Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion Cont. 15) S. Tarafder. Ordinals in an Algebra-Valued Model of a Paraconsistent Set Theory . M. Banerjee and S. Krishna, eds., Logic and Its Applications, 6th International Conference, ICLA 2015, Lecture Notes in Computer Science, Vol. 8923, pp. 195–206, 2015. 16) S. Tarafder and M. K. Chakraborty. A Paraconsistent Logic Obtained from an Algebra-Valued Model of Set Theory . J. Y. Beziau, M. K. Chakraborty and S. Dutta, eds., New Directions in Paraconsistent Logic, 5th WCP, Springer Proceedings in Mathematics & Statistics, Vol. 152, pp. 165–183, 2016. 17) W. Carnielli and M. E. Coniglio. Paraconsistent set theory by predicating on consistency . Journal of Logic and Computation, 26(1): pp. 97–116, 2016. 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion Paraconsistent set theory by predicating on consistency Walter Carnielli and Marcelo E. Coniglio have talked about paraconsistent set theories ZFmbC and ZFCil in this paper. 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion Paraconsistent set theory by predicating on consistency Walter Carnielli and Marcelo E. Coniglio have talked about paraconsistent set theories ZFmbC and ZFCil in this paper. “We propose here a new axiomatic paraconsistent set theory based on the first-order version of some (paraconsistent) LFIs, by admitting that sets, as well as sentences, can be either consistent or inconsistent. A salient feature of such a paraconsistent set theory, inherited from LFIs, is that only consistent and contradictory objects will explode into triviality. Moreover, if we declare that all sets and sentences are consistent, we immediately obtain traditional ZF set theory, and nothing new.” 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion cont. Logics of formal inconsistencies (LFI) are paraconsistent logics containing the unary connectives ◦ and • describing the notions of consistency and inconsistency, respectively. The formula ◦ ϕ can be thought of as the formula ϕ is consistent. 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion cont. In a paraconsistent logic we know that a contradictory pair { ϕ, ¬ ϕ } of sentences are not necessarily explosive. But if ◦ ϕ is included in { ϕ, ¬ ϕ } then the collection will be trivial, i.e., { ϕ, ¬ ϕ, ◦ ϕ } ⊢ ψ, for any formula ψ . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion cont. The propositional logic mbC is one of the basic LFIs, having the following axioms with the one inference rule Modus Ponens. ϕ → ( ψ → ϕ ) ( ϕ → ψ ) → (( ϕ → ( ψ → γ )) → ( ϕ → γ ) ϕ → ( ψ → ( ϕ ∧ γ )) ( ϕ ∧ ψ ) → ϕ ( ϕ ∧ ψ ) → ψ ϕ → ( ϕ ∨ ψ ) ψ → ( ϕ ∨ ψ ) ( φ → γ ) → (( ψ → γ ) → ( ϕ ∨ ψ → γ )) ϕ ∨ ( ϕ → ψ ) ϕ ∨ ¬ ϕ ◦ ϕ → ( ϕ → ( ¬ ϕ → ψ )) 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion cont. QmbC is the first order predicate extension of mbC by adding the following axioms and inference rules. Predicate axioms: ϕ [ x / t ] → ∃ x ϕ , if t is free for x in ϕ ∀ x ϕ → ϕ [ x / t ], if t is free for x in ϕ If ϕ is a variant of ψ then ϕ → ψ is an axiom. Inference rules: ϕ → ψ/ϕ → ∀ x ψ , if x is not free in ϕ ϕ → ψ/ ∃ x ϕ → ψ , if x is not free in ψ . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion cont. The weak negation ¬ in the axiom system of QmbC is not classical. Whereas the strong classical negation ∼ can be defined using ¬ and ◦ as follows. For any formula α define ⊥ α = α ∧ ¬ α ∧ ◦ α . Then we define ∼ ϕ = ϕ → ⊥ p , where p is the formula ∀ x ( x = x ). 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion cont. The set theory ZFmbC consists of the first order logic QmbC containing two binary predicates ‘=’ (for equality) and ‘ ∈ ’ (for membership), and a unary predicate C (for consistency of sets), together with the following axioms, organized in five groups: 1) The Leibniz axiom for equality: ( x = y ) → ( ϕ → ϕ x y ). 2) All the axioms of ZF − . 3) The Regularity Axiom: C ( x ) → ( ∃ y ( y ∈ x ) → ∃ y ( y ∈ x ∧ ∼ ∃ z ( z ∈ x ∧ z ∈ y ))). 4) The Unextensionality Axiom: ( x � = y ) ↔ ∃ z (( z ∈ x ) ∧ ( z / ∈ y )) ∨ ∃ z (( z ∈ y ) ∧ ( z / ∈ x )). 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion cont. 5) Axioms for the consistency predicate: ∀ x ( C ( x ) → ◦ ( x = x )) ∀ x ( ¬ ◦ ( x = x ) → ¬ C ( x )) ∀ x ( x ∈ y → C ( x )) → C ( y )) 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion cont. Some of the results in ZFmbC : Seperation axiom can be derived in ZFmbC . ⊢ ∀ x (( x ∈ x ) ∧ ¬ ( x ∈ x ) → ¬ ( x = x )). ⊢ ∀ x ( C ( x ) →∼ ( x ∈ x )). C ( x ) , x = x , ¬ ( x = x ) ⊢ ϕ , for any formula ϕ . C ( x ) , x ∈ x , ¬ ( x ∈ x ) ⊢ ϕ , for any formula ϕ . Theorem. If ZF is consistent then ZFmbC is non-trivial, i.e., every formula cannot be derived from ZFmbC . