ar X iv :1 91 1. 11 83 3v 2 [ m at h. L O ] 1 D ec 2 01 9 Twist-Valued Models for Three-valued Paraconsistent Set Theory Walter Carnielli and Marcelo E. Coniglio Centre for Logic, Epistemology and the History of ScienceCLE and Department of Philosophy University of Campinas -Unicamp walterac@unicamp.br; coniglio@unicamp.br Abstract Boolean-valued models of set theory were introduced by Scott and Solovay in 1965 (and independently by Vopěnka in the same year), offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific threevalued model called PS3 which satisfies all the axioms of ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent model of ZF. We observe here that (PS3,*) coincides (up to language) with da Costa and D'Ottaviano logic J3, a three-valued paraconsistent logic that have been proposed independently in the literature by several authors and with different motivations: for instance, it was reintroduced as CLuNs, LFI1 and MPT, among others. We propose in this paper a family of algebraic models of ZFC based on a paraconsistent three-valued logic called LPT0, another linguistic variant of J3 and so of (PS3,*) introduced by us in 2016. The semantics of LPT0, as well as of its firstorder version QLPT0, is given by twist structures defined over arbitrary complete Boolean agebras. From this, it is possible to adapt the standard Boolean-valued models of (classical) ZFC to an expansion of ZFC by adding a paraconsistent negation. This paraconsistent set theory is based on QLPT0, hence it is a paraconsistent expansion of ZFC characterized by a class of twist-valued models. We argue that the implication operator of LPT0 considered in this paper is, in a sense, more suitable for a paraconsistent set theory than the implication of PS3: indeed, our implication allows for genuinely inconsistent sets (in a precise sense, [[(w ≈ w)]] = 1 2 for some w). It is to be remarked that our implication does not fall under the definition of the so-called 'reasonable implication algebras' of Löwe and Tarafder. This suggests that 'reasonable implication algebras' are just one way to define a paraconsistent set theory, perhaps not the most appropriate. Our twist-valued models for LPT0 can be easily adapted to provide twist-valued models for (PS3,*); in this way twist-valued models generalize Löwe and Tarafder's three-valued ZF model, showing that all of them (including (PS3,*)) are, in fact, models of ZFC (not only of ZF). This offers more options for investigating independence results in paraconsistent set theory. 1 1 On models of set theory: Gödel shrinks, Cohen expands The interest for – and the overall knowledge about – models for set theory changed dramatically after the famous invention (or discovery) of Paul Cohen's methods of forcing. Cohen was able to show that the notion of cardinal number is elastic and relative, in contrast with the methods of "inner models" that Gödel used. Gödel has shown that, by shrinking the totality of sets in a model, they would turn to be 'well-behaved'. As a consequence, the constructible sets could not be used to prove the relative consistency of the negation of the Axiom of Choice (AC) or of the Continuum Hypothesis (CH). Paul J. Cohen, on the contrary, had the idea of reverting the paradigm, and instead of cutting down the sets within models, found a way to expand a countable standard model into a standard model in which CH or AC can be false, doing this in a minimalist but controlled fashion. Cohen elements are 'bad-behaved', but finely guided so as to make 'logical space' for the independence of AC and CH, As Dana Scott puts in the forward of Bell's book [1], "Cohen's achievement lies in being able to expand models (countable, standard models) by adding new sets in a very economical fashion: they more or less have only the properties they are forced to have by the axioms (or by the truths of the given model)." Cohen's methods, however, are not easy, being regarded by some researchers as somewhat lengthy and tedious – but were the only tool available until the Boolean-valued models of set theory put forward by Scott and Solovay (and independently by Vopěnka) in 1965 offered a more natural and rich alternative for describing forcing. This does not discredit the brilliant idea of Cohen, who did not have the machinery of Boolean-valued models available at his time. What is a Boolean-valued model? The intuitive idea is to pick a suitable Boolean algebra A, and define the set of all A-valued sets in M, generalizing the familiar {0, 1} valued models. Then add to the language one constant symbol for each element of the model. After this, define a map φ 7→ [[φ]]A from the sentences in S to A which obey certain equations so that it should assign 1 to all the axioms of ZFC. The resulting structure MA will not be a standard model of ZFC, because it will consist of "relaxed sets" somehow similar to fuzzy sets, and not sets properly. If we take an arbitrary sentence about sets (for instance, "does Y is a member of X" ?) and ask whether it holds in MA, then the answer may be neither plain "yes" nor "no", but some element of the Boolean algebra A meaning the "degree" to which Y is a member of X . However, MA will satisfy ZFC, and to turn MA into an actual model of ZFC with certain desired properties it is sufficient to take a suitable quotient of MB that eliminates the elements of fuzziness. Boolean-valued models not only avoid tedious details of Cohen's original construction, but permit a great generalization by varying on any Boolean algebra. 2 Losing unnecessary weight: the role of alternative set theories It is a well-known historical fact that the discovery of the paradoxes in set theory and in the foundations of mathematics was the fuse that fired the revolution in contemporary set theory around its efforts to attempt to rescue Cantor's naive theory from triviality. The usual culprit was the Principle of (unrestricted) Abstraction, also known as the Principle of 2 Comprehension. Unrestricted abstraction allows sets to be defined by arbitrary conditions, and this freedom combined with the axiom of extensionality, leads to a contradiction, which by its turn leads to triviality in the sense that "everything goes", when the laws of the underlying logic obey the standard principles that comprise the so-called "classical" logic. But there is a way out from this maze. Paraconsistent set theory is the theoretical move to maintain the freedom of defining sets, while stripping the theory of unnecessary principles so as to avoid triviality, a disastrous consequences of contradictions involving sets in ZF. This philosophical maneuver is in frank opposition to traditional strategies, which deprive the freedom of set theory so appreciated by Cantor, by maintaining the underlying logic and weakening the Principle of Abstraction, An analogy may be instructive. The basic goal of reverse mathematics is to study the relative logical strengths of theorems from ordinary non-set theoretic mathematics. To this end, one tries to find the minimal natural axiom system A that is capable of proving a theorem T . In a perhaps vague, but illuminating analogy, paraconsistent logic tries to find the minimal natural principles that are capable of permitting us to reason in generic circumstances, even in the undesired circumstances of contradictions. This does not mean that contradictions are necessarily real: [4] gives a formal system and a corresponding intended interpretation, according to which true contradictions are not tolerated. Contradictions are, instead, epistemically understood as conflicting evidence. There are indeed many cases of contradictions in reasoning, but the classical principle Ex Contradictione Quodlibet, or Principle of Explosion, is neither used in mathematics in general; it is not, therefore, a characteristic of good reasoning, and has to be abandoned. Some people may be mislead by thinking that Reductio ad Absurdum, which is a useful and robust rule of inference, would be lost by abandoning the Principle of Explosion. This is not so: even if discarding such a principle, proofs by Reductio ad Absurdum get unaffected, as long as one can define a strong negation. This is achieved in many paraconsistent logics, in particular in all the logics of the family of the Logics of Formal Inconsistency (LFIs), see [9, 8, 7]. Reasoning does not necessarily require the full power of Ex Contradictione Quodlibet, because contradictions reached in a Reductio proof are not really used to cause any deductive explosion; what is used is the manipulation of negation. 