[PPT] - Russells Paradox and free zig zag solutions Ludovica Conti FINO- PowerPoint Presentation

SLIDE 1

Russell’s Paradox and free zig zag solutions

Ludovica Conti

FINO- Northwestern Philosophy Consortium (Italy)

Bern, 25 April 2019

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 1 / 55

SLIDE 2

Plan of the talk

1

The debate about Russell’s Paradox Russell’s Paradox Cantorian vs Predicativist explanations Ferreira’s and Boccuni’s zig zag solutions

2

Free zig zag solutions Negative free logic Negative free logic and Russell’s Paradox Free fregean theories

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 2 / 55

SLIDE 3

The debate about Russell’s Paradox Russell’s Paradox

Russell’s Paradox.

1. ∀X∀Y (ǫX = ǫY ↔ ∀x(Xx ↔ Yx))

(BLV)

2. ∃X∀x(Xx ↔ ∃Y (x = ǫY ∧ ¬Yx)). Call this concept R.

(CA)

3. ∀X∃x(x = ǫX)

(AT)

4. ∃x(x = ǫR)

(2,3)

5. ¬RǫR → RǫR

(2,4)

6. RǫR → ∃Y (ǫR = ǫY ∧ ¬Y ǫR)

(2,4)

7. ¬RǫR

(1,6)

8. RǫR ↔ ¬RǫR

(5,7)

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 3 / 55

SLIDE 4

The debate about Russell’s Paradox Russell’s Paradox

Paradox: minimal version of a contradiction’s derivation. → list of all and only necessary premises; → elimination (or relevant change) of each of them is sufficient to avoid the contradiction. Explanation: instruction for a solution.

Expl. 1: selection of the specific guilty premise:

what premise we have to change to solve the paradox.

Expl. 2: indication of the guilt itself:

how we have to change a (selected) premise to solve the paradox. Solution: specific change of the derivation which

follows from an explanation;
is sufficient to avoid the contradiction;
is able to preserve as much as possible the derivational power of the

theory.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 4 / 55

SLIDE 5

The debate about Russell’s Paradox Cantorian vs Predicativist explanations

Traditional debate

Cantorian explanation: The inconsistency follows from the existential assumption of an injective function from concepts to extensions because it imposes a cardinality request which is incompatible with Cantor’s theorem. Expl.1: BLVb ∀X∀Y (ǫX = ǫY → ∀x(Xx ↔ Yx)); Expl.2: injectivity of extensionality function - intended as violation of Cantor’s theorem by the request that object’s domain has (at least) the same cardinality of concept’s domain. Predicativist explanation: The inconsistency follows from the specification of Russell’s concept because of its impredicativity. Expl.1: CA: ∃X∀x(Xx ↔ ∃Y (x = ǫY ∧ ¬Yx)); Expl.2: impredicativity of concepts specification - intended as implicit and vicious circularity, source of indefinite extensibility, lack of definitional guarantees (...).

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 5 / 55

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The debate about Russell’s Paradox Cantorian vs Predicativist explanations

Incomplete conclusion

Cantorian explanation (Expl. 2) fails: the related cantorian solution is not sufficient to avoid the contradiction - there is a derivation of the same contradiction from a restricted version of BLV (Definable- BLV) that is compatible with Cantor’s theorem (Paseau 2015).

1. ∀X(∀x(Xx ↔ φx) → ∀Y (ǫX = ǫY ) ↔ ∀x(Xx ↔ Yx))

(Def.-BLV)

2. ∃X∀x(Xx ↔ ∃Y (x = ǫY ) ∧ ¬Yx)). Call this concept R.

(CA)

3. ∀X∃x(x = ǫX)

(AT)

4. ∃x(x = ǫR)

(2,3)

5. ∀x((Xx ↔ ∃Y (x = ǫY ) ∧ ¬Yx))

→ ∀Y (ǫR = ǫY ↔ ∀x(Rx ↔ Yx)) (1)

6. ∀Y (ǫR = ǫY ↔ ∀x(Rx ↔ Yx))

(2,5)

7. ¬R(ǫR) → R(ǫR)

(2,4)

8. R(ǫR) → ∃Y (ǫR = ǫY ) ∧ ¬Y ǫR)

(2,4)

9. ¬R(ǫR)

(6,8) 10.R(ǫR) ↔ ¬R(ǫR) (7,9)

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 6 / 55

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The debate about Russell’s Paradox Cantorian vs Predicativist explanations

Predicativist explanation is not very strong because there are several other impredicative but consistent abstraction principles ( cfr. HP:∀F∀G(♯F = ♯G ↔ F ≈ g)). Predicativist solution works but is very weak: it consists in a predicative restriction of the comprehension’s formula of CA

avoids the contradiction but
allows to derive only Robinson Arithmetic Q

(prevent the derivation of Peano Arithmetic PA - first goal of the original fregean proposal).

