Nontransitive Approaches to Paradox and Compositional Principles of - - PowerPoint PPT Presentation

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Nontransitive Approaches to Paradox and Compositional Principles of Truth

Jonathan Dittrich LMU Munich, MCMP Jonathan.Dittrich@lmu.de 04.05.2017

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The argument

(1) Non-transitive theories render compositional principles of truth paradoxical in virtue of their notion of paradoxicality.

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Preliminaries Premise 1 Premise 2

The argument

(1) Non-transitive theories render compositional principles of truth paradoxical in virtue of their notion of paradoxicality. (2) If a theory renders compositional principles of truth paradoxical, it is inadequate.

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Preliminaries Premise 1 Premise 2

The argument

(1) Non-transitive theories render compositional principles of truth paradoxical in virtue of their notion of paradoxicality. (2) If a theory renders compositional principles of truth paradoxical, it is inadequate. (C) Non-transitive theories of truth are inadequate.

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Structure

1

Preliminaries

2

Premise 1

3

Premise 2

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Preliminaries

We work in the language L of FOL=, Q and a unary truth-predicate T

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Preliminaries

We work in the language L of FOL=, Q and a unary truth-predicate T #φ is the G¨

• del code of φ

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Preliminaries Premise 1 Premise 2

Preliminaries

We work in the language L of FOL=, Q and a unary truth-predicate T #φ is the G¨

• del code of φ

φ is the numeral of the G¨

• del code of φ

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Preliminaries Premise 1 Premise 2

Preliminaries

We work in the language L of FOL=, Q and a unary truth-predicate T #φ is the G¨

• del code of φ

φ is the numeral of the G¨

• del code of φ

We use a suitable proof system in sequent-calculus style that allows for Cut-elimination

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The operational rules

Γ, φ ⊢ ∆

¬R

Γ ⊢ ∆, ¬φ Γ ⊢ ∆, φ

¬L

Γ, ¬φ ⊢ ∆ Γ ⊢ ∆, φ, ψ

∨R

Γ ⊢ ∆, φ ∨ ψ Γ, φ ⊢ ∆ Γ′, ψ ⊢ ∆′

∨L

Γ, Γ′, φ ∨ ψ ⊢ ∆, ∆′ Γ ⊢ ∆, φ(y)

∀R

Γ ⊢ ∆, ∀x φ(x) Γ, φ(a) ⊢ ∆

∀L

Γ ∀x φ(x) ⊢ ∆

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Identity rules

a=a, Γ ⊢ ∆

=ID

Γ ⊢ ∆ a=b,φ(a),φ(b), Γ ⊢ ∆

=repL

a=b, φ(b), φ(b) Γ ⊢ ∆ a=b, Γ ⊢ ∆, φ(a),φ(b)

=repR

a=b, Γ ⊢ ∆, φ(b), φ(b)

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The structural rules

Reflexivity:

R

φ ⊢ φ

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The structural rules

Reflexivity:

R

φ ⊢ φ Weakening: Γ ⊢ ∆

WL

Γ, φ ⊢ ∆ Γ ⊢ ∆

WR

Γ ⊢ φ, ∆

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The structural rules

Reflexivity:

R

φ ⊢ φ Weakening: Γ ⊢ ∆

WL

Γ, φ ⊢ ∆ Γ ⊢ ∆

WR

Γ ⊢ φ, ∆ Contraction: Γ, φ, φ ⊢ ∆

CL

Γ, φ ⊢ ∆ Γ ⊢ φ, φ, ∆

CR

Γ ⊢ φ, ∆

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The structural rules

Reflexivity:

