A generalization of Twist-Structures Semantics for n-valued logics

A generalization of Twist-Structures Semantics for n -valued logics

V´ıctor Fern´ andez, Carina Murciano

Basic Sciences Institute (Mathematical Area) Philosophy College

National University of San Juan (UNSJ) San Juan, Argentina

E-mail:[email protected], [email protected]

Abstract

In this paper we show a generalization of Twist-Structures Semantics which allows, under certain conditions, to express consequence relations defined by abstract finite matrices, in an alternative way. This general construction (called Discriminant Structures Semantics here) is based on the existence of adiscriminant pair, to be defined. A number of motivating examples are shown, and some technical results are demonstrated. Besides that, we compare this technique with other already existing ones.

1 Preliminaries

The currently called Twist-Structures Semantics were developed by M. Fidel and D. Vakarelov (see [3] and [12]) to obtain an alternative presentation of the algebraic semantics for the Strongly Intuitionistic Nelson’s LogicN, character- ized by a special connective∼: thestrong negation. In its original application, Twist-Structures Semantics consists of a class of algebras whose supports are setsS ⊆H×H^∗. Here,H is a Heyting algebra, meanwhileH^∗ is its dual (in the underlying order). The “torsion” of the second set allows to interpret in a more faithful way the behavior of∼. Besides that, the implication connective

⊃ of N is related with the natural order of H ×H^∗, too. Twist-Structures Semantics were applied to other logics different fromN(see [4], or [6]). In these new applications, a basic idea is preserved: twist-structures are just new (more intuitive) representations for previously existent algebras.

A somewhat different approach to this semantics was given in [5] and [7], wherein twist-structures provide a new characterization ofmatrix logics(in particular, ofn-valued ones). The underlying idea here is not focused on the algebraic (or

lattice-theoretic) characterization of the supports of these structures, but on other considerations. Between them:

•The elements of the structures to be defined are pairs (or, generally,t-uples).

•Since the logics considered are defined byn-valued matrices, Twist-Structures Semantics discriminates the involved truth-values, in a certain sense.

•In the considered logics, a negation connective ¬allows this discrimination.

On the other hand, the process of “torsion” in some axes of the twist-structures is used in the mentioned works, as well (obviously, this is the reason of the name twist-structure).

In the present article we intend to give a generalization of the mentioned cons- truction, in such a way that it can be applied, eventually, to arbitraryn-valued logics (despite the presence of negation connectives). As we will see, the central point here is the separation of the truth-values of the logics to be considered.

The generalization here proposed will be calledDiscriminant Structures Seman- tics, or D.S.S, for short. In addition, besides the definition of this construction, some abstract technical results will be proved, and we will see some examples of this semantics, too. Briefly, the organization of the topics treated in this article, is as follows: after a minimal fixation of definitions and notation about Abstract Logic, an example of Twist-Structures Semantics (already presented in the literature) will be shown in the next section. This motivates the definition of D. S. S., exemplified by the logicL^?, in Section 3. In the same section will be studied in which way an abstract version of Discriminant Structures Semantics can be defined. This discussion will be extended (in a more technical way) in Section 4. We will show here that the key of our construction is the existence of adiscriminant pair. By the way, this existence is not a trivial fact, because not every matrix can define such a pair, as a counter-example shows, at the end of the section.

Considering that discriminant structures have a certain algebraic character, we will relate our construction with some basic ideas of Abstract Algebraic Logic.

It will be proved, in Section 5, that Discriminant Structures Semantics can be defined, even in the case of non-algebraizable logics. This is done by means of examples of logics that admit D.S.S. but are not protoalgebraic (nor algebraiz- able, therefore).

The last two sections are focused on comparisons of D.S.S. with other construc- tions of the same type. In particular, Section 6 relates discriminant structures with Dyadic Semantics (another construction, defined in [1], which shares the

“same spirit” of our generalization). Finally, the last section is based on Twist- Structures Semantics again, comparing it with Discriminant Structures Seman- tics. So, in this way, the present article is placed in context. We prevent here to the reader that the Sections 3 and 4 are strongly related and are similar in a certain sense. But, meanwhile the former deals with D.D.S by means of an example, the latter applies the previously sketched ideas to give a formal treat-

ment to our construction. We have chosen to take the risk of being redundant, in benefit to a more intuitive approach to D. S. S.

With respect to the basic notation and definitions that will be used in this paper, please take into account that we are studying different ways of definition of a same consequence relation. So, we chose to use the traditional formalism of Abstract Logic, applied mainly to the particular case of consequence relations defined by matrices. For that, we are based mainly in [2], with some little notational changes, when necessary.

Definition 1.1. We denote by ω ={0,1,2...}the set of natural numbers; let V be a countable set, called the set of atomic formulas, fixed from now on. The elements of V are denoted by p, q, r . . ., with subscripts if necessary.

With this in mind we define a signature as a pair (C, ρ), where C is a set of symbols (named connectives), and ρ : C−→ω. For every c ∈ C, ρ(c) is the arity of c. Often we denote any signature just by C, if there is no risk of confusion. For every signatureC, thepropositional language generated by C (denoted by L(C)) is the absolutely free algebra, generated by C over V. Besides that, a C-matrix is a pair M = (A, D), where A is an algebra, similar toL(C), beingD ⊆Athe set ofdesignated values of M. For every connective c, its associated operation in A will be called thetruth-function in A, associated to c. In general terms, every operation f : Aⁿ−→A will be called anA-truth-function¹. Note that, by simplicity, we will identify an algebra with its universe. Besides, we identify the sets of truth-functions inA, associated to the connectives ofC, withCitself. This notational abuse will be sustained all along the paper.

EveryC-matrix defines a consequence relation inL(C), as usual:

Definition 1.2. LetM = (A, D) be aC-matrix. AM-valuation is a homo- morphismv :L(C)→A. Theconsequence relation induced byM is|=M, defined in the following way: Γ |=M α iff, for each valuation v : L(C)−→A, if v(Γ)⊆D, then v(α) ∈ D. We say that α is tautology (relatively to M) iff ∅ |=M α(denoting this as |=M α). Thelogic induced by M is the pair L= (C,|=M). If the domain of a C-matrixM is finite we will say thatM is a n-valued matrix and, by extension, that L = (C,|=M) is an-valued logic.

This definition can be generalized to classes: ifK is a class ofC-matrices, the consequence relation|=K is given by: Γ|=Kαiff Γ|=M αfor everyM inK.

Definition 1.3. Given twoC-matricesM1 = (A1, D1) andM2= (A2, D2), we say thath:A1−→A2is amatrix homomorphism fromM toN ²iff verifies:

(1)his homomorphism (in the algebraic sense) fromA toB.

(2)h(D1)⊆D2.

Two matricesM1andM2areisomorphiciff exists a matrix homomorphismh

1The truth-functions not depend onM, but onA.