Cfr. Heck 1996, Wehemeier 1999, Ferreira-Wehemeier 2002.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 7 / 55

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The debate about Russell’s Paradox Ferreira’s and Boccuni’s zig zag solutions

Zig zag proposal

Russell’s zig zag proposal: "In the zigzag theory, we start from the suggestion that propositional functions determine classes when they are fairly simple, and only fail to do so when they are complicated and recondite" General idea: Not all propositional functions (open formulas) determine classes (extensions). In our terms - admitted that every open formulas specifies a concept:

the full second order domain is specified (unlike predicativist solutions);
the correlation between concepts and extensions is injective (unlike

cantorian solutions);

the correlation between concepts and extensions is not total.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 8 / 55

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The debate about Russell’s Paradox Ferreira’s and Boccuni’s zig zag solutions

Carving the correlation. First way.

I) CARVING CORRELATION BY A DISTINCTION ON THE CONCEPTS’ DOMAIN. every open formulas specifies a concept but there are two sort of open formulas:

predicative formulas that specifies concepts related to extensions;
not-predicative formulas that specifies concepts that go zig zag between

the extensions. Simplifying: there are two sort of concepts - defined by formulas - predicative and not-predicative.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 9 / 55

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The debate about Russell’s Paradox Ferreira’s and Boccuni’s zig zag solutions

What it means to be predicative? (Russell) fairly simple (Ferreira - Boccuni*) predicative in modern acception A definition is said to be predicative if it does not quantify over a totality to which the entity being defined belongs. Otherwise the definition is said to be predicative. A comprehension axiom is predicative if the comprehension formula φ(x) contains no bound second-order variables, and impredicatve otherwise.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 10 / 55

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The debate about Russell’s Paradox Ferreira’s and Boccuni’s zig zag solutions

Ferreira’s zig zag theory: PE Predicative Extensions

Two sorted second order language (primitive symbols):

denumerably many first order variables: x, y, z;
denumerably many PREDICATIVE second order variables: F, G, H;
denumerably many IMPREDICATIVE second order variables: F, G, H;
logical constants: ¬, ∧, ∨, →;
quantifiers for each order and sort of variables:∃ x, ∃ F, ∃ F;
operator term-forming ǫ applied to open formulas.

Syntax:

Complex singular terms: if φ(x) is a PREDICATIVE formula, ǫ.φ x is a

(complex) singular term;

Atomic formulae: if Π is a (PREDICATIVE or IMPREDICATIVE) second
rder variable and x is a first order variable, Π(x) is an atomic formulas;
Complex formulae by usual inductive definition.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 11 / 55

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The debate about Russell’s Paradox Ferreira’s and Boccuni’s zig zag solutions

Axioms of PE:

Second order Logic;
Predicative comprehension axiom schema: ∃F∀x(Fx ↔ φ(x)) -

where φ(x) is a PREDICATIVE formula (without F free);

Impredicative comprehension axiom schema: ∃F∀x(Fx ↔ φ(x)) -

where φ(x) is a IMPREDICATIVE formula;

schematic Basic Law V: ǫx.φx = ǫx.ψx ↔ ∀x(φx ↔ ψx)

→ Automatically restricted to PREDICATIVE formulas (φx, ψx).

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 12 / 55

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The debate about Russell’s Paradox Ferreira’s and Boccuni’s zig zag solutions

Boccuni’s theory*: PG Plural Grundgesetze

Second order with two sorted first order language:

denumerably many first order SINGULAR variables: x, y, z;
denumerably many first order PLURAL variables: xx, yy, zz;
denumerably many second order variables (conceptual variables): X, Y, Z;
logical conectives: ¬, ∧, ∨, → - quantifiers for each order and sort of

variables:∃x, ∃xx, ∃X;

relational constant η, between fol SINGULAR and fol PLURAL variables;
operator term-forming ǫ applied to open formulas.

Syntax:

Complex singular terms: if φ(x) includes free sol variables, bounded fol

PLURAL variables, free or bounded fol SINGULAR variables, ǫx.φ(x) is a (complex) singular term;

Atomic formulae: if a and b are terms, aa is a fol PLURAL term and F is

a sol term, a = b, aηaa, Fa are atomic formulae;

Complex formulae by usual inductive definition.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 13 / 55

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The debate about Russell’s Paradox Ferreira’s and Boccuni’s zig zag solutions

Axioms of PG:

Second order Logic;
"Predicative" comprehension axiom schema for sol variables:

∃F∀x(Fx ↔ φ(x)) - where φ(x) includes free sol variables, bounded fol PLURAL variables, free

r bounded fol SINGULAR variables;

*It is more than predicative: φ(x) contain niether bound sol variables nor free fol PLURAL variables.

