjoint, η unit and ɛ counit. 14

15 for each Σ I -morphism h : L S [I] and each a C(L). The definition does not depend on i, because a C(L) implies h(a) C(S [I] ), hence h(a) is a constant function. χ S,L (h) : C(L) S is obviously a Σ-morphism. Define ψ S,L : Alg (Σ,E) (C(L), S) Alg (ΣI,E I )(L, T (S)) by: for each Σ- morphism g : C(L) S, ψ S,L (g) : L S [I] is ψ S,L (g)(a)(i) = g(l ϕi(a)), for all a L, i I. We prove that ψ S,L (g) is a (Σ I, E I )-morphism. Denote Θ = S [I]. - preservation of order: a b implies L ϕi (a) L ϕi (b), which implies, since g is increasing, g(l ϕi (a)) g(l ϕi (b)); - commutation with the ϕ i -s: this is true, since, for each i and a, both ψ S,L (g)(l ϕi (a)) and Θ ϕi (ψ S,L (g)(a)) are the constant g(l ϕi (a)) functions; - commutation with the Σ-operations: let σ Σ w, with w = w 1... w n, a 1,..., a n L and i I. Then ψ S,L (g)l Σ (a 1,..., a n )(i) = gl ϕi L Σ (a 1,..., a n ). (I) On the other hand, Θ Σ (ψ S,L (g)(a 1 ),..., ψ S,L (a n ))(i) = S Σ (y 1,..., y n ), where, for each k {1,..., n}, y k = ψ S,L (a k )(i) = g(l ϕi (a k )) if w k = 1 and y k = ψ S,L (a k )(i) = g(l ϕd(i) (a k )) if w k = 1. This further means, also applying that g is a Σ-morphism, Θ Σ (ψ S,L (g)(a 1 ),..., ψ S,L (a n ))(i) = S Σ (g(z 1 ),..., g(z n )) = g(l σ (z 1 ),..., L σ (z n )), (II) where, for each k {1,..., n}, z k = L ϕi (a k ) if w k = 1 and y k = L ϕd(i) (a k ) if w k = 1. Now, the last member of equality (I) is equal to the last member of (II), because L satisfies (A5). Let us now prove that χ S,L and ψ S,L are inverse to each other. For each g Alg (Σ,E) (C(L), S), a C(L), χ S,L ψ S,L (g)(a) = ψ S,L (g)(a)(i) = g(l ϕi (a)) = g(a)%;. For each h Alg (ΣI,E I )(L, T (S)), a L, i I, ψ S,L χ S,L (h)(a)(i) = ϕ S,L (h)(l ϕi (a)) = h(l ϕi (a))(i) = h(a)(i). The naturality of ϕ (and hence that of ψ) is obvious. So we have an adjunction. The unit η = (η L ) L and counit ɛ = (ɛ S ) S are given, according to general adjunction properties, by η L = ψ C(L),L (1 C(L) ) and ɛ S = χ S,T (S) (1 T (S) ). That is: - η L : L T C(L), η L (a)(i) = ϕ i (a), for each a L, i I; this is precisely the embedding u from Lemma 2.(3); - ɛ S : S CT (S), ɛ S (x) is the constant function x for each x S; this is precisely the isomorphism from Lemma 2.(2). 15

16 (1): Let S be a (Σ, E)-algebra. Because S CT (S), it follows that S = C(L) for some L isomorphic to T (S); so C is surjective on objects. T is obviously injective on objects. T is faithful because, for a Σ-morphism f : S S, the restriction of T (f) to the constant functions from T (S) uniquely determines f. That T is full immediately follows from the fact that, for each Σ-algebra S, CT (S) S. q.e.d. In the next two corollaries, assume that the class Alg (Σ,E) is closed to arbitrary direct products - in particular, this is the case of E consisting only of Horn sentences and infinitary disjunctions of atomic ground (i.e. without variables) sentences; this is mostly the case of algebraic theories that correspond to logics. Corollary 1 Let C be a class of (Σ, E)-algebras such that each (Σ, E)-algebra can be embedded into a direct product of elements from C. The each (Σ I, E I )- algebra can be embedded into a direct product of elements from T (C) = {T (S) / S C}. Proof: Notice first that, according to Lemma 4, Alg ΣI,E I also preserves products. In addition, being full subcategories of Alg Σ and Alg ΣI, the categories Alg Σ,E and Alg ΣI,E I have direct products, which coincide to the usual ones from the big categories. Now, it is immediate that T preserves embeddings. Also, being a right adjoint, it preserves direct products. Let L be a (Σ I, E I )-algebra. Then C(L) is embedded into p P S p, with (S p ) p P C; hence L is embedded into T C(L), ( which is embedded into T p P p) S p P T (S p); thus L is embedded into p P T (S p). q.e.d. Corollary 2 Assume that each (Σ, E)-algebra is simple iff it is directly indecomposable. Then any (Σ I, E I ) algebra is simple iff it is subdirectly irreducible. Proof: Simple implies subdirectly irreducible is always true. Conversely, assume that L is a non-simple (Σ I, E I )-algebra. We want to prove that it is subdirectly reducible. By Lemma 3, C(L) is not simple and hence directly decomposable. This means that we can get two non-trivial (Σ, E)-algebras S 1 and S 2 and an isomorphism f : C(L) S 1 S 2. We make the following denotations and remarks: - pr i : S 1 S 2 S i, i {1, 2}, denote the two canonical projections; - Also, π i : T (S 1 ) T (S 2 ) T (S i ), i {1, 2}, denote the two canonical projections; - T, being a right adjoint, preserves direct products, so we have an isomorphism 16

17 g : T (S 1 S 2 ) T (S 1 ) T (S 2 ) such that π i g = T (pr i ), i {1, 2}; - T (f) : T C(L) T (S 1 S 2 ) is an isomorphism, since any functor preserves isomorphisms; - Define L i to be the image of T (pr i ) T (f) η L, i {1, 2}; L i is a subalgebra of T (S i ), with operations and relation inherited from T (S i ); denote ι i : L i T (S i ), i {1, 2}, the inclusion morphism, which is an embedding; - Denote ι 1 ι 2 : L 1 L 2 T (S 1 ) T (S 2 ) the product of ι 1 and ι 2 ; - Define r : L L 1 L 2 to be the corestriction of g T (f) η L to L 1 L 2 - this is well defined, because of the choice of the L i -s; in addition, r is a morphism and ι 1 ι 2 r = g T (f) η L. We want to show that L is a subdirect product of L 1 and L 2 (by r) and that p i r is not an isomorphism, i {1, 2}, where the p i -s are the canonical projections. Since g, T (f), and η L are embeddings, r is an embedding too. Also, by the choice of L i, p i r is a surjective morphism, i {1, 2}. Finally, suppose, by absurd, that one of these two morphisms, say p 1 r, is an isomorphism. Then ι 1 p 1 r is an embedding. But ι 1 p 1 r = T (pr 1 ) T (f) η L = T (pr 1 f) η L, hence T (pr 1 f) η L is an embedding. Denote, for each S, L, by χ S,L : Alg (Si,E I )(L, T (B)) Alg (Σ,E) (C(L), B) the natural bijection of the adjunction (C, T, η, ɛ). By general adjunctions properties, we have that: - χ C(L),L (η L ) = 1 C(L) ; - χ S1,L(T (pr 1 f) η L ) = pr 1 f; - χ S1,T (S 1)(1 T (S1)) = ɛ S1. Using the naturality of χ w.r.t. the morphisms C(L) pr1 f S 1 and L T (pr1 f) η L T (S 1 ), we obtain ɛ S1 C(T (pr 1 f) η L ) (pr 1 f) = T (pr 1 f) η L to be an embedding. Since ɛ S1 is an isomorphism and C(T (pr 1 f) η L ) is an embedding (because C preserves embeddings), we get that pr 1 f : C(L) S 1 is an embedding. But, f being an isomorphism, pr 1 is an embedding. It immediately follows that S 2 is trivial, which is a contradiction. q.e.d. 4 Logical aspects As already mentioned, we only discuss the case of logics with truth-functional connectors. Let us sketch here some logical implications of the previous section. It is a well established fact [19] that propositional calculi are strongly connected to equational logic - giving an (substitution-free) axiomatization of one propositional calculus comes to giving an equational axiomatization at the truth structures level, while completeness w.r.t. an emphasized class of standard truth 17

18 structures (e.g. {0, 1} in classical logic, continuous t-norms in fuzzy logic etc.) comes to the generation of the quasivariety by that class; this is the reason why a representation theorem at the algebraic level assures completeness at the logical propositional level. In our case, there is however a distinction to be made. By allowing the specification (Σ, E) of the starting propositional logic to contain also non-equational axioms, we had in mind two types of axioms: - those that state the general semantic behavior of logical connectors, translating rules of reasoning that are specific to the considered logic; these axioms are usually finitary Horn clauses; - those that refer specifically to the truth structure itself, reflecting purely semantical principles, that are not syntactically expressible, like the logical true is different from false or the fuzzy conjunction is continuous ; here we also encounter principles that are almost, but not