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A brief introduction to RTT An approach to designing paradoxes

Paradoxes and Revision Theory of Truth

Ming HsiungSouth China Normal University

Week of Mathematical PhilosophyPeking University2019.6.22-6.26

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Main Target

A brief introduction to revision theory of truth

A general approach to designing paradoxesvia revision theory of truth

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Main Target

A brief introduction to revision theory of truth

A general approach to designing paradoxesvia revision theory of truth

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Designer babies

Just as we can (but we should not at least so far) create ababy with preferred traits by genetic selection engineering,

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Designer paradoxes

we can also create a paradox (or any other similarself-referential object) with certain features by use of therevision-theoretic techniques!

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In the field of truth theories, the paradoxes that you hadseen before are just a tip of an iceberg.

This will give you a full view of truth-theoretical paradoxes.

(1) (3)

(2) (4)

(5) …

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In this talk, we only assume

some basic syntax and semantics of first-order logic (forinstance, models and satisfaction),

and some elementary facts about the ordinals (forinstance, successor ordinals and limit ordinals).

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Contents

1 A brief introduction to RTTArithmetic language with T

Revision sequenceAn application of revision sequence

2 An approach to designing paradoxesPrimary period and critical pointDesigner paradoxes

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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence

Contents

1 A brief introduction to RTTArithmetic language with T

Revision sequenceAn application of revision sequence

2 An approach to designing paradoxes

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Revision theory of truth

The revision theory of truth (RTT for short) was put forwardby Gupta (1982) and Herzberger (1982) independently.

The core concept of RTT was called the ‘revision’procedure by Gupta (1982).

The name of RTT is largely due to Belnap’s 1982 paper:Gupta’s Rule of Revision Theory of Truth, JPL, 11(1),103-116.

The title of Gupta and Belnap’s 1993 book is The revisiontheory of truth.

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The revision theory of truth (RTT for short) was put forwardby Gupta (1982) and Herzberger (1982) independently.

The core concept of RTT was called the ‘revision’procedure by Gupta (1982).

The name of RTT is largely due to Belnap’s 1982 paper:Gupta’s Rule of Revision Theory of Truth, JPL, 11(1),103-116.

The title of Gupta and Belnap’s 1993 book is The revisiontheory of truth.

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Herzberger called his theory ‘naive semantics’. When I firstsee this name, my impression is: is this theorysimple-minded or even stupid?

Actually, this theory is ‘both philosophically illuminating andmathematically elegant’. (McGee 1996)

Personally, this is my favorite theory of truth.

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Semantic theories of truth

RTT is a semantic theory of truth, in which people usually‘attempt to characterize truth by defining a suitableinterpretation of the truth predicate in a semanticmetalanguage’. (Fisher, Halbach and Kriener 2015)There are other semantic theories of truth, for instance:

Tarski (1935)Kripke (1975)Field (2002)Leitgeb (2005)……

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Language LT

LT : the language obtained from the first-order arithmeticlanguage by adding T as a unary predicate symbol.

⟨N, X⟩: a ground model for LT , in which N is the standardmodel for Peano arithmetic PA, and X ⊆ N is theextension of T .

V⟨N,X⟩: the valuation in the ground model ⟨N, X⟩.

A, B, δ: sentences of LT , unless otherwise claimed.

V⟨N,X⟩(A) = 1 (resp. 0): A is true (resp. false) in theground model ⟨N, X⟩.

A ≡N B (abbrev. A ≡ B): V⟨N,X⟩(A ↔ B) = 1 for any X.

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Language LT

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Language LT

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Language LT

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Language LT

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Language LT

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⌜A⌝: Gödel number of A.

⌜A⌝: the corresponding numeral.

If no confusion arises, A = ⌜A⌝ = ⌜A⌝. For instance,we will use A⌜δ⌝ rather than A

(⌜δ⌝

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(⌜δ⌝

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(⌜δ⌝

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We can construct a sentence λ of LT such that

λ ≡ ¬T ⌜λ⌝.

The liar sentence

Sentence (λ) is untrue (λ)

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We can construct a sentence λ of LT such that

λ ≡ ¬T ⌜λ⌝.

