Twist and ω-Twist Structures for the logics of the Hierarchy I n P k
Fernando M. Ramos, V´ıctor L. Fern´ andez
Basic Sciences Institute (Mathematical Area), Philosophy College National University of San Juan (UNSJ)
San Juan, Argentina
E-mail:{fmramos,vlfernan}@ffha.unsj.edu.ar
Abstract
In this work we define, in a general way, an algebraic semantics for the logics of the hierarchyInPk. This semantics is defined by means of an al- ternative construction, with respect to the usual algebraic semantics, and it is known in the literature asTwist-structures semantics. Besides that, we modify such construction, defining the so-called ω-Twist-structures here. This adaptation allows us to prove adequacity theorems for every logic of the hierarchyInPk.
Introduction
In 1977 M. Fidel and D. Vakarelov defined, in independient works (see [4]
and [15], respectively), a class of non traditional algebraic structures, having in mind to get an algebraic semantics for (N), the intuitionistic logic with strong negation, defined by D.Nelson. This semantics was an alternative to the one studied by H. Rasiowa in [11]. Their definition was strongly motivated by the meaning of the connective of strong negation in N. Because os this certain subsets R⊆H×H∗ were defined, where H is a Heyting algebra and H∗ its dual. Since the basis of the definition was a Heyting algebra, a lot of properties of the intuitionistic logic were inherited by these new structures. Besides that, it was possible to interpret every formulaα of (N) in a non truth-functional way with respect to the strong negation,∼α.
Both authors claimed that, even when this kind of construction was not usual in the field of Algebraic Logic, it interpreted more faithfully the intuitive meaning of the connective∼. In the literature this construction was called the Fidel-Vakarelov construction and, in the last years, Twist-structures semantics (denomination used in [8], for example). The reason of such nomenclature is based in the fact thatH×H∗ suggests a sort of “torsion” inH. As S. Odinstov says in the mentioned paper: “(the term Twist-structure)is very suggestive. We
define an algebraic structure(...) in such a way that operations are not defined componentwise, but twisted in some sense”. According to this point of view we indicate these algebras as Twist-structures along this article.
The aim of this article is the study of other forms of application of this cons- truction, beyond the logicN. That is, we are interested in the Twist-structures themselves, independently of the logics “descripted” by such structures. In particular in this paper, our purpose is the adaptation of the Twist-structures semantics at the many-valued logics of the hierarchyInPk, defined in [2], and studied in [3]. Moreover, their proof of adequacity for every logic with respect to the Twist-structures semantics is given in a general way. In this point we must indicate that, in the general proof of adequacity mentioned, we define a new notion: the concept ofω-Twist-structure, which is a subset of acountable product of algebras, wherein the algebra of each “axis” is dual to its sucesor one.
This definition is very convenient in the study of different logics at the same time.
For a better reading of this article, its organization is as follows: in the first section we indicate the notation and previous concepts that will be used in the sequel. Section 2 provides an adequate Twist-structures semantics for the logic I1P0, giving some examples of Twist-structures and detailing the proofs. In the next Section we proceed in the same way, but without detailed algebraic proofs.
Instead, we will show an adequacity proof of syntactic style, having in mind to give new techniques of demonstrations.
Section 4 has an informative spirit: it is a brief survey of the main char- acteristics of the Hierarchy InPk, which will allow us in the next section to get an adequate semantics for every logic of this hierarchy, using in this case ω-Twist-structures.
In the last section we will discuss about open problems and future work.
Within this context we will relate the Twist andω-Twist-structures to another non traditional semantics already defined in the literature.
1 Preliminaries
Since we are studying different ways of definition of a same consequence relation, we choose to use the usual formalism within Abstract Logic, applied mainly to the particular case of consequence relations defined by matrices; see [16], for example.
Definition 1.1 We denote by ω={0,1,2...}the set of natural numbers; let V be a countable set1, called theset of propositional variables, fixed from now on. The elements of V are denoted by p, q, r . . ., with subscripts if necessary.
Then:
(1) Asignatureis a familyC={Ck}k∈ωof sets such thatCk∩Cn=∅when k 6= n and Ck ∩ V = ∅ for every k. The elements of every Ck are the connectives of arityk of the signatureC.
1That is, equipotent withω.
(2) The propositional language generated by C, and denoted by L(C), is the absolutely free algebra, generated by C over V, being every Ck the set ofk-ary operations of such algebra.