3 Expanding Cohen's expansion: twist-valued models Boolean-valued models were adapted by Takeuti, Titani, Kozawa and Ozawa to latticevalued models of set theory, with applications to quantum set theory and fuzzy set theory (see [21, 23, 24, 19, 20]). The guidelines of these constructions were taken by Löwe and Tarafder in [18] in order to obtain a three-valued model (in the form of a latticevalued model) for a paraconsistent set theory based on ZF. They propose a class of algebras based on a certain kind of implication, called reasonable implication algebras (see Section 9) which satisfy several axioms of ZF. From this class, they found an especific three-valued model which satisfies all the axioms of ZF, and it can be expanded to an 3 algebra (PS3, *) with a paraconsistent negation *, obtaining so a paraconsistent model of ZF. As we discuss in Section 9, the logic (PS3, *) is the same as the logic MPT introduced in [13], and coincides up to language with the logic LPT0 adopted in the present paper. Here, we will introduce the notion of twist-valued models for a paraconsistent set theory ZFLPT0 based on QLPT0, a first-order version of LPT0. Our models, defined for any complete Boolean algebra A, constitute a generalization of the Boolean-valued models for set theory, at the same time generalizing Löwe and Tarafder's three-valued model. Indeed, in Section 9 the model of ZF based on (PS3, *) will be generalized to twist-valued models over an arbitrary complete Boolean algebra, obtaining so a class of models of ZFC. The structure over (PS3, *) will constitute a particular case, by considering the two-element complete Boolean algebra. As a consequence of this, it follows that Löwe and Tarafder's three-valued structure is, indeed, a model of ZFC. Twist-structure semantics have been independently proposed by M. Fidel [15] and D. Vakarelov [25], in order to semantically characterize the well-known Nelson logic. A twist structure consists of operations defined on the cartesian product of the universe of a lattice, L× L so that the negative and positive algebraic characteristics can be treated separately. In terms of logic, a pair (a, b) in L×L is such that a represents a truth-value for a formula φ while b corresponds to a truth-value for the negation of φ. That is, a is a positive value for φ while b is a negative value for it, thus justifying the name 'twist structures' given for this kind of algebras. This strategy is especially useful for obtaining semantical characterizations for non-standard logics. As a limiting case, a Boolean algebra turns out being a particular case of twist structures when there is no need to give separate attention to negative and positive algebraic characteristics, since the latter are uniquely obtained from the former by the dualizing Boolean complement ∼. In this case, every pair (a, b) is of the form (a,∼a), hence the second coordinate is redundant. Our proposal is based on models for ZF based on twist structures, thus the sentences of the language of ZF will be interpreted as pairs (a, b) in a suitable twist structure, such that the supremum a∨b is always 1, but the infimum a∧b is not necessarily equal to 0. This corresponds to the validity of the third-excluded middle for the non-classical negation of the underlying logic, while the explosion law φ∧¬φ → ψ is not valid in general in the underlying paraconsistent logic LPT0. A somewhat related approach was proposed by Libert in [16]: he proposes models for a naive set theory in which the truth-values are pairs of sets (A,B) of a universe U such that A ∪ B = U where A and B represent, respectively, the extension and the anti-extension of a set a. However, besides this similarity, our approach is quite different: we are interesting in giving paraconsistent models for ZFC and not in new models for Naive set theory. It is important to notice that there exists in the literature several approaches to paraconsistent set theory, under different perspectives. In particular, we propose in [6] a paraconsistent set theory based on several LFIs, but that approach differs from the one in the present paper. First, in the previous paper the systems were presented axiomatically, by means of suitable modifications of ZF. Moreover, in that logics a consistency predicate C(x) was considering, with the intuitive meaning that 'x is a consistent set'. On the other hand, in the present paper a model for standard ZFC will be presented instead of a Hilbert calculus for a modified version of ZF. We will return to this point in Section 10. As mentioned above, twist structures over a Boolean algebra generalize Boolean algebras, and are by their turn generalized by the swap structures introduced in [7, Chapter 6] (a previous notion of swap structures was given in [5]). Swap structures are non4 deterministic algebras defined over the three-fold Cartesian product A×A×A of a given Boolean algebra so that in a triple (a, b, c) the first component a represents the truthvalue of a given formula φ while b and c represent, respectively, possible values for the paraconsistent negation ¬φ of φ, and for the consistency ◦φ of φ. Swap structures are committed to semantics with a non deterministic character, while twist structures are used when the semantics are deterministic (or truth-functional). Definition 4.6 below shows how the definition of twist structures for the three-valued logic LFI1◦ introduced in [10, Definition 9.2] can be adapted to LPT0. As noted in Section 7, the three-valued logic (PS3, *) used in [22] already appears in [13] under the name MPT, and it is equivalent to LPT0 and also to LFI1◦. Variants of this logic have been independently proposed by different authors at with different motivations in several occasions (for instance, as the well-known da Costa and D'Ottaviano's logic J3). The naturalness of this logic is reflected by the fact that the three-valued algebra of LPT0 (see Definition 4.2 below) is equivalent, up to language, to the algebra underlying Lukasiewicz three-valued logic L3. The only difference is that in the former the set of distinguished (or designated) truth values is {1, 1 2 } instead of {1}, and this is why LPT0 is paraconsistent while L3 is paracomplete. Twist-valued models work beautifully as enjoying many properties similar to Booleanvalued models (when restricted to pure ZF-languages). Such similarities lead to a natural proof that ZFC is valid w.r.t. twist-valued models, as our central Theorem 8.21 shows. This paper deals with a paraconsistent set theory named ZFLPT0, defined by using as the underlying logic a first-order version of LPT0, called QLPT0, proposed in [12] under the form of QLFI1◦ (that is, by replacing the strong negation ∼ by the consistency operator ◦). The paraconsistent character of twist-valued models as regarding ZFLPT0 as rival of ZFC is emphasized. Despite having some limitative results, as much as Löwe and Tarafder's model, ZFLPT0 has a great potential as generator of models for paraconsistent set theory. A subtle, but critical advantage of our models is that the implication operator of LPT0 is much more suitable for a paraconsistent set theory than the one of PS3. Indeed, our models allow for inconsistent sets, and this is of paramount importance, as we argue below. Moreover, as pointed out above, our models generalize the three-valued model based on PS3, since they can be defined for any complete Boolean algebra. In this way, we have several models at our disposal, and in principle this can be used to investigate independence results in paraconsistency set theory. Albeit Boolean-valued models and their generalization in the form of twist-valued models are naturally devoted to study independence results, this paper does not tackle this big questions yet. The paper, instead, is dedicated to clarifying such models while establishing their basic properties. 4 The logic LPT0 In this section the logic LPT0 will be briefly discussed, including its twist structures semantics. From now on, if Σ′ is a propositional signature then, given a denumerable set V = {p1, p2, . . .} of propositional variables, the propositional language generated by Σ′ from V will be denoted by LΣ′. The paraconsistent logics considered in this paper belong to the class of logics known as logics of formal inconsistency, introduced in [9] (see also [8, 7]). 