The liar sentence

Sentence (λ) is untrue (λ)

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An Example

Wen’s paradox (2003)

sentence (δ2) is true, but sentence (δ3) is false, (δ1)

either sentence (δ1) is false, or sentence (δ3) is true, (δ2)

both (δ1) and (δ2) are true. (δ3)

Formalization of Wen’s paradox in LTδ1 ≡ T ⌜δ2⌝ ∧ ¬T ⌜δ3⌝δ2 ≡ ¬T ⌜δ1⌝ ∨ T ⌜δ3⌝δ3 ≡ T ⌜δ1⌝ ∧ T ⌜δ2⌝

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All paradoxes in this talk can be formalized by theso-called ‘diagonal method’ in LT .

Don’t worry about this method, if you do not know it.

You just need to accept that these paradoxes do exist inLT .

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A sequence of hypotheses

A hypothesis (or assignment) is a function from the set ofsentences to the truth-value set {1, 0}.

Recall: Σ is identified with {⌜A⌝|A ∈ Σ}.

We define two sequences as follows:

h0 Γ0 = {A|h0(A) = 1}h1 = V⟨N,Γ0⟩ Γ1 = {A|h1(A) = 1}

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h0 Γ0 = {A|h0(A) = 1}h1 = V⟨N,Γ0⟩ Γ1 = {A|h1(A) = 1}

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h0 Γ0 = {A|h0(A) = 1}h1 = V⟨N,Γ0⟩ Γ1 = {A|h1(A) = 1}

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h0 Γ0 = {A|h0(A) = 1}h1 = V⟨N,Γ0⟩ Γ1 = {A|h1(A) = 1}

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h0 Γ0 = {A|h0(A) = 1}h1 = V⟨N,Γ0⟩ Γ1 = {A|h1(A) = 1}

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h0 Γ0 = {A|h0(A) = 1}h1 = V⟨N,Γ0⟩ Γ1 = {A|h1(A) = 1}

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h0 Γ0 = {A|h0(A) = 1}h1 = V⟨N,Γ0⟩ Γ1 = {A|h1(A) = 1}

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h0 Γ0 = {A|h0(A) = 1}h1 = V⟨N,Γ0⟩ Γ1 = {A|h1(A) = 1}

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h0 Γ0 = {A|h0(A) = 1}h1 = V⟨N,Γ0⟩ Γ1 = {A|h1(A) = 1}

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h0 Γ0 = {A|h0(A) = 1}h1 = V⟨N,Γ0⟩ Γ1 = {A|h1(A) = 1}

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h0 Γ0 = {A|h0(A) = 1}h1 = V⟨N,Γ0⟩ Γ1 = {A|h1(A) = 1}

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h0 Γ0 = {A|h0(A) = 1}h1 = V⟨N,Γ0⟩ Γ1 = {A|h1(A) = 1}

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h0 Γ0 = {A|h0(A) = 1}h1 = V⟨N,Γ0⟩ Γ1 = {A|h1(A) = 1}h2 = V⟨N,Γ1⟩ Γ2 = {A|h2(A) = 1}. . . . . . . . . . . . . . . . . .

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Starting from the initial hypothesis h0, we obtain thefollowing two sequences:

What about hω?

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Starting from the initial hypothesis h0, we obtain thefollowing two sequences:

What about hω?

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Example

Let h0, h1, h2, …be a sequence starting from the emptyhypothesis (i.e., h0(A) = 0 for all A).

If hn(A) = 1, we will say A is true (1) at stage n (of thissequence), otherwise A is false (0) at stage n.

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Example

stage 0 = 0

h0(0 = 0) = 0 0 0

h1(0 = 0) = V⟨N,Γ0⟩(0 = 0)

= 1 1 1

h2(0 = 0) = V⟨N,Γ1⟩(0 = 0)

= 1 2 1

. . . . . . . . .

0 1 2 3 4 … ω

0 = 0 0 1 1 1 1 … ?

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Example

stage 0 = 0

h0(0 = 0) = 0 0 0

h1(0 = 0) = V⟨N,Γ0⟩(0 = 0)

= 1 1 1

h2(0 = 0) = V⟨N,Γ1⟩(0 = 0)

= 1 2 1

. . . . . . . . .

0 1 2 3 4 … ω

0 = 0 0 1 1 1 1 … ?

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Example

stage 0 = 0

h0(0 = 0) = 0 0 0

h1(0 = 0) = V⟨N,Γ0⟩(0 = 0)

= 1 1 1

h2(0 = 0) = V⟨N,Γ1⟩(0 = 0)

= 1 2 1

. . . . . . . . .