GivenC, one of the simplest ways to define consequence relations inL(C) is by means oflogical matrices:
Definition 1.2 Fixed a signatureC, aC-matrixis a pairM = (A, D), where Ais an algebra similar to L(C), and D⊆A, called the set of designated values ofM.
Note that, by simplicity, we denote every algebra as the same form as its universe. Besides that, we identify the sets of operations in A with C itself.
This notational abuse will be sustained all along the paper.
Definition 1.3 Given two C-matrices M = (A, D) and N = (B, E), we say thath:M−→N is a matrix homomorphism fromM to N iff verifies:
(1)his homomorphism(in the algebraic sense) fromAtoB.
(2)h(D)⊆E.
Two matrices M = (A, D) y N = (B, E) are called isomorphic iff exists a matrix homomorphismhverifying additionally:
(a) : his a bijective function.
(b) :his an strict homomorphism. That is: h(A−D)⊆B−E.
We indicate thatM and N are isomorphic withM ∼=N.
In the definion of consequence relations using logical matrices we need the notion of valuation, which is very simple for our approach:
Definition 1.4 Let M = (A, D) be a C-matrix. A M-valuation is a homo- morphism2v: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. In this context, we say that αistautology accor- ding to M (or a M-tautology) iff∅ |=M α(denoting this as|=M α). Finally, thelogic LM induced by M is the pairL =hC,|=Mi.
An important result, relating matrix isomorphisms and consequence rela- tions, is the following:
Proposition 1.5 If the C-matrices Mi = (Ai, Di) (i = 1,2) are isomorphic then the following facts are valid:
(a) For each M1-valuation v exists a M2-valuation wv such that, for every α∈L(C),v(α)∈D1 if and only if wv(α)∈D2.
(b) For each M2-valuation w exists a M1-valuation vw such that, for every α∈L(C),vw(α)∈D1 if and only if w(α)∈D2 .
2It is convenient to distinguish between valuations (homomorphisms from L(C) tp the support of theC-matrices) and matrix homomorphisms, cf. Definition 1.3.
Proof: Leth:M1→M2 be a matrix isomorphism. Let us prove (a): given a M1-valuationv : L(C) → A1, consider wv : L(C) → A2 defined by wv(α) = h(v(α)), for every α∈L(C). Then, for everyα∈L(C) we have: ifv(α)∈D1, then h(v(α)) ∈ D2, because h is homomorphism. So, wv(α) ∈ D2. Now, if v(α)6∈D1, since his strict, holds on thatwv(α) =h(v(α))6∈D2. In the proof
of (b) we useg:=h−1. 2
From the proposition above we have the following result:
Proposition 1.6 Consider two C-matrices M1 and M2. If M1 ∼= M2 then
|=M1 =|=M2, and thereforeL1= (C,|=M1) = L2= (C,|=M2).
Proof: Obvious. 2
We conclude this section indicating that along this paper we will use a fixed signatureC, where: C1={¬};C2={⊃};Cm=∅for everym6= 1,m6= 2. So, all the logics of the hierarchy InPk have the same language L(C), generated from a negation and an implication only.
2 Twist-structures for I
1P
0: an algebraic proof
The logicI1P0 was originally defined by A. Sette and W. Carnielli in [14], with the name of I1 (we will explain the reason of its new name in Section 4). It is usually classified as aparacompletelogic, or aweakly intuitionisticone because, as the intuitionistic logics, the excluded-middle principle is not valid in it. This logic is defined by means of a 3-valued matrix, as indicated in the following definition:
Definition 2.1 The logic I1P0 is the pair I1P0 = hC,|=M(1,0)i, with C indicated in the previous section, being|=M(1,0)defined by the matrixM(1,0) = h({T0, F1, F0},{⊃,¬}),{T0}i, where the operations ¬and ⊃are given by:
T0 F1 F0
¬ F0 F0 T0
⊃ T0 F1 F0
T0 T0 F0 F0 F1 T0 T0 T0 F0 T0 T0 T0
Even when in this article we do not use secondary connectives, we just mention that inI1P0it can be defined the binary connectives∨,∧y↔, in such a way that its associated truth tables are the following:
∨ T0 F1 F0
T0 T0 T0 T0 F1 T0 F0 F0
F0 T0 F0 F0
∧ T0 F1 F0
T0 T0 F0 F0 F1 F0 F0 F0
F0 F0 F0 F0
↔ T0 F1 F0
T0 T0 F0 F0 F1 F0 T0 T0
F0 F0 T0 T0
The meaning of the superscripts in the name of this logic should be clear: if we consider thatF0 and T0 are “the classical values” (of falsehood and truth,
resp.), then there is just one alternative value of falsehood (F1, in fact), mean- while there is not any alternative value indicating “truth”. Now, since the values Fi give us the intuitionistic character of this logic (as the different valuesTi in- dicate its paraconsistent characteristics), it is natural to call this logic asI1P0. As we said, this process will be generalized in Section 4. Besides that, note that the binary connectives of I1P0 are working “in a classical way”. That is, the formulas with binary connectives just can have the valuesT0 or F0. Moreover, the truth table of the binary connectives interpret the designated values asT0, and the non designated ones asF0.