2It is convenient to distinguish between valuations (homomorphisms fromL(C), to the support of theC-matrices), cf. Definition 1.2, and matrix homomorphisms.

verifying additionally:

(a)his a bijective function.

(b)his anstrict homomorphism. That is: h(A1−D1)⊆A2−D2.

The symbol∼= denotes isomorphism (between matrices or algebras, depending on the context). Please note that, if M1 and M2 are isomorphic C-matrices, then|=M1 =|=M2. Matrices are one of the simplest ways to furnish consequence relations and, therefore, abstract logics, whose formal definition is the following:

Definition 1.4. Anabstract logic is a pair (CL,`L), whereC is a signature and`L ⊆℘(L(C))×L(C), satisfying, for every Γ∪{α} ⊆L(C),extensiveness, monotonicityandtransitivity(we omit these well-known definitions). Given a logicL = (CL,`L), if there is aCL-matrixM such that`L =|=M, we will say thatLis amatrix logic, and we will use simply|=M, instead of`L. Please recall that a matrix logic is always structural(i.e. closed by substitutions).

In addition, ifM is finite, then|=M verifies finitariness (we omit its definition, too).

We conclude this section with some comments about notation: I.H. means “In- duction Hypothesis”, as usual. The symbolπi denotes the i^th projection of a t-uple. For a formulaα = α(x1, . . . , xn), the expression α(x1/β1, . . . , xn/βn) denotes the uniform substitution, inα, of the variables xi by the formulasβi. If there is not risk of confusion, the expressionα(x1/β1, . . . , xn/βn) will be de- noted simply byα(β1, . . . , βn). With respect to notations related to algebras:

ifB is any algebra, the symbol~adenotest-uples: ~a:=(a1, . . . , at)∈B^t. Besides that, we will use the “classical” two-elements boolean algebra widely through- out this paper. This algebra, as usual, will be denoted as 2 = {0,1}. The 2-truth-functions will be called boolean ones. Finally, the formal equational language that “will talk about algebras”³ is built on the basis ofL(C) (consid- ering its elements asterms, in this context). This implies that the connectives ofC are considered as symbols of functions, in this case. Finally, the symbol≈ will denote the equality predicate of equational languages.

2 An example of Twist-Structures Semantics for n -valued logics

In order to understand the behavior of Twist-Structures Semantics in the con- text ofn-valued logics, consider the following example (taken from [7]), applied to the weakly-intuitionistic logicI¹P⁰, defined in [9]. This logic has as signa- ture the setC[1,0]={¬,⊃}(withρ(¬) = 1,ρ(⊃) = 2), and its definition is the following:

3In particular, we will use equational languages to express in a formal way some boolean equations. Hence, the underlying set of function symbols will be{∧,∨,−,0,1}, with obvious arities.

Example 2.1. The logic I¹P⁰ = (C[1,0],|=M[1,0]) is defined by means of the C[1,0]-matrixM[1,0]= ({F0, F1, T0},{T0}), wherein the truth-functions¬and⊃ are indicated in the tables below:

⊃ F0 F1 T0

F0 T0 T0 T0

F1 T0 T0 T0

T0 F0 F0 T0

¬ F0 T0

F1 F0

T0 F0

Just for a better understanding ofM[1,0], the truth-valuesF0andT0are classical truth and falsehood, respectively; on the other handF1is an “intermediate value of falsehood”.

Considering the definition of I¹P⁰, it is possible to characterize |=M[1,0] using twist-structures. Its definition (taken from [7]) is given in the sequel:

Definition 2.2. For every boolean algebra (B,∨B,∧B,−B,1B,0B)⁴, thetwist- structure associated toB is the algebraR[1,0](B) = (R[1,0](B),¬,⊃), where:

(1) Its support isR1,0(B) ={(x0, x1)∈B×B^∗:x0∧Bx1= 0B}, whereB^∗ is the dual algebra (in its order) ofB.

(2)¬(x0, x1) = (x1,−Bx1).

(3) (x0, x1)⊃(y0, y1)∈R1,0(B) = (x0→By0,−B(x0→By0)).

The class of all the twist-structures of type (1,0) will be denoted byT[1,0]. It is easy to prove that the operations indicated above make sense. On the other hand, the operations in B and B^∗ can be mutually defined. For in- stance, if (x0, x1), (y0, y1)∈R1,0(B), then ((x0, x1)⊃(y0, y1))0:=π0((x0, x1)⊃ (y0, y1)) = x0 →B y0. Also, ((x0, x1) ⊃(y0, y1))1 :=π1((x0, x1)⊃(y0, y1)) =

−B(x0→B x1) = x0∧B−Bx1=x0∨B^∗ −B^∗x1=x1→B^∗ x0. For the sake of simplicity we choose to express all the operations inR1,0(B) in terms ofB. So, B^∗ just remarks the behavior of the negations of any formula, suggesting that the orderin the elements ofB^∗ is inverse to the order inB.

Definition 2.3. Let R[1,0](B) be the twist-structure of type (1,0) associated to B. A R[1,0](B)-valuation is any homomorphismw :V−→R_[1,0](B). Now, theconsequence relation|=T_[1,0] defined inL(C) is given by: Γ|=T_[1,0] αiff, for every twist-structureR[1,0](B) ofT_[1,0], and for every R[1,0](B)-valuationw is valid that:

Ifw(Γ) ={(1B,0B)}thenv(α) = (1B,0B).

In addition, we say thatαistautology (with respect toT[1,0])iff ∅ |=T_[1,0] α (denoted, as usual, by|=T_[1,0] α).

4From now on, the subscript of the operations ofBwill be dropped if there is not risk of confusion.

Remark 2.4. every twist-structureR[1,0](B) inT[1,0]is, in fact, aC[1,0]-matrix whose designated set is{(1B,0B)}. So, the relation|=T_[1,0]can be understood as the consequence relation defined by aclass of C[1,0]-matrices (recall Definition 1.2), which isT_[1,0].

Two essential facts about Twist-Structures Semantics forI¹P⁰are the following:

•InT[1,0] exists a canonical twist-structureM⁰, which is isomorphic toM[1,0].

•Moreover, if6|=T_[1,0] α, then6|=M⁰ α.

These facts (that will be analyzed in more detail later) imply this result:

Theorem 2.5. |=M_[1,0] αiff|=T_[1,0] α, for every α∈L(C[1,0]).

Which are the conditions that allowed us to obtain a Twist-Structures Semantics forI¹P⁰? At first sight, we note that the connective¬was widely used in the definition of the elements of everyR[1,0](B). So, a natural question is if this kind of construction can be applied toany abstract logic, with or without negation.

For that, note that the truth-values ofI¹P⁰can be separated (or discriminated) by means of successive iterations of¬(this is done in a somewhat hidden way).

We will discuss this fact in the next section.