Impredicative comprehension axiom schema for fol PLURAL variables:

∃xx∀x(xηxx ↔ φ(x)) - where φ(x) is a IMPREDICATIVE second order formula (without xx free);

schematic Basic Law V: ǫx.φx = ǫx.ψx ↔ ∀x(φx ↔ ψx)

* Automatically restricted to "predicative" formulas containing neither bound sol variables nor free fol PLURAL variables - the same that specifies sol variables. ** There is, for every concept (specified sol variable), a corresponding complex singular term and vice versa.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 14 / 55

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The debate about Russell’s Paradox Ferreira’s and Boccuni’s zig zag solutions

Some differences: 1) Predicativity:

PE involve a standard definition of predicativity (φ(x) is predicative if and
nly if contains no bound second order variables) and use this notion as

primitive tool to distinguish the two sorts of second order variables.

PG involve two primitive sorts of non-singular variables (fol PLURAL and

sol variables) - allowing to distinguish two forms of reference - and use this richer vocabulary to characterize a finer grained definition of predicativity (φ(x) is predicative if and only if contains no bound second order variables and no free first order free variables). 2) Zig zag:

PE is a classical zig zag theory: some (predicative) concepts are

correlated with extensions and other (impredicative) concepts have not correlated extensions.

PG is less - strictly speaking - zig zag and more fregean theory: every

concepts (which is "predicatively" specified) has a correlated extension.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 15 / 55

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The debate about Russell’s Paradox Ferreira’s and Boccuni’s zig zag solutions

Semantic agreement by a zig zag model (M=<D, I>). D: set of natural numbers = objectual domain. (domain for fol variables of PE and fol SINGULAR variables of PG) Π(D): power set of D = conceptual domain (domain for IMPREDICATIVE sol variables of PE and fol PLURAL variables of PG) π(D) ⊆ Π(D): countable subset of power set of D = predicative subset of conceptual domain (domain for PREDICATIVE sol variables of PE and every sol variables of PG) I(ǫ): partial (injective) function f: π(D) → D = zig zag correlation from conceptual into objectual domain.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 16 / 55

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The debate about Russell’s Paradox Ferreira’s and Boccuni’s zig zag solutions

From the (first) zig zag solutions to an explanation.

First zig zag proposals: theories with two sorts of second order variables, which involve two versions

f second order comprehension axiom schema (CA) and a correspondent

restriction of BLV. Explanatory suggestion: The reason why the full second order domain cannot be the domain of the extensionality function is NOT the specification of its members via CA the classical assumption that each of them is correlated to an extension.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 17 / 55

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The debate about Russell’s Paradox Ferreira’s and Boccuni’s zig zag solutions

From this explanation to another zig zag solution.

Explanatory idea: The domain of the extensionality function is a proper subset of the second

rder domain.

New zig zag solution: a restriction of the zig- zag correlation between concepts and extensions

btained by working on the correlation itself

by substituting classical logic with a negative free logic and by moving the restrictions, traditionally imposed on the comprehension axiom schema, on the right hand of BLV.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 18 / 55

SLIDE 19

Free zig zag solutions Negative free logic

Carving the correlation. Second option.

I) CARVING CORRELATION INTO THE CORRELATION ITSELF. There are some concepts without a correlated extension. Problematic premise: Abstracts’ Theorem ∀X∃x(x = ǫX)

which allows to derive, from Russell’s concept

(∃X∀x(Xx ↔ ∃Y (x = ǫY ∧ ¬Yx)), namely R) Russell’s extension (∃x(x = ǫR), namely r) - is not valid.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 19 / 55

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Free zig zag solutions Negative free logic

1. ∀X∀Y (ǫX = ǫY ↔ ∀x(Xx ↔ Yx))

(BLV)

2. ∃X∀x(Xx ↔ ∃Y (x = ǫY ∧ ¬Yx)). Call this concept R.

(CA)

3. ∀X∃x(x = ǫX)

(AT)

4. ∃x(x = ǫR)

(2,3)

5. ¬RǫR → RǫR

(2,4)

6. RǫR → ∃Y (ǫR = ǫY ∧ ¬Y ǫR)

(2,4)

7. ¬RǫR

(1,6)

8. RǫR ↔ ¬RǫR

(5,7)

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 20 / 55

SLIDE 21

Free zig zag solutions Negative free logic

Abstraction’s Principle: @X = @Y ↔ X ≡ Y Abstracts’ Theorem: ∀X∃x(x = @X): 1) 1) t = t [FOL=] 2) ∀x(x = x) [1, UI] 3) @X = @X [2, UE] 4) ∃x(x = @X) [3, EI] 5) ∀X∃x(x = @X) [4, UI] 2) 1)X ≡ X [FOL=] 2)@(X) = @(X) [1, AP] 3)∃x(x = @(X)) [2, EI] 4)∀X∃x(x = @(X)) [3, UI]

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 21 / 55

SLIDE 22

Free zig zag solutions Negative free logic

Negative free language

Standard second order language (primitive symbols):

denumerably many first order variables: x, y, z;
denumerably many second order variables: X, Y, Z;
logical constants: ¬, ∧, ∨, Σ, ∃;
identity: =;
operator term-forming: ǫ.