Considering the definition of I1P0, our purpose is to characterize |=M(1,0) using Twist-structures. As indicated in the Introduction, a Twist-structure is a subset of a product of algebras,A×A∗, where A∗ is the dual algebra (in its order) ofA. Here, unlike the Twist-structures for N, the underlying algebra A is a boolean algebra, because of the obvious properties of I1P0. From the discussion above, the following definition of Twist-structure of type I1P0 is rather obvious:
Definition 2.2 Given a boolean algebra (A,∨A,∧A,−A,1A,0A) 3, we define the Twist-structure of type (1,0) associated to A, as the algebraR1,0(A)
=(R1,0(A),¬,⊃)such that:
(1) The universe(called usually the Twist setof type(1,0) associated toA) is R1,0(A) = {(x0, x1)∈A×A∗: x0∧Ax1= 0A}, whereA∗ is the dual algebra (in its order) ofA.
(2) ¬(x0, x1) = (x1,−Ax1).
(3) (x0, x1)⊇(y0, y1)∈R1,0(A) = (x0→Ay0,−A(x0→Ay0)).
(where, as usual, x→Ay := −Ax∨Ay). The class of all the Twist-structures of type(1,0)will be denoted asT(1,0).
Remark 2.3 Considering that it is easy to prove that:
R1) R1,0(A)6=∅
R2) If (x0, x1)∈R1,0(A), then (x1,−Ax1)∈R1,0(A)
R3) (x0 →A y0,−A(x0 →Ay0))∈R1,0(A), whenever (x0, x1), (y0, y1) belong toR1,0(A)
the operations indicated above are well defined. On the other hand, note that the operations inA andA∗ can be mutually defined. For instance, if (x0, x1), (y0, y1) ∈ R1,0(A), then ((x0, x1) ⊃ (y0, y1))0 = x0 →A y0 (defined in A).
On the other hand ((x0, x1) ⊃ (y0, y1))1 = −A(x0 →A x1) =x0∧A−Ax1 = x0∨A∗−A∗x1=x1→A∗ x0. For the sake of simplicity we choose to express all the operations inR1,0(A) in terms of A. So, A∗ just remarks the behavior of the negations of any formula, suggesting thatthe orderin the elemens ofA∗ is inverse to the order inA. The following example clarifies this fact.
3Here the superscript of the operations ofAis dropped when there is not risk of confusion.
Example 2.4 Consider the boolean Algebra “diamond”: A={0, a, b,1}, with 0≤a≤1, 0≤b≤1 (aandb are not comparable). ThenA×A∗ is
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(0, b) (b,1)
(b, b) (b, a)
(b,0)
(a,1) (a, a) (a, b)
(a,0)
(1,1) (1, b) (1, a)
(1,0)
and the setR1,0(A)⊆A×A∗ consists of
R1,0(A) ={(0,1),(0, a),(0, b),(0,0),(a,0),(a, b),(0,0),(b, a),(b,0),(1,0)}
whose diagram, according to the graphic above, is:
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that, of course, can be visualized in this simpler form:
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(0,1)
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(a, b) (b, a)
(a,0) (b,0)
(1,0)
Anyway, the more intuitive example of Twist-structure is the following:
Example 2.5 Let be the boolean algebra2:={0,1}. Then the Twist setR1,0(2)
={(0,1),(0,0),(1,0)}is the universe of the Twist-structure whose operations, according to Definition 2.2, can be represented by the following tables.