quite syntactically expressible, like the truth degrees are totally ordered, approximated in fuzzy logic [9] by the BL axiom (ϕ χ) (χ χ). Axioms of the second type are not usually considered when the propositional logic is axiomatized. In order to achieve standard completeness (i.e. w.r.t. to the desired truth structures) one appeals to the mentioned trick of getting a quasivariety generation result. The passing from to (Σ, E) to (E I, Σ I ) is, according to the above discussion, not only algebraic, but also logical: while the (Σ I, E I )-algebras offer semantical interpretations for the propositions of the shaded calculus, the Horn sentences from E I offer logical axioms. The logical meanings of some algebraic results from the previous section are as follows: - Lemma 1.(2): Each proposition implies its strongest shade and is implied by its weakest shade. - Proposition 1.1.(b): all the sound regular (Σ, E)-propositional axioms are inherited by the (Σ I, E I ) propositional logic; for instance, if ϕ Σ, this is the case of ϕ (χ ϕ), [ϕ (χ ψ)] [χ (ϕ ψ)], both sound in almost all fuzzy logics, [ϕ (χ ψ)] [(ϕ χ) (ϕ ψ)], sound in Gödel s logic, (ϕ χ) [(ψ χ) ((ϕ ψ) χ)], sound in all logics with classically interpreted disjunction etc. - Corollary 1: if in (Σ, E) there is provided a completeness theorem based on a representation theorem, then the logic of (Σ I, E I ) is also complete - this was applied in [2] for the case of Lukasiewicz-Moisil theory, where a propositional logic together with a completeness theorem are provided. 18

19 5 Examples of shades 5.1 Boolean algebras We consider the postulates of classical logic in their very basic form. Namely, let Σ consist, besides, of,,,, 0, 1, and E be the following set of axioms: - 0 1; (the principle of consistency); - ( x)x = 0 or x = 1 (the principle of bivalence); - the whole table that defines the logical connectors, in ground (i.e. variable-free) form: 0 0 = 0, 1 = 0 etc. Of course, the only model (up to an isomorphism) is the classical {0, 1} truth value. Let I be a non-empty set, the diagonal and d the identity and let us look at the (Σ I, E I )-algebras B. Notice that, because of Proposition 1.(2), each B satisfies all the axioms from E except the bivalence principle, and, moreover, all the Horn consequences of E; in particular: - the algebraic reduction of : x y iff x y = x; - all the Boolean algebra axioms. This means that a (Σ I, E I )-algebra B is just a Boolean algebra together with a family of [Boolean operations]-preserving functions (ϕ i ) i I, ϕ i : B {0, 1}, such that each b B is determined by its 0, 1-shades. If we define f : B P(I) by f(b) = {i I / ϕ i (b) = 1}, we find, using the Stone representation theorem, that a (Σ I, E I )-algebra is just a Boolean algebra with card(i) ultrafilters and a (Σ I, E I )-morphism is just a Boolean morphism. The above discussion made precise the statement that Boolean algebras are truth structures obtained from {0, 1} by a process of toning Modalities We have already seen how modalities naturally induce complex truth values that are nothing more than instantiations of a formal shade hierarchy. Next, we provide a toy semantical framework inside of which the logical connectors on the complex truth values are precisely those inherited from their Boolean shades. Assume that there is an experiment with a certain degree of randomness in its behavior that takes place over and over again, indefinitely. The sentences are supposed to describe the situation that resulted from the experiment - but we do not consider a present experiment, but the whole series of experiments, globally, hence the evaluation of statements will not depend on a particular moment in time (this could be seen as a flattening of temporal logic [18]). Let the hierarchy (I, ) of shades that compose the truth of a sentence be always 12 We find this (mathematically quite trivial) observation important for pointing out the extensive presence of shades in mathematical logic. 19