Definable symbols:

universal "un-restricted" quantifier FOL Π: ΠxAx =def ¬Σx¬Ax;
universal "restricted" quantifier FOL ∀: ∀xAx =def ¬∃x¬Ax;
predicative monadic existential constant E!: E!a =def ∃x(x = a).

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 22 / 55

SLIDE 23

Free zig zag solutions Negative free logic

Negative free logic (axiomatic version)

Theory:

Impredicative comprehension axiom schema (CA: ∃X∀x(Xx ↔ φ(x)));
standard second order logic (without identity) for second order and

"unrestricted" first order quantification;

free (non inclusive) logic (with identity) for "restricted" quantification

and identity: N1) ∀vα → (E!t → α(t/v)); N2) ∃vE!v; N3) s = t → (α(s) → α(t//s)); N4) ∀v(v = v); N5) Πt1, ...tn → E!ti(with1 ≤ i ≤ n).

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 23 / 55

SLIDE 24

Free zig zag solutions Negative free logic

Negative free logic interpretation

Model M = <D, D0, I>:

D: domain of "restricted" quantification (D ⊆ D0);
D0: domain of "unrestricted" quantification;
I: total interpretation function on D0.

I(ǫ): partial injective function from a subset of the powerset of D0 in D.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 24 / 55

SLIDE 25

Free zig zag solutions Negative free logic and Russell’s Paradox

No Extensions’ Theorem:

Abstracts’ theorem (standard/free version): 1) t = t [FOL=] Not in FL 2) ∀x(x = x) [1, IU] [N4] 3) @(X) = @(X) [2, EU] EU Not in FL 4) ∃x(x = @(X)) 5) ∀X∃x(x = @(X)) [4, IU sol] 1)X ≡ X [FOL=] Not (necessarly) in FL 2)@(X) = @(X) [1, BLVr-l] 3)∃x(x = @(X)) [2, EG] 4)∀X∃x(x = @(X)) [3, UG]

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 25 / 55

SLIDE 26

Free zig zag solutions Negative free logic and Russell’s Paradox

No Russell’s Paradox...

Standard version:

1. ∀X∀Y (ǫX = ǫY ↔ ∀x(Xx ↔ Yx))

(BLV)

2. ∃X∀x(Xx ↔ ∃Y (x = ǫY ∧ ¬Yx)). Call this concept R.

(CA)

3. ∀X∃x(x = ǫX)

(AT)

4. ∃x(x = ǫR)

(2,3)

5. ¬RǫR → RǫR

(2,4)

6. RǫR → ∃Y (ǫR = ǫY ∧ ¬Y ǫR)

(2,4)

7. ¬RǫR

(1,6)

8. RǫR ↔ ¬RǫR

(5,7)

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 26 / 55

SLIDE 27

Free zig zag solutions Negative free logic and Russell’s Paradox

... but less logicist result

In the classical framework (SOL= + BLV), the logical part of the theory is involved in the specification of ǫ’s domain:

BLVa: ∀X∀Y (∀x(Xx ↔ Yx)) → ǫX = ǫY ): ǫ is a functional correlation;
BLVb: ∀X∀Y (ǫX = ǫY → ∀x(Xx ↔ Yx)): ǫ is an injective correlation.
AT → FOL= : ∀X∃x(x = ǫX): ǫ is a total correlation;

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 27 / 55

SLIDE 28

Free zig zag solutions Negative free logic and Russell’s Paradox

In the free framework (FL + BLV), the abstraction principle defines also the ǫ’s domain: Not only:

BLVa: ∀X∀Y (Πx(Xx ↔ Yx)) → ǫX = ǫY ): ǫ is a functional correlation;
BLVb: ∀X∀Y (ǫX = ǫY → Πx(Xx ↔ Yx)): ǫ is an injective correlation.

But also:

1. X ≡ X → ǫ(X) = ǫ(X)

(BLVa)

2. X ≡ X

(A)

3. ǫ(X) = ǫ(X)

(MP)

4. ∃x(x = ǫ(X))

(IE) : ǫ is a partial function.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 28 / 55

SLIDE 29

Free zig zag solutions Negative free logic and Russell’s Paradox

... a new circularity

If Russell’s concept is reflexively co-extensional with itself, (by BLVa) its extension exists, then the contradiction arises. R ≈ R ǫ(R) = ǫ(R) ∃x(x = ǫ(R)) ... ⊥. Now - because ǫ can be a partial function - we can conclude that Russell’s extension does not exist. But, if Russell’s extension does not exist, Russell’s concept (by BLVb) should be not reflexively co-extensional with itself. Instead, if Russell’s extension does not exists, Russell’s concept seems to be reflexively co-extensional with itself and then, again, able to introduce its problematic extension.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 29 / 55

