the multioperations in the multialgebra BmbCA2 , it follows that Θ is a multicongruence over BAmbC2 . It is easy to prove that the multioperations in the quotient multialgebra BmbCA2 /Θ are trivial, that is: for every x, y ∈{a, b} and # ∈{∧,∨,→}, (x#y) =¬x=◦x={a, b}.
Clearly BmbCA2 /Θ is not a swap structure for mbC: otherwise, it would generate a trivial non-deterministic matrix where the set of designated values is the whole domain. This would contradict (CARNIELLI; CONIGLIO, 2016, Proposition 6.4.5(ii)), where it was proven that no non-deterministic matrix in the class M at(KmbC) is trivial. This shows that BmbCA2 /Θ, the homomorphic image of the canonical map p:BmbCA2 →BAmbC2 /Θ, does not belong to the class KmbC, despite its domain BmbCA2 is in KmbC.
From this last result, a question that arises is: Why don’t we change/adapt the homomorphic images or the congruence definitions in order to show that KmbC is a variety of multialgebras? The problem is that, if we do this, we will lose some important results such as the method of completeness that we apply in the main theorems of this Thesis.
Proof. The proof is very similar to that of Proposition 4.2.3.
For every Boolean algebra A there is a unique swap structure BAmbCciw for mbCciw with domainBciwA such that, for every a = (a1, a2, a3) and b= (b1, b2, b3) inBciwA :
(i) (a1, a2, a3)#(b1, b2, b3) = {(c1, c2, c3)∈BciwA : c1 =a1#b1}, for # ∈{∧,∨,→};
(ii) ¬(a1, a2, a3) ={(c1, c2, c3)∈BciwA : c1 =a2}; (iii) ◦(a1, a2, a3) ={(c1, c2, c3)∈BciwA : c1 =a3}.
The full subcategory in SWCPL+e of swap structures for mbCciw will be denoted by SWmbCciw. By the very definitions, SWmbCciw is a full subcategory in SWmbC, and a full subcategory in MAlg(Σ). Hence, the class of objects of SWmbCciw
is KmbCciw, and the morphisms between two given swap structures for mbCciw are just the homomorphisms between them as multialgebras over Σ.
The class M at(KmbCciw) of non-deterministic matrices associated to swap structures for mbCciw is defined analogously to the class M at(KCPL+e) introduced in Definition 4.1.5.
Theorem 4.3.5. (CARNIELLI; CONIGLIO, 2016, Theorem 6.5.4)Let Γ∪{α}⊆F or(Σ) be a set of formulas. Then: Γ�mbCciw α iff Γ|=M at(KmbCciw) α.
Now, stronger extensions of mbC will be analized:
Definition 4.3.6. Consider the following extensions of mbC:
(1) The logic mbCci(CARNIELLI; CONIGLIO, 2016, Definition 3.1.7) is obtained from mbC by adding the axiom schema:
¬◦α→(α∧¬α) (ci)
(2) The logic CPLe is obtained from mbC by adding the axiom schema:
◦α (cons) Proposition 4.3.7. The following holds:
(1) The logic mbCci properly extends mbCciw.
(2) The logic CPLe is an expansion of CPL by a connective ◦ such that ◦α is a valid schema. Thus, it properly extends mbCci, and it is semantically characterized by the usual 2-valued truth-tables for CPL plus the operator ◦(x) = 1 for every x∈{0,1}.
Proof. (1) See (CARNIELLI; CONIGLIO, 2016, Proposition 3.1.10).
(2) Observe that, by (cons), (bc1) and MP, the negation ¬ is explosive in CPLe and so it coincides with the classical negation. Since CPL+ is included in CPLe then, by axiom (Ax10), this logic is nothing more than an expansion of CPL by adding as theorems all
the formulas of the form ◦α.
Definition 4.3.8. A swap structure for mbCci is any B∈KmbCciw such that:
◦(a1, a2, a3) def= {(∼(a1∧a2), a1∧a2,1)}. The class of swap structures formbCci will be denoted byKmbCci.
The class M at(KmbCci) of non-deterministic matrices is defined analogously to the class M at(KCPL+e) introduced in Definition 4.1.5.