(1,0) (0,0) (0,1)
¬ (0,1) (0,1) (1,0)
⊃ (1,0) (0,0) (0,1) (1,0) (1,0) (0,1) (0,1) (0,0) (1,0) (1,0) (1,0) (0,1) (1,0) (1,0) (1,0)
Definition 2.6 Let R1,0(A) be the Twist-structure of type (1,0) associated to A. AR1,0(A)-valuation is any homomorphismw:V−→R1,0(A).
Remark 2.7 The definition ofR1,0(A)-valuation makes sense because the Twist- structures of typeI1P0 have the same similarity type of M(1,0). Anyway, in the literature is more common to describe everyR1,0(A)-valuationwas a pair of “local functions”:w= (w0, w1), wherew0:L(C)−→Aandw1:L(C)−→A∗. Actually, wi(α):=(w(α))i (i = 0,1). Realize that these functions (unlike w) are not homomorphisms, neither in A nor in A∗. So, for example, it is easy to see that, if α, β ∈ L(C), and w = (w0, w1) is a R1,0(A)-valuation, then w0(¬α) =w1(α);w0(α⊃β) =w0(α)→Aw1(β). In a similar way,w1(¬(α)) =
−A(w1(α)), andw1(α→β) = −A(w0(α)→A w0(β)). This formalism will be used along all the paper, without explicit mention.
The classT(1,0), together with the valuations defined above, determines the consequence relation|=T(1,0), defined in the sequel.
Definition 2.8 The consequence relation |=T(1,0) defined in L(C) is given by: Γ|=T(1,0)αiff, for every Twist-structureR1,0(A) of T(1,0), and for every R1,0(A)-valuationw is valid that:
If w(Γ) ={1R1,0(A)}thenv(α) = 1R1,0(A)= (1,0),with1R1,0(A)= (1A,0A) In addition, we say thatαis tautology with respect toT(1,0) (or T(1,0)- tautology) if and only if ∅ |=T(1,0)α(denoted, as usual, by |=T(1,0)α).
Remark 2.9 Please note that, if (x0, x1)∈R(1,0(A), then (x0, x1) = 1R(1,0)(A) iffx0 = 1A. So, the consequence relation can be expressed just in terms of the first components of the valuations: Γ|=T(1,0)αif and only if, for every Twist- structure R1,0(A) of T(1,0) and for every R1,0(A)-valuation w = (w0, w1) is valid that: if w0(γ) = 1A, for every γ ∈ Γ, then w0(α) = 1A. This can be applied to the definition of tautology of |=T(1,0), too. This approach is very convenient and will be used later.
Examples 2.10 Let us see some examples of tautologies accordingT(1,0) (we omit, from now on, the subscripts in the operations ofA):
• p⊃ ¬¬pis a tautology according toT(1,0). In fact: letw:V →R1,0(A) be a valuation, withR1,0(A) inT(1,0) such thatw(p) = (x0, x1). Then w(p⊃ ¬¬p) =w(p)⊃ ¬¬w(p) = (x0, x1)⊃ ¬¬(x0, x1)
(sincew is a homomorphism). So, remembering thatx0∧x1= 0, (x0, x1)⊃ ¬¬(x0, x1) = (x0, x1)⊃ ¬(x1,−x1)
= (x0, x1)⊃(−x1, x1)
= (x0→ −x1,−(x0→ −x1))
= (−x0∨ −x1,−(−x0∨ −x1))
= (−(x0∧x1), x0∧x1)
= (1,0) = 1R(A).
• Let us prove now that 6|=T(1,0)¬¬p⊃p: givenR1,0(A)∈T(1,0), we can define w:V →R1,0(A) such that w(¬¬p⊃p)6= 1R1,0(A). In fact, let w be such thatw(p) = (0,0)(∈R1,0(A)). Hence,
w(¬¬p⊃p) =¬¬w(p)⊃w(p) =¬¬(0,0)⊃(0,0)
=¬(0,1)⊃(0,0) = (1,0)⊃(0,0)
= (1→0,−(1→0)) = (0,1).
• From the previous example we can infer that (¬p∨p) is not a tautology with respect toT(1,0), since¬p∨p=¬¬p⊃p.