SLIDE 30

Free zig zag solutions Negative free logic and Russell’s Paradox

Other Restrictions

The solutions suggested by the zig zag proposals presuppose not only a logic weakening (from classical to free logic) but also a correspondent weakening of the non logical part of the theory (BLVa). We will compare three free zig-zag fregean systems (E-FL, P-FL, T-FL) which share the logical axioms (FL) and distinguish one another by the different restrictions admitted on the right hand of BLV: 1) E-BLV:∀X∀Y (ǫX = ǫY ↔ E!ǫX ∧ E!ǫY ∧ ∀x(Xx ↔ Yx)) 2) P-BLV:∀X∀Y (ǫX = ǫY ↔ φX ∧ φY ∧ ∀x(Xx ↔ Yx)) - where φ means predicative; 3) T-BLV:∀X∀Y (ǫX = ǫY ↔ φX ∧ φY ∧ ∀x(Xx ↔ Yx)) - where φ means positive.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 30 / 55

SLIDE 31

Free zig zag solutions Free fregean theories

E-FL

Theory: FL; E-BLV:∀X∀Y (ǫX = ǫY ↔ E!ǫX ∧ E!ǫY ∧ ∀x(Xx ↔ Yx)). Frege’s arithmetic vocabulary: Definition (Number 0) 0 = ♯([λx.x = x]) = ǫ(λx.∃X(x = ǫ(X) ∧ X ≈ [λx.x = x]) Definition (Numeber 1) 1 = ♯([λx.x = 0]) = ǫ(λx.∃X(x = ǫ(X) ∧ X ≈ [λx.x = 0])) Definition (Predecessor) P(x,y) = ∃X∃z(Xz ∧ y = ♯(X) ∧ x = ♯([λw.Xw ∧ w = z]))

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 31 / 55

SLIDE 32

Free zig zag solutions Free fregean theories

Definition (Hereditary property) H(X, R) = ∀x∀y(Rxy → (Xx → Xy)) Definition (Ancestral) R(xy) =∀X((∀z(Rxz → Xz) ∧ Er(X, R)) → Xy) Definition (Weak Ancestral) R"(x,y) = R(x, y) ∨ x = y Definition (Natural Number) Nx = P"(0, x) However E-BLV is too weak to derive the existence of each number - namely to derive that number terms’ denotations belong to D0. So we are not able to follows Frege’s strategy: deriving HP from E-BLV and deriving Peano axioms - as theorems - from HP.

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 32 / 55

SLIDE 33

Free zig zag solutions Free fregean theories

There is another strategy: deriving Peano Arithmetic considering ǫ as the analogue of singleton in set theory. Set arithmetic vocabulary: Definition 0 = ǫ(λx.x = x) Definition 1 = ǫ(λx.x = 0) Definition Sn = ǫ(λx.x = n) Definition I(X) = X0 ∧ ∀z(Xz → Xǫ(λx.x = z)) Definition Nx = ∀X(I(X) → Xx)

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 33 / 55

SLIDE 34

Free zig zag solutions Free fregean theories

Theorem N0. Proof.

1. E!0

(A caso1)

2. ∀X((I(X) → X0)

(def. I(X))

3. N0

(2, def. N)

3. E!0 → N0

(1,3)

4. ¬E!0

(A caso 2)

5. ∀X(¬I(X))

(4, def. I(X))

6. ∀X((I(X) → X0)

(5)

7. N0

(6, def. N)

8. ¬E!0 → N0

(4,7)

9. (E!0 ∨ ¬E!0) → N0

(3,8)

10. (E!0 ∨ ¬E!0)

(C1)

11. N0

(9, 10, MP)

Ludovica Conti (FINO- Northwestern Philosophy Consortium (Italy)) Russell’s Paradox and free zig zag solutions Bern, 25 April 2019 34 / 55

SLIDE 35

Free zig zag solutions Free fregean theories

Theorem ∀y∀z(ǫ(λx.x = y) = ǫ(λx.x = z) → y = z). Proof.

1. ǫ(λx.x = y) = ǫ(λx.x = z)

(A)

2. ∀x(λx.x = y)(x) ↔ (λx.x = z)(x)

(E-BLVb)

3. (λx.x = y)(a) ↔ (λx.x = z)(a)

(N1, E!a)

4. a = y ↔ a = z

(λ − conv.)

5. y = z

(N3)

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Theorem ∀X(X0 ∧ ∀y(Xy → Xǫ(λx.x = y)) → ∀x(Xx)) Proof. It follows from the definition of N. But we are not able to derive the theorem correspondent to PA2: ∀x(Sx = 0) and PA4:∀y∃x(x = ǫ(λx.x = y))

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P-FL

Theory: FL; P-BLV:∀X∀Y (ǫX = ǫY ↔ φX ∧ φY ∧ ∀x(Xx ↔ Yx)) - where φ means predicative (it does not contains bound second ordered variables). We are not able to define Frege Arithmetic’s vocabulary because Frege definitions of cardinal numbers are impredicative ( 0 =def ♯([λx.x = x]) = ǫ(λx.∃X(x = ǫ(X) ∧ X ≈ [λx.x = x])). We directly adopt the second strategy and define the Set Arithmetic’s vocabulary: 0 =def ǫ(λx.x = x) Sn =def ǫ(λx.x = n) I(X) =def X0 ∧ ∀z(Xz → Xǫ(λx.x = z)) Nx =def ∀X(I(X) → Xx)

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Theorem N0. Proof.

