Paraconsistent Set Theory Sourav Tarafder St. Xaviers College - - PowerPoint PPT Presentation

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion

Paraconsistent Set Theory

Sourav Tarafder

• St. Xavier’s College

Kolkata

Paraconsistent Logic Pre-Conference Workshop, ICLA 2019, IIT Delhi, 1st March, 2019

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

Paraconsistent Logic

What is a paraconsistent logic?

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

Paraconsistent Logic

What is a paraconsistent logic? A set Γ of formulas is inconsistent if there is a formula ϕ in its language such that Γ ⊢ ϕ and Γ ⊢ ¬ϕ.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

Paraconsistent Logic

What is a paraconsistent logic? A set Γ of formulas is inconsistent if there is a formula ϕ in its language such that Γ ⊢ ϕ and Γ ⊢ ¬ϕ. A set Γ of formulas is trivial or explosive if for any formula ϕ of its language, Γ ⊢ ϕ.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

Paraconsistent Logic

What is a paraconsistent logic? A set Γ of formulas is inconsistent if there is a formula ϕ in its language such that Γ ⊢ ϕ and Γ ⊢ ¬ϕ. A set Γ of formulas is trivial or explosive if for any formula ϕ of its language, Γ ⊢ ϕ. In the context of classical logic the above two notions are equivalent.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

Paraconsistent Logic

What is a paraconsistent logic? A set Γ of formulas is inconsistent if there is a formula ϕ in its language such that Γ ⊢ ϕ and Γ ⊢ ¬ϕ. A set Γ of formulas is trivial or explosive if for any formula ϕ of its language, Γ ⊢ ϕ. In the context of classical logic the above two notions are equivalent. A logic is said to be paraconsistent if there exists a set Γ of formulas such that Γ is inconsistent but not explosive.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

Definition

To explain it simply we have the following definition of paraconsistent logic:

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

Definition

To explain it simply we have the following definition of paraconsistent logic: Definition A logic is called paraconsistent if there exist formulas ϕ and ψ such that {ϕ, ¬ϕ} ψ.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

Some Well Known Paraconsistent Logics

Many paraconsistent logics are studied till date. All these logics are developed with various motivations.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

Some Well Known Paraconsistent Logics

Many paraconsistent logics are studied till date. All these logics are developed with various motivations. For example: Jaskow´ ski’s paraconsistent logic, Da Costa’s paraconsistent logic systems Cn where 0 < n < ω, Priest’s logic of paradox, also other paraconsistent logics made by C. Mortensen, R. Brady, J. Marcos, W.A. Carnielli, A. Avron etc.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

Paraconsistent Set Theories- a Survey

Syntactic developments of some paraconsistent set theories are already made by the paraconsistentists like N. C. A. da Costa, Zach Weber, Walter Carnielli, Marcelo E. Coniglio etc. Whereas the semantical developments of this area are not strongly established other than some research work done by Thierry Libert, Olivier Esser etc.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

Cont.

1) A. Church. Set theory with a universal set. In proceedings of the Tarski Symposium, L. Henkin, eds., pp. 297–308, 1974. 2) A. I. Arruda. The Russell paradox in the system NFn. In Proceedings of the Third Brazilian Conference on Mathematical Logic, A. I. Arruda, N. C. A. da Costa and A. M. Sette, eds., pp. 1–12, 1980. 3) A. I. Arruda and D. Batens. Russells set versus the universal set in paraconsistent set theory. Logique et Analyse 98, 121–133, 1998.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

Cont.

4) N. C. A. da Costa. On paraconsistent set theory. Logique et Analyse 29(115): pp. 361–71, 1986. 5) G. Restall. A Note on Naive Set Theory in LP. Notre Dame Journal of Formal Logic, 33(3), 1992. 6) N. C. A. da Costa. Inconsistent formal systems (in Portuguese), Habilitation Thesis, 1963. Republished by Editora UFPR, Curitiba, 1993. 7) N. C. A. da Costa, J.-Y. Bziau and O. Bueno. Elementos de teoria paraconsistente de conjuntos (Elements of paraconsistent set theory, in Portuguese), vol. 23 of Colec¨ ao CLE. CLE-Unicamp, Campinas, 1998.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

Cont.