20 < usually < relevantly often < sometimes and the decreasing involution d : I I be the only one possible, reverting the chain. The obtained complex truth structure is a chain of five truth degrees, that could be called: necessarily false < rarely true < often enough true < usually true < necessarily true. Interpret 0 as the always false sentence and ϕ χ as ϕ brings χ (here brings means in the next moment in time ). Also, interpret negation in the usual way, as saying non ϕ. The starting structure S is the classical Boolean algebra L 2 = ({0, 1},,, 0), while the shaded structure, L [I] 2, is precisely the one with implication, negation and zero described above. Indeed, the (Boolean) value of not usually ϕ is, of course, the same as that of relevantly often ϕ; also, ϕ always brings χ means: if ϕ never happens, than χ is free of duties, but if ϕ happenes sometimes, than χ has to happen all the time - since all the repetitions of the experiment are independent, at each time, you never know, ϕ might actually occur next time, so χ must happen this time. In accordance with Proposition 1 and our interpretation of connectors, the following principles hold here: - ϕ means ϕ brings false; - sometimes ϕ brings χ means always ϕ implies sometimes χ. (Note that, if, at a time, ϕ does not occur, it means that this time, ϕ brought χ. Note also that the shaded implication, brings, is expressible in terms of its Boolean shades, which use the classical implies.) Since ϕ ϕ does not come from a regular Horn sentence (see Proposition 1), this doesn t have to hold (in the shaded structure, and it actually doesn t: hold means here be true w.r.t. all shades, and ϕ ϕ is not necessarily true, since ϕ does not always have to be followed by ϕ. Of course, the above discussed structure is no other than the canonical 5- valued Lukasiewicz-Moisil algebra, L 5 (see below). Our point was to provide a shaded approach to modalities that goes all the way, not using the external Lukasiewicz implication (not definable from the Lukasiewicz-Moisil operations for n 5), but the implication inherited from Booleans. The lattice disjunction and conjunction from Lukasiewicz-Moisil algebras, as well the axioms of their commutation with shades, also make sense in this framework, if we interpret, for instance,, as a two-time disjunction, ϕ χ meaning a fail of phi brings a realization of χ, or (in two times) either ϕ, or χ occurs. As for passing to arbitrary L n -s, this is done following further the Lukasiewicz idea of allowing degrees of necessity and possibility (in our case, by allowing more sensible frequencies between always and occasionally). 5.3 Lukasiewicz-Moisil algebras The following definition is taken from [2] 20

21 Definition 1 Let Θ be the order type of a totally ordered set with least element 0 and suppose Θ 2. We fix J = {0} + I (ordinal sum) of type Θ and a decreasing involution d : I I. A Θ-valued Lukasiewicz-Moisil algebra (with negation), LM Θ for short, is a structure of the form (L,,,, (ϕ i )i I) such that: (1) (L,,,, 0, 1) is a de Morgan algebra, that is a bounded distributed lattice with a decreasing involution satisfying the de Morgan property x y = x ȳ; (2) For each i I, ϕ i : L L is a lattice endomorphism; (3) For each i I, x L, ϕ i (x) is complemented by, that is ϕ i (x) ϕ i (x) = 1, ϕ i (x) ϕ i (x) = 0; (5) For each i, j I, ϕ i ϕ j = ϕ j ; (6) For each i j I, ϕ i ϕ j ; (7) For each i I, x X, ϕ i ( x) = ϕ d(i) (x). (8) (Moisil s determination principle) [ i I, ϕ i (x) = ϕ i (y)] implies x = y. If n 2, J = {0,..., n 1} and hence I = {1,..., n 1}, and d : I I is d(i) = n i, then L is called n-valued Lukasiewicz-Moisil algebra (LM n for short). 