aa r X i v : . [ m a t h . L O ] J u l Leibniz’s law and its paraconsistent models
Aldo Figallo-Orellano
Centre for Logic, Epistemology and the History of Science (CLE),University of Campinas (Unicamp),BrazilE-mail: aldofi[email protected]
Abstract
This paper aims at discussing the importance of Leibniz Law to getting modelsfor Paraconsistent Set Theories.
Contents
Paraconsistency is the study of logic systems having a negation ¬ which is not explosive;that is, there exist formulas α and β in the language of the logic such that β is not derivablefrom the contradictory set { α, ¬ α } . In other words, the logic has contradictory, but non-trivial theories. There are several approaches to paraconsistency in the literature since theintroduction of Jaskowski’s system of Discursive logic such as Relevant logics, Adaptive1ogics, Many-valued logics, and many others in 1948. The well-known 3-valued logic ofParadox (LP) was introduced by Priest with the aim of formalizing the philosophicalperspective underlying Priest and Sylvan’s Dialetheism. As it is well-known, the mainthesis behind Dialetheism is that there are true contradictions, that is, that some sentencescan be both true and false at the same time and in the same way. The logic LP has beenintensively studied and developed by several authors proposing, particularly, extensionsto first-order languages and applications to Set Theory.The 1963 publication of da Costa’s Habilitation thesis Sistemas Formais Inconsis-tentes constitutes a landmark in the history of paraconsistency. In that thesis, da Costaintroduced the hierarchy C n (for n ≤
1) and C ω of C-systems, [12].Recall that C ω is defined over the signature Σ = {→ , ∧ , ∨ , ¬} and the language L Σ determined by the Hilbert calculus from axiom schemas from Intuitionistic Positive Cal-culus , the rule modus ponens and the following axiom schemata: ( Cω α ∨ ¬ α and ( Cω ¬¬ α → α .In 1969, da Costa visited Universidad Nacional del Sur and suggested finding a se-mantics for C n and C ω to Fidel. In that time, they knew that the negation ¬ was notcongruencial. In fact, as we proved in [14], C w is not algebrizable with Blok-Pigozzi’smethod. Fidel overcame this difficulty by means of a presentation of a novel algebraic-relational class of structures called F-structure by adapting Lindenbaum-Tarski methodin order to prove completeness theorems. The F -structure are pairs h A, { N x } x ∈ A i where A is a generalized Heyting algebra and N x is a set of all possible negation of x ∈ A . Thealgebraic part of the structures captures the algebrizable fragment of the system, that isto say, the negation-free fragment.In our paper [17], we apply Fidel’s method in order to prove an adequacy theorem,in the strong version for C ω and we also present models for first-order C ω (Q C ω ) logic byadapting our work developed in [16].The paper is organized as follow: In the section 2, we do a brief review state of art of theclassical Set Theory and non-classical Set Theories in the setting of the existences of theirmodel. In section 3, first we do a summary of the known Paraconsistent Set Theories andwe discuss the importance of Leibniz Law to obtain models for da Costa’s ParaconsistentSet theory. Later on, we analyze the minimal conditions for a models constructed overa Heyting algebras that we need to prove that several Zermelo-Fraenkel’s set-theoreticaxioms are valid in a suitable algebraic-like models and finally, we present models forParaconsistent Nelon’s Set Theory, all this is part of the section 4 and 5.2 Non-classical Set Theory and their models
In this section we review the non-classical set theories in the literature. First, recall thatBoolean-valued models of set theory were introduced by Scott, Solovay and Vopnka in1965; this theory can be found in Bell (2005), see [4].Next, we will make a synthesis of the construction of Bell’s book for Zermelo-Frankealclassical Set Theory, summarizing the fundamental concepts.We fix a model of set theory V and an Booelan algebra A = ( A, ∧ , ∨ , , , → ) andconstruct a universe of names by transfinite recursion: V ξ A = { x : x a function and ran ( x ) ⊆ A and dom ( x ) ⊆ V A ζ for some ζ < ξ } and V A = { x : x ∈ V ξ A for some ξ } The class V A is called the Boolean-valued model over A . We note that this definitiondoes not depend on the algebraic operations in A , but only on the set A , so any expansionof A to a richer language will give the same class of names V A . By L ∈ , we denote thefirst-order language of set theory using only the propositional connectives ∧ , ∨ , ⊥ , and → . We can now expand this language by adding all of the elements of V A as constants;the expanded (class-sized) language will be called L A . The (meta-)induction principle for V A can be proved by a simple induction on the rank function: for every property Φ ofnames, if for all x ∈ V A , we have ∀ y ∈ dom ( x )(Φ( y ) implies Φ( x )) , then all names x ∈ V A have the property Φ. We can now define a map || · || assigningto each negation formula in L A a truth value in A as follows. Definition 2.1
For a given complete Boolean algebra A . If u, v ∈ V A and ϕ formulas,then the mapping || · || : L A → A is defined for closed formulas: ||⊥|| = 0 , || u ∈ v || = W x ∈ dom ( v ) ( v ( x ) ∧ || x ≈ u || ) || u ≈ v || = V x ∈ dom ( u ) ( u ( x )) → || x ∈ v || ) ∧ V x ∈ dom ( v ) ( v ( x ) → || x ∈ u || ) || ϕ ψ || = || ϕ || ˜ || ψ || , for every ∈ {∧ , ∨ , →} , |∃ xϕ || = W u ∈ V A || ϕ ( u ) || , ||∀ xϕ || = V u ∈ V A || ϕ ( u ) || . || ϕ || is called the truth-value of the sentence ϕ in the language L A in V A Boolean-valued model over A . As usual, we abbreviate ∃ x ( x ∈ u ∧ Ψ( x )) by ∃ x ∈ u Ψ( x ) and ∀ x ( x ∈ u → Ψ( x )) by ∀ x ∈ u Ψ( x ) and call these bounded quantifiers. We say that γ is valid in V A if || γ || = 1and write, V A (cid:15) γ . The basic system of Zermelo-Fraenkel set theory here is called ZF andconsists of first order version QCL of Classical logic ( CL ) over the first-order signatureΘ which contains an equality predicate ≈ and a binary predicate ∈ . The system ZF isthe first order theory with equality obtained from the logic QCL over Θ by adding thefollowing set-theoretic axiom schemas:(Extensionality) ∀ x ∀ y [ ∀ z ( z ∈ x ↔ z ∈ y ) → ( z ≈ y )](Pairing) ∀ x ∀ y ∃ w ∀ z [ z ∈ w ↔ ( z ≈ x ∨ z ≈ y )](Colletion) ∀ x [( ∀ y ∈ x ∃ zφ ( y, z )) → ∃ w ∀ y ∈ x ∃ z ∈ wφ ( y, z )](Powerset) ∀ x ∃ w ∀ z [ z ∈ w ↔ ∀ y ∈ z ( y ∈ x )](Separation) ∀ x ∃ w ∀ z [ z ∈ w ↔ ( z ∈ x ∧ φ ( z ))](Empty set) ∃ x ∀ z [ z ∈ x ↔ ¬ ( z ≈ z )]The set satisfying this axiom is, by extensionality, unique and we refer to it withnotation ∅ .(Union) ∀ x ∃ w ∀ z [ z ∈ w ↔ ∃ y ∈ x ( z ∈ y )](Infinity) ∃ x [ ∅ ∈ x ∧ ∀ y ∈ x ( y + ∈ x )]From union and pairing and extensionality, we can note by y + the unique set y ∪ { y } .(Induction) ∀ x [( ∀ y ∈ xφ ( y )) → φ ( x )] → ∀ xφ ( x ).The original intuition of Boolean-valued models was that the names represent objectsand that the equivalence classes of names under the equivalence relation defined by u ∼ v if and only if || u ≈ v || = 1 can serve as the ontology of the new model. In particular,this means that if two names represent the same object, they should instantiate the sameproperties. This is known as indiscernibility of identicals, one of the directions of Leibniz’sLaw. In our setting, we can represent this by a statement of the type || u ≈ v || ∧ || Ψ( u ) || ≤ || Ψ( v ) || . V A verify this Law. So, we have the following Theorem 2.2