8) N. C. A. da Costa. Paraconsistent mathematics. In Frontiers of Paraconsistent Logic, D. Batens, C. Mortensen, G. Priest and J. P. Van Bendegem, eds, pp. 165180., 2000. 9) T. Libert. ZF and the Axiom of Choice in some paraconsistent set theories. Logic and Logical Philosophy, 11(1): pp. 91–114, 2003. 10) O. Esser. A strong model of paraconsistent logic. Notre Dame Journal of Formal Logic, 44(3): pp. 149–156., 2003. 11) T. Libert. Models for a paraconsistent set theory. Journal of Applied Logic, 3(1): pp. 15–41, 2005.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

Cont.

12) Z. Weber. Transfinite numbers in paraconsistent set theory. The Review of Symbolic Logic, 3(1): pp. 71–92., 2010. 13) Z. Weber. Transfinite cardinals in paraconsistent set theory. The Review of Symbolic Logic, 5(2): pp. 269–293., 2012. 14) B. Loewe and S. Tarafder. Generalised algebra-valued models

• f set theory. The Review of Symbolic Logic, 8(1): pp. 192–205.,

2015.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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

• n the first-order version of some (paraconsistent) LFIs, by

admitting that sets, as well as sentences, can be either consistent

• r 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

• btain traditional ZF set theory, and nothing new.”

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 → ϕx

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

The technique of building the model

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

The technique of building the model

Consider a set M.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

The technique of building the model

Consider a set M. Suppose PP(M) := {(A, B) : A ∪ B = M}.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

The technique of building the model

Consider a set M. Suppose PP(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 PP(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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 and a /

∈M b iff a ∈ [b]−

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 ǫM of the membership relation ‘∈’ in M as follows:

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 ǫM of the membership relation ‘∈’ in M as follows: t ∈ ǫM(a, b) iff a ∈M b f ∈ ǫM(a, b) iff a / ∈M b, for all a, b ∈ M.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 ǫM of the membership relation ‘∈’ in M as follows: t ∈ ǫM(a, b) iff a ∈M b f ∈ ǫM(a, b) iff a / ∈M b, for all a, b ∈ M. So, in this way, ǫM(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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

cont.

In this way a structure for a paraconsistent set theory appears as M := M, ǫM.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

cont.

In this way a structure for a paraconsistent set theory appears as M := M, ǫM. The truth degree of the atomic formula ‘x ∈ y’ in a given structure has been defined. More generally, the truth degree

• f 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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

cont.

We are now in a position to define the satisfaction relation | =. Consider a structure M := M, ǫM 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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

∀p0 · · · ∀pn∀x∃y∀z(z ∈ y ↔ z ∈ x ∧ ϕ(z, p0, . . . , pn)) (Separationϕ) ∀p0 · · · ∀pn−1∀x[∀y ∈ x∃zϕ(y, z, p0, . . . , pn−1) → ∃w∀v ∈ x∃u ∈ w ϕ(v, u, p0, . . . , pn−1)] (Replacementϕ) ∀p0 · · · ∀pn∀x[∀y ∈ x ϕ(y, p0, . . . , pn) → ϕ(x, p0, . . . , pn)] → ∀zϕ(z, p0, . . . , pn) (Foundationϕ)

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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

• ver the standard model V of ZFC.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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

• ver 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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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

• ver 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,

V(B)

α

= {x : Func(x)∧ran(x) ⊆ B ∧∃ξ < α(dom(x) ⊆ V(B)

ξ

)}

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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

• ver 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,

V(B)

α

= {x : Func(x)∧ran(x) ⊆ B ∧∃ξ < α(dom(x) ⊆ V(B)

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models 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

• f 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 =

• x∈dom(v)

(v(x) ∧ x = u) u = v =

• x∈dom(u)

(u(x) ⇒ x ∈ v)∧

• y∈dom(v)

(v(y) ⇒ y ∈ u)

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models 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∗ ∀xϕ(x) =

• x∈V(B)

ϕ(x) ∃xϕ(x) =

• x∈V(B)

ϕ(x)

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models 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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

Bounded Quantification Property

The following property BQϕ named after the bounded quantification property for the formula ϕ, played a very important role in proving the above two theorems. ∀x ∈ uϕ(x) =

• x∈dom(u)

(u(x) ⇒ ϕ(x)) (BQϕ) where ∀x ∈ uϕ(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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Paraconsistent Logic and set theories Axioms of ZF set theory Boolean and Heyting valued models

Bounded Quantification Property

The following property BQϕ named after the bounded quantification property for the formula ϕ, played a very important role in proving the above two theorems. ∀x ∈ uϕ(x) =

• x∈dom(u)

(u(x) ⇒ ϕ(x)) (BQϕ) where ∀x ∈ uϕ(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 BQϕ 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 PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Algebraic Properties Negation Free Formulas & Algebra-Valued Models

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 PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Algebraic Properties Negation Free Formulas & Algebra-Valued Models

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 PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Algebraic Properties Negation Free Formulas & Algebra-Valued Models

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 PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Algebraic Properties Negation Free Formulas & Algebra-Valued Models

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 PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Algebraic Properties Negation Free Formulas & Algebra-Valued Models

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 − BQϕ). 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¨

• we, B., and S. Tarafder, Generalized Algebra-Valued Models of Set Theory, Review
• f Symbolic Logic, Cambridge University Press, 8(1), pp. 192–205, 2015.)