3 A generalization: Semantics of Discriminant Structures

To motivate the definition that will allow us to deal with the generalization suggested in the previous section, consider the following example:

Example 3.1. LetL^? be the {⊃,¬}-fragment of the well-known Lukasiewicz three-valued logic L3, but considering {¹₂} as a designated value. Formally speaking, L^? = (CL^?,|=M^?), where CL^? = {⊃,¬} (with obvious arities) and

|=M^? is defined byM^? = (A^?, D^?), withA^?={0,¹₂,1}andD^? ={₂¹}. InM^?, the truth-functions associated to⊃and¬are:

⊃ 0 ¹₂ 1

0 1 1 1

1 2

1

2 1 1

1 0 ¹₂ 1

¬ 0 1

1 2

1 2

1 0

Of course, even when the truth-values (and, moreover, the operations) are the same as in the matrix that defines L3, the change in the set of designated values produces different tautologies in both logics. In fact, we will prove later that L^? has not tautologies at all. Also, an adequate Twist-Structures Semantics forL^? will be constructed later. For that, we use some basic and well-known facts about Algebraic Logic (see [8]). Besides that, recall that we consider a twist-structure for a logicL= (CL,`L) simply as a particularCL-matrix. We apply this idea in the following definition.

Definition 3.2. Given a boolean algebra (B,∨,∧,−,1,0), theDiscriminant Structure(associated to B) is the followingCL^?-matrix:

R(B) = ((R(B),¬,⊃),{(1,0)}), where:

(1)R(B) ={(x0, x1)∈B²:x0∧x1= 0}.

(2) (x0, x1)⊃(y0, y1) := −x1∧[(x0↔ −y0)∧(x0↔y1)],−(x0∨x1∨y0)∧y1 (3)¬(x0, x1) := x0,−(x0∨x1)

.

The class of all the discriminant structures forL^?is denoted byD^?. In addition, we define the consequence relation given by the class of discriminant structures D^? as the matrix consequence relation|=D^? ⊆℘(L(CL^?))×L(CL^?). That is, Γ|=D^?α iff, for every discriminant structure R(B), it is valid that, for every R(B)-valuation⁵ w:L(CL^?)−→R(B), w(Γ)⊆ {(1,0)}impliesw(α) = (1,0).

About the previous definition it can be easily proved that:

Proposition 3.3. The operations ⊃and¬are well defined. That is, every set R(B) is closed by applications of⊃and ¬.

Definition 3.4. The canonical discriminant structure for L^? is R(2).

According to Definition 3.2, we have that, in this structure, the functions⊃and

¬behave as depicted in the following tables.

⊃ (0,1) (1,0) (0,0) (0,1) (0,0) (0,0) (0,0) (1,0) (1,0) (0,0) (0,0) (0,0) (0,1) (1,0) (0,0)

¬

(0,1) (0,0) (1,0) (1,0) (0,0) (0,1)

It should be clear thatR(2) is a matrix isomorphic toM^?, cf. Definition 1.3.

Besides that, it is easy to prove:

Proposition 3.5. The discriminant structures ofL^? verify:

(a) If B1 andB2 are isomorphic boolean algebras, thenR(B1) andR(B2) are isomorphic matrices.

(b) Considering every R(B)-valuationw : L(CL^?)−→R(B) as a pair of (non- homomorphical) functions: w = (w0, w1) (where wi:=πi◦w) then, for every valuationw:L(CL^?)−→R(A),w(α) = (1,0) iffw0(α) = 1.

On the other hand, the basic tools taken from Algebraic Logic that will be used in the sequel are related to prime filters. We summarize the needed results about it:

Proposition 3.6. LetB be any boolean algebra. Then:

(a) For every a6= 1, exists a prime filter ∇such thata /∈ ∇.

(b) For every prime filter∇ inB, the binary relation ≡∇ defined by: x≡∇ y iff {x → y, y → x} ⊆ ∇ is a congruence, and its quotient B/∇ ={∇,∆} is isomorphic to the boolean algebra2(being ∆:=B− ∇the prime ideal induced by∇).

5That is, for every homomorphismwfromL(CL?) toR(B).

For Propositions 3.5(a) and 3.6(b) we have:

Corollary 3.7. For every boolean algebra B, for every prime filter ∇ ⊆B, R(B/∇)∼=R(2) ,where the operations inR(B/∇) are given as in Definition 3.4 (replacing the element 1 by∇ and the 0 by ∆, of course).

In the sequel we will prove that|=_D?α iff |=_{M ?}α. For that, we will relate the canonical structure to prime filters of boolean algebras. First of all:

Proposition 3.8. [Trichotomy]: Let B be any boolean algebra, and let ∇ be any prime filter ofB. Then, for every pair (x0, x1) ∈ R(B), is valid one and only one of the following conditions:

-x0∈ ∇ andx1∈∆.

-x0∈∆ andx1∈∆.

-x0∈∆ andx1∈ ∇.

Proof: Straightforward, from the definitions ofR(B) and prime filters. 2 Proposition 3.9.Every prime filter∇of a boolean algebraBinduces a matrix epimorphisme∇:R(B)−→R(B/∇) such thate∇(x0, x1) = (∇,∆) iffx0∈ ∇.

Proof: Define (for every (x0, x1) ∈R(B)) e∇(x0, x1) := (x0, x1) (where xi is the equivalence class ofxi inB/∇). By Proposition 3.8, e∇ is a well-defined surjective function. Besides, x0 ∈ ∇ iff x0 = ∇ iff (x0, x1) = (∇,∆) (again, by Proposition 3.8), iffe∇(x0, x1) = (∇,∆). This also implies thate∇(1,0) = (∇,∆), verifying thus the preservation of designated values. So, we just need to prove thate∇ is a homomorphism. That is:

(A):e∇(¬(x0, x1)) =¬(e∇(x0, x1)). Consider these cases (by Proposition 3.8):

Case A.1: x0∈ ∇, x1∈∆. So, e∇(x0, x1) = (∇,∆), and then ¬(e∇(x0, x1)) = (∇,∆), too (by Corollary 3.7). Besides that,e∇(¬(x0, x1)) =e∇(x0,−(x0∨x1))

= (x0,−x0∨x1). Realizing thatx0 ∈ ∇implies (x0∨x1)∈ ∇(which implies

−(x0∨x1)∈∆), we have thate∇(¬(x0, x1)) = (∇,∆), too.

Case A.2: x0∈∆,x1∈∆. Case A.3: x0∈∆,x1∈ ∇. Proceed as in Case A.1.

(B): e∇((x0, x1)⊃(y0, y1)) =e∇(x0, x1)⊃e∇(y0, y1). In our (combinatorial) proof we should consider nine possibilities. However, by the definition of the connective¬inR(B/∇) (see Corollary 3.7), we can consider simply six cases:

Case B.1: x0∈∆,x1∈ ∇.

Case B.2: y0 ∈∆,y1 ∈∆.