5 Definition 4.1. Let L = 〈Σ′,⊢〉 be a Tarskian, finitary and structural logic defined over a propositional signature Σ′, which contains a negation ¬, and let ◦ be a (primitive or defined) unary connective. The logic L is said to be a logic of formal inconsistency (LFI) with respect to ¬ and ◦ if the following holds: (i) φ,¬φ 0 ψ for some φ and ψ; (ii) there are two formulas φ and ψ such that (ii.a) ◦α, φ 0 ψ; (ii.b) ◦α,¬φ 0 ψ; (iii) ◦φ, φ,¬φ ⊢ ψ for every φ and ψ. Recall the logic MPT0 presented in [7] as a linguistic variant of the logic MPT introduced in [13]. Definition 4.2. (Modified Propositional logic of Pragmatic Truth MPT0, [7, Definition 4.4.51]) Let MPT0 = 〈M,D〉 be the three-valued logical matrix over Σ = {∧,∨,→ ,∼,¬} with domain M = {1, 1 2 , 0} and set of designated values D = {1, 1 2 } such that the operators are defined as follows: ∧ 1 1 2 0 1 1 1 2 0 1 2 1 2 1 2 0 0 0 0 0 ∨ 1 1 2 0 1 1 1 1 1 2 1 1 2 1 2 0 1 1 2 0 → 1 1 2 0 1 1 1 2 0 1 2 1 1 2 0 0 1 1 1 ∼ 1 0 1 2 0 0 1 ¬ 1 0 1 2 1 2 0 1 The logic associated to the logical matrix MPT0 is called MPT0. The three-valued algebra underlying MPT0 will be called APT0. Observe that x → y = ∼x ∨ y for every x, y. Recall that, by definition, the consequence relation MPT0 of MPT0 is given as follows: for every Γ ∪ {φ} ⊆ LΣ, Γ MPT0 φ iff, for every homomorphism v : LΣ →M of algebras over Σ, if v[Γ] ⊆ D then v(φ) ∈ D. From [7] a sound and complete Hilbert calculus for MPT0, called LPT0, can be defined. This calculus is an axiomatic extension of a Hilbert calculus for classical propositional logic CPL over the signature Σc = {∧,∨,→,∼}. From now on, φ ↔ ψ will be an abbreviation for the formula (φ→ ψ) ∧ (ψ → φ). Definition 4.3. (The calculus LPT0, [7, Definition 4.4.52]) The Hilbert calculus LPT0 over Σ is defined as follows:1 1To be rigorous, in [7, Theorem 4.4.56] an additional axiom schema is required: ¬∼φ → φ. However, it is easy to prove that this axiom is derivable from the others, by using MP. 6 Axiom Schemas: (Ax1) φ→ (ψ → φ) (Ax2) (φ→ (ψ → γ)) → ((φ→ ψ) → (φ→ γ)) (Ax3) φ→ (ψ → (φ ∧ ψ)) (Ax4) (φ ∧ ψ) → φ (Ax5) (φ ∧ ψ) → ψ (Ax6) φ→ (φ ∨ ψ) (Ax7) ψ → (φ ∨ ψ) (Ax8) (φ→ γ) → ((ψ → γ) → ((φ ∨ ψ) → γ)) (Ax9) φ ∨ (φ→ ψ) (TND) φ ∨ ∼φ (exp) φ→ ( ∼φ→ ψ ) (TND¬) φ ∨ ¬φ (dneg) ¬¬φ↔ φ (neg∨) ¬(φ ∨ ψ) ↔ (¬φ ∧ ¬ψ) (neg∧) ¬(φ ∧ ψ) ↔ (¬φ ∨ ¬ψ) (neg →) ¬(φ→ ψ) ↔ (φ ∧ ¬ψ) Inference rule: (MP) φ φ→ ψ ψ It is worth noting that axioms (Ax1)-(Ax9), (TND) and (exp), together with (MP), constitute an adequate Hilbert calculus for classical propositional logic CPL in the signature Σc = {∧,∨,→,∼}. Moreover, (Ax1)-(Ax9) plus (MP) is an adequate Hilbert calculus for classical positive popositional logic CPL+ in the signature Σcp = {∧,∨,→}. Theorem 4.4. ([7, Theorem 4.4.56]) The logic LPT0 is sound and complete w.r.t. the matrix logic of MPT0: Γ ⊢LPT0 φ iff Γ MPT0 φ, for every Γ ∪ {φ} ⊆ LΣ. The latter result can be extended to twist-structures semantics, as shown in [10]. Indeed, LPT0 coincides (up to signature) with LFI1◦, an LFI defined over the signature Σ◦ = {∧,∨,→,¬, ◦} such that the consistency operator ◦ is defined as ◦ 1 1 1 2 0 0 1 7 In LFI1◦ the strong negation ∼ is defined as ∼φ =def φ → ⊥φ such that ⊥φ =def (φ ∧ ¬φ) ∧ ◦φ. On the other hand, the consistency operator ◦ is defined in LPT0 as ◦φ =def ∼(φ ∧ ¬φ). The twist-structures semantics for LFI1◦ introduced in [10, Definition 9.2] can be adapted to LPT0 as follows: Definition 4.5. Let A = 〈A,∧,∨,→,∼, 0, 1〉 be a Boolean algebra.2 The twist domain generated by A is the set TA = {(z1, z2) ∈ A×A : z1 ∨ z2 = 1}. Definition 4.6. Let A be a Boolean algebra. The twist structure for LPT0 over A is the algebra TA = 〈TA, ∧, ∨, →, ∼, ¬〉 over Σ such that the operations are defined as follows, for every (z1, z2), (w1, w2) ∈ TA: (i) (z1, z2) ∧ (w1, w2) = (z1 ∧ w1, z2 ∨ w2); (ii) (z1, z2) ∨ (w1, w2) = (z1 ∨ w1, z2 ∧ w2); (iii) (z1, z2) → (w1, w2) = (z1 → w1, z1 ∧ w2); (iv) ∼(z1, z2) = (∼z1, z1); (v) ¬(z1, z2) = (z2, z1). By recalling that the consistency operator ◦ is defined in LPT0 as ◦φ =def ∼(φ ∧ ¬φ), it follows that ◦(z1, z2) = (∼(z1 ∧ z2), z1 ∧ z2). 3 Definition 4.7. The logical matrix associated to the twist structure TA is MT A = 〈TA, DA〉 where DA = {(z1, z2) ∈ TA : z1 = 1} = {(1, a) : a ∈ A}. The consequence relation associated to MT A will be denoted by TA. Let MLPT0 = {MT A : A is a Boolean algebra} be the class of twist models for LPT0. The twist-consequence relation for LPT0 is the consequence relation MLPT0 associated to MLPT0, namely: Γ MLPT0 φ iff Γ TA φ for every Boolean algebra A. Remark 4.8. In [10, Theorem 9.6] it was shown that LPT0 is sound and complete w.r.t. twist structures semantics, namely: Γ ⊢LPT0 φ iff Γ MLPT0 φ, for every set of formulas Γ ∪ {φ}. On the other hand, if A2 is the two-element Boolean algebra with domain {0, 1} then TA2 consists of three elements: (1, 0), (1, 1) and (0, 1). By identifying these elements with 1, 1 2 and 0, respectively, then TA2 coincides with the three-valued algebra APT0 underlying the matrix MPT0 (recall Definition 4.2). Moreover, MT A2 coincides with MPT0. Taking into consideration Theorem 4.4, this situation is analogous to the semantical characterization of CPL w.r.t. Boolean algebras: it is enough to consider the two-element Boolean algebra A2. 2In this paper the symbol ∼ will be used for denoting the strong negation of LPT0 as well as for denoting the classical negation and its semantical interpretation (the Boolean complement in a Boolean algebra). The context will avoid possible confusions 3This is why in [10, Definition 9.2] clause (v) was replaced by this clause defining ◦. 8 5 The logic QLPT0 A first-order version of LPT0, called QLPT0, was proposed in [12] under the equivalent (up to language) form of QLFI1◦. 4 For convenience, we reproduce here the main features of QLPT0. Definition 5.1. Let V ar = {v1, v2, . . .} be a denumerable set of individual variables. A first-order signature Θ for QLPT0 is given as follows: a set C of individual constants; for each n ≥ 1, a set Fn of function symbols of arity n, for each n ≥ 1, a nonempty set Pn of predicate symbols of arity n. The sets of terms and formulas generated by a signature Θ will be denoted by Ter(Θ) and For(Θ), respectively. The set of closed formulas (or sentences) and the set of closed terms (terms without variables) over Θ will be denoted by Sen(Θ) and CTer(Θ), respectively. The formula obtained from a given formula φ by substituting every free occurrence of a variable x by a term t will be denoted by φ[x/t]. Definition 5.2. Let Θ be a first-order signature. The logic QLPT0 is obtained from LPT0 by adding the following axioms and rules: Axioms Schemas: (Ax∃) φ[x/t] → ∃xφ, if t is a term free for x in φ (Ax∀) ∀xφ→ φ[x/t], if t is a term free for x in φ (Ax¬∃) ¬∃xφ ↔ ∀x¬φ (Ax¬∀) ¬∀xφ ↔ ∃x¬φ Inference rules: (∃-In) φ→ ψ ∃xφ→ ψ , where x does not occur free in ψ (∀-In) φ→ ψ φ→ ∀xψ , where x does not occur free in φ The consequence relation of QLPT0 will be denoted by ⊢QLPT0. 6 Twist structures semantics for QLPT0 In [12] a semantics of first-order structures based on twist structures for LFI1◦ was proposed for QLFI1◦. That semantics will be briefly recalled here, adapted to QLPT0. From now on, only complete Bolean algebras will be considered. 4That is, by taking ◦ instead of ∼ as a primitive connective. 9 Definition 6.1. let A be a complete Boolean algebra. Let MT A be the logical matrix associated to a twist structure TA for LPT0, and let Θ be a first-order signature (see Definition 5.1). A (first-order) structure over MT A and Θ (or a QLPT0-structure over Θ) is pair A = 〈U, IA〉 such that U is a nonempty set (the domain or universe of the structure) and IA is an interpretation function which assigns: an element IA(c) of U to each individual constant c ∈ C; a function IA(f) : U n → U to each function symbol f of arity n; a function IA(P ) : U n → TA to each predicate symbol P of arity n. Notation 6.2. From now on, we will write cA, fA and PA instead of IA(c), IA(f) and IA(P ) to denote the interpretation of an individual constant symbol c, a function symbol f and a predicate symbol P , respectively. Definition 6.3. Given a structure A over MT A and Θ, an assignment over A is any function μ : V ar → U . Definition 6.4. Given a structure A over MT A and Θ, and given an assignment μ : V ar → U we define recursively, for each term t, an element [[t]]Aμ in U as follows: - [[c]]Aμ = c A if c is an individual constant; - [[x]]Aμ = μ(x) if x is a variable; - [[f(t1, . . . , tn)]] A μ = f A([[t1]] A μ , . . . , [[tn]] A μ) if f is a function symbol of arity n and t1, . . . , tn are terms. Definition 6.5. Let A be a structure over MT A and Θ. The diagram language of A is the set of formulas For(ΘU), where ΘU is the signature obtained from Θ by adding, for each element a ∈ U , a new individual constant ā . Definition 6.6. The structure Â = 〈U, I Â 〉 over ΘU is the structure A over Θ extended by I Â (ā) = a for every a ∈ A. It is worth noting that sÂ = sA whenever s is a symbol (individual constant, function symbol or predicate symbol) of Θ. Notation 6.7. The set of sentences or closed formulas (that is, formulas without free variables) of the