BQϕ return 1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Matrix of PS3 PS3-valued model The Logic LPS3

Is there any algebra other than complete Boolean algebras and complete Heyting algebras which is a deductive reasonable implication algebra satisfying the NFF − BQϕ?

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Matrix of PS3 PS3-valued model The Logic LPS3

Three Valued Matrix PS3

Let us consider the three valued matrix PS3 = {1, 1/2, 0}, ∧, ∨, ⇒,∗ , 1, 0 where the truth tables for the

• perators are given below:

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Matrix of PS3 PS3-valued model The Logic LPS3

Three Valued Matrix PS3

Let us consider the three valued matrix PS3 = {1, 1/2, 0}, ∧, ∨, ⇒,∗ , 1, 0 where the truth tables for the

• perators are given below:

∧ 1

1/2

1 1

1/2 1/2 1/2 1/2

∨ 1

1/2

1 1 1 1

1/2

1

1/2 1/2

1

1/2

⇒ 1

1/2

1 1 1

1/2

1 1 1 1 1

∗

1

1/2 1/2

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Matrix of PS3 PS3-valued model The Logic LPS3

The Answer

It can be verified that PS3 is a deductive reasonable implication algebra and BQϕ holds in V(PS3) for all negation free formula ϕ.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Matrix of PS3 PS3-valued model The Logic LPS3

So we can conclude that Extensionality, Pairing, Infinity, Union, Power Set, NFF-Separation and NFF-Replacement are valid in V(PS3).

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Matrix of PS3 PS3-valued model The Logic LPS3

So we can conclude that Extensionality, Pairing, Infinity, Union, Power Set, NFF-Separation and NFF-Replacement are valid in V(PS3). Moreover it is proved separately that NFF-Regularity is also valid in V(PS3).

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Matrix of PS3 PS3-valued model The Logic LPS3

Logic Corresponding to PS3

What is the logic corresponding to PS3? Is it paraconsistent?

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Matrix of PS3 PS3-valued model The Logic LPS3

Logic Corresponding to PS3

What is the logic corresponding to PS3? Is it paraconsistent? We have found a logic LPS3 which is sound and complete with respect to PS3. More interestingly it can be proved that LPS3 is a paraconsistent logic.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Matrix of PS3 PS3-valued model The Logic LPS3

Axioms for LPS3

The following formulas are taken as the axioms for LPS3: (Ax1) ϕ → (ψ → ϕ) (Ax2) (ϕ → (ψ → γ)) → ((ϕ → ψ) → (ϕ → γ)) (Ax3) ϕ ∧ ψ → ϕ (Ax4) ϕ ∧ ψ → ψ (Ax5) ϕ → ϕ ∨ ψ (Ax6) (ϕ → γ) ∧ (ψ → γ) → (ϕ ∨ ψ → γ) (Ax7) (ϕ → ψ) ∧ (ϕ → γ) → (ϕ → ψ ∧ γ)

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Matrix of PS3 PS3-valued model The Logic LPS3

(Ax8) ϕ ↔ ¬¬ϕ (Ax9) ¬(ϕ ∧ ψ) ↔ (¬ϕ ∨ ¬ψ) (Ax10) (ϕ ∧ ¬ϕ) → (¬(ψ → ϕ) → γ) (Ax11) (ϕ → ψ) → (¬(ϕ → γ) → ψ) (Ax12) (¬ϕ → ψ) → (¬(γ → ϕ) → ψ) (Ax13) ⊥ → ϕ (Ax14) (ϕ ∧ (ψ → ⊥)) → ¬(ϕ → ψ) (Ax15) (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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Matrix of PS3 PS3-valued model The Logic LPS3

Rules for LPS3

The rules for LPS3 are the following:

1

ϕ, ψ ϕ ∧ ψ

2

ϕ, ϕ → ψ ψ

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Matrix of PS3 PS3-valued model The Logic LPS3