Case B.3: x0∈ ∇,x1∈∆;y0∈∆,y1∈ ∇.

Case B.4: x0∈ ∇,x1∈∆;y0∈ ∇,y1∈∆;

Case B.5: x0∈∆,x1∈∆;y0∈∆,y1∈ ∇.

Case B.6: x0∈∆,x1∈∆;y0∈ ∇,y1∈∆.

For the analyisis of all these possibilities, as in (A), we use repeatedly Corollary 3.7, as the definition of the operation ⊃ in every R(B). Also, we use basic properties of prime filters and ideals without explicit mention. We will give the demonstrations of cases B.1 and B.6 as examples (the other ones are similar):

Case B.1: x0∈∆,x1∈ ∇. So,e∇(x0, x1)⊃e∇(y0, y1) = (∆,∆) (the behaviors of y0 and y1 are not important here). From this, we have−x1 ∈∆ and then

−x1∧[(x0↔ −y0)∧(x0 ↔y1)]∈∆ (∗). On the other hand,x0∨x1∨y0∈ ∇, and so−(x0∨x1∨y0)∈∆. Hence,−(x0∨x1∨y0)∧y1∈∆ (∗∗). From (∗) and (∗∗), e∇((x0, x1)⊃(y0, y1)) = (−x1∧[(x0↔ −y0)∧(x0↔y1)],−(x0∨x1∨y0)∧y1)

= (∆,∆).

Case B.6: x0 ∈ ∆, x1 ∈ ∆; y0 ∈ ∇, y1 ∈ ∆. Then, e∇(x0, x1) ⊃ e∇(y0, y1)

= (∇,∆). Besides that, note that −y0 →x0 =y0∨x0 ∈ ∇, and x0 → −y0

=−x0∨ −y0 ∈ ∇, too. So,x0 ↔ −y0 ∈ ∇. In a similar way, y1 →x0 ∈ ∇, x0→y1∈ ∇, and thenx0↔y1∈ ∇. So,−x1∧[(x0↔ −y0)∧(x0↔y1)]∈ ∇.

Finally,x1∨x0∨y0 ∈ ∇(since y0 ∈ ∇), which implies−(x0∨x1∨y0)∈∆.

Thus,−(x0∨x1∨y0)∧y1∈∆ and, therefore, e∇((x0, x1)⊃(y0, y1)) = (∇,∆).

As was said, the rest of the cases are proved in a similar way. And, sincee∇ is a homomorphism, the proof of the theorem is completed. 2 Theorem 3.10. For every α∈L(CL^?),|=_{M ?}αiff|=_D?α.

Proof: On one hand, since M^? is isomorphic to R(2), we get that |=_D? α implies |=_{M ?}α. For the reciprocal, suppose that exists a boolean algebra B and a homomorphism w : L(CL^?)−→R(B) such that w(α) 6= (1,0). Then, w0(α)6= 1B (recall Proposition 3.5(b)). Hence, there is a prime filter ∇ inB withw0(α)∈ ∇, by Proposition 3.6(a). From this and Proposition 3.9, there is a/ matrix epimorphisme∇:R(B)−→R(B/∇), withe∇(w(α))6= (∇,∆) (because w0(α)∈ ∇). So,/ e∇◦w:L(CL^?)−→R(B/∇) is aR(B/∇)-valuation that verifies (e∇◦w)(α) 6= (∇,∆), the designated value or R(B/∇). Since this matrix is isomorphic to R(2) (and therefore isomorphic to M^?), we have 6|=_{M ?}α. This

concludes the proof. 2

Based on the construction forL^?given in the previous example (and motivated by Example 2.1 too), let us try to explain, using informal terms, the process to would allows to find a Discriminant Structures Semantics for a given matrix logic. For that, consider as a starting point a signatureC, and a C-matrixM

= (A, D). In addition, take in account the functionh:A−→2defined as: h(x)

= 1 iffx∈D (that is,h =χD, relatively toA). Also, we will abbreviate the composition of truth-functionsf ◦ · · · ◦f (ktimes) byf^k. Of course, ifk = 0, thenf^k =id. With these conventions, the basis of our previous construction is the existence of adiscriminant pairforM, which is defined in the sequel.

Definition 3.11. Given aC-matrixM = (A, D), adiscriminant pair forM is a pair (β, ~a) , whereβ =β(p, q1, . . . , qm)∈L(C),~a= (a1, . . . , am)∈A^mand theA-truth-functionf(β,~a)(x):=β(x, ~a), isdiscriminant (by iterations). That is, there isk∈ω such that the boolean functionhk :A→2^k+1 is injective, being hk(x) := [h(x), h(f(β,~a)(x)), h(f_(β,~² _a)(x)), . . . , h(f_(β,~^k _a)(x))].

With the previous definition in mind, the sketch of a construction of a suitable discriminant structure for an arbitraryC-matrixM = (A, D) is given below:

1)Find, forM, a discriminant pair (β, ~a).

2) Of course, if such a pair exists, we can define a matrix M⁰ ∼= M, where M⁰ = (hk(A), hk(D)), and the truth-functions ofM⁰ are defined “copying” the

behavior of the truth-functions of M. M⁰ will be considered the canonical discriminant structure.

3)Characterizehk(A) (considering that it is a subset of2^k+1), by means of the lattice-theoretical behavior of its components.

4) Explain the truth-functions of M⁰, associated to every connective c ∈ C, again considering the lattice-theoretical properties used in the truth-functionc, in the matrixM⁰.

5) Define a class of subsets of algebras that generalize hk(A). This class is formed byC-algebrasR(B), with its domainR(B)⊆B^k+1(where the algebras Bare ranging over a certain fixed class). In every algebraR(B), the operations c ∈ C would be defined according to the considerations of 4). On the other hand, for everyR(B), consider a special subset D(B), which is, of course, the generalization of hk(D). Every pair (R(B), D(B)) will be called a discrimi- nant structureand is, actually, a C-matrix. In addition, define the classDM, constituted by all the discriminant structures induced byM.

6) Finally, define the relation |=_DM ⊆℘(L(C))×L(C) as usual: Γ |=_DMα iff, for every discriminant structure (R(B), D(B)), for every homomorphism w:L(C)−→R(B),w(Γ)⊆D(B) impliesw(α)∈D(B). Here we note that the sets D(B) (of designated values) often are characterized by equations⁶. This relation is the same as the relation between standard matrix semantics with algebraic semantics, where the set of designated values of a matrixM can be (sometimes) characterized by equations.

The construction sketched above was applied in Example 2.1, as in Example 3.1. In the last case, note that the discriminant pair (β, ~a) is given by:

β =q1⊃p,~a=a1= ¹₂ (∗)

Hence, f(β,~a) = ¹₂ ⊃xand, so, f(β,~a)(0) = ¹₂, f(β,~a)(¹₂) = 1, andf(β,~a)(1) = 1.