Remark 2.11 Note that every Twist-structure R1,0(A) inT(1,0) can define a local consequence relationR1,0(A), in the obvious way:
Γ R1,0(A) α iff, for every R1,0(A)-valuation w, if w(Γ) ⊆ {1R1,0(A)}, then
w(α) = 1R1,0(A). So, we can visualizeR1,0(A) as the consequence relation in- duced by the C-matrix MR(A) = (R(1,0)(A),{1R1,0(A)}). Hence, the relation
|=T(1,0) can be understood as the intersection of all the relations R1,0(A) in- duced by the matrices of the formMR1,0(A).
Proposition 2.12 If|=T(1,0)α, then |=M(1,0)α, for everyα∈L(C).
Proof: Consider the Twist-structureR1,0(2), of Example 2.5: it can be under- stood as the matrixMR1,0(2), cf. Remark 2.11. Comparing this matrix with the one of Definition 3.1, we can see that the functionh: {F0, F1, T0}−→R1,0(2), where h(T0) = (1,0),h(F1) = (0,0) andh(F0) = (0,1), is an isomorphism of matrices. Then,6|=M(1,0)αimplies6R1,0(2)α, and therefore 6|=T(1,0)α, by Re-
mark 2.11 again. 2
Our purpose now is to demonstrate the reciprocal of the proposition above.
That is, we must prove that, if|=M(1,0)α, then|=T(1,0)α. We will give a proof of this fact based in simple very algebraic concepts, that allow us to understand the algebraic behavior of the Twist-structures. To develope such proof, recall certain well-known notions and results, that can be found in any textbook of algebraic logic (see [12], for example )4.
Definition 2.13 In a latticeL= (L,∨,∧), an ideal is a set∆⊆Lverifying:
(i) If x∈∆,y∈∆, thenx∨y∈∆.
(ii) If x∈∆and y≤x, theny∈∆.
We say that an ideal ∆ is maximal if it is proper and maximal with the in- clusion order. On the other hand, a proper ideal∆⊆Los primeif it is valid that: ifx∧y∈∆, then x∈∆oy∈∆.
Recall now that the concept of (maximal, prime) filter, in lattices, is dual to the definition of (maximal, prime) ideal.
If, in additon, we are considering maximal ideals inboolean algebrasinstead of lattices, we have the following results:
Proposition 2.14 LetA =(A,∨,∧,−,1,0)be a boolean algebra, and let∆be an ideal ofA. Then the following facts are equivalent:
a) ∆is a maximal ideal.
b) ∆es a prime ideal.
c) ∇:=A−∆is a prime filter.
d) For everyx∈A, x∈∆or−x∈∆, but not both cases at the same time.
4Of course, the reader with algebraic knowledge can skip these results, from Definition 2.13 to Proposition 2.15.
e) For everyx∈A, x∈∆orx∈ ∇, but not both cases at the same time.
Finally, remember the following proposition:
Proposition 2.15 LetA be a boolean algebra. Ifx∈A,x6= 1, then there is a maximal ideal∆x, such that x∈∆x.
Let us relate this results to the Twist-structures:
Proposition 2.16 (Tricotomy): LetA be a boolean algebra and let R1,0(A) be the Twist-structure of type I1P0 associated to A. Then, for every element (x0, x1)∈R(1,0)(A), for every maximal ideal ∆⊆A, one and only one of the following cases is verified :
(i) x0∈∆and x1∈ ∇.
(ii) x0∈ ∇and x1∈∆.
(iii) x0∈∆and x1∈∆.
Proof: The three cases are pairwise disjoint: (i) and (ii) cannot be both valid because x0; the same fact happens with (ii) and (iii); finally (i) and (iii) are not simultaneously valid becausex1.
Besides that, if (i) or (ii) are not valid then we have that x0∈ ∇orx2∈∆ and, at the same time,x0∈∆ orx1∈ ∇. From this we have that some of these cases must be valid: (a)x0∈ ∇andx0∈∆; (b)x0∈ ∇andx1∈ ∇; (c)x0∈∆ andx1∈∆; (d)x1∈∆ andx1∈ ∇. Now: (a) and (d) are obviously imposible.
But (x0, x1)∈R(A), and therefore x0∧x1= 0∈∆. So, since ∆ is prime, we have thatx0 ∈∆ orx1∈∆. But this fact contradicts (b). Hence, (c) (that is, (iii)), it must be valid. With this we have demonstrated that (i), (ii) and (iii)
are exhaustive. 2
Proposition 2.17 Let R1,0(A) be a Twist-structure of type I1P0, let α be a formula in L(C) and consider a R1,0(A)-valuation w : L(C)−→R1,0(A) such thatw0(α)6= 1. Let ∆w0(α) be the maximal ideal induced by w0(α) (cf. Propo- sition 2.15). Then, exists a valuation q(w,α) : L(C)−→{F0, F1, T0} (the uni- verse of M(1,0)) such that, for every γ ∈L(C), q(w,α)(γ) = T0 if and only if w0(γ)∈ ∇w0(α).