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion Transfinite numbers in paraconsistent set theory The author of this paper, Zach Weber used a fist order logic TLQ for the base logic to form a paraconsistent set theory. To introduce the set theoretic axioms the author said the following: “Our first-order formal language is now augmented with a variable binding term forming operator { . : −} ; it remains open how to conservatively add term-forming symbols in relevance contexts, and is not a problem addressed here.” 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion cont. The set concept is now characterized by two axioms. Abstraction : x ∈ { z : ϕ ( z ) } ↔ ϕ ( x ). Extensionality : ∀ z ( z ∈ x ↔ z ∈ y ) ↔ ( x = y ). 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion cont. The set concept is now characterized by two axioms. Abstraction : x ∈ { z : ϕ ( z ) } ↔ ϕ ( x ). Extensionality : ∀ z ( z ∈ x ↔ z ∈ y ) ↔ ( x = y ). Theorem. (Cmprehension):For any formula ϕ ( x ) having one free variable x , ⊢ ∃ y ∀ x ( x ∈ y ↔ ϕ ( x )). 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion cont. Some of the other derived results in this set theory. All the axioms of ZF − can be proved. ⊢ ∃ x ( x � = x ). ⊢ ∃ x ( x ∈ a ∧ x / ∈ a ) → a / ∈ a . There exists a universal set. There is a set of all ordinals. 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion Some models of paraconsistent set theory There is a common technique mainly used to build the models of paraconsistent set theory, before the technique of algebra-valued models of some paraconsistent set theory came into existence. Thierry Libert, Olivier Esser, Walter Carnielli, Marcelo Coniglio, etc. have given models of some paraconsistent set theories using the above mentioned technique. 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion The technique of building the model 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion The technique of building the model Consider a set M . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion The technique of building the model Consider a set M . Suppose P P ( M ) := { ( A , B ) : A ∪ B = M } . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion The technique of building the model Consider a set M . Suppose P P ( M ) := { ( A , B ) : A ∪ B = M } . A structure M for a paraconsistent set theory is defined by a non-empty set M together with a function [ . ] M from M into P P ( M ), which associate any a ∈ M simultaneously to its positive extension [ a ] + M and its negative extension [ a ] − M , i.e., [ a ] M = ([ a ] + M , [ a ] − M ) . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion Cont. Note that there may exists a ∈ M such that [ a ] + M ∩ [ a ] − M is non-empty. 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion Cont. Note that there may exists a ∈ M such that [ a ] + M ∩ [ a ] − M is non-empty. By setting a ∈ M b iff a ∈ [ b ] + ∈ M b iff a ∈ [ b ] − M and a / M , for any a , b ∈ M , the structure M can equivalently be defined as M := � M , ∈ M , / ∈ M � where ∈ M ∪ / ∈ M = M x M . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion cont. Accordingly, ‘ x ∈ y ’ can be interpreted as being both ‘true’ and ‘false’ for some x , y in M . To formalize this, we define the truth function ǫ M of the membership relation ‘ ∈ ’ in M as follows: 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion cont. Accordingly, ‘ x ∈ y ’ can be interpreted as being both ‘true’ and ‘false’ for some x , y in M . To formalize this, we define the truth function ǫ M of the membership relation ‘ ∈ ’ in M as follows: t ∈ ǫ M ( a , b ) iff a ∈ M b f ∈ ǫ M ( a , b ) iff a / ∈ M b , for all a , b ∈ M . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion cont. Accordingly, ‘ x ∈ y ’ can be interpreted as being both ‘true’ and ‘false’ for some x , y in M . To formalize this, we define the truth function ǫ M of the membership relation ‘ ∈ ’ in M as follows: t ∈ ǫ M ( a , b ) iff a ∈ M b f ∈ ǫ M ( a , b ) iff a / ∈ M b , for all a , b ∈ M . So, in this way, ǫ M ( a , b ) takes exactly one of the following truth values : 0 := { f } , 1 := { t } , i := { t , f } . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion cont. In this way a structure for a paraconsistent set theory appears as M := � M , ǫ M � . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion cont. In this way a structure for a paraconsistent set theory appears as M := � M , ǫ M � . The truth degree of the atomic formula ‘ x ∈ y ’ in a given structure has been defined. More generally, the truth degree of any formula ϕ interpreted within a given structure M is denoted by | ϕ | M . Incidentally, whenever we write | ϕ | M , it will be assumed that an assignment has been given to the free variables of ϕ into M so that the truth degree of ϕ in M is computable, inductively. 