0 1 2 3 4 … ω

0 = 0 0 1 1 1 1 … ?

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Example

stage 0 = 0

h0(0 = 0) = 0 0 0

h1(0 = 0) = V⟨N,Γ0⟩(0 = 0)

= 1 1 1

h2(0 = 0) = V⟨N,Γ1⟩(0 = 0)

= 1 2 1

. . . . . . . . .

0 1 2 3 4 … ω

0 = 0 0 1 1 1 1 … ?

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Example

stage 0 = 0

h0(0 = 0) = 0 0 0

h1(0 = 0) = V⟨N,Γ0⟩(0 = 0)

= 1 1 1

h2(0 = 0) = V⟨N,Γ1⟩(0 = 0)

= 1 2 1

. . . . . . . . .

0 1 2 3 4 … ω

0 = 0 0 1 1 1 1 … ?

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Example

stage 0 = 0

h0(0 = 0) = 0 0 0

h1(0 = 0) = V⟨N,Γ0⟩(0 = 0)

= 1 1 1

h2(0 = 0) = V⟨N,Γ1⟩(0 = 0)

= 1 2 1

. . . . . . . . .

0 1 2 3 4 … ω

0 = 0 0 1 1 1 1 … ?

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Example

stage 0 = 0

h0(0 = 0) = 0 0 0

h1(0 = 0) = V⟨N,Γ0⟩(0 = 0)

= 1 1 1

h2(0 = 0) = V⟨N,Γ1⟩(0 = 0)

= 1 2 1

. . . . . . . . .

0 1 2 3 4 … ω

0 = 0 0 1 1 1 1 … ?

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Example

stage 0 = 0

h0(0 = 0) = 0 0 0

h1(0 = 0) = V⟨N,Γ0⟩(0 = 0)

= 1 1 1

h2(0 = 0) = V⟨N,Γ1⟩(0 = 0)

= 1 2 1

. . . . . . . . .

0 1 2 3 4 … ω

0 = 0 0 1 1 1 1 … ?

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Example

stage 0 = 0

h0(0 = 0) = 0 0 0

h1(0 = 0) = V⟨N,Γ0⟩(0 = 0)

= 1 1 1

h2(0 = 0) = V⟨N,Γ1⟩(0 = 0)

= 1 2 1

. . . . . . . . .

0 1 2 3 4 … ω

0 = 0 0 1 1 1 1 … ?

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Example

stage 0 = 0

h0(0 = 0) = 0 0 0

h1(0 = 0) = V⟨N,Γ0⟩(0 = 0)

= 1 1 1

h2(0 = 0) = V⟨N,Γ1⟩(0 = 0)

= 1 2 1

. . . . . . . . .

0 1 2 3 4 … ω

0 = 0 0 1 1 1 1 … ?

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Example

stage 0 = 0

h0(0 = 0) = 0 0 0

h1(0 = 0) = V⟨N,Γ0⟩(0 = 0)

= 1 1 1

h2(0 = 0) = V⟨N,Γ1⟩(0 = 0)

= 1 2 1

. . . . . . . . .

0 1 2 3 4 … ω

0 = 0 0 1 1 1 1 … ?

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Example

stage 0 = 0

h0(0 = 0) = 0 0 0

h1(0 = 0) = V⟨N,Γ0⟩(0 = 0)

= 1 1 1

h2(0 = 0) = V⟨N,Γ1⟩(0 = 0)

= 1 2 1

. . . . . . . . .

0 1 2 3 4 … ω

0 = 0 0 1 1 1 1 … ?

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Example

stage T ⌜0 = 0⌝h0(T ⌜0 = 0⌝) = 0 0 0

h1(T ⌜0 = 0⌝) = V⟨N,Γ0⟩(T ⌜0 = 0⌝)= 0 (∵ 0 = 0 /∈ Γ0) 1 0

h2(T ⌜0 = 0⌝) = V⟨N,Γ1⟩(T ⌜0 = 0⌝)= 1 (∵ 0 = 0 ∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

T ⌜0 = 0⌝ 0 0 1 1 1 … ?

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Example

stage T ⌜0 = 0⌝h0(T ⌜0 = 0⌝) = 0 0 0

h1(T ⌜0 = 0⌝) = V⟨N,Γ0⟩(T ⌜0 = 0⌝)= 0 (∵ 0 = 0 /∈ Γ0) 1 0

h2(T ⌜0 = 0⌝) = V⟨N,Γ1⟩(T ⌜0 = 0⌝)= 1 (∵ 0 = 0 ∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

T ⌜0 = 0⌝ 0 0 1 1 1 … ?