diagram language For(ΘU) is denoted by Sen(ΘU), and the set of terms and of closed terms over ΘU will be denoted by Ter(ΘU) and CTer(ΘU), respectively. If t is a closed term we can write [[t]]A instead of [[t]]Aμ , for any assignment μ, since it does not depend on μ. Notation 6.8. From now on, if z ∈ TA then (z)1 and (z)2 (or simply z1 and z2) will denote the first and second coordinates of z, respectively. Definition 6.9 (QLPT0 interpretation maps). Let A be a complete Boolean algebra, and let A be a structure over MT A and Θ. The interpretation map for QLPT0 over A and MT A is a function [[*]] A : Sen(ΘU) → TA satisfying the following clauses (using Notation 6.8 in clauses (iv) and (v)): (i) [[P (t1, . . . , tn)]] A = PA([[t1]] Â, . . . , [[tn]] Â), if P (t1, . . . , tn) is atomic; 10 (ii) [[#φ]]A = #[[φ]]A, for every # ∈ {¬,∼}; (iii) [[φ#ψ]]A = [[φ]]A # [[ψ]]A, for every # ∈ {∧,∨,→}; (iv) [[∀xφ]]A = (∧ a∈U([[φ[x/ā]]] A)1, ∨ a∈U([[φ[x/ā]]] A)2 ) . (v) [[∃xφ]]A = (∨ a∈U([[φ[x/ā]]] A)1, ∧ a∈U([[φ[x/ā]]] A)2 ) . Remark 6.10. A partial order can be naturally introduced in TA as follows: z ≤ w iff z1 ≤ w1 and z2 ≥ w2. It is easy to see that, with this order, TA is a complete lattice (since A is a complete Boolean algebra), in which ∧ i∈I zi = (∧ i∈I(zi)1, ∨ i∈I(zi)2 ) , and ∨ i∈I zi = (∨ i∈I(zi)1, ∧ i∈I(zi)2 ) . Note that 1 =def (1, 0) and 0 =def (0, 1) are the top and bottom elements of TA, respectively. These considerations justify the definition of the interpretation of the quantifiers given in Definition 6.9(iv) and (v). Recall the notation stated in Definition 6.5. The interpretation map can be extended to arbitrary formulas as follows: Definition 6.11. Let A be a complete Boolean algebra, and let A be a structure over MT A and Θ. Given an assignment μ over A, the extended interpretation map [[*]] A μ : For(ΘU) → TA is given by [[φ]] A μ = [[φ[x1/μ(x1), . . . , xn/μ(xn)]]] A, provided that the free variables of φ occur in {x1, . . . , xn}. Definition 6.12. Let A be a complete Boolean algebra, and let A be a structure over MT A and Θ. Given a set of formulas Γ ∪ {φ} ⊆ For(ΘU), φ is said to be a semantical consequence of Γ w.r.t. (A,MT A), denoted by Γ |=(A,MT A) φ, if the following holds: if [[γ]]Aμ ∈ D, for every formula γ ∈ Γ and every assignment μ, then [[φ]] A μ ∈ D, for every assignment μ. Definition 6.13 (Semantical consequence relation in QLPT0 w.r.t. twist structures). Let Γ∪{φ} ⊆ For(Θ) be a set of formulas. Then φ is said to be a semantical consequence of Γ in QLPT0 w.r.t. first-order twist structures, denoted by Γ |=QLPT0 φ, if Γ |=(A,MT A) φ for every pair (A,MT A). Theorem 6.14 (Adequacy of QLPT0 w.r.t. first-order twist structures ([12])). For every set Γ ∪ {φ} ⊆ For(Θ): Γ ⊢QLPT0 φ if and only if Γ |=QLPT0 φ. 5 In Remark 4.8 was observed that TA2 , the twist structure for LPT0 defined over the twoelement Boolean algebra A2, coincides (up to names) with the three-valued algebra APT0 underlying the matrix MPT0 and, moreover, MT A2 coincides with the three-valued characteristic matrix MPT0 of LPT0. In [12] it was proven that QLPT0 can be characterized by first-order structures defined over MPT0. 6 5As observed above, in [12] the logic QLFI1◦ was analyzed instead of QLPT0. However, both logics are equivalent, the only difference being the use of ◦ instead of ∼ as primitive connective. The adaptation of the adequacy result for QLFI1◦ given in [12] to the logic QLPT0 is straightforward. 6Once again, it is worth observing that the result obtained in [12] concerns the logic QLFI1◦ instead of QLPT0. 11 Theorem 6.15 (Adequacy of QLPT0 w.r.t. first-order structures over MPT0 ([12])). For every set Γ ∪ {φ} ⊆ For(Θ): Γ ⊢QLPT0 φ iff Γ |=(A,MPT0) φ for every structure A over Θ and MPT0. Remark 6.16. It is worth observing that Theorem 6.15 constitutes a variant of the adequacy theorem of first-order J3 w.r.t. first-order structures given in [14]. Indeed, both logics are the same (up to language), and the semantic structures are the same, up to presentation. 7 Twist-valued models for set theory As mentioned before, a three-valued model for a paraconsistent set theory based on latticevalued models for ZF, as a non-classical variant of the well-known Scott-Solovay-Vopěnka Boolean-valued models for ZF, was proposed by Löwe and Tarafder in [18]. Specifically, they introduce a three-valued logic called PS3 which can be expanded with a paraconsistent negation ¬ (which they denote by ∗) and then a model for ZF is constructed over the three-valued algebra PS3, as well as over its expansion (PS3,¬), along the same lines as the traditional Boolean-valued models. It is known that the logic (PS3,¬), introduced in [13] as MPT, coincides up to language with LPT0. We will return to this point in Section 9. In this section, a twist-valued model for a paraconsistent set theory ZFLPT0 based on QLPT0 will be defined, for any complete Boolean algebra A. It will be shown that this models constitute a generalization of the Boolean-valued models for set theory, as well as of Löwe-Tarafder's three-valued model. Our constructions, as well as the proof of their formal properties, are entirely based on the exposition of Boolean-valued models given in the book [1], which constitutes a fundamental reference to this subject. Consider the first order signature ΘZF for set theory ZF which consists of two binary predicates ǫ (for membership) and ≈ (for identity). The logic ZFLPT0 will be defined over the first-order language L generated by ΘZF based on the signature of QLPT0, that is: the set of connectives is Σ = {∧,∨,→,∼,¬}, together with the quantifiers ∀ and ∃ and the set V ar = {v1, v2, . . .} of individual variables. As usual, dom(f) and ran(f) will thenote the domain and image (or rank) of a given function f . Definition 7.1. Let A be a complete Boolean algebra, and let α be an ordinal number. Define, by transfinite recursion on α, the following: VTAα = {x : x is a function and ran(x) ⊆ TA and dom(x) ⊆ V TA ξ for some ξ < α}; VTA = {x : x ∈ VTAα for some α}. The class VTA is called the twist-valued model over the complete Boolean algebra A. Definition 7.2. Expand the language L by adding a constant ū to each element u of VTA, obtaining a language denoted by L(TA). The fragments of L and L(TA) without the connective ¬ will be denoted by Lp and Lp(TA), respectively. They will be called the pure ZF-languages. Observe that L(TA) and Lp(TA) are proper classes. Finally, a formula φ in Lp is called restricted if every occurrence of a quantifier in φ is of the form ∀x(x ∈ y → . . .) or ∃x(x ∈ y∧ . . .), or if it is proved to be equivalent in ZFC to a formula of this kind. 12 Notation 7.3. By simplicity, and as it is done with Boolean-valued models, we will identify the element u of VTA with its name ū in L(TA), simply writting u. Moreover, if φ is a formula in which x is the unique variable (possibly) occurring free, we will write φ(u) instead of φ[x/u] or φ[x/ū]. Remark 7.4 (Induction principles). Recall that, from the regularity axiom of ZF, the sets Vα = {x : x ⊆ Vξ for some ξ < α} are definable for every ordinal α. Moreover, in ZF every set x belongs to some Vα. This induces a function rank(x) =def least α such that x ∈ Vα. Since rank(x) < rank(y) is well-founded, it induces a principle of induction on rank: Let Ψ be a property over sets. Assume, for every set x, the following: if Ψ(y) holds for every y such that rank(y) < rank(x) then Ψ(x) holds. Hence, Ψ(x) holds for every x. From this, the following Induction Principle (IP) holds in VTA (similar to the one for Boolean-valued models): Let Ψ be a property over individuals in VTA. Assume, for every x ∈ VTA, the following: if Ψ(y) holds for every y ∈ dom(x) then Ψ(x) holds. Hence, Ψ(x) holds for every x ∈ VTA. Both induction principles are fundamental tools in order to prove properties in VTA. Definition 7.5. Define by induction on the complexity in L(TA) a mapping [[*]] VTA (or simply [[*]]) assigning to each closed formula in L(TA) a value in TA as follows: [[u ǫ v]] = ∨ x∈dom(v) (v(x) ∧ [[x ≈ u]]) = ( ∨ x∈dom(v) ((v(x))1 ∧ [[x ≈ u]]1), ∧ x∈dom(v) ((v(x))2 ∨ [[x ≈ u]]2) ) [[u ≈ v]] = ∧ x∈dom(u) (u(x) → [[x ǫ v]]) ∧ ∧ x∈dom(v) (v(x) → [[x ǫ u]]) = ( ∧ x∈dom(u) ((u(x))1 → [[x ǫ v]]1), ∨ x∈dom(u) ((u(x))1 ∧ [[x ǫ v]]2) ) ∧ ( ∧ x∈dom(v) ((v(x))1 → [[x ǫ u]]1), ∨ x∈dom(v) ((v(x))1 ∧ [[x ǫ u]]2) ) [[φ#ψ]] = [[φ]]#[[ψ]] for # ∈ {∧,∨,→} [[#ψ]] = #[[ψ]] for # ∈ {∼,¬} [[∀xφ(x)]] = ∧ u∈VTA [[φ(u)]] = ( ∧ u∈VTA [[φ(u)]]1, ∨ u∈VTA [[φ(u)]]2 ) [[∃xφ(x)]] = ∨ u∈VTA [[φ(u)]] = ( ∨ u∈VTA [[φ(u)]]1, ∧ u∈VTA [[φ(u)]]2 ) . [[φ]]V TA is called the twist truth-value of the sentence φ ∈ L(TA) in the twist-valued model VTA over the complete Boolean algebra A. 13 Remark 7.6. Observe