All the axioms, hence all the theorems, of ZF are valid in V A . Now, replacing the Boolean algebra by a Heyting algebra, one obtains a Heyting-valuedmodel. The proofs of the Boolean case transfer to the Heyting-valued, where the logic ofthe Heyting algebra determines the logic of the Heyting-valued model of set theory. Thisidea was further generalized by Takeuti & Titani (1992); Titani (1999); Titani & Kozawa(2003); and Ozawa (2017), replacing the Heyting algebra by appropriate lattices thatallow models of quantum set theory or fuzzy set theory. After this, L¨owe and Tarafderproposed a class of reasonable implication algebra in order to construct algebraic-valuedmodels that validate all axioms of the negation-free fragment of Zermelo-Fraenkel settheory, [27, 26, 25, 24]. From now on, we shall call this Set Theories as non-classical setTheories.It is important to note that there are different kinds of models for the above non-classical Set Theories, see for instance, [19, 18, 6].Zermelo-Fraenkel type set theories with models are the based on Intuitionistic, Fuzzy,Quantum logics and the family of set theories based on intermediate logics between theclassical logics and the mentioned logics. Such non-classical set theories are particularlybased on algebraizable logics and the models can be constructed over algebras that aresemantics of such logics.Other models, such as the ones constructed over sheaf, topoi, possible world Kripke’ssemantics, or topological spaces are possible to be constructed due to the algebrizebilityof the associated logics. While it is possible that these models need very sophisticatedtechnical work, the existence of such models is strongly based on the fact that these settheories already have models of different nature.Zermelo-Fraenkel’s axioms are valid in models of these set theories; that is to say, theset theories are sound from the logics point of view. Even though we have not seen aproof the correctness, we think it is not possible to give it due to a consequence of G¨odelsincompletness theorem.In general, the models of logic systems live in what philosophers call Meta-Mathematicsand it is here where all mathematics mainly live; namely, analysis, algebra, topology andwhichever branch we know. This is exactly where G¨odel’s proof lives.Although we can formalize new set theories, the models are construted over the setof recursive fucntions and the ordinal numbers of Meta-Mathematics. Thus, we thinkthat G¨odels proof is possible to be given to each set theory; hence, these set theories areincomplete from this point of view.Recall that G¨odel’s proof of the famous incompletness theorem proved the indecibilityof certain formulas of Russell and Whiteheads
Principia Mathematica . Specifically, G¨odelproves that there are properties of natural numbers that are true, but they can not5e formally proved in the logical system. Russell and Whitehead’s logical system ofPrincipia Mathematica was elaborated with the intention to make Hilbert’s dream areality. That is to say, to find a logical system where each mathematical theorem (of theMeta-Mathematics) has a logical theorem that reflects it.G¨odels proof shows the impossibility of that dream in the most simple mathematics,i.e. that of the natural numbers where certain recursive functions are possible to bedefined. Now is the right moment to ask ourselves: What did Hilbert look for? A modelfor all mathematics? The logic provides the models to other disciplines such as Philosophy,Computer Science, Economics, Physics, and so on. In this sense, Is it possible to have amodel of all? We do not think so. Mathematical models are not more than simplificationsof the area we want to model; indeed, very useful ones which allow a measure forwardin knowledge, but the reality to be modeled is much more complex than the model canexpress and mathematics is not the exception.It is worth mentioning that in the non-classical Set Theories mentioned above, thenegation formulas are positive formulas; indeed, ¬ x := x → ⊥ . Leibniz’s law is verifiedfor the positive formulas for each of these Set Theories. It is easy to see that all thesesystems can not produce paraconsistency ; i.e., either only a formula, or its negation isvalid.Our reflexions over Meta-Mathematics are part of our research, allowing an undertand-ing and development of our objectives. Initially, we believed that to work in ParaconsistentSet Theory we would have to develop a kind of theory of recursive funtions and transfiniterecursion. As above said, we have at our disposal these tools in Meta-Mathematics. Going back to our topic, type ZF Paraconsistent Set Theories (PSTs) can be constructedin two groups; namely, one where Russell’s paradox is accepted and another where thetheories can be constructed using a paraconsistent logic.Paraconsistent set theories of the first group have been studied by many authors(Brady, 1971; Brady & Routley, 1989; Restall, 1992; Libert, 2005; Weber, 2009, 2010,2013); all of these accounts start from the observation that ZF was created to avoid thecontradiction that can be obtained from the axiom scheme of Comprehension ∃ x ∀ y ( y ∈ x ↔ Ψ( y ))via Russells paradox. Arguing that contradictions are not necessarily devastating ina paraconsistent setting, these authors reinstate the axiom scheme of Comprehension asacceptable, allow the formation of the Russell set R, and conclude that both R ∈ R and R R are true, see [8, 23, 31, 32, 33, 34]. 6ow, on PST of the second group the following papers can be mentioned [9, 10].In these works, the authors try to present models for certain PSTs type ZF (that wedenote, PS3-ZF) in both papers models for positive fragment of PS3-ZF are presented;i.e, formulas without paraconsistent negation. Besides, in these papers the philosophicalconcept called Leibniz Law (LL) is studied, observing that the LL is not verified forformulas with negation for PS3-ZF. Eve more, in [29], the author proved that the axiomSEPARATION of ZF is not valid for formulas with negation. This fact shows that it isnot possible to provide models in this way, but that does not mean that PS3-ZF has nomodels.Moreover, we have worked in other paraconsistent algebrizable logics without success,aiming to prove Leibniz law on a PST. Particularly, Priest’s paraconsistent logics thathe called da Costa logic were studied by us, see [21, 22]. This logic is algebrizable withthe Blok-Pigozzi’s method. We had realized that LL is an essential technical resource toprove the soundness of all axioms of ZF. Later on, we decided to change the strategy usingnon-determinism to finding models, we had in our hands two kinds of non-determinism,one from the Nmatrix of Avron and the other from Fidel’s structures. Avron’s non-determinism is more unstable that of Fidel’s . The former is a non-determinism viamultialgebras. In the paper [11], we see that the formulas of first-order logic have a dis-connect with his corresponding interpretations, producing a technical difficulty to giveproofs. However, the latter, Fidel’s non-determinism strongly uses the algebraic fragmentof the system and formulas with negation have a valoration belonging to a certain al-gebra. These formulas with negation do not have an associated interpretation, but wecan always assign a value of truth to them. Hence, as we can see in the paper [17],Fidel’s non-determinism is more stable. The associated interpretation of negation-freefirst-order formulas works exactly in the same way as the algebraic case. These formulasverify Leibniz law as the intuitionistic case. Constructing F -structures-valued models forParaconsistent Set Theory type ZF based on da Costas Logic C ω (ZF C ω ), we can see thatLeibniz law is verified by formulas with negation that allow proving that all axioms of ZFare valid on Fidel’s models . It is possible to assign a value of truth belonging to a certainHeyting algebra to the formulas with negation. We do not known which is the real value,but we know there exists and it verifies the law.On the other hand, we show that C ω is not algebrizable in the Blok-Pigozzis sence;besides, we present a family of non-algebraic extentions of C ω with Fidel’s models for eachof them. For each extention of C ω , we associate it a Paraconsistent Set Theory and presenta full model for each one. It is worth mentioning that a very important philosophicalconcept as Leibniz law is definitely the only technical obstacle to getting full models;this fact will be part the our future studies. Another interesting aspect of these PSTsdo not permit Russell’s paradox. Besides, the same strategy to use non-determinism canbe applied for PS3-ZF. Actually, the logic PS3 seems to have two different F-strutures;7amely, one based on Boolean algebras and another based on Heyting alegbras. Moreover,we can treat the algebrizable paraconsistent logics introduced by Priest in this way.Now, we shall briefly present our results about ZF C ω for the details the reader canconsult our paper [17]. We fix a model of set theory V and a completed C ω -structure( A, N ). Let us construct a universe of names by transfinite recursion on (