13 Let now BOOLE denote both the specification and the variety itself of Boolean algebras, in the usual axiomatization using operations,,, 0, 1: distributive bounded lattice, de Morgan, and complementation properties; we also take inside BOOLE, with its postulated relation to infima. By Proposition 1.1, the shaded specification (Σ I, E I ) can be taken such that E I contains, among others, all the regular axioms of Boolean algebras - but notice that these are precisely the de Morgan algebra axioms, the only non-regular Boolean axioms being x x = 1 and x x = 0. So these two should be stated in E I as point (3) of Definition 1. If we also remark that, because the ϕ i -s commute with, the determination principle could be equivalently stated using = or, we find that the LM Θ algebras are precisely the (Σ I, E I )-algebras. The standard LM n -algebra is L n = {0, 1/(n 1),..., (n 2)/(n 1), 1}, having the usual number order, i/(n 1) = (n i)/(n 1), and ϕ i : L n {0, 1}, ϕ i (j/(n 1)) = 0 iff j i. If n = 2, we find the Boolean algebra {0, 1} with ϕ 1 the identity. Notice that, because T ({0, 1}) L n, the representation theorem of LM n -algebras, saying that each is embedded into a power of L n, is a direct consequence Corollary 1, via the Stone representation theorem (this was already pointed out in [7]). On the other hand, Corollary 2 does not provide us the whole information here, since it was proved [2] that the simple LM n -s coincide with the directly indecomposable ones, like Boolean algebras. 13 These were in fact the structures firstly introduced by Moisil in 1941 [14], after he had previously considered, in 1940, the 3 and 4-valued cases [13]. 21

22 5.4 Shaded fuzzy algebras We have already pointed out that shaded, in this paper s acceptance, does not mean fuzzy. While fuzzy refines truth (and its continuity) by allowing more degrees, shaded also refines truth, but in the direction of considering (perhaps hierarchirised) tonalities. The passage from simple to shaded is not always suitable in a componentwise fashion for all the logical operators (although for most of them). For instance, in the case of Lukasiewicz-Moisil algebras, unlike,,, the implication does not suit much being imported from Boolean algebras. Still, the inherited good Boolean properties assures residuation of conjunction, hence the existence of an implication, called the Heyting implication. 14 Thus, sometimes, when toning a logic, we may want to leave some of the operators (that are uniquely determined by others) implicit rather than explicit. This shall be illustrated by looking at a structure fuzzy by nature, the residuated lattice (see [8]). Definition 2 A residuated lattice (RL) is a structure (L,,,, 0, 1) where (L,,, 0, 1) is a bounded lattice, (L,, 1) is a commutative monoid, and admits a residuum (or implication), that is one of the following equivalent conditions holds: (1) ( x, y)( max{z / x z y}; (2) there exists a binary operation on L,, such that x y z iff x y z. If is taken as an operation in the RL-signature, we have a quasivariety (immediately turning up to be a variety) of algebras - let E be its axiomatization (consisting of the bounded lattice axioms and condition (2) from the above definition). On the other hand, residuated lattices can be axiomatized by a first-order theory E without taking in the starting signature. Let (I,, d) be arbitrary. The shaded theory E I inherits some implicational looking properties from E (namely the regular ones), like x (y z) = y (x z) = (x y) z. Among the axioms that are inherited from diverse subvarieties E F of residuated lattices, like MTL, BL, Product, Gödel, MV, Heyting algebras, to the corresponding F I -algebras we find: - any commutativity, associativity, idempotence, cancelation, neutral or anihilator element axiom, distributivity between any isotone binary connectors (like, +,, etc.); - x y [(y z) (x z)]; - x (y z) (x y) (y z); - (x y) (x z) x (y z). But other axioms, like the residuation itself, are not inherited (as mentioned above, not even in the case of Lukasiewicz-Moisil algebras). However, when the number of shades is finite, the E I -algebras, that we call shaded residuated lattices, are residuated, in the sense of satisfying condition (1) of Definition 14 This is because, with it, one gets a Heyting algebra. 22