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion cont. We are now in a position to define the satisfaction relation | =. Consider a structure M := � M , ǫ M � and a formula ϕ from the set theoretic language corresponding to M , define M | = ϕ iff t ∈ | ϕ | M . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion Axioms of Zermelo-Fraenkel set theory ∀ x ∀ y [ ∀ z ( z ∈ x ↔ z ∈ y ) → x = y ] (Extensionality) ∀ x ∀ y ∃ z ∀ w ( w ∈ z ↔ ( w = x ∨ w = y )) (Pairing) ∃ x [ ∃ y ( ∀ z ( z / ∈ y ) ∧ y ∈ x ) ∧ ∀ w ∈ x ∃ u ∈ x ( w ∈ u )] (Infinity) ∀ x ∃ y ∀ z ( z ∈ y ↔ ∃ w ∈ x ( z ∈ x )) (Union) ∀ x ∃ y ∀ z ( z ∈ y ↔ ∀ w ∈ z ( w ∈ x )) (Power Set) 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion ∀ p 0 · · · ∀ p n ∀ x ∃ y ∀ z ( z ∈ y ↔ z ∈ x ∧ ϕ ( z , p 0 , . . . , p n )) (Separation ϕ ) ∀ p 0 · · · ∀ p n − 1 ∀ x [ ∀ y ∈ x ∃ z ϕ ( y , z , p 0 , . . . , p n − 1 ) → ∃ w ∀ v ∈ x ∃ u ∈ w ϕ ( v , u , p 0 , . . . , p n − 1 )] (Replacement ϕ ) ∀ p 0 · · · ∀ p n ∀ x [ ∀ y ∈ x ϕ ( y , p 0 , . . . , p n ) → ϕ ( x , p 0 , . . . , p n )] → ∀ z ϕ ( z , p 0 , . . . , p n ) (Foundation ϕ ) 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion Construction of Boolean Valued Model In the following steps it will be discussed briefly that how a Boolean valued model is constructed and in which sense it becomes a model of ZFC . The whole construction will take place over the standard model V of ZFC . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion Construction of Boolean Valued Model In the following steps it will be discussed briefly that how a Boolean valued model is constructed and in which sense it becomes a model of ZFC . The whole construction will take place over the standard model V of ZFC . 1 Let us take a complete Boolean algebra, B = � B , ∧ , ∨ , ⇒ , ∗ , 0 , 1 � . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion Construction of Boolean Valued Model In the following steps it will be discussed briefly that how a Boolean valued model is constructed and in which sense it becomes a model of ZFC . The whole construction will take place over the standard model V of ZFC . 1 Let us take a complete Boolean algebra, B = � B , ∧ , ∨ , ⇒ , ∗ , 0 , 1 � . 2 For any ordinal α we define, = { x : Func ( x ) ∧ ran ( x ) ⊆ B ∧∃ ξ < α ( dom ( x ) ⊆ V ( B ) V ( B ) ) } α ξ 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion Construction of Boolean Valued Model In the following steps it will be discussed briefly that how a Boolean valued model is constructed and in which sense it becomes a model of ZFC . The whole construction will take place over the standard model V of ZFC . 1 Let us take a complete Boolean algebra, B = � B , ∧ , ∨ , ⇒ , ∗ , 0 , 1 � . 2 For any ordinal α we define, = { x : Func ( x ) ∧ ran ( x ) ⊆ B ∧∃ ξ < α ( dom ( x ) ⊆ V ( B ) V ( B ) ) } α ξ 3 Using the above we get a Boolean valued model as, V ( B ) = { x : ∃ α ( x ∈ V ( B ) α ) } 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion 4 Extend the language of classical ZFC by adding a name corresponding to each element of V ( B ) , in it. 5 Associate every formula of the extended language with a value of B by the map � . � . First we give the algebraic expressions which associate the two basic well-formed formulas with values of B . For any u , v in V ( B ) , � � u ∈ v � = ( v ( x ) ∧ � x = u � ) x ∈ dom ( v ) � � � u = v � = ( u ( x ) ⇒ � x ∈ v � ) ∧ ( v ( y ) ⇒ � y ∈ u � ) x ∈ dom ( u ) y ∈ dom ( v ) 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion 6 Then for any sentences σ and τ of the new language we define, � σ ∧ τ � = � σ � ∧ � τ � � σ ∨ τ � = � σ � ∨ � τ � � σ → τ � = � σ � ⇒ � τ � � ¬ σ � = � σ � ∗ � � ∀ x ϕ ( x ) � = � ϕ ( x ) � x ∈ V ( B ) � � ∃ x ϕ ( x ) � = � ϕ ( x ) � x ∈ V ( B ) 7 A sentence σ will be called valid in V ( B ) or V ( B ) will be called a model of a sentence σ if � σ � = 1. It will be denoted as V ( B ) | = σ . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion 8 Then we ultimately get the following celebrated result: Theorem For any complete Boolean algebra B , V ( B ) | = ZFC , i.e., all the classical logic axioms and ZFC axioms are valid in V ( B ) . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion Heyting-Valued Model Instead of a complete Boolean algebra if we take a complete Heyting algebra H by the similar construction one can conclude Theorem V ( H ) | = IZF , where IZF stands for the intuitionistic Zermelo-Fraenkel set theory, a set theory based on the intuitionistic logic. 