A, N ):We fix a model of set theory V and a completed C ω -structure h A, N i . Let us constructa universe of names by transfinite recursion: V ξ h A,N i = { x : x a function and ran ( x ) ⊆ A and dom ( x ) ⊆ V h A,N i ζ for some ζ < ξ } and V h A,N i = { x : x ∈ V ξ h A,N i for some ξ } The V h A,N i is called the C ω -structure-valued model over h A, N i . Let us observe thatwe only need set A in order to define V h A,N i ξ . By L ∈ , we denote the first-order languageof set theory which consists of the propositional connectives {→ , ∧ , ∨ , ¬} of the C ω andtwo binary predicates ∈ and ≈ . We can expand this language by adding all the elementsof V h A,N i ; the expanded language we will denote L h A,N i . Induction principles:
The sets V ζ = { x : x ⊆ V ξ , for some ξ < ζ } are definablefor every ordinal ξ and then, every set x belongs to V α for some α .So, this fact induce a function rank ( x ) = least ordinal ξ such that x ∈ V ξ . Since rank ( x ) < rank ( y ) is well-founded we induce a principle of induction on rank : let Ψbe a property over sets. Assume, for every set x , if Ψ( y ) holds for every y such that rank ( y ) < rank ( x ), then Ψ( x ) holds. Thus, Ψ( x ) for every x . From the latter, thefollowing (meta-)Induction Principles (IP) holds in V h A,N i : Let us consider a property Ψ over sets. Assume, for every x ∈ V h A,N i , if Ψ( y ) holds forevery y ∈ dom ( x ) , then Ψ( x ) holds. Hence, Ψ( x ) holds for every x ∈ V h A,N i . By simplicity, we note every set u ∈ V h A,N i by its name u of L h A,N i . Besides, we willwrite ϕ ( u ) instead of ϕ ( x/u ). Now, we are going to define a valuation by induction onthe complexity of a closed formula in L h A,N i . Definition 3.1
For a given complete C ω -structure h A, N i , the mapping || · || : L h A,N i →h A, N i is defined as follows: || u ∈ v || = W x ∈ dom ( v ) ( v ( x ) ∧ || x ≈ u || ) || u ≈ v || = V x ∈ dom ( u ) ( u ( x )) → || x ∈ v || ) ∧ V x ∈ dom ( v ) ( v ( x ) → || x ∈ u || )8 | ϕ ψ || = || ϕ || ˜ || ψ || , for every ∈ {∧ , ∨ , →} , ||¬ α || ∈ N || α || and ||¬¬ α || ≤ || α || , ||∃ xϕ || = W u ∈ V h A,N i || ϕ ( u ) || and ||∀ xϕ || = V u ∈ V h A,N i || ϕ ( u ) || . || ϕ || is called the truth-value of the sentence ϕ in the language L h A,N i in the C ω -structure-valued model over h A, N i . Definition 3.2
A sentence ϕ in the language L h A,N i is said to be valid in V h A,N i , whichis denoted by V h A,N i (cid:15) ϕ , if || ϕ || = 1 . It is important to note that for every completed C ω -structure h A, N i , the element V x ∈ A x is the first element of A and so, A is a complete Heyting algebra, we denote by ”0”this element. Besides, for every closed formula φ of L h A,N i we have || φ || ∈ A . Then, thefollowing lemma has the same proof as intuitionistic set theory. Lemma 3.3
For a given completed C ω -structure h A, N i . Then, || u ≈ u || = 1 , u ( x ) ≤|| x ∈ u || for every x ∈ dom ( u ) , and || u = v || = || v = u || , for every u, v ∈ V h A,N i The identity of indiscernibles is an ontological principle that states that there cannot beseparate objects or entities that have all their properties in common. To suppose thattwo things indiscernible is suppose they are the same thing under different names.A form of the principle indiscernibility of identicals is attributed to the Germanphilosopher
Gottfried Wilhelm Leibniz . In the non-classical set theories, we have thatthe manes represent objects and if we have identical objects they would have to have thesame properties. This is known as indiscernibility of identicals and it could be consideredas Leibniz’s law by the following axiom: u ≈ v ∧ ϕ ( u ) → ϕ ( v )In the next, we are going to consider complete C ω -structures which verify the Leibniz’slaw. It is important to note that we have C ω -structures that verify this law, it is enoughto require 1 ∈ N x for all x ∈ A for every x = 1 and 0 ∈ N .We will adopt the following notation, for every formula ϕ ( x ) and every u ∈ V h A,N i : ∃ x ∈ uϕ ( x ) = ∃ x ( x ∈ u ∧ ϕ ( x )) and ∀ x ∈ uϕ ( x ) = ∀ x ( x ∈ u → ϕ ( x )). Thus, we have thefollowing 9 emma 3.4 Let h A, N i be a complete Leibniz C ω -structure, for every formula ϕ ( x ) andevery u ∈ V h A,N i we have ||∃ x ∈ uϕ ( x ) || = _ x ∈ dom ( u ) ( u ( x ) ∧ || ϕ ( x ) || ) , ||∀ x ∈ uϕ ( x ) || = ^ x ∈ dom ( u ) ( u ( x ) → || ϕ ( x ) || ) . The basic system of paraconsistent set theory here is called ZF C ω and consists of firstorder version Q C ω of C ω over the first-order signature Θ ω which contains an equalitypredicate ≈ and a binary predicate ∈ . Definition 3.5
The system ZF C ω is the first order theory with equality obtained from thelogic Q C ω over Θ ω by adding the following set-theoretic axiom schemas: (Extensionality),(Pairing), (Colletion), (Powerset), (Separation), (Empty set), (Union), (Infinity) and(Induction). Theorem 3.6
Let h A, N i be a complete C ω -structure such that V h A,N i satisfies Leibniz’sLaw. Then, the all set-theoretic axioms of ZF C ω are valid in V h A,N i (cid:15) ϕ . Corollary 3.7
The axiom of scheme Comprehension is not valid in V h A,N i . It is enough to see that ||∃ x ∀ y ( y ∈ x ) || = 0, this formula ∃ x ∀ y ( y ∈ x ) is an instanceof Comprehension. In this section we shal analyze the minimal conditions for a models constructed overa Heyting algebras that we need to prove that several Zermelo-Fraenkel’s set-theoreticaxioms are valid in a suitable algebraic-like models.We fix a model of set theory V and a completed reasonable implication algebra A .Let us construct a universe of names by transfinite recursion: V ξA = { x : x a function and ran ( x ) ⊆ A and dom ( x ) ⊆ V Aζ for some ζ < ξ } V A = { x : x ∈ V ξA for some ξ } V A is called the algebraic-valued model over A . Let us observe that we onlyneed the set A in order to define V ξ h A,N i . By L ∈ , we denote the first-order language ofset theory which consists of only the propositional connectives {→ , ∧ , ∨ , ¬} of the C ω andtwo binary predicates ∈ and =. We can expand this language by adding all the elementsof V h A,N i ; the expanded language we will denote L h A,N i . For this construction of modelswe also have Induction principles as the case above.Now, we shall consider a minimal requirement for defining value of truth of formulas inorder to prove some of set-theoretic axiom of Zermelo-Freankel for Set Theory are valid.Now, for a given completed Heyting algebra A , the mapping || · || : L A → A is defined asfollow: || u ∈ v || = W x ∈ dom ( v ) ( v ( x ) ∧ || x ≈ u || ) || u ≈ u || = 1 || u ≈ v || ≤ || φ ( u ) → φ ( v ) || for every formula φ , ||¬ ϕ || = || ϕ || ∗ , || ϕ ψ || = || ϕ || ˜ || ψ || , for every ∈ {∧ , ∨ , →} , ||∃ xϕ || = W u ∈ V A || ϕ ( u ) || and ||∀ xϕ || = V u ∈ V A || ϕ ( u ) || . || ϕ || is called the truth-value of the sentence ϕ in the language L A in thealgebraic-valued model over A . Definition 4.1
A sentence ϕ in the language L A is said to be valid in V A , which isdenoted by V A (cid:15) ϕ , if || ϕ || = 1 . Lemma 4.2
For a given completed reasonable implication algebra A . Then, u, v ∈ V A we have (i) || u = v || = || v = u || , (ii) u ( x ) ≤ || x ∈ u || for every x ∈ dom ( u ) . Proof. (i) Let us consider the formula φ ( z ) := u = z , then || u = v || ≤ || φ ( u ) → φ ( v ) || = || u = u || → || v = u || = 1 → || v = u || = || v = u || . Analogously, we have || v = u || ≤ || u = v || .(ii) || x ∈ u || = W z ∈ dom ( u ) ( u ( z ) ∧ || z = x || ) ≥ u ( x ) ∧ || x = x || = u ( x ). (cid:3)
11e will adopt the following notation, for every formula ϕ ( x ) and every u ∈ V h A,N i : ∃ x ∈ uϕ ( x ) = ∃ x ( x ∈ u ∧ ϕ ( x )) and ∀ x ∈ uϕ ( x ) = ∀ x ( x ∈ u → ϕ ( x )).Now, we recall that for a given Heyting algebra A we have the following propertieshold: (P1) x ∧ y ≤ z implies x ≤ y ⇒ z and (P3) y ≤ z implies z ⇒ x ≤ y ⇒ x for any x, y, z ∈ A .Thus, we have the following Lemma 4.3
Let A be a Heyting algebra algebra, for every formula ϕ ( x ) and every u ∈ V A we have ||∃ x ∈ uϕ ( x ) || = _ x ∈ dom ( u ) ( u ( x ) ∧ || ϕ ( x ) || ) , ||∀ x ∈ uϕ ( x ) || = ^ x ∈ dom ( u ) ( u ( x ) → || ϕ ( x ) || ) . Proof.