Considering thatDM^? ={¹₂}, we can discriminate the truth-values just using one iteration. From this,h1(A^?) ={h1(0), h1(¹₂), h1(1)} ={(0,1),(1,0),(0,0)}

(this is the reason of Definition 3.4).

Once we got a discriminant pair for L^?, we need to characterize h1(A^?) (in algebraic terms). Here note thath1(A^?) = {(x, y)∈2²:x∧y= 0}(∗∗).

Considering that h1(D^?) = {(1,0)}, we must define operations ⊃ and ¬ in h1(A^?) (in such a way thatM⁰:=(h1(A^?), h1(D^?)) be isomorphic to M^?). For that, recall that, cf. Definition 3.2,

(x0, x1)⊃(y0, y1) := −x1∧[(x0↔ −y0)∧(x0↔y1)],−(x0∨x1∨y0)∧y1

¬(x0, x1) := x0,−(x0∨x1) .

6For instance, in Example 2.1, givenR_[1,0](A), (x0, x₁)∈D_[1,0](A) iffx₀= 1A. The same characterization can be used in Example 3.1

What is the “hidden” reason for the definition of⊃and¬inh1(A^?)? To clarify this, denote (x0, x1)⊃(y0, y1):=(z0, z1) (and, similarly,¬(x0, x1):=(r0, r1)). We can see that, for instance,⊃is simply defined componentwise and considering, in each componentz0 andz1, convenient boolean expressions ⁷ that relate the values of all the components involved in the domains withz0and withz1. Please check this idea, considering that, for⊃, we get two functions: f₀^⊃:2⁴−→2, and f₁^⊃ : 2⁴−→2 (since in the domain we have truth-values forx0, x1, y0 and y1, and in the images for z0 and z1, resp.). In the case of ¬, each component is obtained by truth-functions from2² to2.

The definition of the (class of) discriminant structures for M^? can be done now. For that, note that the lattice-theoretical characterization ofh1(A^?) can be adapted toevery boolean algebra. Also, the boolean expressions used for the definition of the2-truth-functions can be applied to any boolean algebra, too.

This fact allowed us, in Definition 3.2, to obtain the class D^? of discriminant structures, as we said. As we have seen in Theorem 3.10,|=D^? α iff |=M^? α.

But we do not know (up to the moment) if, in general terms, we can define a relation|=_DM in such a way that the mentioned theorem is, in fact, a particular case of a general result. This question has an affirmative answer, as we will see later.

4 Some technical results

Summarizing the previous discussion, given aC-matrixM = (A, D), a D.S.S.

associated to it would be obtained paying attention to:

(i) The discriminant pair (β, ~a).

(ii) The lattice-theoretical characterization of the sets hk(A) andhk(D).

(iii) An adequate definition of the truth-functions (inhk(A)), corresponding to the connectives ofC.

The fundamental technical result about discriminant structures, to be proved in the sequel, establishes that the existence of a discriminant pair of the form (β, ~a) is sufficient to guarantee the Discriminant Structures Semantics as a whole. For its demonstration, we need some technical results, starting with some simple facts about boolean algebras:

Proposition 4.1. Every subset S of a finite boolean algebra B can be char- acterized (relatively toB) by an equation, defined in the language of boolean algebras (that is,∨,∧,−, 0 and 1).

Proof: Since B is finite, it is isomorphic to 2^r for a certain r ∈ ω. So, we can consider the elements ofB as r-tuples conformed by 0 and/or 1. On the other hand, every S ⊆B can be identified with a set S⊆2^r. Consider now

7That is, an expression built on the boolean language, using the symbols∧,∨,−, 1, 0 and

≈. In our example (which motivates the characterization of Definition 3.2), we have started considering a normal form, and then we have simplified it.

the characteristic function χS :2^r−→2. Since χS is a boolean truth-function, admits a conjunctive normal form (say,α1∧....∧αp, with 1 ≤p≤2^r). The equation that characterizesS is, actually,α1∧....∧αp≈1. 2 Proposition 4.2. LetM = (A, D) be aC-matrix which admits a discriminant pair (β, ~a), by means of k-iterations, and let hk : A−→2^k+1 be the function obtained by (β, ~a). Then there is a matrixM⁰ isomorphic toM, whose domain ishk(A).

Proof: Define M⁰ := (A⁰, D⁰), where A⁰:=hk(A) and D⁰:= hk(D), as it was suggested in Section 3. Sincehk is injective, hk(A) is equipotent with A(and hk(D) is equipotent with D). Now, for every n-ary operation c define the operation c⁰ : (A⁰)ⁿ−→A⁰, by: c⁰(x1, . . . , xn):= hk(c(hk

−1(x1), . . . , hk

−1(xn)).

This definition makes sense becausehk is injective. From this, it can be easily

proved thatM⁰∼=M. 2

Remark 4.3. Note here the following fact about M⁰: ifc⁰ : (A⁰)ⁿ −→ A⁰ is the truth-function associated toc∈C, it can be defined componentwise. This entails that, for every 0≤i≤k, exists a truth-functionfi:2^n(k+1)−→2defined as fi(~x1, . . . , ~xn):=πi(c⁰(~x1, . . . , ~xn)) (where~x1, . . . , ~xn belong to 2^k+1). Every functionfi has an equivalent conjunctive normal form (c.n.f.) fi0:2^n(k+1)−→2.

So, the operations inA⁰ can be defined now in this alternative way: for every n-ary connective c ∈ C, its corresponding operation cA⁰ : A⁰ⁿ−→A⁰ is given by: c_A⁰(~x1, . . . , ~xn) = (f⁰i(~x1, . . . , ~xn))0≤i≤k, where f⁰i : (A⁰)^n(k+1)−→A⁰ is the c.n.f. found previously ⁸. Formally, if c is a n-ary connective of C, and {~x1, . . . , ~xn} ⊆A⁰, thencA⁰(~x1, . . . , ~xn):= (f₀⁰(~x1, . . . , ~xn), . . . , f_k⁰(~x1, . . . , ~xn)).

The previous proposition suggests the following:

Definition 4.4. Consider M = (A, D) admitting a discriminant pair, and the equationseqA,eqD, which characterize hk(A) (resp. hk(D)) as subsets of2^k+1, cf. Proposition 4.1. For every boolean algebraB, thediscriminant structure associated toB is the C-matrix (R(B), D(B)), with:

R(B) :={~x∈B^k+1:~xsatisfieseqA}, D(B) :={~x∈B^k+1:~xsatisfieseqD}⁹. On the other hand, for anyn-ary connective c, for every {~x1, . . . , ~xn} ⊆R(B), cR(B)(~x1, . . . , ~xn):= (f₀⁰(~x1, . . . , ~xn), . . . , f_k⁰(~x1, . . . , ~xn)), where f00, . . . , fk0 are the c.n.f found in Remark 4.3,but applied to R(B), in each case.