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion Bounded Quantification Property The following property BQ ϕ named after the bounded quantification property for the formula ϕ , played a very important role in proving the above two theorems. � � ∀ x ∈ u ϕ ( x ) � = ( u ( x ) ⇒ � ϕ ( x ) � ) ( BQ ϕ ) x ∈ dom ( u ) where ∀ x ∈ u ϕ ( x ) is the abbreviation for ∀ x ( x ∈ u → ϕ ( x )). Back to the theorem 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Paraconsistent Logic and set theories The Three Valued Matrix PS 3 and its logic Axioms of ZF set theory Ordinals in V ( PS 3) Boolean and Heyting valued models Non-Classical Behaviour of V ( PS 3) Conclusion Bounded Quantification Property The following property BQ ϕ named after the bounded quantification property for the formula ϕ , played a very important role in proving the above two theorems. � � ∀ x ∈ u ϕ ( x ) � = ( u ( x ) ⇒ � ϕ ( x ) � ) ( BQ ϕ ) x ∈ dom ( u ) where ∀ x ∈ u ϕ ( x ) is the abbreviation for ∀ x ( x ∈ u → ϕ ( x )). Back to the theorem It can be proved that for any complete Boolean algebra B and complete Heyting algebra H the property BQ ϕ holds in both V ( B ) and V ( H ) for all formula ϕ . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS 3 and its logic Algebraic Properties Ordinals in V ( PS 3) Negation Free Formulas & Algebra-Valued Models Non-Classical Behaviour of V ( PS 3) Conclusion Reasonable Implication Algebra Definition An algebra A = � A , ∧ , ∨ , 1 , 0 , ⇒� is called a reasonable implication algebra if � A , ∧ , ∨ , 1 , 0 � is a complete distributive lattice and ⇒ has the following properties: 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS 3 and its logic Algebraic Properties Ordinals in V ( PS 3) Negation Free Formulas & Algebra-Valued Models Non-Classical Behaviour of V ( PS 3) Conclusion Reasonable Implication Algebra Definition An algebra A = � A , ∧ , ∨ , 1 , 0 , ⇒� is called a reasonable implication algebra if � A , ∧ , ∨ , 1 , 0 � is a complete distributive lattice and ⇒ has the following properties: P1: x ∧ y ≤ z implies x ≤ y ⇒ z . P2: y ≤ z implies x ⇒ y ≤ x ⇒ z . P3: y ≤ z implies z ⇒ x ≤ y ⇒ x . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS 3 and its logic Algebraic Properties Ordinals in V ( PS 3) Negation Free Formulas & Algebra-Valued Models Non-Classical Behaviour of V ( PS 3) Conclusion Reasonable Implication Algebra Definition An algebra A = � A , ∧ , ∨ , 1 , 0 , ⇒� is called a reasonable implication algebra if � A , ∧ , ∨ , 1 , 0 � is a complete distributive lattice and ⇒ has the following properties: P1: x ∧ y ≤ z implies x ≤ y ⇒ z . P2: y ≤ z implies x ⇒ y ≤ x ⇒ z . P3: y ≤ z implies z ⇒ x ≤ y ⇒ x . A reasonable implication algebra is said to be deductive if it satisfies ( x ∧ y ) ⇒ z = x ⇒ ( y ⇒ z ) . return 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS 3 and its logic Algebraic Properties Ordinals in V ( PS 3) Negation Free Formulas & Algebra-Valued Models Non-Classical Behaviour of V ( PS 3) Conclusion Negation Free Formulas, NFF If L is any first-order language including the connectives ∧ , ∨ , ⊥ and → and Λ any class of L -formulas, we denote closure of Λ under ∧ , ∨ , ⊥ , ∃ , ∀ , and → by Cl (Λ) and call it the negation-free closure of Λ. A class Λ of formulas is negation-free closed if Cl (Λ) = Λ. By NFF we denote the negation-free closure of the atomic formulas; its elements are called the negation-free formulas . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS 3 and its logic Algebraic Properties Ordinals in V ( PS 3) Negation Free Formulas & Algebra-Valued Models Non-Classical Behaviour of V ( PS 3) Conclusion An Algebra-Valued Model of a Set Theory Following the above mentioned constructions we have proved: Theorem Let A be a deductive reasonable implication algebra such that V ( A ) satisfies the NFF -bounded quantification property ( NFF − BQ ϕ ) . Then Extensionality, Pairing, Infinity, Union, Power Set, NFF -Separation and NFF -Replacement are valid in V ( A ) . NFF -(...) stands for the instances of (...) only for the negation free formulas. (L¨ owe, B., and S. Tarafder, Generalized Algebra-Valued Models of Set Theory, Review of Symbolic Logic , Cambridge University Press, 8(1), pp. 192–205, 2015.) BQ ϕ return 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Matrix of PS 3 The Three Valued Matrix PS 3 and its logic PS 3 -valued model Ordinals in V ( PS 3) The Logic LPS 3 Non-Classical Behaviour of V ( PS 3) Conclusion Is there any algebra other than complete Boolean algebras and complete Heyting algebras which is a deductive reasonable implication algebra satisfying the NFF − BQ ϕ ? 