Form the definition of || · || we have: ||∃ x ∈ uϕ ( x ) || = ||∃ x ( x ∈ u ∧ ϕ ( x )) || = W v ∈ V A ( || v ∈ u || ∧ || ϕ ( v ) || ) = W v ∈ V A W x ∈ dom ( u ) ( u ( x ) ∧|| x = v || ∧ || ϕ ( v ) || ) = W x ∈ dom ( u ) u ( x ) ∧ W v ∈ V A ( || x = v || ∧ || ϕ ( v ) || ).Now, we have || v = x ∧ ϕ ( v ) || ≤ || ϕ ( x ) || and || x = x ∧ ϕ ( x ) || = || ϕ ( x ) || . Therefore, W z ∈ dom ( u ) u ( x ) ∧ W v ∈ V A ( || z = u || ∧ || ϕ ( v ) || ) = W x ∈ dom ( u ) || u ( x ) ∧ ϕ ( x ) || .On the other hand, ||∀ x ∈ uϕ ( x ) || = ||∀ x ( x ∈ u → ϕ ( x )) || = V v ∈ V A || v ∈ u || → || ϕ ( v ) || Then, we have V x ∈ dom ( u ) [ u ( x ) → || ϕ ( x ) || ] ∧ || v ∈ u || = V x ∈ dom ( u ) [ u ( x ) → || ϕ ( x ) || ] ∧ W x ∈ dom ( u ) ( u ( x ) ∧ || v = x || ) = W x ∈ dom ( u ) ( V x ∈ dom ( u ) [ u ( x ) → || ϕ ( x ) || ] ∧ u ( x ) ∧ || v = x || ) ≤ ( P W x ∈ dom ( u ) || ϕ ( x ) || ∧ || v = x || ≤ || ϕ ( v ) || Form the latter and (P1), we can conclude that V x ∈ dom ( u ) [ u ( x ) → || ϕ ( x ) || ] ≤ || v ∈ u || →|| ϕ ( v ) || .Now using Lemma 2.3 (ii) and (P3) we obtain V v ∈ V A || v ∈ u || → || ϕ ( v ) || ≤ V v ∈ dom ( u ) || v ∈ u || → || ϕ ( v ) || ≤ V v ∈ dom ( u ) || u ( v ) || → || ϕ ( v ) || . (cid:3) efinition 4.4 Let A be a complete Heyting algebra. Given collection of sets { u i : i ∈ I } ⊆ V A and { a i : i ∈ I } ⊆ A , then mixture Σ i ∈ I a i · u i is the fucntion u with dom ( u ) = S i ∈ I dom ( u i ) and u ( x ) = W i ∈ I a i ∧ || x ∈ u i || . The following result is known as
Mixing Lemma and its proof is exactly the same forintuitionistic case because it is an assertion about positive formulas.
Lemma 4.5
Let u be the mixture Σ i ∈ I a i · u i . If a i ∧ a j ≤ || u i = u j || for all i, j ∈ I , then a i ≤ || u i = u || . A set B refines a set A if for all b ∈ B there is some a ∈ A such that b ≤ a . A Heytingalgebra H is refinable if every subset A ⊆ H there exists some anti-chaim B in H thatrefines A and verifies W A = W B . Theorem 4.6
Let A be a complete Heyting algebra such that A is refinable. If V A (cid:15) ∃ xψ ( x ) , then there is u ∈ V A such that V A (cid:15) ψ ( u ) . Now, given a complete Heyting A ′ of A , we have the associated models V A ′ and V A .Then, it is easy to see that V A ′ ⊆ V A .On the other hand, we say that a formula ψ is restricted if all quantifiers are of theform ∃ y ∈ x or ∀ y ∈ x , then we have Lemma 4.7
For any complete Heyting algebra A ′ of A and any restricted negation-freeformula ψ ( x , · · · , x n ) with variables in V A ′ the equality || ψ ( x , · · · , x n ) || A ′ = || ψ ( x , · · · , x n ) || A . Next, we are going to consider the Boolean algebra = ( { , } , ∧ , ∨ , ¬ , ,
1) and thenatural mapping ˆ · : V A → V defined by ˆ u = {h ˆ v, i : v ∈ u } . This is well defined byrecursion on v ∈ dom ( u ). Then, we have the following lemma holds: Lemma 4.8 (i) || u ∈ ˆ v || = W x ∈ v || u = ˆ x || for all v ∈ V and u ∈ V A , (ii) u ∈ v ↔ V A (cid:15) ˆ u ∈ ˆ v and u = v ↔ V A (cid:15) ˆ u = ˆ v , (iii) for all x ∈ V there exists a unique v ∈ V such that V (cid:15) x = ˆ v , (iv) for any formula negation-free formula ψ ( x , · · · , x n ) and any x , · · · , x n ∈ V , wehave ψ ( x , · · · , x n ) ↔ V (cid:15) ψ ( ˆ x , · · · , ˆ x n ) . Moreover for any restricted negation-free formula φ , we have φ ( x , · · · , x n ) ↔ V A (cid:15) φ ( ˆ x , · · · , ˆ x n ) . The proof of the last theorem is the same for intuitionistic case because we considerrestricted negation-free formulas and it will be used to prove the validity of axiom Infinity.13 .1 Validating axioms
New, we are going to prove the validity of several set-theoretical axioms of ZF and let usconsider a fix model V A . Then: Pairing
Let u, v ∈ V A and consider the function w = {h u, i , h v, i} . Thus, we have that || z ∈ w || = ( w ( u ) ∧ || z = u || ) ∨ ( w ( v ) ∧ || z = v || ) = || z = u || ∨ || z = v || = || z = u ∨ z = v || . Union
Given u ∈ V A and consider tha function w with dom ( w ) = S v ∈ dom ( u ) dom ( v ) and w ( x ) = W v ∈ A x v ( x ) where A x = { v ∈ dom ( u ) : x ∈ dom ( v ) } . Then, || y ∈ w || = _ x ∈ dom ( w ) ( || x = y || ∧ _ v ∈ A x v ( x ))= _ x ∈ dom ( w ) _ v ∈ A x ( || x = y || ∧ v ( x ))= _ v ∈ dom ( u ) _ x ∈ dom ( v ) ( || x = y || ∧ v ( x ))= ||∃ v ∈ u ( y ∈ v ) || . Separation
Given u ∈ V A and suppose dom ( w ) = dom ( u ) and w ( x ) = || x ∈ u || ∧ || φ ( x ) || then || z ∈ w || = _ x ∈ dom ( w ) ( || y ∈ w || ∧ || φ ( y ) || ∧ || y = z || ) ≤ _ x ∈ dom ( w ) ( || φ ( z ) || ∧ || y = z || ) . Besides, 14 | φ ( z ) || ∧ || y = z || = _ y ∈ dom ( u ) ( u ( y ) ∧ || z = y || ∧ || φ ( z ) || ) ≤ _ y ∈ dom ( u ) ( || y ∈ u || ∧ || z = y || ∧ || φ ( y ) || )= _ y ∈ dom ( u ) ( w ( y ) ∧ || z = y || ) = || z ∈ w || . Infinity