The following results are valid in D.S.S. First of all, they are well defined.

Proposition 4.5. For every boolean algebraB, the set R(B) is closed by the operationscR(B). That is,R(B) is well defined as an algebra.

8The existence of a conjunctive normal form is not essential itself. The relevant point is that the functionsfican be expressed in a boolean language. We also could use disjunctive normal forms, for example.

9If~xsatisfieseqD, then~xsatisfieseqA, too. Hence,D(B)⊆R(B).

Proof: Our claim is just a reformulation of the following fact: Let S⊆2^k+1, characterized by the equationχS ≈1, and letn be inω. Consider the truth- functionsfi :2^(k+1)n−→2(expressed in boolean terms by f⁰i), and define the functionf :2^(k+1)n−→2by f(x~1, . . . , ~xn):=(f⁰i(x~1, . . . , ~xn))0≤i≤k. Define, for every boolean algebra B, the set R(B):={~x ∈ B^k+1 : χS(~x) ≈ 1}, and the functionf(x~1, . . . , ~xn) as before, but interpreting in B the boolean connectives involved. Then, if ~x1, . . . , ~xn ∈ S implies f(~x1, . . . , ~xn) ∈ S, it is valid that

~

x1, . . . , ~xn ∈ R(B) implies f(~x1, . . . , ~xn) ∈R(B). And this result is valid be- cause we are defining our sets and functions by means of equations, and because

every equation valid in2is valid inB, too. 2

In addition, the setsD(B) can be equationally characterized in a simpler way, in the context ofR(B):

Proposition 4.6. For every~x= (x0, . . . , xk) in R(B),~x∈D(B) iffx0= 1B. Proof: SinceM ∼=M⁰=(R(2), D(2)), we have that, for every~x= (x0, . . . , x1) in R(2), ~x satisfies eqD iff there is a ∈ D such that hk(a) = ~x. But a ∈ D impliesπ0(h(a)) = 1. This proves that, for every~x∈R(2), are equivalent facts:

-~xsatisfieseqD.

-~xsatisfies the equationeqR(2)≈1 andx0= 1.

-~xsatisfieseqR(2)∧p0≈1.

So, we have proved our claim for R(2), which can be extrapolated to every R(B), since the required properties are characterized by boolean equations. 2 Besides, it is always possible to “recover” the canonical discriminant structure, by means of prime filters. For that, note some obvious facts: obviously, ifB1and B2 are isomorphic boolean algebras, then (R(B1), D(B1)) and (R(B2), D(B2)) are isomorphicC-matrices. From this, and noting that, for every boolean al- gebra B, for every prime filter ∇ of B, B/∇ is a boolean algebra (which is isomorphic to 2) we have that (R(B/∇), D(B/∇)) is a matrix isomorphic to M⁰, given in Proposition 4.2. From this, we have:

Corollary 4.7. Let M = (A, D) a finite matrix with r elements. Then, for every boolean algebraB, for every prime filter∇ofB, it holds:

(a) The setR(B/∇) has exactlyrelements.

(b) Every (b0, . . . , bk)∈R(B) belongs to exactly one class ofR(B/∇). All the classes of this last set are defined as (b0, . . . , bk) = (b0, . . . , bk).

(c) The functione∇:R(B)−→R(B/∇) given bye∇(b0, . . . , bk) = (b0, . . . , bk) is a matrix epimorphism. Moreover,e∇(b0, . . . , bk)∈D(B/∇) iffb0∈ ∇.

Proof:SinceR(2) hasrelements we have (a), and (b) follows straightforwardly.

For (c), the proof is as in Proposition 3.9¹⁰, using items (a) and (b). 2 Definition 4.8. The class of all the discriminant M-structureswill be denoted byDM. This class defines the “local consequence relations”|=R(B)and

10The results stated in (a) and (b) are the abstractions of Proposition 3.8. On the other hand, (c) is, of course, a generalization of Proposition 3.9.

the “global consequence relation”|=D_M as in Definition 1.2. That is: if R(B) is inDM, then Γ|=R(B)αiff, for everyR(B)-valuationw:L(C)−→R(B) such thatw(γ) satisfieseqD for every γ∈Γ, it is valid thatw(α) satisfieseqD. And

|=DM:=T|=R(B) (withR(B) ranging inDM).

Theorem 4.9. If a finite C-matrix M = (A, D) admits a discriminant pair, then the previously defined semanticsDM verifies |=D_M αiff|=M α, for every α∈L(C).

Proof:Suppose that there is a formulaβ(p, q1, . . . , qm), a t-uple~a= (a1, . . . , am) of elements ofA(with|A|=r∈ω) and a numberk∈ωsuch that the function hk :A−→2^k+1is injective. By Proposition 4.1, the setshk(A) andhk(D) can be characterized by means of equations (which we will denote, respectively, aseqA

andeqD). These equations define the class of discriminant structures DM, cf.

Definition 4.8. We will prove that|=M αiff|=DMα. First of all, if we consider M⁰= (hk(A), hk(D)) of Proposition 4.2 (which is, actually, a discriminant struc- ture), we have that|=DM αimplies|=M⁰ α, which is equivalent to|=M α. For the other implication, suppose that exists a boolean algebraBand aR(B)-valuation w : L(C)−→R(B) such that w(α) ∈/ D(B). So, w0(α):=π0(w(α)) 6= 1B, by Proposition 4.6. There is a prime filter∇ofB, withw0(α)∈ ∇. From this and/ Corollary 4.7 (c), there is a matrix epimorphisme∇ : R(B)−→R(B/∇), with e∇(w(α))∈/ D(B/∇) (because the first component of the elements ofD(B/∇) is always∇, and w0(α) 6=∇). So, e∇◦w :L(C)−→R(B/∇) is the R(B/∇)- valuation that verifies (e∇◦w)(α)∈/ D(B/∇). Hence,6|=R(B/∇)α. Equivalently, 6|=M⁰ α(sinceR(B/∇)∼=R(2)∼=M⁰), and so6|=M α. 2 Remark 4.10. Note that Theorem 4.9 establishes a weakadequacity between the relations|=M and |=D_M. It is part of a future work the proof of a strong adequacity (that is, that Γ|=M αiff Γ|=D_M α).

At this point we have seen that semantics of the formDM are, in fact, a sort of

“convenient representations” of certain matrix semanticsM. Moreover, Theo- rem 4.9 establishes that this kind of semantics can be obtained when a discrim- inant pair is found. We will see now, however, thatit is not always possibleto obtain such a pair, for a given matrixM. The next example shows a particular case of it:

Example 4.11. Consider the Urquhart logicU rq, characterized by the {∗}- matrix MU rq = (A, D) = (({I, II, III, IV, V},{∗}),{I}), where ∗ is a binary operation, indicated in the following truth-table:

∗ I II III IV V

I V V I I V

II V V II I V

III V V V V V

IV V V V V V

V V V V V V

This logic was defined by A. Urquhart in [11], to show an example of a matrix logic that cannot be axiomatized by a finite set of structural rules. It is possible to prove thatMU rq not admits a discriminant pair. This fact is demonstrated in the sequel.