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Example

stage T ⌜0 = 0⌝h0(T ⌜0 = 0⌝) = 0 0 0

h1(T ⌜0 = 0⌝) = V⟨N,Γ0⟩(T ⌜0 = 0⌝)= 0 (∵ 0 = 0 /∈ Γ0) 1 0

h2(T ⌜0 = 0⌝) = V⟨N,Γ1⟩(T ⌜0 = 0⌝)= 1 (∵ 0 = 0 ∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

T ⌜0 = 0⌝ 0 0 1 1 1 … ?

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Example

stage T ⌜0 = 0⌝h0(T ⌜0 = 0⌝) = 0 0 0

h1(T ⌜0 = 0⌝) = V⟨N,Γ0⟩(T ⌜0 = 0⌝)= 0 (∵ 0 = 0 /∈ Γ0) 1 0

h2(T ⌜0 = 0⌝) = V⟨N,Γ1⟩(T ⌜0 = 0⌝)= 1 (∵ 0 = 0 ∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

T ⌜0 = 0⌝ 0 0 1 1 1 … ?

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Example

stage T ⌜0 = 0⌝h0(T ⌜0 = 0⌝) = 0 0 0

h1(T ⌜0 = 0⌝) = V⟨N,Γ0⟩(T ⌜0 = 0⌝)= 0 (∵ 0 = 0 /∈ Γ0) 1 0

h2(T ⌜0 = 0⌝) = V⟨N,Γ1⟩(T ⌜0 = 0⌝)= 1 (∵ 0 = 0 ∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

T ⌜0 = 0⌝ 0 0 1 1 1 … ?

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Example

stage T ⌜0 = 0⌝h0(T ⌜0 = 0⌝) = 0 0 0

h1(T ⌜0 = 0⌝) = V⟨N,Γ0⟩(T ⌜0 = 0⌝)= 0 (∵ 0 = 0 /∈ Γ0) 1 0

h2(T ⌜0 = 0⌝) = V⟨N,Γ1⟩(T ⌜0 = 0⌝)= 1 (∵ 0 = 0 ∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

T ⌜0 = 0⌝ 0 0 1 1 1 … ?

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Example

stage T ⌜0 = 0⌝h0(T ⌜0 = 0⌝) = 0 0 0

h1(T ⌜0 = 0⌝) = V⟨N,Γ0⟩(T ⌜0 = 0⌝)= 0 (∵ 0 = 0 /∈ Γ0) 1 0

h2(T ⌜0 = 0⌝) = V⟨N,Γ1⟩(T ⌜0 = 0⌝)= 1 (∵ 0 = 0 ∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

T ⌜0 = 0⌝ 0 0 1 1 1 … ?

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Example

stage T ⌜0 = 0⌝h0(T ⌜0 = 0⌝) = 0 0 0

h1(T ⌜0 = 0⌝) = V⟨N,Γ0⟩(T ⌜0 = 0⌝)= 0 (∵ 0 = 0 /∈ Γ0) 1 0

h2(T ⌜0 = 0⌝) = V⟨N,Γ1⟩(T ⌜0 = 0⌝)= 1 (∵ 0 = 0 ∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

T ⌜0 = 0⌝ 0 0 1 1 1 … ?

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Example

stage T ⌜0 = 0⌝h0(T ⌜0 = 0⌝) = 0 0 0

h1(T ⌜0 = 0⌝) = V⟨N,Γ0⟩(T ⌜0 = 0⌝)= 0 (∵ 0 = 0 /∈ Γ0) 1 0

h2(T ⌜0 = 0⌝) = V⟨N,Γ1⟩(T ⌜0 = 0⌝)= 1 (∵ 0 = 0 ∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

T ⌜0 = 0⌝ 0 0 1 1 1 … ?

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Example

stage T ⌜0 = 0⌝h0(T ⌜0 = 0⌝) = 0 0 0

h1(T ⌜0 = 0⌝) = V⟨N,Γ0⟩(T ⌜0 = 0⌝)= 0 (∵ 0 = 0 /∈ Γ0) 1 0

h2(T ⌜0 = 0⌝) = V⟨N,Γ1⟩(T ⌜0 = 0⌝)= 1 (∵ 0 = 0 ∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

T ⌜0 = 0⌝ 0 0 1 1 1 … ?