Proof:ConsiderR1,0(A),α∈L(C) andw:L(C)−→R1,0(A) verifying the con- ditions of the proposition. We define the functionq(w,α):L(C)−→{T0, F1, F0}, in this way: ifγ∈L(C) then
q(w,α)(γ) =
T0 iff w0(γ)∈ ∇w0(α) andw1(α)∈∆w0(α) F1 iff w0(γ)∈∆w0(α) andw1(α)∈∆w0(α)
F0 iff w0(γ)∈∆w0(α) andw1(α)∈ ∇w0(α)
Taking into account that (w0(γ), w1(γ))∈R1,0(A), and that ∆w0(α)is maximal, we have thatq(w,α)(γ) is, actually, a function (because Proposition 2.16). On the other hand, by its very definition,q(w,α)(γ) =T0 if and only ifw0(γ)∈ ∇w0(α). So, we just need to prove thatq(w,α)is, in fact, a homomorphism fromL(C) to ({F0, F1, T0},{¬,⊃}).
(I) Let us prove thatq(w,α)(¬δ) =¬q(w,α)(δ), for every δ∈L(C):
(I.1) Suppose that q(w,α)(¬δ) = T0. Then, w0(¬δ) ∈ ∇w0(α). That is, w1(δ) ∈ ∇w0(α). Hence, w1(δ) ∈/ ∆w0(α). Now, since ∆w0(α) is prime, and w0(δ)∧w1(δ) = 0 ∈∆w0(α), then w0(δ) ∈∆w0(α). By our definition,q(w,α)(δ) =F0. Then¬q(w,α)(δ) =T0.
(I.2) Suppose that q(w,α)(¬δ) = F0; then, w0(¬δ) ∈ ∆w0(α). That is, w1(δ) ∈ ∆w0(α). Then, q(w,α)(δ) = F1 or q(w,α)(δ) = T0. In both cases ¬q(w,α)(δ) =F0.
(I.3) Suppose that q(w,α)(¬δ) = F1. This is impossible since we would have thatw0(¬δ) =w1(δ)∈∆w0(α). Moreover, we would have that w1(¬δ) = −w1(δ)∈∆w0(α), which is absurd.
(II) Let us prove that q(w,α)(δ ⊃ε) =q(w,α)(δ)⊃q(w,α)(ε)
(II.1) Suppose that q(w,α)(δ ⊃ ε) = T0. We have then that w0(δ ⊃ ε)
= w0(δ) → w0(ε) ∈ ∇w0(α). Let us relate w0(δ) with ∇w0(α): if w0(δ)∈ ∇w0(α), thenw0(ε)∈ ∇w0(α)because ∇w0(α)is filter. Thus, q(w,α)(δ) =T0 andq(w,α)(ε) =T0. Then,q(w,α)(δ)⊃q(w,α)(ε) =T0. Now, ifw0(δ)∈ ∇/ w0(α), thenw0(δ)∈∆w0(α) and (by the definition ofq(w,α)) we have thatq(w,α)(δ) =F0orq(w,α)(δ) =F1 (the value of q(w,α)(ε) is not relevant here). Hence,q(w,α)(δ)⊃q(w,α)(ε) =T0. (II.2) Suppose thatq(w,α)(δ ⊃ε) =F0. From the definition of q(w,α), we
have thatw0(δ ⊃ ε) =w0(δ) →w0(ε) = −w0(δ)∨w0(ε) ∈∆w0(α). Since ∆w0(α)is an ideal, then−w0(δ)∈∆w0(α)andw0(ε)∈∆w0(α). Since ∆w0(α)is maximal (and using Proposition 2.14, items (d) and (e)), then w0(δ)∈ ∇w
0(α). Then,q(w,α)(δ) =T0 andq(w,α)(ε)6=T0. Therefore,q(w,α)(δ)⊃q(w,α)(ε) =F0.
(II.3) Suppose that q(w,α)(δ ⊃ ε) = F1. It is not analyzed, by similar considerations to (I.3).