Proposition 4.12. For every formula α /∈ V, for every t-uple~a of elements of {I, II, III, IV, V}, there are two elements a0, a1 ∈/ D, a0 6= a1, such that α(~a, a0) = α(~a, a1) = V.

Proof: We will prove our claim by induction onn, the number of ocurrences of the connective ∗ inα(where n≥1, because α /∈ V). When n = 1, then α

=p∗q1 orα=q1∗p¹¹, and~a∈A. The five truth-functions obtained for the first case aref(α,I) =I∗x,...,f(α,V) =V ∗x. For these functions, considera0

=II, anda1 =V, and so the result is valid. For the second case we have the truth-functionsg(α,I)=x∗I,...,g(α,V)=x∗V. Herea0 =IV, a1 =V. Now, suppose that the result is valid for everym≤n, and supposeα=α(p, q1, . . . , qt) and~a = (a1, . . . , at) is anyt-uple of A^t, where αhasn ocurrences of∗. Then α=β∗γ. Ifβ, γ ∈ V we return to m= 1, which was already analyzed. So, suppose (without loss of generality), thatβ /∈ V. By I. H., there area0 6= a1

such thatβ(~a, a0) = β(~a, a1) =V. Now, realizing thatV ∗x=x∗V =V, for everyx∈A, we have thatα(~a, a0) =β(~a, a0)∗γ(~a, a0) =V∗γ(~a, a0) =V, and α(~a, a1) = V using the same argument. Ifβ∈ V, then γ /∈ V and use the same

reasoning. This concludes the proof. 2

The previous conclusion entails, obviously:

Corollary 4.13. The matrixMU rq does not admit discriminant pairs.

Proof: Note that Proposition 4.12 implies that there are not pairs (β, ~a) that can discriminate all the values of A by one iteration. But, noting that V is an absorbent element of MU rq we have that, for every pair (β, ~a), for every k∈ω, there are two different elementsa0,a1∈Asuch thathk(a0) =hk(a1) = (0,0, . . . ,0) (k+ 1 times), and so they cannot be discriminated. 2 This last result shows that, despite the simplicity of the process to obtain a D.S.S., the basis of it (that is, the existence of a discriminant pair) is not trivial.

5 Discriminant Structures and Algebraizability

The expressibility of a matrix semantics by means of discriminant structures entails certain nice properties in algebraic terms, according to our point of view. The basic idea is that, cf. Theorem 4.9, a discriminant structure is

11Of course, in both casesα isthe same formula, but we distinguish such variables inα that will be instanced by~ausingq₁, because the truth-functions obtained are not the same, considering that∗is not commutative.

just a matrix such that: (i) its support is defined on a subset of a boolean algebra and so it can be characterized by means of boolean equations. (ii) its operations are defined by means of boolean functions. Moreover, the tautologies can be explained using boolean equations, too. These facts give to discriminant structures a certain “algebraic character”, in a broad sense. But also deserves to be remarked that this kind of semantics can be applied to logics that are not algebraizable(that is, that not have an equivalent algebraic semantics)¹². This definition is based on the notion of protoalgebraizability. We recall both concepts in the sequel.

Definition 5.1. A logicL = (CL,`L) is protoalgebraicif and only if exists a setP R(p1, p2) ={φi(p1, p2)}i∈I ⊆L(CL) such that verifies ¹³:

(R)`Lφi(p1, p1), for everyφi∈P R(p1, p2).

(M P)p1, P R(p1, p2)`Lp2.

On the other hand, L is algebraizable iff exists a class K of algebras that constitutes anequivalent algebraic semanticsforL. For instance, classical logic is algebraizable (being K the class of boolean algebras). Also, Intuitionistic Logic and Lukasiewicz logic are algebraizable (its equivalent algebraic semantics are the class of Heyting Algebras andM V-algebras, resp.).

It must be remarked that every algebraizable logic is also protoalgebraic. This fact will allow us to show a logic that verifies: (a) It admits Discriminant Struc- tures Semantics. (b) It is not algebraizable. In fact, this logic isL^?, the basis of our Example 3.1. As it was already proved,L^? verifies (a). With respect to (b), our proof is based on the following fact: L^? has not tautologies, as we will see now. For that, consider the following definitions and notations. First, recall the functionP ar:ω²−→2defined in the obvious way: P ar(i, j) = 1 iffi and j are both odd or are both even. Besides that, the orderof a formulaαis the number of different atomic formulas that appear inα. With this in mind, we have:

Proposition 5.2.For everyα∈L(CL^?),αof the form¬^kp,αis not a tautology.

Proof: Obvious.

Proposition 5.3. For every α = α(p) ∈ L(CL^?), if α contains at least one symbol⊃, then, for every valuationv,v(α)6=¹₂.

Proof: By induction on k ≥ 1, the number of the implications in α. If k = 1, then αis the form ¬ⁱp ⊃ ¬^jp. So, by the truth-table of ¬ we have that, for everyv, v(α) = 0 orv(α) = 1 (depending of P ar(i, j)). Suppose that our assumption holds for every k < n, and considerα containing n ocurrences of

⊃. We can see αas ¬^t(β ⊃γ), with 0≤t. By I.H.,v(β)6= ¹₂ 6=v(γ). By the truth-table of⊃,v(β ⊃γ)6= ¹₂, and so (by the truth-table of¬),v(α)6=¹₂. 2

12The relation between an algebraizable logic and its equivalent algebraic semantics is similar to the one existing between classical logic and the class of Boolean Algebras, or the relation between Intuitionistic Logic and the class of Heyting algebras. Again, a very complete text about Algebraizability and Abstract Algebraic Logic in general terms is [2].

13Usually,P R(p1, p2) is denoted asp1P Rp2

The previous proposition entails two results. First of all, considering Proposi- tions 5.2 and 5.3, we have the following fact:

Corollary 5.4. For every formulaαof order 1 (α=α(p)), it holds that6|=M^?α.

Proposition 5.5. For everyα∈L(CL^?),6|=M^? α.

Proof: Let α = α(x1, . . . , xn) be in L(CL^?), and consider now the formula α⁰:=α(x1, x2/x1, . . . , xn/x1). So, α⁰ is of order 1. By Corollary 5.4, there is a valuationv, and an elementa∈ A^?, such thatv(x1) = a, andv(α⁰(x1))6= ¹₂. Defining now the valuationwv:V−→A^? such thatwv(x1) =· · ·=wv(xn) =a, it is easy to see thatwv(α) =v(α⁰)6= ¹₂, and therefore6|=M^?α. 2 On the other hand, we can relate logics without tautologies and protoalgebraiz- ability in this way:

Proposition 5.6. If a logicL = (CL,`L) is a logic without tautologies, and

`L can be defined by a matrix M = (A, D), where ∅ 6=D 6=A, thenL is not protoalgebraic (and, so, is not algebraizable).