that VTA can be seen as a structure for QLPT0 over MT A and ΘZF in a wide sense, given that its domain is a proper class. Under this identification, the twist truth-value [[φ]]V TA of the sentence φ in VTA is exactly the value assigned to φ by the interpretation map for QLPT0 over VTA and MT A (recall Definition 6.9). In this case we assume that the mappings (* ǫ *)V TA and (* ≈ *)V TA are as in Definition 7.5. Recall the notion of semantical consequence relation in QLPT0 (see Definitions 6.12 and 6.13). This motivates the following: Definition 7.7. A sentence φ in L(TA) is said to be valid in V TA, which is denoted by VTA |= φ, if [[φ]]V TA ∈ DA. The semantical notions introduced above can easily be generalized to formulas with free variables. Recall from Notation 7.3 that u is identified with u in VTA. Then: Definition 7.8. Let φ be a formula in L whose free variables occur in {x1, . . . , xn}. Given a twist-valued model VTA and an assignment μ : V ar → VTA, the twist truth-value of φ in VTA and μ is defined as follows: [[φ]]V TA μ =def [[φ[x1/μ(x1), . . . , xn/μ(xn)]]] VTA . The formula φ is valid in VTA if [[φ]]V TA μ ∈ DA for every μ. Definition 7.9. ZFLPT0 is the logic of the class of twist-valued models, seen as QLPT0structures over the signature ΘZF. That is, ZFLPT0 is the set of formulas of L which are valid in every twist-valued model VTA. 8 Boolean-valued models versus twist-valued models In this section, the relationship between twist-valued models and Boolean-valued models will be briefly analized. It will be shown that these models enjoy similar properties than the Boolean-valued models (when restricted to pure ZF-languages). These similarities will be fundamental in order to prove that ZFC is valid w.r.t. twist-valued models (see Theorem 8.21 below). The following basic results for twist-valued models are analogous to the corresponding ones for Boolean-valued models obtained in [1, Theorem 1.17]. All these results will be proven by using the Induction Principle (IP) (recall Remark 7.4). From now on we assume that the reader is familiar with the book [1]. Lemma 8.1. Let A be a complete Boolean algebra, and let u ∈ VTA. Then [[u ∈ u]]1 = 0. Proof. Assume the inductive hypothesis [[y ∈ y]]1 = 0 for every y ∈ dom(u). Note that [[u ǫ u]]1 = ∨ y∈dom(u) ((u(y))1 ∧ [[y ≈ u]]1). Let y ∈ dom(u). Then (u(y))1 ∧ [[y ≈ u]]1 ≤ (u(y))1 ∧ ∧ x∈dom(u) ((u(x))1 → [[x ǫ y]]1) ≤ (u(y))1 ∧ ((u(y))1 → [[y ǫ y]]1) ≤ [[y ǫ y]]1 = 0. Then u(y)1 ∧ [[y ≈ u]]1 = 0 for every y ∈ dom(u), hence [[u ∈ u]]1 = 0. 14 Theorem 8.2. Let A be a complete Boolean algebra, and let u, v, w ∈ VTA. Then: (i) [[u ≈ u]]1 = 1. (ii) u(x)1 ≤ [[x ǫ u]]1, for every x ∈ dom(u). (iii) [[u ≈ v]]1 = [[v ≈ u]]1. (iv) [[u ≈ v]]1 ∧ [[v ≈ w]]1 ≤ [[u ≈ w]]1. (v) [[u ≈ v]]1 ∧ [[u ǫw]]1 ≤ [[v ǫw]]1. (vi) [[v ≈ w]]1 ∧ [[u ǫ v]]1 ≤ [[u ǫw]]1. (vii) [[u ≈ v]]1 ∧ [[φ(u)]]1 ≤ [[φ(v)]]1 for every formula φ(x) in Lp(TA). Proof. The proof of items (i)-(vi) is analogous to the proof of the corresponding items found in [1, Theorem 1.17]. The proof of item (vii) is easily done by induction on the complexity of φ(x) by observing that: the proof when φ is atomic uses Lemma 8.1, for φ = (x ǫ x), and items (i)-(vi) for the other cases. For complex formulas the result follows easily by induction hypothesis. Lemma 8.3. Let A be a complete Boolean algebra. Then, for every formula φ(x) in Lp(TA) and every u ∈ V TA: [[∃y((u ≈ y) ∧ φ(y))]]1 = [[φ(u)]]1. Proof. It follows from Theorem 8.2 items (i), (iii) and (viii). Indeed, [[∃y((u ≈ y) ∧ φ(y))]]1 = ∨ x∈dom(u) ([[u ≈ y]]1 ∧ [[φ(y)]]1) ≤ [[φ(u)]]1 = [[u ≈ u]]1 ∧ [[φ(u)]]1 ≤ [[∃y((u ≈ y) ∧ φ(y))]]1. Notation 8.4. The following notation from [1] will be adopted from now on: ∃x ǫ u φ(x) =def ∃x(x ǫ u ∧ φ(x)); ∀x ǫ u φ(x) =def ∀x(x ǫ u → φ(x)). Theorem 8.5. Let A be a complete Boolean algebra. Then, for every formula φ(x) in Lp(TA) and every u ∈ V TA: [[∃x ǫ u φ(x)]]1 = ∨ x∈dom(u) ((u(x))1 ∧ [[φ(x)]]1) and [[∀x ǫ u φ(x)]]1 = ∧ x∈dom(u) ((u(x))1 → [[φ(x)]]1). Proof. The proof is similar to that for [1, Corollary 1.18], taking into account Theorem 8.2 and Lemma 8.3 Recall that a complete Boolean algebra A' is a complete subalgebra of the complete Boolean algebra A provided that A' is a subalgebra of A and ∨ A′ X = ∨ A X and ∧ A′ X =∧ A X for every X ⊆ |A′|. Analogously, we say that a twist-structure TA′ is a complete subalgebra of the twist-structure TA if TA′ is a subalgebra of TA and ∨ TA′ X = ∨ TA X and∧ TA′ X = ∧ TA X for every X ⊆ |TA′ |, recalling Remark 6.10. 15 Proposition 8.6. If A' is a complete subalgebra of A then TA′ is a complete subalgebra of TA. Proof. If follows from Definition 4.6 and Remark 6.10. Theorem 8.7. Let A' be a complete subalgebra of the complete Boolean algebra A. Then: (i) VTA′ ⊆ VTA. (ii) for every u, v ∈ VTA′ : [[u ǫw]]V T A′ = [[u ǫw]]V TA , and [[u ≈ w]]V T A′ = [[u ≈ w]]V TA . Corollary 8.8. Suppose that A' is a complete subalgebra of A. Then, for any restricted formula φ(x1, . . . , xn) in Lp (recall Definition 7.2) and for every u1, . . . , un ∈ TA′: [[φ(u1, . . . , un)]] V T A′ = [[φ(u1, . . . , un)]] VTA . Proof. The proof is analogous to that for [1, Corollary 1.21]. Remark 8.9. Recall from Remark 4.8 that TA2 , the twist structure for LPT0 defined over the two-element Boolean algebra A2, coincides (up to names) with the three-valued algebra APT0 underlying the matrix MPT0, where 1, 1 2 and 0 are identified with (1, 0), (1, 1) and (0, 1), respectively. Hence, the twist-valued structure VTA2 will be denoted by VAPT0 Since A2 is a complete subalgebra of any complete Boolean algebra A then V APT0 is a complete subalgebra of VTA, for any TA. By Theorem 8.7, [[u ǫ v]] VAPT0 = [[u ǫ v]]V TA and [[u ≈ v]]V APT0 = [[u ≈ v]]V TA for every u, v ∈ VAPT0 and every TA. As happens with the Boolean-valued model VA2, the twist-valued model VAPT0 is, in some sense, isomorphic to the standard universe V, as it will be shown in Theorem 8.13 below. Definition 8.10. Define by transfinite recursion on the well-founded relation y ∈ x the following, for each x ∈ V: x =def {〈ŷ, 1〉 : y ∈ x}. It is clear that x ∈ VAPT0 and so x ∈ VTA for every TA. Hence, if φ(v1, . . . , vn) is a restricted formula in Lp and x1, . . . , xn ∈ V then [[φ(x1, . . . , xn)]] VAPT0 = [[φ(x1, . . . , xn)]] VTA for every TA, by Corollary 8.8. Lemma 8.11. Let φ(v1, . . . , vn) be a formula in Lp, and let x1, . . . , xn ∈ V. Then, [[φ(x1, . . . , xn)]] VAPT0 ∈ {0, 1}. Proof. The result is proven by induction on the complexity of φ. Corollary 8.12. Let φ(v1, . . . , vn) be a restricted formula in Lp, and let x1, . . . , xn ∈ V. Then, [[φ(x1, . . . , xn)]] VTA ∈ {0, 1} for every A. Proof. It follows by Lemma 8.11 and by Corollary 8.8. Theorem 8.13. (i) For every x ∈ V and u ∈ VTA: [[u ǫ x]] = ∨ y∈x [[u ≈ ŷ]]. (ii) For x, y ∈ V: x ∈ y holds in ZFC iff VTA |= (x ǫ ŷ) for every A; x = y holds in ZFC iff VTA |= (x ≈ ŷ) for every A. 16 (iii) The function x 7→ x is one-to-one from V to VAPT0. (iv) For every u ∈ VAPT0 there is a (unique) x ∈ V such that VTA |= (u ≈ x) for all A. (v) For every formula φ(v1, . . . , vn) in Lp and every x1, . . . , xn ∈ V: φ(x1, . . . , xn) holds in ZFC iff V APT0 |= φ(x1, . . . , xn). In addition if φ is restricted (recall Definition 7.2) then, for every x1, . . . , xn ∈ V: φ(x1, . . . , xn) holds in ZFC iff V TA |= φ(x1, . . . , xn), for every A. Proof. It follows by an easy adaptation of the proof of [1, Theorem 1.23]. The only points to be considered are the following: (i) Note that 1 ∧ a = a for every a ∈ |TA|. Then, the adaptation of the proof of this item is immediate. (ii) Both assertions are simultaneously proven by induction on rank(y) (see Remark 7.4), where the induction hypothesis is: for every z with rank(z) < rank(y), x ∈ z iff VTA |= (x ǫ ẑ) for every x and A; x = z iff VTA |= (x ≈ ẑ) for every x and A; and z ∈ x iff VTA |= (ẑ ǫ x) for every x and A. For the first assertion, Corollary 8.12 should be used. For the second assertion, note that 1 → a = a for every a ∈ |A|. Hence ([[x ≈ ẑ]]V TA )1 =∧ y∈x ([[ŷ ǫ ẑ]]V TA )1 ∧ ∧ y∈z ([[ŷ ǫ x]]V TA )1. Use then the first assertion, induction hypothesis and the axiom of extensionality. (iii) It follows from (ii). (iv) By adapting the proof of [1, Theorem 1.23(iv)], at some point of the proof the set v = {y ∈ V : u(x) = 1 and ([[x ≈ ŷ]]V TA )1 = 1, for some x ∈ dom(u)} of V must be considered. (v) In order to adapt the proof of [1, Theorem 1.23(v)] it should be noted that, if ∅ 6= X ⊆ |APT0| is such that ∨ APT0 X = 1, then 1 ∈ X . From this, the inductive step φ = ∃xψ can be treated analogously to the proof of [1, Theorem 1.23(v)]. In addition, the use of the Leibniz rule (see [1, Theorem 1.17(vii)]) at this point of the proof can be adapted here to an application of Theorem 8.2(vii) as follows: 1 = ([[ψ(x, x1, . . . , xn]] VAPT0 )1 ∧ ([[x ≈ ŷ]] VAPT0 )1 ≤ ([[ψ(ŷ, x1, . . . , xn]] VAPT0 )1. Hence ([[ψ(ŷ, x1, . . . , xn]] VAPT0 )1 = 1, and the rest of the proof follows from here. Now it will be shown the Maximum Principle of Boolean-valued models (see [1, Lemma 1.27]) is also valid in twist-valued models. The adaptation to our framework of the proof of this result found in [1] is straightfoward. Definition 8.14. Let A be a complete Boolean algebra. Given sets E = {ai : i ∈ I} ⊆ |A| and F = {ui : i ∈ I} ⊆ V TA, the twist mixture of F with respect to E is the element u = ∑ i∈I ai ⊙ ui of V TA defined as follows:7 dom(u) = ⋃ i∈I dom(ui), and 7It is worth observing that the definition of the second coordinate of u(z) will be irrelevant. 