Soundness and completeness of LPS3

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 PS3.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Matrix of PS3 PS3-valued model The Logic LPS3

Soundness and completeness of LPS3

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 PS3. 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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Matrix of PS3 PS3-valued model The Logic LPS3

An Algebra-Valued Model of a Paraconsistent Set Theory

Hence we have reached to the fact that V(PS3) is an algebra-valued model of a paraconsistent set theory.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Naive Definitions α-like Elements Properties of α-like Elements α-like Elements as the Ordinals in V(PS3)

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Naive Definitions α-like Elements Properties of α-like Elements α-like Elements as the Ordinals in V(PS3)

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

• rder 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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Naive Definitions α-like Elements Properties of α-like Elements α-like Elements as the Ordinals in V(PS3)

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

• rder 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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Naive Definitions α-like Elements Properties of α-like Elements α-like Elements as the Ordinals in V(PS3)

α-like Elements

For each α ∈ ORD the α-like names in V(PS3) are defined by transfinite recursion as follows. Definition An element x ∈ V(PS3) 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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Naive Definitions α-like Elements Properties of α-like Elements α-like Elements as the Ordinals in V(PS3)

α-like Elements Meets Our Expectations

Theorem Let x ∈ V(PS3) be α-like for some α ∈ ORD. For any y ∈ V(PS3), x = y = 1 if and only if y is α-like.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Naive Definitions α-like Elements Properties of α-like Elements α-like Elements as the Ordinals in V(PS3)

α-like Elements Meets Our Expectations

Theorem Let x ∈ V(PS3) be α-like for some α ∈ ORD. For any y ∈ V(PS3), x = y = 1 if and only if y is α-like. Theorem Let x ∈ V(PS3) be α-like for some non-zero α ∈ ORD. For any y ∈ V(PS3), 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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Naive Definitions α-like Elements Properties of α-like Elements α-like Elements as the Ordinals in V(PS3)

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Naive Definitions α-like Elements Properties of α-like Elements α-like Elements as the Ordinals in V(PS3)

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Naive Definitions α-like Elements Properties of α-like Elements α-like Elements as the Ordinals in V(PS3)

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Naive Definitions α-like Elements Properties of α-like Elements α-like Elements as the Ordinals in V(PS3)

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Naive Definitions α-like Elements Properties of α-like Elements α-like Elements as the Ordinals in V(PS3)

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Naive Definitions α-like Elements Properties of α-like Elements α-like Elements as the Ordinals in V(PS3)

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 The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Naive Definitions α-like Elements Properties of α-like Elements α-like Elements as the Ordinals in V(PS3)

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(PS3). Then the following hold:

1 V(PS3) |

= Trans(u)

2 V(PS3) |

= LO(u)

3 V(PS3) |

= WO∈(u)

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Naive Definitions α-like Elements Properties of α-like Elements α-like Elements as the Ordinals in V(PS3)

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(PS3). Then V(PS3) | = ORD(u).

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion Naive Definitions α-like Elements Properties of α-like Elements α-like Elements as the Ordinals in V(PS3)

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(PS3). Then V(PS3) | = ORD(u). Like the classical set theory we also have Theorem There is no set of all ordinals: V(PS3) ∃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 PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) 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(PS3) for all formula ϕ.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) 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(PS3) 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 PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) 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(PS3)?

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) 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(PS3)? 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(PS3).

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion

Summary

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) 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 PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) 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 PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) 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 PS3 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 PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) 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 PS3 which is neither Boolean nor

Heyting but a deductive reasonable implication algebra.

go to 4 The logic LPS3 which is sound and complete with respect to

PS3 is a paraconsistent logic.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) 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 PS3 which is neither Boolean nor

Heyting but a deductive reasonable implication algebra.

go to 4 The logic LPS3 which is sound and complete with respect to

PS3 is a paraconsistent logic.

5 As a consequence, V(PS3) 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 PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) 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 PS3 which is neither Boolean nor

Heyting but a deductive reasonable implication algebra.

go to 4 The logic LPS3 which is sound and complete with respect to

PS3 is a paraconsistent logic.

5 As a consequence, V(PS3) is a model of some paraconsistent

set theory.

6 Defined ordinal-like elements inside V(PS3) and studied some

classical and non-classical properties of them.

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory

Introduction Generalised Algebra-Valued Models The Three Valued Matrix PS3 and its logic Ordinals in V(PS3) Non-Classical Behaviour of V(PS3) Conclusion

Thank You

1st March, 2019

• S. Tarafder

Paraconsistent Set Theory