Assume the formula ψ ( x ) is ∅ ∈ x ∧∀ y ∈ x ( y + ∈ x ). Then, the axiom in question is thesentence ∃ xψ ( x ). Now, it is clear that the negation-free formula ∅ ∈ x ∧ ∀ y ∈ x ( y + ∈ x )is restricted and certainly ψ ( ω ) is true. Hence, by Lemma 4.8 (iv), we get || ψ (ˆ ω ) || = 1,and so, ||∃ xψ ( x ) || = 1. Collection
Given u ∈ V A and x ∈ dom ( u ) there exists by Axiom of Choice some ordinal α x suchthat W y ∈ V A || φ ( x, y ) || = W y ∈ V A αx || φ ( x, y ) || . For α = { α x : x ∈ dom ( u ) } and v the functionwith domain V A and range { } , we have ||∀ x ∈ u ∃ yφ ( x, y ) || = ^ x ∈ dom ( u ) ( u ( x ) → _ y ∈ V h A,N i || φ ( x, y ) || )= ^ x ∈ dom ( u ) ( u ( x ) → _ y ∈ V h A,N i α || φ ( x, y ) || )= ^ x ∈ dom ( u ) ( u ( x ) → ||∃ y ∈ vφ ( x, y ) || )= ||∀ x ∈ u ∃ y ∈ vφ ( x, y ) ||≤ ||∃ w ∀ x ∈ u ∃ y ∈ wφ ( x, y ) || . Paracosinsistent Nelson’s logic, for short PNL, was studied systematically by Odintsov.For more details and information of the issue the reader can consult Odintsov’s book [28].15n the paper [1], Akama considered at the first time the PNL in 1999.In this part of the paper, we shall present F -structures as semantics for first-order ver-sion of paracosinsistent Nelson’s logic. First, assume the propositional signature proposi-tional languages L = {∨ , ∧ , → , ¬ , ⊥} , where ¬ is a symbol for strong negation as well as,the symbol ∀ , universal quantifier, and ∃ , existential quantifier, together with punctua-tion marks, commas and parentheses. Besides, let V ar = { v , v , ... } be a numerable setof individual variables. A first-order signature Θ is also composted by the pair hP , F i ,where P denotes a non-empty set of predicate symbols and F is a set of function symbols.The notions of bound and free variables, closed terms, sentences, and substitutability aredefined as usual. We denote by Fm Θ over the set of all formulas of Θ and by T er theabsolutely free algebra of terms. Sometimes, we say that Fm Θ is the language over Θ.By ϕ ( x/t ) we denote the formula that results from ϕ by replacing simultaneously allthe free occurrences of the variable x by the term t . The connectives of equivalence ↔ and of strong equivalence ⇔ are defined as follows: ψ ↔ φ := ( ψ ↔ φ ) ∧ ( φ ↔ ψ ), ψ ⇔ φ := ( ψ ↔ φ ) ∧ ( ¬ ψ ↔ ¬ φ ). As above, logics will be defined via Hilbert-styledeductive systems with only the rules of substitution and modus ponens. In this way, todefine a logic it is enough to give its axioms. First-order version of paraconsistent Nelsonslogic N4, for short QN4, is a logic in the language L characterized by the following list ofaxioms: Axioms (N1) α → ( β → α ),(N2) ( α → ( β → γ )) → (( α → β ) → ( α → γ )),(N3) ( α ∧ β ) → β ,(N4) ( α ∧ β ) → α ,(N5) α → ( β → ( α ∧ β )),(N6) α → ( α ∨ β ),(N7) β → ( α ∨ β ),(N8) ( α → γ ) → (( β → γ ) → (( α ∨ β ) → γ )),(N9) ∼ ( α → β ) ↔ α ∧ ∼ β ,(N10) ∼ ( α ∧ β ) ↔ ∼ α ∨ ∼ β ,(N11) ∼ ( α ∨ β ) ↔ ∼ α ∧ ∼ β , 16N12) ∼ ( ¬ α ) ↔ α ,(N13) ∼ ( ∼ α ) ↔ α ,(A1) ϕ ( x/t ) → ∃ xϕ , if t is a term free for x in ϕ ,(A2) ∀ xϕ → ϕ ( x/t ), if t is a term free for x in ϕ , Inference rules (MP) α, α → ββ ,(R3) α → β ∃ xα → β , and x does not occur free in β ,(R4) α → βα → ∀ xβ , and x does not occur free in α .It is worth mentioning that in the propositional setting if we take the axioms from(N1) to (N13) with the rule modus ponens we have the propositional logic N4. Besides, ifwe add the axiom (N14) ∼ α → ( α → β ) we have Nelson logics that is known as N3, see[28, Section 8.2]. Now, we introduce a class of F -structures that will serve as semanticsfor QN4. First, recall that Fidel presented for the first time F -structures as semantics forN3 in [15].Now, recall that an algebra A = h A, ∨ , ∧ , → , , i is said to be a Heyting algebra ifthe reduct h A, ∨ , ∧ , , i is a bounded distributive lattice and the condition x ∧ y ≤ z iff x ≤ y → z ( ∗ ) holds. Besides, the algebra A = h A, ∨ , ∧ , → , i is said to be generalizedHeyting algebra if the reduct A = h A, ∨ , ∧i it is a distributive lattice and ∗ is verified. Definition 5.1 A F -structure for N4 is a system h A, { N x } x ∈ A i where A is a generalizedHeyting algebra and { N x } x ∈ A is a family of set of A such that the following conditionshold: (i) for any x ∈ A , N x = ∅ , (ii) for any x, y ∈ A , x ′ ∈ N x and y ′ ∈ N y , the following relations hold x ′ ∨ y ′ ∈ N x ∧ y and x ′ ∧ y ′ ∈ N x ∨ y , x ∈ N x ′ , (iii) for any x, y ∈ A , y ′ ∈ N x , we have x ∧ y ′ ∈ N x → y .