17 u(z) = (∨ i∈I (ai ∧ [[z ǫ ui]]1),∼ ∨ i∈I (ai ∧ [[z ǫ ui]]1) ) , for every z ∈ dom(u). Lemma 8.15 (Mixing Lemma). Let {ai : i ∈ I} ⊆ |A| and {ui : i ∈ I} ⊆ V TA, and let u = ∑ i∈I ai ⊙ ui. Suppose that, for every i, j ∈ I, ai ∧ aj ≤ [[ui ≈ uj]]1. Then ai ≤ [[u ≈ ui]]1 for every i ∈ I. Proof. It can be proved by a straightforward adaptation of the proof of [1, Lemma 1.25], taking into account Theorem 8.2 items (ii), (iii) and (vi). The next fundamental result shows that the set of pure ZF-sentences validated by each twist-valued structure VTA is a Henkin theory: Lemma 8.16 (The Maximum Principle). Let A be a complete Boolean algebra. Then, for every formula φ(x) in Lp(TA), there is u ∈ V TA such that [[∃xφ(x)]]1 = [[φ(u)]]1. In particular, if VTA |= ∃xφ(x) then VTA |= φ(u) for some u ∈ VTA. Proof. The proof is obtained by a straightforward adaptation of the proof of [1, Lemma 1.27]. The collection X = {[[φ(u)]] : u ∈ VTA} is a set, since TA is a set. By the Axiom of Choice, there is an ordinal α and a set {uξ : ξ < α} ⊆ V TA} such that X = {[[φ(uξ)]] : ξ < α}, hence [[∃xφ(x)]]1 = ∨ ξ<α[[φ(uξ)]]1. For each ξ < α let aξ = [[φ(uξ)]]1∧∼ ∨ η<ξ[[φ(uη)]]1, and let u = ∑ ξ<α aξ⊙uξ. By the Mixing Lemma 8.15 and by Theorem 8.2 items (ii) and (vii) it follows that [[∃xφ(x)]]1 = [[φ(u)]]1. Corollary 8.17. Let φ(x) be a formula in Lp(TA) such that V TA |= ∃xφ(x). Then: (i) For any v ∈ VTA there exists u ∈ VTA such that [[φ(u)]]1 = 1 and [[φ(v)]]1 = [[u ≈ v]]1. (ii) Let ψ(x) be a formula in Lp(TA) such that V TA |= φ(u) implies that VTA |= ψ(u), for every u ∈ VTA. Then VTA |= ∀x(φ(x) → ψ(x)). Proof. Is an easy adaptation of the proof of [1, Corollary 1.28], taking into account Lemma 8.16 and Theorem 8.2 items (ii) and (vii). The notion of core for a Boolean-valued set (see [1]) can be easily adapted to twist-valued sets: Definition 8.18. Let u ∈ VTA. A core for u is a set v ⊆ VTA such that: (i) [[x ǫ u]]1 = 1 for every x ∈ v; and (ii) for every y ∈ VTA such that [[y ǫ u]]1 = 1, there is a unique x ∈ v such that [[x ≈ y]]1 = 1. Lemma 8.19. Any u ∈ VTA has a core. Proof. Is an easy adaptation of the proof of [1, Lemma 1.31]. Let ∅ be the empty element of VTA. As happens with Boolean-valued models, if u ∈ VTA is such that VTA |= ∼(u ≈ ∅) then, by the Maximum Principle, any core of u is nonempty. Corollary 8.20. Let u ∈ VTA such that VTA |= ∼(u ≈ ∅), and let v be a core for u. Then, for any x ∈ VTA there exists y ∈ v such that [[x ≈ y]]1 = [[x ǫ u]]1. 18 Proof. Is follows from Corollary 8.17. From the results obtained above, one of the main results of the paper can be established: Theorem 8.21. All the axioms (hence all the theorems) of ZFC, when restricted to pure ZF-languages Lp(TA) (recall Definition 7.2), are valid in V TA, for every A. Proof. It is a relatively easy (but arduous) adaptation of the proof of [1, Theorem 1.33], taking into account the auxiliary results obtained within this section, which are similar to the ones required in [1]. 9 Twist-valued models for (PS3,¬) In this section the three-valued model for set theory introduced by Löwe and Tarafder in [18] will be extended to a class of twist-valued models. As observed in Section 7, the three-valued logic (PS3,¬) (denoted as (PS3, ∗) in [18]) was already considered in [13] under the name MPT. Indeed, this logic has been independenly proposed by different authors at several times, and with different motivations.8 For instance, the same logic was proposed in 1970 by da Costa and D'Ottaviano's as J3. It was reintroduced in 2000 by Carnielli, Marcos and de Amo as LFI1 and by Batens and De Clerq as the propositional fragment of the first-order logic CLuNs, in 2014. As observed by Batens, this logic was firstly proposed by Karl Scütte in 1960 under the name Φv (see [7] for details and specific references). Each of the three-valued algebras above is equivalent, up to language, to the three-valued algebra of Lukasiewicz three-valued logic L3. Hence, these logics are equivalent to L3 with {1, 1 2 } as designated values. Moreover, as it was shown by Blok and Pigozzi in [2], the class of algebraic models of J3 (and so the class of twist structures for LPT0) coincides with the agebraic models of Lukasiewicz's three-valued logic L3. More remarks about these three-valued equivalent logics can be found in [7], Chapters 4 and 7. As shown in [13, p. 407], the implication ⇒ given by ⇒ 1 1 2 0 1 1 1 0 1 2 1 1 0 0 1 1 1 (which is the same implication ⇒ of PS3 and the primitive implication of MPT) can be defined in the language of LFI1 (hence in the language of LPT0) as follows: φ⇒ ψ =def ¬∼(φ→ ψ). From this, it is easy to adapt Definition 4.6 of twist-structures for LPT0 to (PS3,¬) (see Definition 9.1 below). Hence, the logic (PS3,¬) will be considered as defined over the signature Σ⇒ = {∧,∨,⇒,¬}. As observed in [13, pp. 395 and 407], the strong negation ∼ can be defined as ∼φ =def φ⇒ ¬(φ⇒ φ), while φ→ ψ =def ∼φ ∨ ψ. Definition 9.1. Let A be a complete Boolean algebra, and let TA as in Definition 4.5. The twist structure for (PS3,¬) over A is the algebra TA∗ = 〈TA, ∧, ∨, ⇒, ¬〉 over Σ⇒ such that the operations ∧, ∨ and ¬ are defined as in Definition 4.6, and ⇒ is defined as follows, for every (z1, z2), (w1, w2) ∈ TA: 8As mentioned in Section 3, LFI1◦ is another presentation of this logic. 19 (z1, z2) ⇒ (w1, w2) = (z1 → w1, z1 ∧ ∼w1). By considering (as mentioned above) ∼ and → as derived connectives in TA∗ , it is clear that ∼(z1, z2) = (∼z1, z1) and (z1, z2) → (w1, w2) = (z1 → w1, z1∧w2). Hence, the original operations of Definition 4.6 can be recovered in TA∗ . As it will be discussed below, we will adopt a technique different to the one used in [18] in order to show the satisfaction of ZFC in the twist-valued models based on TA∗ . However, it is interesting to observe that a nice property of (PS3,¬) is preserved by any TA∗ . Indeed, in [18] the following notion of reasonable implication algebras was proposed in order to provide suitable lattice-valued for ZF: Definition 9.2. An algebra A = 〈A,∧,∨,⇒, 0, 1〉 is an reasonable implication algebra if the reduct 〈A,∧,∨, 0, 1〉 is a complete lattice with bottom 0 and top 1, and ⇒ is a binary operator satisfying the following, for every z, w, u ∈ A: (P1) z ∧ w ≤ u implies that z ≤ (w ⇒ u); (P2) z ≤ w implies that (u⇒ z) ≤ (u⇒ w); (P3) z ≤ w implies that (w ⇒ u) ≤ (z ⇒ u). Proposition 9.3. For every complete Boolean algebra A, the twist structure TA∗ for (PS3,¬) is a reasonable implication algebra such that 0 = (0, 1) and 1 = (1, 0). 9 Proof. Let (z1, z2), (w1, w2), (u1, u2) ∈ TA. (P1): Assume that (z1, z2) ∧ (w1, w2) ≤ (u1, u2). That is, (z1 ∧ w1, z2 ∨ w2) ≤ (u1, u2). Then z1 ∧ w1 ≤ u1 and z2 ∨ w2 ≥ u2. From z1 ∧ w1 ≤ u1 it follows that z1 ≤ w1 → u1. Besides, since z1∨z2 = 1 then ∼z2 ≤ z1 ≤ w1 → u1. Hence z2 ≥ ∼(w1 → u1) = w1∧∼u1. From this, (z1, z2) ≤ (w1 → u1, w1 ∧ ∼u1) = (w1, w2) ⇒ (u1, u2). (P2): Assume that (z1, z2) ≤ (w1, w2). Then z1 ≤ w1, hence u1 → z1 ≤ u1 → w1 and so u1 ∧∼z1 = ∼(u1 → z1) ≥ ∼(u1 → w1) = u1 ∧∼w1. This means that (u1, u2) ⇒ (z1, z2) ≤ (u1, u2) ⇒ (w1, w2). (P3): It is proved analogously, but now taking into account that z1 ≤ w1 implies that w1 → u1 ≤ z1 → u1. Now, the three-valued model of set theory presented in [18] will be generalized to twistvalued models over any complete Boolean algebra. The structure VTA∗ is defined as the structure VTA given in Definition 7.1. This does not come as a surprise, given that the domain of TA and TA∗ is the same, the set TA. However, V TA and VTA∗ are different as first-order structures, namely, the way in which the formulas are interpreted. The only difference, besides using different implications in the underlying logics, will be in the form in which the predicates ǫ and ≈ are interpreted. Thus, the twist truth-value [[φ]]V T A∗ of a sentence φ in VTA∗ will be defined according to the recursive clauses in Definition 7.5, with the following difference: any occurrence of the operator → must be replaced by the operator ⇒ Note that the clause interpreting ∼φ is now derived from the others, taking into account the observation after Definition 9.1. 