Theorem 4.3.9. (CARNIELLI; CONIGLIO, 2016, Theorem 6.5.11) LetΓ∪{α}⊆F or(Σ) be a set of formulas. Then: Γ�mbCci α iff Γ|=M at(KmbCci) α.
Proposition 4.3.10. The following holds:
KmbCci = {B∈KmbCciw : |=M(B)(ci)}
= {B∈KmbC : |=M(B) (ci)}
= {B∈KCPL+e : |=M(B)(Ax10)∧(bc1)∧(ci)}. Proof. The proof is very similar to that of Proposition 4.2.3.
Definition 4.3.11. Let A be a Boolean algebra with domain A. The universe of swap structures for CPLe over A is the set:
BCPLA e ={(c1, c2, c3)∈BciwA : c2 =∼c1}={(a,∼a,1) : a∈A}�A.
Definition 4.3.12. Aswap structure for CPLeis anyB ∈KmbCcisuch that |B|⊆BCPLA e. The class of swap structures forCPLe will be denoted by KCPLe.
The class M at(KCPLe) of non-deterministic matrices is defined analogously to the class M at(KCPL+e) introduced in Definition 4.1.5.
Proposition 4.3.13. The following holds:
KCPLe = {B ∈KmbCci : |=M(B)(cons)}
= {B ∈KmbC : |=M(B) (cons)}
= {B ∈KCPL+e : |=M(B) (Ax10)∧(bc1)∧(cons)}.
For every Boolean algebra Athe swap structure BmbCciA formbCciandBCPLA e forCPLeare defined as expected. The full subcategory inSWCPL+e of swap structures for mbCci and for CPLe will be denoted by SWmbCci and SWCPLe, respectively. By the very definitions, they are full subcategories inSWmbC, and full subcategories inMAlg(Σ).
Remark 4.3.14.
(1) If B ∈ KCPLe then B can be seen as a Boolean algebra isomorphic to the Boolean algebra π1[|B|].
(2) Observe that
KCPLe ⊂KmbCci ⊂KmbCciw ⊂KmbC ⊂KCPL+e
while
CPLe ⊃mbCci⊃mbCciw⊃mbC⊃CPL+e.
As analyzed in (CARNIELLI; CONIGLIO, 2016, Chapter 6), the logicmbCciw can be characterized by a single 3-valued non-deterministic matrix, by considering the two-element Boolean algebraA2. Indeed the non-deterministic matrixMmbCciw3 induced by the swap structure BmbCciwA2 , that isMmbCciw3 =M�BAmbCciw2
�, was originally considered by A. Avron in (AVRON, 2005) obtaining so a semantical characterization of mbCciw. The domain of the multialgebra BAmbCciw2 is the set BmbCciwA2 = �t, I, f� such that t= (1,0,1), I = (1,1,0) and f = (0,1,1), where D3 ={t, I} is the set of designated elements of the non-deterministic matrix MmbCciw3 . The multioperations are defined as follows:
∧ t I f
t {t, I} {t, I} {F} I {t, I} {t, I} {F} f {F} {F} {F}
∨ t I f
t {t, I} {t, I} {t, I} I {t, I} {t, I} {t, I} f {t, I} {t, I} {F}
→ t I f
t {t, I} {t, I} {f} I {t, I} {t, I} {f} f {t, I} {t, I} {t, I}
¬ t {f} I {t, I} f {t, I}
◦ t {t, I} I {f} f {t, I}
It is clear thatBAmbCciw2 is a submultialgebra of BmbCA2 . Moreover, by an analysis similar to the one presented above, it is possible to prove representation theorem for KmbCciw analogous to that for KmbC (recall Theorem 4.2.15). In order to do this, consider the following lemma:
Lemma 4.3.15. Let an assignment FmbCciw :BAlg→SWmbCciw, with FmbCciw(A) = BAmbCciw, and FmbCciw(f) = f∗ for every morphism f : A → A� in BAlg, where f∗ : BAmbCciw → BmbCciwA� and f∗(z) = (f(z1), f(z2), f(z3)), for every z ∈ BmbCciwA . Then FmbCciw is a functor which preserves monomorphisms and arbitrary products.