From (I) and (II),q(w,α) is a homomorphism. 2
Corollary 2.18 Let R1,0(A) be a Twist-structure in T(1,0). Then, for every formula α ∈ L(C) and every R1,0(A)-valuation w verifying the conditions of Proposition 2.17, we have that q(w,α)(α)6=T0.
Proof: Consider the definition ofq(w,α), and note thatw0(α)∈∆w0(α). 2 From the previous results, and using the characterization of tautology w.r.t T(1,0) given in Remark 2.9, we have:
Theorem 2.19 If|=M(1,0)αthen|=T(1,0)α.
Proof: Suppose that 6|=T(1,0) α. Then, exists a Twist-structure R1,0(A) in T(1,0) and aR1,0(A)-valuationwsuch thatw(α)6= (1,0). That is,w0(α)6= 1.
From Proposition 2.17, there is aM(1,0)-valuation such that q(w,α)(α)6= T0,
by Corollary 2.18. So,6|=I1P0 α. 2
Remark 2.20 Note that the demonstrations given above, despite its combi- natorial character, are relatively simple. However, they are not the only pos- sible proofs. In fact, it can be adapted a very common demonstration within the context of algebraic logic (for the case of boolean algebras), based in the isomorphism between the algebra 2 and the quotient boolean algebras A/∆, when ∆ is a maximal ideal. In this case, we define a “canonical application”
h:R1,0(A)−→R1,0(A/∆w0(α)) such thath◦wis a valuation on a Twist-structure isomorphic to the universe ofM(1,0). We leave the details to the reader.
3 Twist-structures for I
0P
1: a syntactic proof
In this section we give a Twist-structures semantics for the logicI0P1. This logic was defined by A. Sette in [13], and was the motivation for the definition ofI1P0, studied in the previous section. In fact, there is a kind of “simmetry”
between both logics, as will be explained in the next section. We choose the presentation of these logics in different sections becauseI0P1has an additional property: the matrixM(0,1) that defines itpossues more than one designated value. Formally:
Definition 3.1 ThelogicI0P1is the pairI0P1=(C,|=M(0,1)), where|=M(0,1) is defined by the matrixM(0,1) =h({T1, T0, F0},{⊃,¬}),{T1, T0}i. Here, the operations¬ y⊃are given by the following truth tables:
T0 T1 F0
¬ F0 T0 T0
⊃ T0 T1 F0 T0 T0 T0 F0
T1 T0 T0 F0
F0 T0 T0 T0
The Twist-structures that “codify” the matrixM(0,1) can be defined in an obvious way:
Definition 3.2 Let (A,∨A,∧A,−A,1A,0A) be a boolean algebra. The Twist- structure of type (0,1) associated to A is R0,1(A) = (R0,1(A),¬,⊃), the algebra such that:
(1) Its universe (that is, the Twist set of type (0,1) associated to A) is R0,1(A) ={(x0, x1)∈A×A∗ :x0∨Ax1= 1A}.
(2) ¬(x0, x1) = (x1,−Ax1).
(3) (x0, x1)⊃(y0, y1)∈R0,1(A) = (x0→Ay0,−A(x0→Ay0)).
For preservation of the notation in the previous section, the class of the Twist- structures of type(0,1)will be denoted asT(0,1).
The operations in R0,1(A) are well defined, as well as the Twist-structures inT(1,0) of the previous section.
Example 3.3 Again, given the boolean algebra2, the canonical Twist-structure of type I0P1 isR0,1(2) = {(0,1),(1,1),(1,0)}. The operations in the algebra R0,1(2) are indicated below:
(1,0) (1,1) (0,1)
¬ (0,1) (1,0) (1,0)
⊃ (1,0) (1,1) (0,1) (1,0) (1,0) (1,0) (0,1) (1,1) (1,0) (1,0) (0,1) (0,1) (1,0) (1,0) (1,0)
The definition of theR0,1(A)-valuation is similar to the given for the Twist- structures ofT(1,0), and we can visualize every R0,1(A)-valuationwas a pair w= (w0, w1), as was suggested in Remark 2.7.
However, to define the consequence relation|=T(0,1)we must previously give the algebraic analogy for the set of designated values inM(0,1) (already sug- gested in Remark 2.9):
Definition 3.4 LetR0,1(A)be a Twist-structure of type(0,1). ThesetDR0,1(A) is defined by: DR0,1(A) ={(x0, x1)∈R0,1(A) :x0= 1A}.