9To be rigorous, the ¬-less reduct of TA∗ expanded with 0 and 1 is a reasonable implication algebra. 20 In Theorem 9.4 below it is stated that every twist-valued structure VTA∗ is a model of ZFC. This constitutes a generalization of [18, Corollary 11]. Indeed, instead of taking just a three-valued model (generated by the two-element Boolean algebra), we obtain a class of models, one for each complete Boolean algebra. Moreover, we also prove that these generalized models (including, of course, the original Löwe-Tarafder model) satisfy, in addition, the Axiom of Choice. The proof of validity of ZF given in [18, Corollary 11] is strongly based on the particularities of the three-valued algebra of (PS3,¬). 10 This forces us to adapt, to this setting, the proof for twist-valued models over TA given in the previous sections (which, by its turn, is adapted from the proof for Boolean-valued sets). Such adaptations from TA to TA∗ are immediate, and all the results and definitions proposed in the previous sections work fine for TA∗ . Hence, we obtain the second main result of the paper: Theorem 9.4. All the axioms (hence all the theorems) of ZFC, when restricted to pure ZF-languages Lp(TA), are valid in V TA∗ , for every A. Remark 9.5. Oberve that, in [18, Corollary 11], it was proved that PS3 is a model of ZF, not of ZFC. Thus, Theorem 9.4 improves the above mentioned result in two ways: it is generalized to arbitary Boolean algebras and, in addition, it proves that the Axiom of Choice AC is also satisfied by all that models, including the original three-valued structure PS3. 10 ZFLPT0 as a paraconsistent set theory After proving that the two classes of twist-valued models proposed here are models of ZFC, in this section the paraconsistent character of both classes of models will be investigated. It will be shown that twist-valued models over TA (that is, over the logic LPT0) are "more paraconsistent" that the ones over TA∗ (that is, defined over (PS3,¬)). Recall from Theorem 8.2(i) that [[u ≈ u]] ∈ DA for every u in every twist-valued model VTA. The interesting fact of ZFLPT0 is that it allows "inconsistent" sets, that is, elements of VTA such that the value of (u 6≈ u) is also designated. Observe that 1 = (1, 0), 1 2 = (1, 1) and 0 = (0, 1) are defined in every TA. Since z ∈ DA iff z = (1, a) for some a ∈ A it follows that 1 2 ≤ z for every z ∈ DA (recalling the partial order for TA considered in Remark 6.10). Proposition 10.1. There exists u ∈ VTA such that [[u ≈ u]] = 1 2 . Proof. Let w be any element of VTA, and let u = {〈w, 1 2 〉}. Since [[w ≈ w]] ∈ DA then [[w ǫ u]] = u(w) ∧ [[w ≈ w]] = 1 2 ∧ [[w ≈ w]] = 1 2 . From this, [[u ≈ u]] = u(w) → [[w ǫ u]] = 1 2 → 1 2 = 1 2 . From the last result it can be proven that ZFLPT0 is strongly paraconsistent, in the sense that there is a contradiction which is valid in the logic: Corollary 10.2. Let σ = ∀x(x ≈ x). Then VTA |= σ ∧ ¬σ. 10For instance, the fact that expressions like [[u ≈ v]] ⇒ [[u ǫw]] can only take either the value 0 or 1 is used several times in [18]. Observe that, in TA∗ , the value of z ⇒w is always of the form (a,∼a) for some a ∈ |A|. Hence [[u ≈ v]]V T A∗ is always of the form (a,∼a) for some a ∈ |A|. 21 Proof. Let VTA be a twist-valued model for ZFLPT0. As observed above, 1 2 ≤ z for every z ∈ DA. By Theorem 8.2(i), [[v ≈ v]] ∈ DA for every v in V TA and so 1 2 ≤ [[v ≈ v]] for every v, that is, 1 2 ≤ [[∀x(x ≈ x)]], by Definition 7.5. On the other hand, [[∀x(x ≈ x)]] ≤ [[u ≈ u]] = 1 2 for u as in Proposition 10.1. This shows that [[σ]] = [[∀x(x ≈ x)]] = 1 2 and so [[¬σ]] = ¬ [[σ]] = 1 2 . Hence [[σ ∧ ¬σ]] = [[σ]] ∧ [[¬σ]] = 1 2 , a designated value. Since the extensionality axiom of ZF is satisfied by every twist-valued model VTA for ZFLPT0, [[u ≈ v]] ∈ DA iff u and v have the same elements, that is: for every w in VTA, [[w ǫ u]] ∈ DA iff [[w ǫ v]] ∈ DA. However, nothing guarantees that u and v will have the same 'non-elements', namely: it could be possible that [[¬(w ǫ u)]] ∈ DA but [[¬(w ǫ v)]] /∈ DA, for some w in V TA, even when [[u ≈ v]] ∈ DA. Given such w, consider the property φ(x) := ¬(w ǫ x), meaning that "w is a non-element of x". Then, this situation shows that VTA 6|= ((u ≈ v)∧φ(u)) → φ(v), which constitutes a violation of the Leibniz rule for the equality predicate ≈ in ZFLPT0. Theorem 10.3. The formula φ(x) := ¬(w ǫ x) is such that the Leibniz rule fails for it in every VTA, namely: VTA 6|= ∀x∀y((x ≈ y) ∧ φ(x) → φ(y)). Proof. Let VTA be a twist-valued model for ZFLPT0, and let ∅ be the empty element of VTA. Observe that w = {〈∅, 1〉}, u = {〈w, 1 2 〉} and v = {〈w, 1〉} belong to every model VTA. Now, [[∅ ǫ w]] = w(∅) ∧ [[∅ ≈ ∅]] = 1 ∧1 = 1. From this, [[w ≈ w]] = w(∅) → [[∅ ǫ w]] = 1 →1 = 1 and so [[w ǫ u]] = u(w) ∧ [[w ≈ w]] = 1 2 ∧1 = 1 2 . On the other hand, [[w ǫ v]] = v(w) ∧ [[w ≈ w]] = 1 ∧1 = 1. This implies that [[u ≈ v]] = (u(w) → [[w ǫ v]]) ∧ (v(w) → [[w ǫ u]]) = (1 2 →1) ∧ (1 → 1 2 ) = 1 2 . But [[φ(u)]] = [[¬(w ǫ u)]] = ¬ [[w ǫ u]] = ¬ 1 2 = 1 2 and [[φ(v)]] = [[¬(w ǫ v)]] = ¬ [[w ǫ v]] = ¬1 = 0. Thus, [[((u ≈ v) ∧ φ(u)) → φ(v)]] = (1 2 ∧ 1 2 ) →0 = 0, which implies that VTA 6|= ∀x∀y((x ≈ y) ∧ φ(x) → φ(y)). It is important to observe that the failure of the Leiniz rule in VTA shown in Theorem 10.3 does not contradict Theorem 8.2(viii): indeed, what Theorem 8.2(viii) states is the validity of the Leibniz rule in VTA for every formula φ(x) in the pure ZF-language Lp(TA). On the other hand, the formula φ(x) found in Theorem 10.3 which violates the Leibniz rule in VTA contains an occurrence of the paraconsistent negation ¬, that is, it does not belong to Lp(TA). In that example, two sets which are equal have different 'non-elements', where 'non' refers to the paraconsistent negation ¬. Besides the failure of the Leibniz rule for the full language, ZFLPT0 does not validate the so-called bounded quantification properties. Definition 10.4. For any formula φ and every u ∈ VTA, the universal bounded quantification property UBQuφ and the existential bounded quantification property EBQ u φ are defined as follows: (UBQuφ) [[∀x(x ǫ u → φ(x))]]1 = ∧ x∈dom(u)((u(x))1 → φ(x)) (EBQuφ) [[∃x(x ǫ u ∧ φ(x))]]1 = ∨ x∈dom(u)((u(x))1 ∧ [[φ(x)]]1) By simplicity, formulas on the left-hand size of UBQuψ and EBQ u φ will be written as [[∀x ǫ u φ(x)]]1 and [[∃x ǫ u φ(x)]]1, respectively. By adapting the proof of [1, Corollary 1.18] it can be proven the following: 22 Theorem 10.5. For any negation-free formula φ (i.e., φ ∈ Lp(TA)) and every u ∈ V TA, the bounded quantification properties UBQuφ and EBQ u φ hold in V TA. However, for formulas containing the paraconsistent negation the latter result does not holds in general: Proposition 10.6. There is u ∈ VTA and formulas φ(x) and ψ(x) such that the bounded quantification properties UBQuψ and EBQ u φ fail in V TA. Proof. It is enough to prove the falure of EBQuφ given that the failure of UBQ u ψ is obtained from it by using ψ(x) := ∼φ(x) and the duality between infimum and supremum through the Boolean complement ∼. Thus, let VTA and let w = {〈∅, 1〉}, v = {〈w, 1 2 〉}, y = {〈w, 1〉} and u = {〈y, 1〉}. Let φ(x) := ¬(w ǫ x). As in the proof of Theorem 10.3 it can be proven that [[v ≈ y]] = [[φ(v)]] = 1 2 and [[φ(y)]] = 0. Hence ∨ x∈dom(u)((u(x))1 ∧ [[φ(x)]]1) = (u(y))1 ∧ [[φ(y)]]1 = 0 while [[∃x ǫ u φ(x)]]1 = [[∃x(x ǫ u ∧ φ(x))]]1 = ∨ v′∈VTA ∨ x∈dom(u)((u(x))1 ∧ [[v ′ ≈ x]]1 ∧ [[φ(v′)]]1) = ∨ v′∈VTA ((u(y))1 ∧ [[v ′ ≈ y]]1 ∧ [[φ(v ′)]]1) ≥ (u(y))1 ∧ [[v ≈ y]]1 ∧ [[φ(v)]]1 = 1. This means that [[∃x ǫ u φ(x)]]1 = 1 6= 0 = ∨ x∈dom(u)((u(x))1 ∧ [[φ(x)]]1). It is worth noting that the limitations of ZFLPT0 pointed out above (namely, the Leibniz rule and the bounded quantification property for formulas containing the paraconsistent negation) are also present in Löwe-Tarafder's model [18]. As mentioned in Section 3, in [6] was presented a family of paraconsistent set theories based on diverse LFIs, such that the original ZF axioms were slightly modified in order to deal with a unary predicate C(x) representing that 'the set x is consistent'. The consistency connective ◦ is primitive in mbC, but it is definable as ◦φ := ∼(φ ∧ ¬φ) in any axiomatic extension of mbC which proves the schema (ciw): ◦φ ∨ (φ ∧ ¬φ) such as LPT0. In the same way, the consistency predicate C(x) can be expressed, in extensions of ZFmbC, in terms of a formula of ZFmbC without using the predicate C, and the same happens with the inconsistency predicate ¬C(x). For instance, ZFmCi is based on mCi, an extension