Proof. From BAmbCciw⊆BAmbC (as submultialgebra) and by proof of the Proposition 4.2.8 (in the case ofF :BAlg→SWmbC), it is easy to see thatf∗ is a morphism in SWmbCciw.
So, FmbCciw is a functor.
By Proposition 4.2.6 and the fact that SWmbCciw is a full subcategory of SWmbC that is a full subcategory of MAlg(Σ), then the functor FmbCciw : BAlg → SWmbCciw preserves arbitrary products.
Let f : A → A� be a monomorphism in BAlg. It is well-known that every monomorphism in BAlg is an injective function, and then f is injective. From this it is immediate to see that f∗ is also an injective function. As a consequence of Proposi-tion 2.5.4, f∗ is a monomorphism in the categoryMAlg(Σ). Given that SWmbCciw is a full subcategory of SWmbC that is a full subcategory of MAlg(Σ), it follows that f∗ is a monomorphism in SWmbCciw. So, the functor FmbCciw:BAlg →SWmbCciw preserves monomorphisms.
Theorem 4.3.16 (Representation Theorem for KmbCciw). Let B be a swap structure for mbCciw. Then, there exists a set I and a monomorphism of multialgebras ˆh:B →
�
i∈IBAmbCciw2 .
Proof. Let B be a swap structure formbCciw. Then, there is a Boolean algebra A such that B⊆BAmbCciw. Let g :B→BmbCciwA be the inclusion monomorphism inSWmbCciw. Using Birkhoff’s representation theorem for Boolean algebras7, there exists a set I and a monomorphism h:A→�i∈I A�i of Boolean algebras, whereA�i =A2, for every i∈I.
Consider the functor FmbCciw:BAlg →SWmbCciw such that FmbCciw(h) = h∗ which preserves monomorphisms (by Lemma 4.3.15). So, there is a monomorphism h∗ :BAmbCciw→BmbCciw�
i∈IA�i.
Recall that SWmbCciw is a full subcategory in SWmbC and SWmbC is a full subcategory in MAlg(Σ), and we have that fG : �i∈IBmbCciwA�i → B�mbCciw
i∈IA�i is an isomorphism in MAlg(Σ), where G = {A�i}i∈I (by Proposition 4.2.6). By definition of A�i it follows that BmbCciwA�i =BAmbCciw2 , for every i ∈I. Then ˆh :B → �i∈I BAmbCciw2 is a monomorphism in MAlg(Σ), where ˆh=fG−1◦h∗◦g.
7 In (DUNN; HARDEGREE, 2001, Theorem 8.11.7, p. 307).
Concerning mbCci and CPLe, similar results can be obtained. Indeed, A.
Avron has proven in (AVRON, 2005) thatmbCcican be characterized by a single 3-valued non-deterministic matrix MmbCci3 .
In (CARNIELLI; CONIGLIO, 2016, Chapter 6) was proved that MmbCci3 is the one obtained by the 3-valued swap structure BAmbCci2 , that is MmbCci3 =M�BmbCciA2
�, with the same domain and multioperations than BAmbCciw2 , but now the multioperator ◦ is single-valued, and it is defined as follows:
◦ t {t} I {f} f {t}
Clearly, BAmbCci2 is a submultialgebra ofBAmbCciw2 and so of BAmbC2 . Moreover:
Theorem 4.3.17 (Representation Theorem for KmbCci). Let B be a swap structure for mbCci. Then, there exists a set I and a monomorphism of multialgebras ˆh : B →
�
i∈IBAmbCci2 .
Proof. It is enough to observe that, let f : A → A� be a monomorphism in BAlg and let FmbCci : BAlg → SWmbCci such that FmbCci(f) = f∗ be a functor that preserves monomorphisms (by similar proof of the Lemma 4.3.15). If a ∈ BmbCciA , then ◦a def= {(∼(a1 ∧ a2), a1 ∧ a2,1)}. So, f∗[◦a] = {f∗(∼(a1 ∧ a2), a1 ∧ a2,1)} = {(f(∼(a1∧a2)), f(a1∧a2), f(1))}={(∼(f(a1)∧f(a2)),(f(a1)∧f(a2)), f(1))}=◦f∗(a) and therefore f∗ is a monomorphism in SWmbCci. The remainder of this proof is similar to proof of Theorem 4.3.16.