Proof: Suppose L protoalgebraic, such that`L = |=M. Then, there is a set P R(p, q)⊆L(CL^?) satisfyingP R(p, p)⊆CnM(∅), andq∈CnM(P R(p, q)∪p).

SinceL has not tautologies,P R(p, q) = ∅. So, p|=M q. Now, if we consider any valuationv such thatv(p)∈D meanwhilev(q)∈/ D (v exists because ∅ 6=

D6=A), we havep6|=M q, which is absurd. So,Lcannot be nor protoalgebraic.

Neither algebraizable, therefore. 2

So, from Definition 5.1 and Propositions 5.5 and 5.6, and realizing that the set of designated values ofL^?is{¹₂}, we have:

Corollary 5.7. L^? is not algebraizable.

6 Separation of truth-values

As presented above, semantics based on discriminant structures are motivated by a recurrent idea in the field of many-valued logics which can be stated as follows: the truth-values of a matrix M = (A, D) can be often separated (or discriminated), according to its relation with the designated set D. This ap- proach was already applied, with several purposes. For example, the separation of truth-values can determine whether certain formulas aresynonymous or not (see [10], for example). In the same line, the Leibniz Operator Ω (which provides an alternative definition to protoalgebraic and algebraizable logics), is defined by means of the notion ofindiscernible formulas (relatively to the designated set D), which in a certain sense is based on separation of truth-values, too.

Also, several definitions of two-valued (non truth-functional) semantics make use of this notion. One example of such construction is Dyadic Semantics (see

[1]), which has certain similarities with Discriminant Structures Semantics, ac- cording to our point of view. So, we will discuss briefly here the relationship between both constructions. For this, the notation of [1] was modified, for a better comparison with this article.

Roughly speaking, a dyadic semantics for a given matrix logicL = (CL,|=M) induced by an-valued matrixM = (A, D) is built on a basis of:

•A set of formulas{φi}1≤i≤k, where (for every i),φi =φi(p)∈L(CL).

•A function h:A−→2such that, for every φi, for every a∈D, h(φi(a)) = 1 iffφi(a)∈D.

In addition, the truth-functions associated to the set{φi}0≤i≤k (together with h) separates, actually, the values of A, by means of (k+ 1)-tuples of elements of 2, and therefore it is a generalization of the formulaβ of our discriminant pair. The scope of Dyadic Semantics is in fact stronger of the D.D.S, wherein just only formulaβ (eventually iterated) is allowed.

On the other hand, Dyadic Semantics is based in the existence of one-variable formulas of L(CL), instead of the truth-functions used in this article. In fact, in D.S.S., the t-uple ~a of elements in the discriminant pair (β, ~a) allows to construct the (one variable-depending) truth-functionsf(β,~a), without the need of associated formulas inL(CL). In other words, meanwhile Dyadic Semantics is focused more strictly on the involved languages, D.S.S analyze mainly the matrices used. It must be noted that if a logic is functionally complete, then every truth-function has an associated formula that describes it, and so it would be possible to “jump” from the matrices to the formal languages themselves.

Besides that, an “hybrid method” of separation of truth-values can be done.

Consider simply that every truth-value of a matrixM = (A, D) is tested by aset f(βi, ~ai) of discriminant (non-iterated) pairs. An informal example of this will be applied to the logicU rq.

Proposition 6.1. The truth-values of the logic U rq of Example 4.11 can be discriminated by a set{fi}0≤i≤4(withfi =f(β_i, ~a_i)) of truth-functions.

Proof: The following schema shows the formulas which discriminate every truth-value of U rq, and the indentification of each truth-value by means of the functionh4 :A−→2⁵, given byh4(a) =(χD(fi(a)))0≤i≤k, for every a∈A:

f0=x f1=x∗I f2=x∗II f3 =I∗x f4=II∗x h4

I I I V V (1,1,1,0,0)

II V I V V (0,0,1,0,0)

III V V I V (0,0,0,1,0)

IV V V I I (0,0,0,1,1)

V V V V V (0,0,0,0,0)

So, all the truth-values of U rq can be discriminated by a convenient set of

truth-functions. 2

It must be indicated that in this example is not developed the method that

“explains” the operation∗ofMU rq. Moreover, it is not clear here (as in Dyadic Semantics in general terms) in which way an algebraic treatment can be done, but of course these topics deserve a deeper treatment, in the future.

7 Twist-Structures Semantics, revisited

We conclude this paper returning to our motivating construction, which is Twist-Structures Semantics. Note at this point that in our proposed gener- alization a key concept is missed: the eventual torsion of some axes (which, as it was already said, suggests the name of the twist-structures). This process is not considered in the general definition of Discriminant Structures Semantics.

Why is the reason of this torsion? Mainly, twist-structures, even being consid- ered as C-matrices, usually were analyzed w.r.t its lattice-theoretic behavior.

Under this perspective, the designated elements of every structure of the form R(A) are usually interpreted as the last elements (according to certain conve- nient order relation≤R(A) inR(A)). For that, it is necessary that the second axis will be considered with a inverted order: if not, the designated elements inR(A) are not the last elements according to≤R(A). See the mentioned ref- erences [3], [4], [7] and [12], as motivating examples of this idea. Besides that, twist-structures were usually applied to logics with certain common properties.

Between them, the existence of an implication connective⊃which is related to the structure R(A) as usual: x ≤_R(A) y iff x ⊃ y = 1R(A). Again, this last condition can be better explained when 1R(A) is the last element with respect to≤_R(A).

In addition, the consequence relations that can be expressed by Twist-Structures Semantics are defined on languages wherein⊃is related at some extent with a

“negation”-connective (∼, or¬). This forces the “torsion” of the second axis in such structures. For example, in the works of M. Fidel and D. Vakarelov about the logicN, this new semantics explains the behavior of counter-examples (that are obviously related with the negations). In this context, it seems like an obvious presentation that the second components of every pair (that, in fact,

“explain” the counter-examples) will be considered with its dual order. A similar fact happens in the case of the logicI¹P⁰, sketched in Example 2.1. In this last case, in addition, the operations of the second axisB^∗ can be explained in terms ofB and, therefore, the definition of the operations in the twist-structures can be simplified in some sense. This does not happen in the case ofN, because its Twist-Structures Semantics is defined on Heyting Algebras (whose dualis nota Heyting Algebra). Thus, in the case ofN, the “torsion” is more necessary than in the case ofI¹P⁰.

Turning back to D.S.S., note that they can be applied to a greater class of logics, since they are not depending neither on considerations of order relations,