17e are going to use the convention of algebraic logic, we will write sometimes h A, N i instead of h A, { N x } x ∈ A i . Besides, we call the F -structures for N4 by N4-structures. Asexample of N4-structure, we can take a generalized Heyting algebra A and the set N sx = { y ∈ A : x ∨ y = 1 } . The structure h A, { N sx } x ∈ A i will be said to be a saturated N4-structure.The N4-structure h A, { N x } x ∈ A i is said to be a substructure of the N4-structure h B, { N ′ x } x ∈ B i if A is a subalgebra of B and N x ⊆ N ′ x holds for x ∈ A . It is easy to see all N4-structure h A, { N x }i is a substructure of h A, { N sx }i defined before. Definition 5.2 A Θ -structure A is a pair ( h A, { N x } x ∈ A i , S ) where h A, { N x } x ∈ A i is acompleted N4-structure; i.e., A is a completed generalized Heyting algebra. Besides, S = h S, { P S } P ∈P , { f S } f ∈F i is composted by a non-empty domain S , a function P S : S n →h A, { N x } x ∈ A i , for each n -ary predicate symbol P ∈ P , and a function f S : S n → S , foreach n -ary function symbol f ∈ F . We are going to consider the usual notion of derivation of a formula α form Γ in QN4and we denote by Γ ⊢ α . Besides, for a given Θ-structure A = ( h A, { N x } x ∈ A i , S ), we saythat a mapping v : V ar → S is a A -valuation, or simply a valuation. By v [ x → a ] wedenote the A -valuation where v [ x → a ]( x ) = a and v [ x → a ]( y ) = v ( y ) for any y ∈ V ar such that y = x . Definition 5.3
Let A = ( h A, { N x } x ∈ A i , S ) be a Θ -structure and v a A -valuation from V ar into S . We define the truth values of the terms and the formulas in A for a valuation v as follows: || x || A v = v ( x ) , || f ( t , · · · , t n ) || A v = f S ( || t || A v , · · · , || t n || A v ) , for any f ∈ F , || P ( t , ..., t n ) || A v = P S ( || t || A v , ..., || t n || A v ) , for any P ∈ P , || ϕ ψ || A = || ϕ || A || ψ || A , for every ∈ {∧ , ∨ , →} , ||¬ α || A v ∈ N || α || A v and ||¬¬ α || A v = || α || A v , ||¬ ( α ∨ β ) || A v = ||¬ α || A v ∧ ||¬ β || A v and ||¬ ( α ∧ β ) || A v = ||¬ α || A v ∨ ||¬ β || A v , ||¬ ( α → β ) || A v = || α || A v ∧ ||¬ β || A v , ||∀ xα || A v = V a ∈ S || α || A v [ x → a ] , ||∃ xα || A v = W a ∈ S || α || A v [ x → a ] , || α ( x/t ) || A v = || α || A v [ x →|| t || A v ] , if t is a term free for x in ϕ .
18t worth mentioning that the substitution condition || ϕ ( x/t ) || A v = || ϕ || A v [ x →|| t || A v ] canbe proved for first order algebrizable logics. In our setting using F -structures for QN4the negation-free formulas works exactly as the algebrizable case and the sustitutionconditions holds, but for the atomic formulas with negation do not have an interpretationassociated of them. Hence, we need to impose the substitution condition as axiom as itwas done for for da Costa’s non-algebrizable paracosnistent logic C ω in [17].Now, we say that A and v satisfy a formula ϕ , denoted by A (cid:15) ϕ [ v ], if || ϕ || A v = 1.Besides, we say that ϕ is true A if || ϕ || A v = 1 for each a A -valuation v and we denote by A (cid:15) ϕ . We say that ϕ is a semantical consequence of Γ in QN4, if, for any structure A : if A (cid:15) γ for each γ ∈ Γ, then A (cid:15) ϕ . For a given set of formulas Γ, we say that the structure A is a model of Γ iff A (cid:15) γ for each γ ∈ Γ.Recall that a logic defined over a language S is a system L = h F or, ⊢i where F or isthe set of formulas over S and the relation ⊢⊆ P ( F or ) × F or ( P ( A ) is the set of allsubsets of A ). The logic L is said to be a tarskian if it satisfies the following properties,for every set Γ ∪ Ω ∪ { ϕ, β } of formulas:(1) if α ∈ Γ, then Γ ⊢ α ,(2) if Γ ⊢ α and Γ ⊆ Ω, then Ω ⊢ α ,(3) if Ω ⊢ α and Γ ⊢ β for every β ∈ Ω, then Γ ⊢ α .A logic L is said to be finitary if it satisfies the following:(4) if Γ ⊢ α , then there exists a finite subset Γ of Γ such that Γ ⊢ α . Definition 5.4
Let L be a tarskian logic and let Γ ∪ { ϕ } be a set of formulas, we say that Γ is a theory. Besides, Γ is said to be a consistent theory if there is ϕ such that Γ L ϕ .Besides, we say that Γ is a maximal consistent theory if Γ , ψ ⊢ L ϕ for any ψ / ∈ Γ and inthis case, we say Γ non-trivial maximal respect to ϕ . A set of formulas Γ is closed in L if the following property holds for every formula ϕ :Γ ⊢ L ϕ if and only if ϕ ∈ Γ. It is easy to see that any maximal consistent theory is closedone.
Lemma 5.5 (Lindenbaum- Los)
Let L be a tarskian and finitary logic. Let Γ ∪ { ϕ } bea set of formulas such that Γ ϕ . Then, there exists a set of formulas Ω such that Γ ⊆ Ω with Ω maximal non-trivial with respect to ϕ in L . roof. It can be found [35, Theorem 2.22]. (cid:3)
It is clear that QN4 is a finitary and tarskian logic. So, we are in conditions to showthe following adequacy theorem. First, we can observe that for given a formula ϕ andsuppose { x , · · · , x n } is the set of variable of ϕ , the universal closure of ϕ is defined by ∀ x · · · ∀ x n ϕ . Thus, it is clear that if ϕ is a sentence then the universal closure of ϕ isitself. Theorem 5.6
Let Γ ∪ { ϕ } ⊆ Fm Θ . Then, Γ ⊢ ϕ iff Γ (cid:15) ϕ . Proof.
We are going to consider a fixed structure M = h ( A, N ) , S i . Let ϕ be a formulasuch that Γ ⊢ ϕ . Then, there exists α , · · · , α n a derivation of ϕ from Γ. If n = 1 then ϕ is an axiom or ϕ ∈ Γ. If ϕ ∈ Γ, then it is easy to see that Γ (cid:15) ϕ . Besides, to provethe first-order version of each propositional axioms from N4 are valid is a routine, see forinstance [16]. Now, for the sake of brevity we shall denote || ϕ || v instead of || ϕ || M v .(A1) Suppose that ϕ is α ( t/x ) → ∃ xα . Then, || ϕ || v = || α || v [ x →|| t || v ] → ||∃ xα || v . Itis clear that || α || v [ x →|| t || v ] ≤ W a ∈ S || α || v [ x → a ] and then, || α || v [ x →|| t || v ] ≤ ||∃ xα || v . Therefore || α ( t/x ) → ∃ xα || v = 1 and this holds for every valuation v . (A2) is analogous to (A1).Suppose now that || α j || v = 1 for each j < n .If there exists { j, k , · · · , k m } ⊆ { , · · · , j − } such that α k , · · · , α k m is a derivationof α → β . Let us suppose that ϕ is ∃ xα → β , where x is not free in β , and it is obtainedby applying ( ∃ − In ). From induction hypothesis || α → β || v = 1 for every valuation v .Now, consider ||∃ xα → β || v = ||∃ xα || v → || β || v = W a ∈ S || α || v [ x → a ] → || β || v . On the otherhand, since || α → β || v = || α || v → || β || v = 1, then we have that || α || v ≤ || β || v for eachvaluation v . Hence, || α || v [ x → a ] ≤ || β || v [ x → a ] = || β || v for every a ∈ S because x is free in β .So, W a ∈ S || α || v [ x → a ] → || β || v = ||∃ xα → β || v = || ϕ || v = 1. The rest of the proof is left to thereader.Conversely, let us suppose Γ (cid:15) ϕ and Γ ϕ . Then, from the definition of (cid:15) , (A2) and( ∀ − In ), we have ∀ Γ (cid:15) ∀ ϕ and ∀ Γ