Finally, the case of CPLe is quite simple. The swap structure BACPL2 e has domain {t, f} where t= (1,0,1) and f = (0,1,1). The multiperations are single-valued, producing a Boolean algebra isomorphic to A2.
Clearly BACPL2 e ⊆BmbCciA2 ⊆BmbCciwA2 ⊆BmbCA2 ⊆BACPL2 +e. Additionally:
Theorem 4.3.18 (Representation Theorem for KCPLe). Let B be a swap structure for CPLe. Then, there exists a set I and a monomorphism of multialgebras ˆh : B →
�
i∈IBACPL2 e.
Proof. It is similar to proof of Theorem 4.3.16.
The last theorem is just the original Birkhoff’s theorem for Boolean algebras published under differen
Final Considerations
This Thesis proposes a general study of non-deterministic matrices applied to logic systems from the perspective of Universal algebra and category theory. The non-deterministic matrices differ from usual matrices (deterministic) by the use of what is called today “multioperations”. That is, operations which assign, to some element of the domain, a non-empty subset of that domain. When this Thesis was conceived while a research project, we aimed to develop a formal theory for non-deterministic matrices semantics in order to better understand the scope of the original proposal by Avron and his collaborators. The main goal of the research was to propose an alternative way for algebraization of logic systems, in order to deal with logics in which the usual algebraization methods cannot (or it is very hard to) be applied. Starting from this, the concept of
“non-deterministic algebra” (or “ND-algebra”)8 arose naturally. However, soon we realize that this notion was already proposed and intensively studied in the literature, under different names: “hyperalgebras”, “multialgebras” and “non-deterministic algebras”, among others.
Since non-deterministic algebras constitute a generalization of standard algebras, it is natural that the generalization of the usual concepts for this framework is not unique.
Being so, at least three different definitions of multialgebra were proposed, and this situation is similar for more specific concepts, such as homomorphism, for which at least five distinct definitions are available. Thus, after adapting and organizing the diverse nomenclature for these concepts, we present them in Chapter 2. Since there are many possibilities for each concept, the choice of the “right” notion in each case was motivated by purely pragmatical reasons: the notions better suited to our proposals (namely, its application to the study of logic systems) were adopted.
It is important to note that most of the authors have studied and developed the theory of hyperstructures independently, which could justify the different names for similar concepts. From this point of view, the historical analysis obtained in Chapter 1 was extremely useful for us. In fact, this historical research leads us to the discovery of some interesting facts: for instance, that Marty in 1935 (MARTY, 1935) already introduced a definition of homomorphism between hyperstructures and so, probably he was the first to present this concept. We also discovered that the Brazilian logician and mathematician Antonio Antunes Mario Sette (from theCentre for Logic, Epistemology and the History of Science –CLE at the University of Campinas – UNICAMP) already used in his Master’s thesis from 1971 the concept of hyperlattice under the name of “reticuloide”. We also found
8 By analogy with the terms “non-deterministic matrix” (or ND-matrix) and “Nmatrix” coined by Avron
that the concept of non-deterministic matrices was applied by other authors much before the works of Avron and his colaborators (AVRON; LEV, 2001). This study provided several possibilities for several future developments in the study of multialgebras applied to Logic.
For instance, the concepts presented in Chapter 2 enabled the development in Chapter 4 of an incipient algebraic theory of swap structures (a particular class of multialgebras introduced by Carnielli and Coniglio in (CARNIELLI; CONIGLIO, 2016, Chapter 6)), by adapting concepts of universal algebra to multialgebras in a suitable way. Although we found in the literature a large amount of theory and research about multialgebras, its application to Logic and especially to the algebraization of logic systems was not enough explored, as far as we know.