of mbC in which ¬◦φ is equivalent to φ∧¬φ. Thus, ¬C(x) was defined to be equivalent to (x ≈ x)∧¬(x ≈ x) in ZFmCi. From this, ¬C(x) is equivalent to ¬◦(x ≈ x) in ZFmCi. Given that LPT0 is an extension of mCi, if a consistency predicate for sets were added to the language of ZFLPT0 then it seems reasonable to require the equivalence between ¬C(x) and ¬◦(x ≈ x) in ZFLPT0. But ◦C(x) is derivable ZFmCi, so it would be valid in ZFLPT0 (indeed, the proof in ZFmCi of ◦C(x) given in [6, Proposition 3.10] holds in QLPT0, assuming the axioms for C from ZFmCi). From this C(x) ↔ ◦(x ≈ x) would be also derivable in QLPT0 and so it would be valid in ZFLPT0 expanded with a suitable predicate C denoting 'consistency for sets'. This motivates the following: Definition 10.7. Define in ZFLPT0 the consistency predicate for sets, C(x), as follows: C(x) =def ∼¬(x ≈ x). According to the previous discussion, C(x) should be equivalent to ◦(x ≈ x) in ZFLPT0. But ◦φ is equivalent to ∼(φ∧¬φ) in LPT0, and (x ≈ x) is valid in ZFLPT0, hence C(x) should be equivalent to ∼¬(x ≈ x) in ZFLPT0, which justifies Definition 10.7. Proposition 10.8. The consistency predicate C(x) is non-trivial: there exist v, w ∈ VTA such that [[C(v)]] = 1 and [[C(w)]] = 0. Moreover, [[C(u)]] 6= 1 2 for every u in VTA. 23 Proof. Let VTA be a twist-valued model for ZFLPT0, and consider v = {〈∅, 1〉} and w = {〈∅, 1 2 〉} in VTA. It is easy to see that [[C(v)]] = 1 and [[C(w)]] = 0. On the other hand, for every u in VTA it is the case that [[C(u)]] = ∼z for z = [[¬(u ≈ u)]]. Hence [[C(u)]] = (∼z1, z1) 6= 1 2 , for every u. Finally, we can show now that twist-valued models over TA (that is, over the logic LPT0) are "more paraconsistent" than the ones over TA∗ (that is, defined over (PS3,¬)). Indeed, as we have seen, ZFLPT0 allow us to define in every twist-valued model V TA an "inconsistent set", namely u, such that (u ≈ u)∧¬(u ≈ u) holds. In fact, any u = {〈w, 1 2 〉} is such that [[u ≈ u]] = 1 2 →1 2 = 1 2 . The difference, of course, rests on the nature of the implication operator considered in each case: in (PS3,¬) the value of (u ≈ u) is always 1, since 1 2 ⇒1 2 = 1. Hence, ¬(u ≈ u) always gets the value 0. The same holds in any model over reasonable implicative algebras considered by Löwe and Tarafder (see [18, Proposition 1]). 10.1 Discussion: ZFLPT0 and the failure of the Leibniz rule At first sigth, having a (paraconsistent) set theory as ZFLPT0 in which the Leibniz rule is not satisfied for every formula φ(x) that represents a property could seem to be a bit disappointing. After all, ZF is defined as a first-order theory with equality, which pressuposes the validity of the Leibniz rule. The Leibniz rule states that the equality predicate preserves logical equivalence, namely: (a ≈ b) → (φ(a) ↔ φ(b) for every formula φ(x) (clearly this can be generalized to formulas with n ≥ 1 free variables, assuming ∧n i=1(ai ≈ bi)). In first-order theories based on classical logic, such as ZF, it is enough to require that this property holds for every atomic formula, and so the general case is proven by induction on the complexity of φ. Of course this proof cannot be reproduced in QLPT0, since ¬ is not congruential: φ(a) ↔ φ(b) does not imply ¬φ(a) ↔ ¬φ(b) in general (and this is the key step in the proof by induction). The solution is requiring the validity of the Leibniz rule for every φ from the beginning, adjusting accordingly the class of interpretations for QLPT0 expanded with equality (see [12]). However, the situation for ZFLPT0 is quite different: because of the extensionality axiom, the definition of the interpretation of the equality predicate depends strongly on the interpretation of the membership predicate. In fact, the interpretation of both predicates is simultaneously defined by transfinite recursion, according to Definition 7.5. The validity of the Leibniz rule, in the case of Boolean-set models for ZFC, is proven as a theorem. The simultaneous definition of the equality and membership predicates is designed to fit exactly the requirements of the extensionality axiom: two individuals (sets) are identical provided that they have the same elements. From this, it is proven by induction of the complexity of φ(x) that [[u ≈ v]] ∧ [[φ(u)]] ≤ [[φ(v)]] in every Booleanvalued model. As we have seen in Theorem 8.2(vii), the same holds in twist-valued models w.r.t. the first coordinate, namely: [[u ≈ v]]1 ∧ [[φ(u)]]1 ≤ [[φ(v)]]1. But then, it is required that this property just holds for 'classical' formulas, that is, formulas φ without occurrences of the paraconsistent negation ¬. The explanation for this fact is simple, from the technical point of view: assuming that the property above holds for φ then, when considering ¬φ, the value of [[¬φ(u)]]1 is [[φ(u)]]2, and we don't have enough information about the relationship between [[φ(u)]]2, and [[φ(v)]]2. The example given in the proof of Theorem 10.3 shows that it is impossible to satisfy the Leibniz rule in ZFLPT0 24 for formulas containing the paraconsistent negation, hence this is an unsolvable problem with the current definitions. Within the present approach, paraconsistent situations such as the existence of 'inconsistent' sets u satisfying ¬(u ≈ u) or the existence of a set being simultaneously an element and a non-element of another set seems to be irreconcilable with the fullfillment of the Leibniz rule for formulas behind the 'classical' language. Because of this, the predicate ≈ in ZFLPT0 should be considered as representing 'indiscernibility by pure ZF-properties', exactly as happens with Boolean-valued models for ZF. In this manner (u ≈ v) implies that, besides having the same elements, u and v have, for instance, the same 'non∗-elements', where 'non∗' stands for the classical negation ∼. That is, ∀w(∼(w ǫ u) ↔ ∼(w ǫ v)) is a consequence of (u ≈ v). On the other hand, as it was shown in Theorem 10.3, (u ≈ v) does not imply (in general) that u and v have the same 'nonelements', where 'non' stands for the paraconsistent negation ¬: ∀w(¬(w ǫ u) ↔ ¬(w ǫ v)) is not a consequence of (u ≈ v). Instead of being regarded as discouraging, the fact that (u ≈ v) does not necessarily imply that u and v have the same 'non-elements' (for 'non' the paraconsistent negation ¬) can be seen as an auspicious property, because it can be a way to circumvent undesirable consequences of 'non-elements', as it happens with the well-known Hempel's Ravens Paradox: evidence, differently from proof, for instance, has its own idiosyncratic properties. This point, however, will be left for further discussion. 11 Concluding remarks In this paper, we introduce a generalization of Boolean-valued models of set theory to a class of algebras represented as twist-structures, defining a class of models for ZFC that we called twist-valued models. This class of algebras characterizes a three-valued paraconsistent logic called LPT, which was extensively studied in the literature of paraconsistent logics under different names and signatures as, for example, as the well-known da Costa and D'Ottaviano's logic J3 and as the logic LFI1 (cf. [3]) . As it was shown by Blok and Pigozzi in [2], the class of algebraic models of J3 (hence, the class of twist structures for LPT0) coincides with the agebraic models of Lukasiewicz three-valued logic L3. With small changes, in Section 9 the twist-valued models for LPT0 were adapted in order to obtain twist-valued for (PS3,¬), the three-valued paraconsistent logic studied by Löwe and Tarafder in [18] as a basis for paraconsistent set theory. Thus, their three-valued algebraic model of ZF was extended to a class of twist-valued models of ZF, each of them defined over a complete Boolean algebra. In addition, it was proved that these models (including the three-valued model over (PS3,¬)) satisfy, in addition, the Axiom of Choice. Moreover, it was shown that the implication operator → of LPT0 is, in a sense, more suitable for a paraconsistent set theory than the one ⇒ of PS3: it allows inconsistent sets (i.e., [[(w ≈ w)]] = 1 2 for some w, see Proposition 10.1). It is worth noting that → does not characterize a 'reasonable implication algebra' (recall Definition 9.2): indeed, 1 ∧ 1 2 ≤ 1 2 but 1 6≤ 1 2 → 1 2 = 1 2 . This shows that reasonable implication algebras are just one way to define a paraconsistent set theory, not the best. Despite having the same limitative results than Löwe-Tarafder's model (that is, the debatable failure of Leibniz rule and the bounded quantification property for formulas containing the paraconsistent negation, recall Section 10) we believe that ZFLPT0 has a great potential as a paraconsistent set theory. 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