R

φ ⊢ φ Weakening: Γ ⊢ ∆

WL

Γ, φ ⊢ ∆ Γ ⊢ ∆

WR

Γ ⊢ φ, ∆ Contraction: Γ, φ, φ ⊢ ∆

CL

Γ, φ ⊢ ∆ Γ ⊢ φ, φ, ∆

CR

Γ ⊢ φ, ∆ Cut: Γ ⊢ φ, Π Σ, φ ⊢ Ω

Cut

Γ, Σ ⊢ Π, Ω

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Q1

Γ, 0=Sa ⊢ ∆ Γ, a=b, Sa=Sb ⊢ ∆

Q2

Γ, Sa=Sb ⊢ ∆ Γ, a=0 ⊢ ∆ Γ, a=Sy ⊢ ∆

Q3

Γ ⊢ ∆ Γ, a+0 = a ⊢ ∆

Q4

Γ ⊢ ∆

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Γ, a+Sb = S(a+b) ⊢ ∆

Q5

Γ ⊢ ∆ Γ,a×0=0 ⊢ ∆

Q6

Γ ⊢ ∆ Γ, a ×Sb=(a × b) + a ⊢ ∆

Q7

Γ ⊢ ∆

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Fully transparent truth-predicate T expressed by the T-rules: Γ ⊢ φ, ∆ Γ ⊢ T(φ), ∆ Γ ⊢ T(φ), ∆ Γ ⊢ φ, ∆ Γ, φ ⊢ ∆ Γ, T(φ) ⊢ ∆ Γ, T(φ) ⊢ ∆ Γ, φ ⊢ ∆

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The Liar(s): Weak Liar λ

There is a recursive diagonalisation function fd(#φ(x)) = #∀x(x = φ(x) → φ(x)) for any φ(x) with x free.

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The Liar(s): Weak Liar λ

There is a recursive diagonalisation function fd(#φ(x)) = #∀x(x = φ(x) → φ(x)) for any φ(x) with x free. We further know that Q includes a predicate D(x, y) that strongly represents this function.

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The Liar(s): Weak Liar λ

There is a recursive diagonalisation function fd(#φ(x)) = #∀x(x = φ(x) → φ(x)) for any φ(x) with x free. We further know that Q includes a predicate D(x, y) that strongly represents this function. This strong representation allows us to prove a weak diagonal lemma: Weak Diagonal Lemma ∀x(x = D(x, y) → φ(y) → ∀y(D(x, y) → φ(y)) ↔ φ(¯ n)

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The Liar(s): Weak Liar λ

By appropriate choice of φ, we get ∀x(x = D(x, y) → ¬T(y) → ∀y(D(x, y) → ¬T(y)) ↔ ¬T(¯ n)).

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The Liar(s): Weak Liar λ

By appropriate choice of φ, we get ∀x(x = D(x, y) → ¬T(y) → ∀y(D(x, y) → ¬T(y)) ↔ ¬T(¯ n)). By letting ∀x(x = D(x, y) → ¬T(y) → ∀y(D(x, y) → ¬T(y)) be λ, we get the usual The Liar λ ↔ ¬T(λ)

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The Liar(s): Weak Liar λ

By appropriate choice of φ, we get ∀x(x = D(x, y) → ¬T(y) → ∀y(D(x, y) → ¬T(y)) ↔ ¬T(¯ n)). By letting ∀x(x = D(x, y) → ¬T(y) → ∀y(D(x, y) → ¬T(y)) be λ, we get the usual The Liar λ ↔ ¬T(λ) By the invertibility of our rules, we know that also ⊢ λ, T(λ) and λ, T(λ) ⊢ are provable.

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Proving absurdity from λ

λ, T(λ) ⊢ T(λ), T(λ) ⊢

CL

T(λ) ⊢ ⊢ λ, T(λ) ⊢ T(λ), T(λ)

CR

⊢ T(λ)

Cut

⊢

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Proving absurdity from λ

λ, T(λ) ⊢ T(λ), T(λ) ⊢

CL

T(λ) ⊢ ⊢ λ, T(λ) ⊢ T(λ), T(λ)

CR

⊢ T(λ)

Cut

⊢ But the empty sequent is not derivable without Cut! This motivates non-transitive solutions which give up Cut (see e.g. Ripley, Tennant).

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Non-transitive theories: Fully transparent, classical (yet inconsistent but non-trivial) theory of truth.

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Non-transitive theories: Fully transparent, classical (yet inconsistent but non-trivial) theory of truth. Thesis: Although non-transitive theories can deal with a fully transparent truth-predicate, they are inadequate with respect to stronger demands to the truth-predicate: The compositional principles of truth.

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Preliminaries Premise 1 Premise 2

The argument

(1) Non-transitive theories render compositional principles of truth paradoxical in virtue of their notion of paradoxicality.

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Preliminaries Premise 1 Premise 2

The argument

(1) Non-transitive theories render compositional principles of truth paradoxical in virtue of their notion of paradoxicality. (2) If a theory renders compositional principles of truth paradoxical, it is inadequate.