1. E!0

(P-BLVa)

2. ∀X((I(X) → X0)

(def. I(X))

3. N0

(2, def. N)

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Theorem ∀x(Sx = 0). Proof.

1. E!0

(P-BLVa)

2. ∃y(E!Sy ∧ y = a)

(def. S, P-BLVa)

3. a = 0

(A)

4. ǫ(λx.x = y) = ǫ(λx.x = x)

(def. S, def. 0)

5. ∀z([λx.x = y](z) ↔ [λx.x = x](z)),

(P-BLVb)

6. ∀z(z = y ↔ z = z).

(λ − conv)

7. ∀zE!y(z = y ↔ z = z)

(T1)

8. y = y ↔ y = y ⊥

(N1)

9. ¬∃y(Sy = 0)

(3,8)

10. ∀y¬(Sy = 0)

(9)

11. ∀y(Sy = 0)

(10, def. =)

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Theorem ∀y∀z(ǫ(λx.x = y) = ǫ(λx.x = z) → y = z). Proof.

1. ǫ(λx.x = y) = ǫ(λx.x = z)

(A)

2. ∀x(λx.x = y)(x) ↔ (λx.x = z)(x)

(P-BLVb)

3. (λx.x = y)(a) ↔ (λx.x = z)(a)

(N1, E!a)

4. a = y ↔ a = z

(λ − conv)

5. y = z

(N3) Theorem ∀X(X0 ∧ ∀y(Xy → Xǫ(λx.x = y)) → ∀x(Xx)). Proof. It follows from the definition of N.

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Theorem ∀y∃x(x = ǫ([λx.x = y)) Proof. 1) ∀y∃X∀x(Xx ↔ x = y). We call this concept [λx.x = y] [AC] 2) φ([λx.x = y]) ∧ Πx([λx.x = y](x) ↔ [λx.x = y](x)) → ǫ(λx.x = y) = ǫ(λx.x = y)) [P-BLVa] 3) φ([λx.x = y]) ∧ Πx([λx.x = y](x) ↔ [λx.x = y](x)) [free FOL=] 4) ǫ(λx.x = y) = ǫ(λx.x = y)) [2,3, MP] 5) ∃x(x = ǫ(λx.x = y)) [4,T2] 6) ∀y∃x(x = ǫ(λx.x = y)) [1-5, IU]

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T-FL

Theory: FL; T-BLV:∀X∀Y (ǫX = ǫY ↔ φX ∧ φY ∧ ∀x(Xx ↔ Yx)) - where φ means positive (it contains bound second ordered variables only in a even number

f negation symbols - considering formulas reduced to its primitive form:

∃X∃x(¬Xx) is not positive; ∃X∃x(Xx ∧ ¬(x = x)) is positive).

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We are able to define Frege Arithmetic’s vocabulary because Frege definitions of cardinal numbers are positive: 0 = ♯([λx.x = x]) = ǫ(λx.∃X(x = ǫ(X) ∧ X ≈ [λx.x = x]) 1 = ♯([λx.x = 0]) = ǫ(λx.∃X(x = ǫ(X) ∧ X ≈ [λx.x = 0])) P(x,y) = ∃X∃z(Xz ∧ y = ♯(X) ∧ x = ♯([λw.Xw ∧ w = z])) H(X, R) = ∀x∀y(Rxy → (Xx → Xy)) R(xy) =∀X((∀z(Rxz → Xz) ∧ Er(X, R)) → Xy) R"(x,y) = R(x, y) ∨ x = y Nx = P"(0, x)

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Definition x ∈ y = ∃X(y = ǫ(X) ∧ Xx) - (o ∀X(y = ǫ(X) → Xx)) Definition X ≈ Y = ∃R(Πx(Xx → ∃!y(Yy ∧ Rxy)) ∧ Πy(Yy → ∃!x(Xx ∧ Ryx))) Lemma ∀X∀Y (Πx(Xx ↔ Yx) → X ≈ Y ) Lemma ∀X(X ≈ X) Lemma ∀X∀Y (X ≈ Y → Y ≈ X) Lemma ∀X∀Y ∀Z(X ≈ Y ∧ Y ≈ Z → X ≈ X)

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Derivation of HP from T-BLV

Theorem (First theorem about extensions - weak positive version) ∀X(φ(X) → ∃x(x = ǫ(X)). Proof.

1. φ(X)

(A)

2. Πx(Xx ↔ Xx) ∧ φ(X) → ǫ(X) = ǫ(X)

(T-BLVa)

3. Πx(Xx ↔ Xx)

(rifl. co-ext)

4. ǫ(X) = ǫ(X)

(1, 2, 3, MP)

5. ∃x(x = ǫ(X))

(4, T2)

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Lemma ∀X(∃x(x = ǫ(X) → φ(X))). Proof.