In Chapter 3 we introduced an original class of swap structures as a suitable semantics for a family of non-normal modal systems. This is the first example of swap structures defined for logics outside the scope ofLogics of Formal Inconsistency (LFIs), the class of paraconsistent logics for which swap structures were originally proposed. Based on the ideas of the completeness proofs of LFIs with respect to Fidel structures semantics found in (CARNIELLI; CONIGLIO, 2016, Chapter 6), a quotient swap structure that we called of Lindenbaum-Tarski swap structure, as well as a canonical valuation over it, were proposed.
After defining a suitable notion of swap structures, as well as the class of non-deterministic matrices naturally associated to them, it was possible to prove the soundness theorems of the Hilbert calculi defining these non-normal modal systems with respect to such Nmatrix semantics. In order to obtain the completeness theorem, the method of Lindenbaum-Tarski swap structures mentioned above was employed.
The notion of Lindenbaum-Tarski swap structures proposed here generalizes in a quite natural way the classical Lindenbaum-Tarski method, allowing to deal with logics which are not algebraizable in the usual sense. We conjecture that this technique could be applied to a wide class of non-algebraizable logics (even by the general techniques of Blok-Pigozzi), allowing an interesting and new paradigm for algebraizing logics by means of multialgebras.
The Lindenbaum-Tarski swap structures method was also applied in Chapter 4 in order to obtain completeness theorem for the system CPL+e (see Theorems 4.1.8 and 4.1.13). Its application to all theLFIs studied there should be immediate (observe that the adequacy of such LFIs with respect to swap structures semantics was already stated in (CARNIELLI; CONIGLIO, 2016, Chapter 6), by proving its equivalence with the Fidel structures semantics). Besides this, in Chapter 4 other important results were obtained, concerning the algebraic and categorial properties of several categories of swap structures (seen as multialgebras), including representation theorems analogous to Birkhoff’s theorem
for
Finally, some suggestions for future work that arise naturally from what has been developed in the present Thesis will be presented:
• In this Thesis, the method of Lindenbaum-Tarski swap structures was applied to two distinct family of logic systems (namely, non-normal modal logics and LFIs). They have in common the existence of at least one non-congruential operator, justifying so this non-deterministic algebraic approach. We believe that the same method can be applied to other logical systems, with similar conditions (that is, where the application of the usual algebraic methods is hard or even impossible).
• The behavior of the logics presented in Chapter 3 and the definitions of swap structures and of Lindenbaum-Tarski swap structures for them suggests that it is possible to obtain algebraic and categorial results, similar to those obtained in Chapter 4 for the family of LFIs, to the family of non-normal modal systems presented in Chapter 3 of this thesis. That is, a modular treatment of the algebraic theory of swap structures for these modal logics could be obtained in this way.
• In the paper (SCHWEIGERT, 1985) by Schweigert the proof of the Birkhoff’s theorem for multialgebras9 is clearly incomplete. Indeed, in that paper the author does not specify the basic definitions that are being adopted in order to obtain the main result. Given that, as discussed above (and in Chapter 2 of this Thesis), each concept from ordinary algebra can be generalized in several ways to the realm of multialgebras, this is not a minor issue. By its turn, Hansoul (HANSOUL, 1983) presented a detailed proof of a version of the Birkhoff’s theorem for multialgebras.
However, the definitions adopted there are too restrictive for our purpose, namely, its application to Logic. For instance, the author assumes that any multioperation must return afinite set of possible values for each argument. In view of this, it would be interesting to investigate the validity of the Birkhoff’s theorem for multialgebras using the definitions presented in Chapter 2 of this thesis.
• Some results have been obtained for the category of multialgebras proposed in Chapter 2. In Chapter 1 we presented some other general proposals and, in Chapter 4, we developed theoretic studies applied to someLFIs. We can still develop an algebraic and categorial study of swap structures (and other categories of multialgebras associated to logic systems) by changing the definition of homomorphism, similar to what Nolan did in (NOLAN, 1979).
As mentioned above, it is important to observe that the theory of multialgebras was not explored from the point of view of its logical application. In this sense, the present
9 Every multialgebra can be represented as a sub-direct product of sub-directly irreducible multialgebras.
research could be considered as pioneering, and the results presented in chapters 3 and 4 speak by themselves. Thus, this field has several open possibilities. This promises to be a fruitful ground for new researches.
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