6⊢ ∀ ϕ ( ∗ ). From the latter and Lindenbaum- Los lemma,there exists Ω maximal consistent theory such that ∀ Γ ⊆ Ω and Ω ϕ . Let’s consider thequotient algebra A := Fm Θ / Ω where [ α ] Γ = { β ∈ Fm Θ : Ω ⊢ α → β, Ω ⊢ β → α } is theclass of α by Ω. So, it is not hard to see 1 = [ β ] Γ = Γ for every β ∈ Γ (i.e. Γ ⊢ β ). It isclear that A is a generalized Heyting algebra, and the a canonical projection q : Fm → A such that q ( α ) = [ α ] Ω is a homomorphism such that q − ( { } ) = Ω. Let us consider theΘ-structure A = ( h A, { N x } x ∈ A i , T er ) and let v : V ar → T er be the identity function.So, we can consider || . || v : Fm → h A, { N x } x ∈ A i defined by || α || v = [ α ] Γ . Now, we haveto prove ||∀ xα || v = V a ∈ T er || α || v [ x → a ] and ||∃ xα || v = W a ∈ T er || α || v [ x → a ] . Indeed, for any term20 we denote ˆ t the new constant. Now, from (A1) we have ⊢ ψ ( x/ ˆ t ) → ∃ xψ for everyterm t free for x in ψ . So, Γ ⊢ ψ ( x/ ˆ t ) → ∃ xψ . Thus, || ψ ( x/ ˆ t ) → ∃ xψ || v = || ψ ( x/ ˆ t ) || v →||∃ xψ || v = || ψ || v [ x →|| ˆ t || v ] → ||∃ xψ || v = || ψ || v [ x → t ] → ||∃ xψ || v = 1 for every t ∈ T er . Now,let us suppose there is sentence φ such that || ψ ( x/ ˆ t ) || v ≤ || φ || v for every term in the somebefore condition; that is to say, || φ || v is a upper bound of the set {|| ψ ( x/ ˆ t ) || v } and x isfree in φ . Thus, || ψ ( x/ ˆ t ) → φ || v = 1. and therefore, Γ ⊢ ψ ( x/ ˆ t ) → φ for every t in thesame condition.In particular for ˆ x , we have || α ( x ) || v ≤ || β || v where v ( x ) = ˆ x . Therefore, Γ ⊢ ψ ( x ) → φ . So, from (R3), we can infer that Γ ⊢ ∃ xψ ( x ) → φ and then, ||∃ xψ || v ≤ || φ || v .Therefore, ||∃ xα || v = W a ∈ T er || α || v [ x → a ] . The rest of proof is completely analogous, but nowby using (A2) and (R4). Therefore, || · || v is a valuation such that [ ψ ] Ω = 1 iff Ω ⊢ ψ . Now,consider the complete lattice A ∗ by MacNeille completion of A , see [3]. Thus, considerthe Θ-structure A ∗ = ( h A ∗ , { N x } x ∈ A ∗ i , T er ). Now, since ∀ Γ is a set of sentences then || γ || v = || γ || µ for every valuation µ and each γ ∈ ∀ Γ. Therefore, by definition (cid:15) , weobtain that A ∗ (cid:15) γ for each γ ∈ ∀ Γ but A ∗ (cid:15) ∀ ϕ which contradicts the statement ( ∗ ). (cid:3) The basic system of paraconsistent set theory here is called ZF-N4 and consists of firstorder version QN4 of N4 over the first-order signature Θ ω which contains an equalitypredicate ≈ and a binary predicate ∈ . The system ZF-N4 is the first order theory withequality obtained from the logic QN4 over Θ ω by adding the following set-theoretic axiomschemas: (Extensionality), (Pairing), (Colletion), (Powerset), (Separation), (Empty set),(Union), (Infinity) and (Induction), see Section 2.Now, we construct the class V h A,N i of N4-structure-valued model over h A, N i followingSection 3. By L ∈ , we denote the first-order language of set theory which consists of thepropositional connectives {→ , ∧ , ∨ , ¬} of the N4 and two binary predicates ∈ and ≈ . Wecan expand this language by adding all the elements of V h A,N i ; the expanded languagewe will denote L h A,N i . Now, we are going to define a valuation by induction on thecomplexity of a closed formula in L h A,N i . Then, for a given complete N4-structure h A, N i ,the mapping || · || : L h A,N i → h A, N i is defined as follows: || u ∈ v || = W x ∈ dom ( v ) ( v ( x ) ∧ || x ≈ u || ) || u ≈ v || = V x ∈ dom ( u ) ( u ( x )) → || x ∈ v || ) ∧ V x ∈ dom ( v ) ( v ( x ) → || x ∈ u || ) || ϕ ψ || = || ϕ || ˜ || ψ || , for every ∈ {∧ , ∨ , →} ,21 |¬ ϕ || A v ∈ N || ϕ || A v and ||¬¬ ϕ || A v = || α || A v , ||¬ ( ϕ ∨ ψ ) || A v = ||¬ ϕ || A v ∧ ||¬ ψ || A v and ||¬ ( ϕ ∧ ψ ) || A v = ||¬ ϕ || A v ∨ ||¬ ψ || A v , ||¬ ( ϕ → ψ ) || A v = || ϕ || A v ∧ ||¬ ψ || A v , ||∃ xϕ || = W u ∈ V h A,N i || ϕ ( u ) || and ||∀ xϕ || = V u ∈ V h A,N i || ϕ ( u ) || . || u ≈ v || ≤ ||¬ φ ( u ) || → ||¬ φ ( v ) || for any formula φ || ϕ || is called the truth-value of the sentence ϕ in the language L h A,N i in the C ω -structure-valued model over h A, N i .Now, we say that a sentence ϕ in the language L h A,N i is said to be valid in V h A,N i ,which is denoted by V h A,N i (cid:15) ϕ , if || ϕ || = 1.For every completed N4-structure h A, N i , the element V x ∈ A x is the first element of A and so, A is a complete Heyting algebra, we denote by ”0” this element. Besides, for everyclosed formula φ of L h A,N i we have || φ || ∈ A and so the proof of the following Lemma canbe given ins the exactly same way that was done in Lemmas 4.2 Lemma 5.7
For a given completed N4-structure h A, N i . Then, || u ≈ u || = 1 , u ( x ) ≤|| x ∈ u || for every x ∈ dom ( u ) , and || u = v || = || v = u || , for every u, v ∈ V h A,N i The following fact can be checked by induction on the structure of formulas.
Lemma 5.8
For any complete N4-structure the following Leiniz law: || u ≈ v || ≤ || φ ( u ) → φ ( v ) || for any formula φ . From the Lemmas 4.3 and 5.8, we have proven the following central result:
Lemma 5.9
Let h A, N i be a complete Leibniz N4-structure, for every formula ϕ ( x ) andevery u ∈ V h A,N i we have ||∃ x ∈ uϕ ( x ) || = _ x ∈ dom ( u ) ( u ( x ) ∧ || ϕ ( x ) || ) , ||∀ x ∈ uϕ ( x ) || = ^ x ∈ dom ( u ) ( u ( x ) → || ϕ ( x ) || ) . Taking into account the content of section 4, we have proven the following Theorem.
Theorem 5.10
Let h A, N i be a complete N4-structure. Then, the set-theoretic axioms(Pairing), (Colletion), (Separation), (Empty set), (Union), (Infinity) and (Induction) arevalid in V h A,N i (cid:15) ϕ . Theorem 5.11
Let h A, N i be a complete N4-structure. Then, the set-theoretic axioms(Extensionality), (Powerset) and (Empty set) are valid in V h A,N i (cid:15) ϕ . Proof.