1. ∃x(x = ǫ(X))

(A)

2. ǫ(X) = ǫ(X)

(1, N4)

3. Πx(ǫ(X) = ǫ(X) → Xx ↔ Xx ∧ φ(X))

(T-BLVb)

4. φ(X)

(2,3 MP)

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Theorem (Secondo Teorema sulle estensioni -versione debole positiva) ∀X(φ(X) → Πx(x ∈ ǫ(X) ↔ Xx)). Proof.

1. φ(F)

(A)

2. a ∈ ǫ(F)

(A)

3. ∃X(ǫ(F) = ǫ(H) ∧ Ha)

(2, def. ∈)

4. ǫ(F) = ǫ(G) ∧ Ga

(3, Lemma 2.2, EE)

5. Πx(Fx ↔ Gx) ∧ φ(F) ∧ φ(G)

(4, T-BLVb)

6. Fa

(4, 5)

7. Fa

(A)

8. ∃x(x = ǫ(F))

(1, 7, Teorema 2.1)

9. ǫ(F) = ǫ(F)

(8, N4)

10. ǫ(F) = ǫ(F)

(7, 9, I∧)

11. ∃X(ǫ(F) = ǫ(X) ∧ Xa)

(10, IE)

12. a ∈ ǫ(F)

(11, def. ∈)

13. ∀X(φ(X) → Πx(x ∈ ǫ(X) ↔ Xx))

(1, 2-6, 7-12)

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Lemma ∀X∀Y (ǫ(Y ) ∈ ♯(X) ↔ [λx.∃Z(x = ǫ(Z) ∧ Z ≈ X)]ǫ(Y )). Proof.

1. ∀X(φ(X) → x(x ∈ ǫ(X) ↔ Xx))
2. ∀X(φ([λx.∃Z(x = ǫ(Z) ∧ Z ≈ X])))
3. ∀XΠx(x ∈ ǫ([λx.∃Z(x = ǫ(Z) ∧ Z ≈ X]) ↔ [λx.∃Z(x = ǫ(Z) ∧ Z ≈ X]x
4. ∀XΠx(x ∈ ♯(X)) ↔ [λx.∃Z(x = ǫ(Z) ∧ Z ≈ X]x))
5. ∀X(ǫ(F) ∈ ♯(X) ↔ [λx.∃Z(x = ǫ(Z) ∧ Z ≈ X)]ǫ(F))
6. ∀X∀Y (ǫ(Y ) ∈ ♯(X) ↔ [λx.∃Z(x = ǫ(Z) ∧ Z ≈ X)]ǫ(Y ))

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Lemma ∀Y (φ(Y ) → ∀X(ǫ(Y ) ∈ ♯(X) ↔ Y ≈ X)). Proof. 1, φ(G) (A)

2. ǫ(G) ∈ ♯(F)

(A)

3. [λx.∃Z(x = ǫ(Z) ∧ Z ≈ F)](ǫ(G)

(2, Lemma 2.4)

4. ∃Z(ǫ(G) = ǫ(Z) ∧ Z ≈ F)

(3, λ − conv.)

5. ǫ(G) = ǫH ∧ H ≈ F

(4, Lemma 2.2, EE)

6. Πx(Gx ↔ Hx) ∧ φ(G) ∧ φ(H)

(5, T-BLVb)

7. G ≈ H

(6, Lemma 1.1)

8. G ≈ F

(5, 7 trans. ≈)

9. G ≈ F

(A)

10. ǫ(G) = ǫ(G) ∧ G ≈ F

(1, N4, 9, I∧)

11. ∃Z(ǫ(G) = ǫ(Z) ∧ Z ≈ F)

(IE)

12. [λx.∃Z(x = ǫ(Z) ∧ Z ≈ F)](ǫ(G)

(11, λ − conv.)

13. ǫ(G) ∈ ♯(F)

(12, Lemma 2.4)

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Theorem X ≈ Y → ♯(X) = ♯(Y ) Proof.

1. F ≈ G
2. [λx.∃X(x = ǫ(X) ∧ X ≈ F]a
3. ∃X(a = ǫX ∧ X ≈ F)

(2,

4. a = ǫ(H) ∧ H ≈ F

(3, Lemma

5. a = ǫ(H) ∧ H ≈ G

(1,

6. ∃X(a = ǫX ∧ X ≈ G)
7. [λx.∃X(x = ǫ(X) ∧ X ≈ G]a

(6,

8. [λx.∃X(x = ǫ(X) ∧ X ≈ F]a → [λx.∃X(x = ǫ(X) ∧ X ≈ G]a

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9. [λx.∃X(x = ǫ(X) ∧ X ≈ G]a
10. ∃X(a = ǫX ∧ X ≈ G)
11. a = ǫ(H) ∧ H ≈ G