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Matrix of PS 3 The Three Valued Matrix PS 3 and its logic PS 3 -valued model Ordinals in V ( PS 3) The Logic LPS 3 Non-Classical Behaviour of V ( PS 3) Conclusion Three Valued Matrix PS 3 Let us consider the three valued matrix PS 3 = �{ 1 , 1 / 2 , 0 } , ∧ , ∨ , ⇒ , ∗ , 1 , 0 � where the truth tables for the operators are given below: 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Matrix of PS 3 The Three Valued Matrix PS 3 and its logic PS 3 -valued model Ordinals in V ( PS 3) The Logic LPS 3 Non-Classical Behaviour of V ( PS 3) Conclusion Three Valued Matrix PS 3 Let us consider the three valued matrix PS 3 = �{ 1 , 1 / 2 , 0 } , ∧ , ∨ , ⇒ , ∗ , 1 , 0 � where the truth tables for the operators are given below: ∧ 1 1 / 2 0 ∨ 1 1 / 2 0 1 1 1 / 2 0 1 1 1 1 1 / 2 1 / 2 1 / 2 0 1 / 2 1 1 / 2 1 / 2 0 0 0 0 0 1 1 / 2 0 ∗ ⇒ 1 1 / 2 0 1 1 1 0 1 0 1 / 2 1 1 0 1 / 2 1 / 2 0 1 1 1 0 1 and { 1 , 1 / 2 } is taken as the designated set. return 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Matrix of PS 3 The Three Valued Matrix PS 3 and its logic PS 3 -valued model Ordinals in V ( PS 3) The Logic LPS 3 Non-Classical Behaviour of V ( PS 3) Conclusion The Answer It can be verified that PS 3 is a deductive reasonable implication algebra and BQ ϕ holds in V ( PS 3 ) for all negation free formula ϕ . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Matrix of PS 3 The Three Valued Matrix PS 3 and its logic PS 3 -valued model Ordinals in V ( PS 3) The Logic LPS 3 Non-Classical Behaviour of V ( PS 3) Conclusion So we can conclude that Extensionality, Pairing, Infinity, Union, Power Set, NFF -Separation and NFF -Replacement are valid in V ( PS 3 ) . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Matrix of PS 3 The Three Valued Matrix PS 3 and its logic PS 3 -valued model Ordinals in V ( PS 3) The Logic LPS 3 Non-Classical Behaviour of V ( PS 3) Conclusion So we can conclude that Extensionality, Pairing, Infinity, Union, Power Set, NFF -Separation and NFF -Replacement are valid in V ( PS 3 ) . Moreover it is proved separately that NFF -Regularity is also valid in V ( PS 3 ) . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Matrix of PS 3 The Three Valued Matrix PS 3 and its logic PS 3 -valued model Ordinals in V ( PS 3) The Logic LPS 3 Non-Classical Behaviour of V ( PS 3) Conclusion Logic Corresponding to PS 3 What is the logic corresponding to PS 3 ? Is it paraconsistent? 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Matrix of PS 3 The Three Valued Matrix PS 3 and its logic PS 3 -valued model Ordinals in V ( PS 3) The Logic LPS 3 Non-Classical Behaviour of V ( PS 3) Conclusion Logic Corresponding to PS 3 What is the logic corresponding to PS 3 ? Is it paraconsistent? We have found a logic LPS 3 which is sound and complete with respect to PS 3 . More interestingly it can be proved that LPS 3 is a paraconsistent logic. 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Matrix of PS 3 The Three Valued Matrix PS 3 and its logic PS 3 -valued model Ordinals in V ( PS 3) The Logic LPS 3 Non-Classical Behaviour of V ( PS 3) Conclusion Axioms for LPS 3 The following formulas are taken as the axioms for LPS 3 : ( Ax 1) ϕ → ( ψ → ϕ ) ( Ax 2) ( ϕ → ( ψ → γ )) → (( ϕ → ψ ) → ( ϕ → γ )) ( Ax 3) ϕ ∧ ψ → ϕ ( Ax 4) ϕ ∧ ψ → ψ ( Ax 5) ϕ → ϕ ∨ ψ ( Ax 6) ( ϕ → γ ) ∧ ( ψ → γ ) → ( ϕ ∨ ψ → γ ) ( Ax 7) ( ϕ → ψ ) ∧ ( ϕ → γ ) → ( ϕ → ψ ∧ γ ) 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Matrix of PS 3 The Three Valued Matrix PS 3 and its logic PS 3 -valued model Ordinals in V ( PS 3) The Logic LPS 3 Non-Classical Behaviour of V ( PS 3) Conclusion ( Ax 8) ϕ ↔ ¬¬ ϕ ( Ax 9) ¬ ( ϕ ∧ ψ ) ↔ ( ¬ ϕ ∨ ¬ ψ ) ( Ax 10) ( ϕ ∧ ¬ ϕ ) → ( ¬ ( ψ → ϕ ) → γ ) ( Ax 11) ( ϕ → ψ ) → ( ¬ ( ϕ → γ ) → ψ ) ( Ax 12) ( ¬ ϕ → ψ ) → ( ¬ ( γ → ϕ ) → ψ ) ( Ax 13) ⊥ → ϕ ( Ax 14) ( ϕ ∧ ( ψ → ⊥ )) → ¬ ( ϕ → ψ ) ( Ax 15) ( ϕ ∧ ( ¬ ϕ → ⊥ )) ∨ ( ϕ ∧ ¬ ϕ ) ∨ ( ¬ ϕ ∧ ( ϕ → ⊥ )) where ϕ, ψ, γ are any well formed formulas and ⊥ is the abbreviation for ¬ ( θ → θ ) for any arbitrary formula θ . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Matrix of PS 3 The Three Valued Matrix PS 3 and its logic PS 3 -valued model Ordinals in V ( PS 3) The Logic LPS 3 Non-Classical Behaviour of V ( PS 3) Conclusion Rules for LPS 3 The rules for LPS 3 are the following: ϕ, ψ 1 ϕ ∧ ψ ϕ, ϕ → ψ 2 ψ 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Matrix of PS 3 The Three Valued Matrix PS 3 and its logic PS 3 -valued model Ordinals in V ( PS 3) The Logic LPS 3 Non-Classical Behaviour of V ( PS 3) Conclusion Soundness and completeness of LPS 3 Let ⊢ and | = be the syntactic and semantic consequence relations respectively defined in the usual way with respect to the above mentioned axiom system and the matrix PS 3 . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Matrix of PS 3 The Three Valued Matrix PS 3 and its logic PS 3 -valued model Ordinals in V ( PS 3) The Logic LPS 3 Non-Classical Behaviour of V ( PS 3) Conclusion Soundness and completeness of LPS 3 Let ⊢ and | = be the syntactic and semantic consequence relations respectively defined in the usual way with respect to the above mentioned axiom system and the matrix PS 3 . Theorem Soundness: For any formula ϕ and a set of formulas Γ , if Γ ⊢ ϕ then Γ | = ϕ . Theorem Completeness: For any formula ϕ , if | = ϕ then ⊢ ϕ . (Tarafder, S., & M. K. Chakraborty, A Paraconsistent Logic Obtained from an Algebra-Valued Model of Set Theory, to appear in: New Directions in Paraconsistent Logic , Springer, 2015.) 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Matrix of PS 3 The Three Valued Matrix PS 3 and its logic PS 3 -valued model Ordinals in V ( PS 3) The Logic LPS 3 Non-Classical Behaviour of V ( PS 3) Conclusion An Algebra-Valued Model of a Paraconsistent Set Theory Hence we have reached to the fact that V ( PS 3 ) is an algebra-valued model of a paraconsistent set theory. 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Naive Definitions The Three Valued Matrix PS 3 and its logic α -like Elements Ordinals in V ( PS 3) Properties of α -like Elements Non-Classical Behaviour of V ( PS 3) α -like Elements as the Ordinals in V ( PS 3) Conclusion Some Classical Definitions in Metalanguage Definition A set x is said to be transitive if every element of x is a subset of x , or equivalently, if y ∈ z and z ∈ x implies y ∈ x . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Naive Definitions The Three Valued Matrix PS 3 and its logic α -like Elements Ordinals in V ( PS 3) Properties of α -like Elements Non-Classical Behaviour of V ( PS 3) α -like Elements as the Ordinals in V ( PS 3) Conclusion Some Classical Definitions in Metalanguage Definition A set x is said to be transitive if every element of x is a subset of x , or equivalently, if y ∈ z and z ∈ x implies y ∈ x . Definition A set A is said to be well-ordered by a relation R if R is a linear order on A and any non-empty subset of A has a least element with respect to R . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Naive Definitions The Three Valued Matrix PS 3 and its logic α -like Elements Ordinals in V ( PS 3) Properties of α -like Elements Non-Classical Behaviour of V ( PS 3) α -like Elements as the Ordinals in V ( PS 3) Conclusion Some Classical Definitions in Metalanguage Definition A set x is said to be transitive if every element of x is a subset of x , or equivalently, if y ∈ z and z ∈ x implies y ∈ x . Definition A set A is said to be well-ordered by a relation R if R is a linear order on A and any non-empty subset of A has a least element with respect to R . Definition An ordinal number is a transitive set well-ordered by ∈ . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Naive Definitions The Three Valued Matrix PS 3 and its logic α -like Elements Ordinals in V ( PS 3) Properties of α -like Elements Non-Classical Behaviour of V ( PS 3) α -like Elements as the Ordinals in V ( PS 3) Conclusion α -like Elements For each α ∈ ORD the α -like names in V ( PS 3 ) are defined by transfinite recursion as follows. Definition An element x ∈ V ( PS 3 ) is called 1 0 -like if for every y ∈ dom ( x ), we have that x ( y ) = 0; and 2 α -like if for each β ∈ α there exists y ∈ dom ( x ) which is β -like and x ( y ) ∈ { 1 , 1 / 2 } , and for any z ∈ dom ( x ) if it is not β -like for any β ∈ α then x ( z ) = 0. 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Naive Definitions The Three Valued Matrix PS 3 and its logic α -like Elements Ordinals in V ( PS 3) Properties of α -like Elements Non-Classical Behaviour of V ( PS 3) α -like Elements as the Ordinals in V ( PS 3) Conclusion α -like Elements Meets Our Expectations Theorem Let x ∈ V ( PS 3 ) be α -like for some α ∈ ORD . For any y ∈ V ( PS 3 ) , � x = y � = 1 if and only if y is α -like. 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Naive Definitions The Three Valued Matrix PS 3 and its logic α -like Elements Ordinals in V ( PS 3) Properties of α -like Elements Non-Classical Behaviour of V ( PS 3) α -like Elements as the Ordinals in V ( PS 3) Conclusion α -like Elements Meets Our Expectations Theorem Let x ∈ V ( PS 3 ) be α -like for some α ∈ ORD . For any y ∈ V ( PS 3 ) , � x = y � = 1 if and only if y is α -like. Theorem Let x ∈ V ( PS 3 ) be α -like for some non-zero α ∈ ORD . For any y ∈ V ( PS 3 ) , � y ∈ x � ∈ { 1 , 1 / 2 } if and only if y is β -like for some β ∈ α . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Naive Definitions The Three Valued Matrix PS 3 and its logic α -like Elements Ordinals in V ( PS 3) Properties of α -like Elements Non-Classical Behaviour of V ( PS 3) α -like Elements as the Ordinals in V ( PS 3) Conclusion Ordinals in First Order Language As promised earlier, the definitions of transitive set, linear-ordered set, well-ordered set and ordinal number is written below in the set theoretic language. 