Given x, y ∈ V h A,N i , then ||∀ z ( z ∈ x ↔ z ∈ y ) || = ||∀ z (( z ∈ x → z ∈ y ) ∧ ( z ∈ y → z ∈ x ) || = ^ z ∈ V h A,N i ( || z ∈ x || → || z ∈ y || ) ∧ ^ z ∈ V h A,N i ( || z ∈ y || → || z ∈ x || ) ≤ ^ z ∈ dom ( x ) ( || z ∈ x || → || z ∈ y || ) ∧ ^ z ∈ dom ( y ) ( || z ∈ y || → || z ∈ x || ) ≤ ^ z ∈ dom ( x ) ( x ( z ) → || z ∈ y || ) ∧ ^ z ∈ dom ( y ) ( y ( z ) → || z ∈ x || )= || x = y || Assume u ∈ V h A,N i and suppose w a function such that dom ( w ) = { f : dom ( u ) → A : f function } and w ( x ) = ||∀ y ∈ x ( y ∈ u ) || . Therefore, || v ∈ w || = _ x ∈ dom ( w ) ( ||∀ y ∈ x ( y ∈ u ) || ∧ || x = v || ) ≤ ||∀ y ∈ v ( y ∈ u ) || . . Thus, axiom Extensionality is valid.On the other hand, given v ∈ V h A,N i and consider the function a such that dom ( a ) = dom ( u ) and a ( z ) = || z ∈ u || ∧ || z ∈ v || . So, it is clear that a ( z ) → || z ∈ v || = 1 for every z ∈ dom ( a ), therefore 23 |∀ y ∈ v ( y ∈ u ) || = ^ y ∈ dom ( v ) ( v ( y ) → || y ∈ u || )= ^ y ∈ dom ( v ) ( v ( y ) → ( || y ∈ u || ∧ v ( y ))) ≤ ^ y ∈ dom ( v ) ( v ( y ) → a ( y )) ≤ ^ y ∈ dom ( v ) ( v ( y ) → || y ∈ a || ) ∧ ^ z ∈ dom ( a ) ( a ( z ) → || z ∈ v || )= || v = a || Since a ( y ) ≤ || y ∈ u || for every y ∈ dom ( a ) then we have ||∀ y ∈ a ( y ∈ u ) || = 1. Now byconstruction we have that a ∈ dom ( w ) and so, ||∀ y ∈ v ( y ∈ u ) || ≤ ||∀ y ∈ a ( y ∈ u ) || ∧|| v = a || = w ( a ) ∧ || v = a || ≤ || v ∈ w || . Therefore, the axiom (Powerset) holds.Now, we show that (Empty set) is valid. Indeed, first let us note that || u = u || = 1for all u ∈ V A and then, ||¬ ( u = u ) || ∈ N . Therefore, let us consider a function w ∈ V A such that u ∈ dom ( w ) and ran ( w ) ⊆ {||¬ ( u = u ) ||} , then it is clear that || u ∈ w || = W x ∈ dom ( w ) ( w ( x ) ∧ || u = x || ) = ||¬ ( u = u ) || which completes the proof. (cid:3) It is worth mentioning that for proving the (Extensionality) and (Powerset) axiomswe only need the definition of valuation for atomic formulas formed with the predicates ∈ and ≈ . For non-classical Set Theories this expression of the valuations permits to provethe Leibniz law, but if one treat with a different negation; that is to say, a negation thatis not a positive formula, this law is not valid, then it is almost impossible to have moredifferent algebraic models for the law. What show that the non-determism is inherentfor Paraconsistent Set Theories. On the other hand, is it interesting or practical to havea logical system that does not verify the law? We do not think so. What means tohave identical object that they have no the same properties? The answer is in the Meta-Matematics, where the models to live, and it is there where the indentical object havethe same properties. This show us that to understanding what the logical systems canexpress we need to have ”right” models. Acknowledgments
The author acknowledges the support of a post-doctoral grant 2016/21928-0 from S˜aoPaulo Research Foundation (FAPESP), Brazil.24 eferences [1] S. Akama,
Nelson’s Paraconsistent logics , Logic and Logical Philosophy, V. 7, 101–115, 1999.[2] A. Avron,
Non-deterministic Matrices and Modular Semantics of Rules , Logica Uni-versalis, J.-Y. Beziau ed., Birkh¨user Verlag, 149–167, 2005.Dame Journal of Formal Logic, vol. 27 (1986), pp. 52327.[3] A. Balbes and P. Dwinger,
Distributive lattices , Univ. of Missouri Press, Columbia,1974.[4] J. Bell, Set theory,
Boolean valued models and independence proofs , Oxford SciencePubblications, 2005.[5] J. L. Bell. Intuitionistic set theory. College Publications, 2014.[6] J. Bell, Toposes and Local Set Theories : An Introduction. Oxford Logic Guides,Vol. 14. Clarendon Press, New York–Oxford, 1988.[7] R. Brady,
The consistency of the axioms of abstraction and extensionality in a threevalued logic , Notre Dame Journal of Formal Logic, 12, 447–453, 1971.[8] R. Brady and R. Routley,
The non-triviality of extensional dialectical set theory , InPriest, G., Routley, R., and Norman, J., editors. Paraconsistent Logic: Essays on theInconsistent. Analytica. Munich: Philosophia Verlag, pp. 415–436, 1989.[9] Benedikt L¨owe and Sourav Tarafder,
Generalized algebra-valued models of set theory ,Review of Symbolic Logic, 8(1):192205, 2015.[10] W. Carnielli and M. E. Coniglio,
Twist-Valued Models for Three-valued Paraconsis-tent Set Theory ,arXiv:1911.11833, math.LO, 2019.[11] M.E. Coniglio; A. Figallo-Orellano; A. C. Golzio,
First-order swap structures seman-tics for some Logics of Formal Inconsistency , Journal of Logic and Computation,2020.[12] N. da Costa,
On the theory of inconsistent formal systems , Notre Dame Journal ofFormal Logic, vol. 15, 497–510, 1974.[13] F. Esteva, A. Figallo-Orellano, L. Godo and T. Flaminio, Logics Preserving Degreesof Truth from the class of Nelson residated lattice expanded with a consistency op-erator, work in progress. 2514] M. Fidel,
The decidability of the calculi C n . Reports on Mathematical Logic, 8:31–40,1977.[15] M. Fidel, An algebraic study of logic with constructive negation, Proc. of the ThirdBrazilian Conf. on Math. Logic, Recife 1979, 1980, 119–129.[16] A. Figallo-Orellano and J. Slagter, Algebraic Monteiro’s notion of maximal consistenttheory for tarskian logics , Submitted, 2019.[17] A. Figallo-Orellano and J. Slagter,
Fidel-structure-valued models that verify Leibniz’slaw are models of a paraconsistent Set Theory , CLE e-Prints Vol. 19 No. 2 (2020).[18] M. Fitting, Intuitionistic Logic, Model Theory and Forcing. NorthHolland Publ.Comp., Amsterdam, 1969. (Ph. D. Thesis)[19] M. P. Fourman, Sheaf models for set theory. Journal of Pure and Applied Algebra,19:91101, 1980.[20] H. Omori,
Remarks on naive set theory based on LP . The Review of Symbolic Logic,8(2):279–295, 2015.[21] G. Priest,
Dualising intuitionistic negation , Principia, 13, 165–184, 2009.[22] G. Priest,
First-order da Costa Logic , Studia Logica 97(1):183–198, 2011.[23] G. Restall,
A note on nave set theory in LP , Notre Dame Journal of Formal Logic,33(3), 422–432, 1992.[24] G. Takeuti and S. Titani,
Fuzzy logic and fuzzy set theory , Archive for MathematicalLogic, 32(1), 1–32, 1992.[25] S.Titani,
A lattice-valued set theory , Archive for Mathematical Logic, 38(6), 395–421,1999.[26] S. Titani and H. Kozawa,
Quantum set theory , International Journal of TheoreticalPhysics, 42(11), 2575–2602, 2003.[27] M. Ozawa,
Orthomodular-valued models for Quantum Set Theory , The Review ofSymbolic Logic, 10(4), 782–807, 2017.[28] S. Odintsov,
Constructive Negation and Paraconsistency , volume 26 of Trends inLogic. Springer, 2008[29] G. Venturi, a personal communication , 2019.2630] Z. Weber,
Transfinite numbers in paraconsistent set theory . The Review of SymbolicLogic 3(1):71–92, 2010.[31] Z. Weber,
Extensionality and restriction in naive set theory , Studia Logica, 94(1),87–104, 2010.[32] Z. Weber,
Transfinite numbers in paraconsistent set theory , Review of Symbolic Logic,3(1), 71–92, 2010.[33] Weber, Z., Notes on inconsistent set theory, In Tanaka, K., Berto, F., Mares, E., andPaoli, F., editors. Paraconsistency: Logic and Applications, Logic, Epistemology, andthe Unity of Science, Vol. 26. Dordrecht: Springer-Verlag, pp. 315–328, 2013.[34] Z. Weber,
Paradox and Foundation , Ph. D. School of Philosophy, Anthropology andSocial Inquiry, The University of Melbourne, 2009. Advisors: Graham Priest andGreg Restall.[35] R. W´ojcicki,