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Preliminaries Premise 1 Premise 2

The argument

(1) Non-transitive theories render compositional principles of truth paradoxical in virtue of their notion of paradoxicality. (2) If a theory renders compositional principles of truth paradoxical, it is inadequate. (C) Non-transitive theories of truth are inadequate.

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The compositional principles of truth

(Neg) ∀x(Sent(x) → (T(¬ . x) ↔ ¬T(x))) (Con) ∀x∀y(Sent(x) ∧ Sent(y) → (T(x∧ . y) ↔ T(x) ∧ T(y))) (Dis) ∀x∀y(Sent(x) ∧ Sent(y) → (T(x∨ . y) ↔ T(x) ∨ T(y))) (Cond) ∀x∀y(Sent(x) ∧ Sent(y) → (T(x→ . y) ↔ T(x) → T(y)))

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The compositional principles of truth

(Neg) ∀x(Sent(x) → (T(¬ . x) ↔ ¬T(x))) (Con) ∀x∀y(Sent(x) ∧ Sent(y) → (T(x∧ . y) ↔ T(x) ∧ T(y))) (Dis) ∀x∀y(Sent(x) ∧ Sent(y) → (T(x∨ . y) ↔ T(x) ∨ T(y))) (Cond) ∀x∀y(Sent(x) ∧ Sent(y) → (T(x→ . y) ↔ T(x) → T(y))) Note that although their instances are provable, the universally quantified principles are not. However, they are typically added to formal theories of truth.

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In support of (1)

By its solution to the paradoxes in terms of Cut-inadmissibility, NT is committed to the following notion of paradoxicality:

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In support of (1)

By its solution to the paradoxes in terms of Cut-inadmissibility, NT is committed to the following notion of paradoxicality: The nontransitivist’s notion of paradoxicality (NTP): φ in sequents Γ ⊢ φ, ∆ and Γ′, φ ⊢ ∆′ is paradoxical relative to a rule system R iff an application of Cut to the sequents Γ ⊢ φ, ∆ and Γ′, φ ⊢ ∆′ is not eliminable with respect to R.

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Note that although the empty sequent is not provable without Cut, the Liar and its negation remain provable: ⊢ λ, T(λ) ⊢ T(λ), T(λ) ⊢ T(λ) λ, T(λ) ⊢ T(λ), T(λ) ⊢ T(λ) ⊢ ⊢ ¬T(λ)

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In support of (1)

These proofs can then be used to show that various compositional principles are paradoxical according to NTP:

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In support of (1)

These proofs can then be used to show that various compositional principles are paradoxical according to NTP: λ, T(λ) ⊢ λ, λ ⊢ λ ⊢ ⊢ ¬λ ⊢ T(¬λ) ⊢ λ, T(λ) ⊢ T(λ), T(λ) ⊢ T(λ) ¬T(λ) ⊢ T(¬λ) → ¬T(λ) ⊢ ∀x(Sent(x) → (T(¬ . x) → ¬T(x))) ⊢

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In support of (1)

Given that we add the respective compositional principle to our non-transitive theory of truth:

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In support of (1)

Given that we add the respective compositional principle to our non-transitive theory of truth: ⊢ ∀x(Sent(x) → (T(¬ . x) → ¬T(x)))

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In support of (1)

Given that we add the respective compositional principle to our non-transitive theory of truth: ⊢ ∀x(Sent(x) → (T(¬ . x) → ¬T(x))) We could then Cut on it to infer the empty sequent. As this Cut is not eliminable, the compositional principle counts as paradoxical by NTP.

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In support of (2)

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In support of (2)

A notion of paradoxicality P is adequate iff it satisfies Universality: If φ is a paradox, then P applies to φ.

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In support of (2)

A notion of paradoxicality P is adequate iff it satisfies Universality: If φ is a paradox, then P applies to φ. Innocence: If P applies to φ, then φ is paradoxical.

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In support of (2)

A notion of paradoxicality P is adequate iff it satisfies Universality: If φ is a paradox, then P applies to φ. Innocence: If P applies to φ, then φ is paradoxical. Intuitively, the compositional principles are not paradoxical.

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In support of (2)

A notion of paradoxicality P is adequate iff it satisfies Universality: If φ is a paradox, then P applies to φ. Innocence: If P applies to φ, then φ is paradoxical. Intuitively, the compositional principles are not paradoxical. Thus NTP is inadequate as it violates Innocence wrt the compositional principles.

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Thank you!

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