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Naive Definitions The Three Valued Matrix PS 3 and its logic α -like Elements Ordinals in V ( PS 3) Properties of α -like Elements Non-Classical Behaviour of V ( PS 3) α -like Elements as the Ordinals in V ( PS 3) Conclusion Ordinals in First Order Language As promised earlier, the definitions of transitive set, linear-ordered set, well-ordered set and ordinal number is written below in the set theoretic language. Trans ( x ) = ∀ y ∀ z ( z ∈ y ∧ y ∈ x → z ∈ x ) 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Naive Definitions The Three Valued Matrix PS 3 and its logic α -like Elements Ordinals in V ( PS 3) Properties of α -like Elements Non-Classical Behaviour of V ( PS 3) α -like Elements as the Ordinals in V ( PS 3) Conclusion Ordinals in First Order Language As promised earlier, the definitions of transitive set, linear-ordered set, well-ordered set and ordinal number is written below in the set theoretic language. Trans ( x ) = ∀ y ∀ z ( z ∈ y ∧ y ∈ x → z ∈ x ) LO ( x ) = ∀ y ∀ z (( y ∈ x ∧ z ∈ x ) → ( y ∈ z ∨ y = z ∨ z ∈ y )) 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Naive Definitions The Three Valued Matrix PS 3 and its logic α -like Elements Ordinals in V ( PS 3) Properties of α -like Elements Non-Classical Behaviour of V ( PS 3) α -like Elements as the Ordinals in V ( PS 3) Conclusion Ordinals in First Order Language As promised earlier, the definitions of transitive set, linear-ordered set, well-ordered set and ordinal number is written below in the set theoretic language. Trans ( x ) = ∀ y ∀ z ( z ∈ y ∧ y ∈ x → z ∈ x ) LO ( x ) = ∀ y ∀ z (( y ∈ x ∧ z ∈ x ) → ( y ∈ z ∨ y = z ∨ z ∈ y )) WO ∈ ( x ) = LO ( x ) ∧∀ y ( y ⊆ x ∧¬ ( y = ∅ ) → ∃ z ( z ∈ y ∧ z ∩ y = ∅ )) 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Naive Definitions The Three Valued Matrix PS 3 and its logic α -like Elements Ordinals in V ( PS 3) Properties of α -like Elements Non-Classical Behaviour of V ( PS 3) α -like Elements as the Ordinals in V ( PS 3) Conclusion Ordinals in First Order Language As promised earlier, the definitions of transitive set, linear-ordered set, well-ordered set and ordinal number is written below in the set theoretic language. Trans ( x ) = ∀ y ∀ z ( z ∈ y ∧ y ∈ x → z ∈ x ) LO ( x ) = ∀ y ∀ z (( y ∈ x ∧ z ∈ x ) → ( y ∈ z ∨ y = z ∨ z ∈ y )) WO ∈ ( x ) = LO ( x ) ∧∀ y ( y ⊆ x ∧¬ ( y = ∅ ) → ∃ z ( z ∈ y ∧ z ∩ y = ∅ )) ORD ( x ) = Trans ( x ) ∧ WO ∈ ( x ) 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Naive Definitions The Three Valued Matrix PS 3 and its logic α -like Elements Ordinals in V ( PS 3) Properties of α -like Elements Non-Classical Behaviour of V ( PS 3) α -like Elements as the Ordinals in V ( PS 3) Conclusion Ordinals in First Order Language The following abbreviations are used in WO ∈ ( x ): y ⊆ x := ∀ t ( t ∈ y → t ∈ x ) , ¬ ( y = ∅ ) := ∃ z ( z ∈ y ) , ( z ∩ y = ∅ ) := ¬ ∃ w ( w ∈ z ∧ w ∈ y ) . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Naive Definitions The Three Valued Matrix PS 3 and its logic α -like Elements Ordinals in V ( PS 3) Properties of α -like Elements Non-Classical Behaviour of V ( PS 3) α -like Elements as the Ordinals in V ( PS 3) Conclusion Some Results on Ordinal-Like Elements Finally, we can connect the notion of α -like name to the set theoretic notion of ordinals: Lemma Let α ∈ ORD and u be an α -like element in V ( PS 3 ) . Then the following hold: 1 V ( PS 3 ) | = Trans ( u ) 2 V ( PS 3 ) | = LO ( u ) 3 V ( PS 3 ) | = WO ∈ ( u ) 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Naive Definitions The Three Valued Matrix PS 3 and its logic α -like Elements Ordinals in V ( PS 3) Properties of α -like Elements Non-Classical Behaviour of V ( PS 3) α -like Elements as the Ordinals in V ( PS 3) Conclusion Some Results on Ordinal-Like Elements Hence we conclude the following theorem: Theorem Let α ∈ ORD and u be an α -like element in V ( PS 3 ) . Then V ( PS 3 ) | = ORD ( u ) . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models Naive Definitions The Three Valued Matrix PS 3 and its logic α -like Elements Ordinals in V ( PS 3) Properties of α -like Elements Non-Classical Behaviour of V ( PS 3) α -like Elements as the Ordinals in V ( PS 3) Conclusion Some Results on Ordinal-Like Elements Hence we conclude the following theorem: Theorem Let α ∈ ORD and u be an α -like element in V ( PS 3 ) . Then V ( PS 3 ) | = ORD ( u ) . Like the classical set theory we also have Theorem There is no set of all ordinals: V ( PS 3 ) � ∃ O ∀ x ( ORD ( x ) → x ∈ O ) . (Tarafder, S., Ordinals in an algebra-valued model of a paraconsistent set theory, Logic and Its Applications , LNCS, Vol. 8923. Berlin: Springer-Verlag, pp. 195–206, 2015.) 