(10,

12. a = ǫ(H) ∧ H ≈ F

(1, 11,

13. ∃X(a = ǫX ∧ X ≈ F)
14. [λx.∃X(x = ǫ(X) ∧ X ≈ F]a
15. [λx.∃X(x = ǫ(X) ∧ X ≈ G]a → [λx.∃X(x = ǫ(X) ∧ X ≈ F]a
16. Πx(∃x(x = a) →

[λx.∃X(x = ǫ(X) ∧ X ≈ F]x ↔ [λx.∃X(x = ǫ(X) ∧ X ≈ G]x

17. Πx(¬∃x(x = a) →

[λx.∃X(x = ǫ(X) ∧ X ≈ F]x ↔ [λx.∃X(x = ǫ(X) ∧ X ≈ G]x

18. Πx([λx.∃X(x = ǫ(X) ∧ X ≈ F]x ↔ [λx.∃X(x = ǫ(X) ∧ X ≈ G]x

19 ∀Z(φ([λx.∃X(x = ǫ(X) ∧ X ≈ Z])

20. ǫ([λx.∃X(x = ǫ(X) ∧ X ≈ F]) = ǫ([λx.∃X(x = ǫ(X) ∧ X ≈ G])
21. ♯(F) = ♯(G)

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Theorem φ(X) ∧ φ(Y ) → (♯(X) = ♯(Y ) → X ≈ Y ) Proof.

1. φ(F) ∧ φ(G)

(A)

2. ♯(F) = ♯(G)

(A)

3. ∃x(x = ♯(F))

(2, T1)

4. ∃x(x = ♯(F))

(2, T1)

5. F ≈ F

(rifl. ≈)

6. ǫ(F) ∈ ♯(F)

(1, 5, Lemma 2.5)

7. ǫ(F) ∈ ♯(G)

(5, N3)

8. F ≈ G

(1, 7, Lemma 2.5)

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Theorem ♯(X) = ♯(Y ) → X ≈ Y Proof.

1. ♯(F) = ♯(G)
2. ∃x(x = ♯(F))
3. ∃x(x = ♯(F))
4. ǫ([[λx.∃X(x = ǫ(X) ∧ X ≈ F]) = ǫ([[λx.∃X(x = ǫ(X) ∧ X ≈ G])
5. Πx(([λx.∃X(x = ǫ(X) ∧ X ≈ F)](x) ↔ [λx.∃X(x = ǫ(X) ∧ X ≈ G)](x))

∧φ([λx.∃X(x = ǫ(X) ∧ X ≈ F])) ∧ φ([[λx.∃X(x = ǫ(X) ∧ X ≈ G])))

6. Πx(∃X(x = ǫ(X) ∧ X ≈ F) ↔ ∃X(x = ǫ(X) ∧ X ≈ G))
7. Πx(x = ǫ(H) ∧ H ≈ F ↔ x = ǫ(H) ∧ H ≈ G)
8. Πx(H ≈ F ↔ H ≈ G)
9. F ≈ G

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Bibliography: Antonelli, A and May R. (2005). Frege’s Other Program, Notre Dame Journal of Formal Logic, Vol. 46, 1, 1-17. Boccuni, F. (2010). Plural Grundgesetze, Studia Logica, 96, 2, 3015-330. Boccuni, F. (2011). On the Consistency of a Plural Theory of Frege’s Grundgesetze, Studia Logica, 97, 3, 329-345. Boolos, G. (1987). Saving Frege from the Contradiction, Proceedings of Aristotelian Society, 87, 137-151. Burgess, J. (2005). Fixing Frege. Princeton University Press, Princeton. Cocchiarella, N. B. (1992). Cantor’s power-set Theorem versus Frege’s double correlation Thesis, History and Philosophy of logic, 13, 179-201. Dummett, M. (1991). Frege, Philosophy of Mathematics, Oxford University Press, Oxford. Ferreira, F. (2018). Zigzag and Fregean Arithmetic, in Tahiri, H. (eds), The Philosophers and Mathematics, Springer, Cham, vol. 43, 81-100.

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Ferreira, F. and K. F. Wehmeier (2002). On the Consistency of the Fragment of Frege’s Grundgesetze. Journal of Philosophical Logic. Frege, G. (1903). Grundgesetze der Arithmetik, II, Verlag Hermann Pohle. Heck, R. (1996). The Consistency of Predicative Fragments CA1

1 of Frege’s

Grundgesetze der Arithmetik. History and Philosophy of Logic 17, 209–220. Payne, J. (2013). Abstraction relations need not be reflexive, Thought, 2, 137 - 147. Paseau, A. C. (2015). Did Frege commit a Cardinal Sin?, Analysis 75 (3), 379–386. Quine W. V. (1955). On Frege’s way out, Mind, 64, 145-159. Russell, B. (1903). On Some Difficulties in the Theory of Transfinite Numbers and Order Types, in Russell B. (1973). Essays in Analysys, New York. Uzquiano, G. (forthcoming). Impredicativity and Paradox. Wehmeier, K. F. (1999). Consistent Fragments of Grundgesetze and the Existence of Non-Logical Objects. Synthese 121(3), 309–328.

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