Definition 3.5 The consequence relation |=T(0,1) defined in L(C) is given by: Γ|=T(0,1)αif and only if, for every Twist-structureR0,1(A)inT(0,1) and for everyR0,1(A)-valuation wit is valid:
If w(Γ)⊆DR0,1(A) thenw(α)∈DR0,1(A)
As in the previous case,αis tautology with respect to T(0,1) if and only if|=T(0,1)α.
Note that, in the case of the Twist-structures of type (0,1),|=T(0,1)αif and only if, for every boolean algebraA, for everyR0,1(A)-valuationw,w0(α) = 1A. Proposition 3.6 If |=T(0,1)α, then|=M(0,1)α
Proof: Similar to Proposition 2.12. The matrix isomorphism between M(0,1) andR0,1(2) is here defined byh(T0) = (1,0),h(T1) = (1,1) yh(F0) = (0,1).2 Now, to prove that the Twist-structure semantics is adequate for |=M(0,1) we just need to demonstrate that, if|=M(0,1)α, then |=T(1,0)α. For this case, we will give another proof of such a result, based on a syntactic consequence relation adequate for I0P1. In fact, this proof (presented previously in [10]) is based on the Hilbert-style axiomatics forI0P1. This axiomatics is presented in the next definition:
Definition 3.7 Given the signature C, the relation `(0,1) denotes the conse- quence relation defined in the usual way by means of Hilbert - style axioms, whose only inference rule is Modus Ponens, and whose schema axioms5 are:
A1) φ⊃(ψ⊃φ)
A2) [φ⊃(ψ⊃θ)]⊃[(φ⊃ψ)⊃(φ⊃θ)]
A3) [¬φ⊃ ¬ψ]⊃[(¬φ⊃ ¬¬ψ)⊃φ]
A4) ¬(φ⊃ ¬¬φ)⊃φ A5) (φ⊃ψ)⊃ ¬¬(φ⊃ψ)
The axiomatics presented above is adequate for the logicI0P1. That is:
Theorem 3.8 `(0,1)α if and only if |=M(0,1)α.
Proof: It is the main result of [13]. 2
So, to demonstrate that |=T(0,1) is adequate w.r.t|=M(0,1), we just need to relate|=T(0,1)to`(0,1). We have as first result:
Proposition 3.9 If `(0,1)α, then |=T(0,1)α
Proof:We will prove that, ifαis an instace of the axioms of Definition 3.7, then
|=T(0,1)α. After that, we will prove that the T(0,1)-tautologies are preserved by the application of Modus Ponens. For this, letwbe aR0,1(A)-valuation for an arbitrary boolean algebraA. For every schema axiom (fromA1) toA5)), we have:
A1): w(φ⊃(ψ⊃φ)) =w(φ)⊃w(ψ⊃φ)
= (w0(φ), w1(φ))⊃((w0(ψ), w1(ψ))⊃(w0(φ), w1(φ)))
= (w0(φ), w1(φ))⊃(w0(ψ)→w1(φ),−(w0(ψ)→w1(φ)))
= (w0(φ)→(w0(ψ)→w0(φ)),−(w0(φ)→(w0(ψ)→w1(φ))))
= (1,0)∈DR0,1(A) A2): proceed asA1).
A3): w([¬φ⊃ ¬ψ]⊃[(¬φ⊃ ¬¬ψ)⊃φ]) =
= [¬w(φ)⊃ ¬w(ψ)]⊃[(¬w(φ)⊃ ¬¬w(ψ))⊃w(φ)] =
= [¬(w0(φ), w1(φ))⊃ ¬(w0(ψ), w1(ψ))]⊃
[(¬(w0(φ), w1(φ))⊃ ¬¬(w0(ψ), w1(ψ)))⊃(w0(φ), w1(φ))] =
= [(w1(φ),−w1(φ))⊃(w1(ψ),−w1(ψ))]⊃
[((w1(φ),−w1(φ))⊃(−w1(ψ), w1(ψ)))⊃(w0(φ), w1(φ))].
following this analysis we can see that the first component of this pair is:
(w1(φ)→w1(ψ))→((w1(φ)→ −w1(ψ))→w0(φ)) which, recalling thatw0(φ)∨w1(φ) = 1A, is just
5That is,φ,ψyθare indicating any formula ofL(C).