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS 3 and its logic Ordinals in V ( PS 3) Non-Classical Behaviour of V ( PS 3) Conclusion Leibniz’s Law of the Indiscernibility of Identicals It can be proved that Leibniz’s law of the indiscernibility of identicals ∀ x ∀ y ( x = y ∧ ϕ ( x ) → ϕ ( y )) is not valid in V ( PS 3 ) for all formula ϕ . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS 3 and its logic Ordinals in V ( PS 3) Non-Classical Behaviour of V ( PS 3) Conclusion Leibniz’s Law of the Indiscernibility of Identicals It can be proved that Leibniz’s law of the indiscernibility of identicals ∀ x ∀ y ( x = y ∧ ϕ ( x ) → ϕ ( y )) is not valid in V ( PS 3 ) for all formula ϕ . On the other hand it is also proved that for any instantiations of Leibniz’s law with NFF -formulas ϕ is valid. 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS 3 and its logic Ordinals in V ( PS 3) Non-Classical Behaviour of V ( PS 3) Conclusion Paraconsistency in the Set Theory Can we identify a formula ϕ in the language of set theory so that both ϕ and ¬ ϕ are true in V ( PS 3 ) ? 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS 3 and its logic Ordinals in V ( PS 3) Non-Classical Behaviour of V ( PS 3) Conclusion Paraconsistency in the Set Theory Can we identify a formula ϕ in the language of set theory so that both ϕ and ¬ ϕ are true in V ( PS 3 ) ? Let ϕ := ∃ x ∃ y ∃ z ( x = y ∧ z ∈ x ∧ z / ∈ y ). Then it can be proved that � ϕ � = 1 / 2 and hence � ¬ ϕ � = 1 / 2 ∗ = 1 / 2 which leads to the fact that both ϕ and ¬ ϕ are valid in V ( PS 3 ) . 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS 3 and its logic Ordinals in V ( PS 3) Non-Classical Behaviour of V ( PS 3) Conclusion Summary 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS 3 and its logic Ordinals in V ( PS 3) Non-Classical Behaviour of V ( PS 3) Conclusion Summary 1 We have found an algebra, A called deductive reasonable implication algebra. go to 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS 3 and its logic Ordinals in V ( PS 3) Non-Classical Behaviour of V ( PS 3) Conclusion Summary 1 We have found an algebra, A called deductive reasonable implication algebra. go to 2 Proved that V ( A ) is an algebra valued model of the set theory NFF − ZF − . go to 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS 3 and its logic Ordinals in V ( PS 3) Non-Classical Behaviour of V ( PS 3) Conclusion Summary 1 We have found an algebra, A called deductive reasonable implication algebra. go to 2 Proved that V ( A ) is an algebra valued model of the set theory NFF − ZF − . go to 3 Found a 3-valued matrix PS 3 which is neither Boolean nor Heyting but a deductive reasonable implication algebra. go to 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS 3 and its logic Ordinals in V ( PS 3) Non-Classical Behaviour of V ( PS 3) Conclusion Summary 1 We have found an algebra, A called deductive reasonable implication algebra. go to 2 Proved that V ( A ) is an algebra valued model of the set theory NFF − ZF − . go to 3 Found a 3-valued matrix PS 3 which is neither Boolean nor Heyting but a deductive reasonable implication algebra. go to 4 The logic LPS 3 which is sound and complete with respect to PS 3 is a paraconsistent logic. 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS 3 and its logic Ordinals in V ( PS 3) Non-Classical Behaviour of V ( PS 3) Conclusion Summary 1 We have found an algebra, A called deductive reasonable implication algebra. go to 2 Proved that V ( A ) is an algebra valued model of the set theory NFF − ZF − . go to 3 Found a 3-valued matrix PS 3 which is neither Boolean nor Heyting but a deductive reasonable implication algebra. go to 4 The logic LPS 3 which is sound and complete with respect to PS 3 is a paraconsistent logic. 5 As a consequence, V ( PS 3 ) is a model of some paraconsistent set theory. 1st March, 2019 S. Tarafder Paraconsistent Set Theory
Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS 3 and its logic Ordinals in V ( PS 3) Non-Classical Behaviour of V ( PS 3) Conclusion Summary 1 We have found an algebra, A called deductive reasonable implication algebra. go to 2 Proved that V ( A ) is an algebra valued model of the set theory NFF − ZF − . go to 3 Found a 3-valued matrix PS 3 which is neither Boolean nor Heyting but a deductive reasonable implication algebra. go to 4 The logic LPS 3 which is sound and complete with respect to PS 3 is a paraconsistent logic. 5 As a consequence, V ( PS 3 ) is a model of some paraconsistent set theory. 6 Defined ordinal-like elements inside V ( PS 3 ) and studied some classical and non-classical properties of them. 1st March, 2019 S. Tarafder Paraconsistent Set Theory
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