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Example

stage T ⌜0 = 0⌝h0(T ⌜0 = 0⌝) = 0 0 0

h1(T ⌜0 = 0⌝) = V⟨N,Γ0⟩(T ⌜0 = 0⌝)= 0 (∵ 0 = 0 /∈ Γ0) 1 0

h2(T ⌜0 = 0⌝) = V⟨N,Γ1⟩(T ⌜0 = 0⌝)= 1 (∵ 0 = 0 ∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

T ⌜0 = 0⌝ 0 0 1 1 1 … ?

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Example

stage T ⌜0 = 0⌝h0(T ⌜0 = 0⌝) = 0 0 0

h1(T ⌜0 = 0⌝) = V⟨N,Γ0⟩(T ⌜0 = 0⌝)= 0 (∵ 0 = 0 /∈ Γ0) 1 0

h2(T ⌜0 = 0⌝) = V⟨N,Γ1⟩(T ⌜0 = 0⌝)= 1 (∵ 0 = 0 ∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

T ⌜0 = 0⌝ 0 0 1 1 1 … ?

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Example

0 1 2 3 4 … ω

0 = 0 0 1 1 1 1 … ?T ⌜0 = 0⌝ 0 0 1 1 1 … ?

0 = 0 is stably true before stage ω: 0 = 0 is always trueafter some stage n onwards (but before stage ω).

T ⌜0 = 0⌝ is also stably true before stage ω in the samesense.

Some sentences are stably false before stage ω.

0 1 2 3 4 … ω

¬T⌜0 = 0⌝ 0 1 0 0 0 … ?

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Example

0 1 2 3 4 … ω

0 = 0 0 1 1 1 1 … ?T ⌜0 = 0⌝ 0 0 1 1 1 … ?

0 = 0 is stably true before stage ω: 0 = 0 is always trueafter some stage n onwards (but before stage ω).

T ⌜0 = 0⌝ is also stably true before stage ω in the samesense.

Some sentences are stably false before stage ω.

0 1 2 3 4 … ω

¬T⌜0 = 0⌝ 0 1 0 0 0 … ?

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Stable before a limit stage

Gupta 1982; Herzberger 1982; Belnap 1982A sentence A is stably true before stage ω: A is alwaystrue after some stage n onwards (but before stage ω).

A sentence A is stably true before a limit stage γ, if A isalways true after some stage α < γ onwards (but beforestage γ).

A sentence A is stably false before a limit stage γ, if ……

A sentence A is stable before a limit stage γ, if it is eitherstably true or stably false before stage γ; otherwise, A isunstable before stage γ.

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The limit rule for stable case

0 1 2 3 4 … ω

0 = 0 0 1 1 1 1 … 1T ⌜0 = 0⌝ 0 0 1 1 1 … 1¬T⌜0 = 0⌝ 0 1 0 0 0 … 0

Gupta 1982; Herzberger 1982; Belnap 1982a sentence A will be true (resp. false) at a limit stage γ, ifA is stably true (resp. stably false) before stage γ.

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Example

Recall: λ is the liar sentence. And λ satisfies: λ ≡ ¬T ⌜λ⌝.The initial hypothesis h0 satisfies:

h0(A) = 1, iff A is λ.

stage λ

h0(λ) = 1 0 1

h1(λ) = V⟨N,Γ0⟩(¬T ⌜λ⌝)= 0 (∵ λ ∈ Γ0) 1 0

h2(λ) = V⟨N,Γ1⟩(¬T ⌜λ⌝)= 1 (∵ λ /∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

λ 1 0 1 0 1 … ?

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Example

stage λ

h0(λ) = 1 0 1

h1(λ) = V⟨N,Γ0⟩(¬T ⌜λ⌝)= 0 (∵ λ ∈ Γ0) 1 0

h2(λ) = V⟨N,Γ1⟩(¬T ⌜λ⌝)= 1 (∵ λ /∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

λ 1 0 1 0 1 … ?

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Example

stage λ

h0(λ) = 1 0 1

h1(λ) = V⟨N,Γ0⟩(¬T ⌜λ⌝)= 0 (∵ λ ∈ Γ0) 1 0

h2(λ) = V⟨N,Γ1⟩(¬T ⌜λ⌝)= 1 (∵ λ /∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

λ 1 0 1 0 1 … ?

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Example

stage λ

h0(λ) = 1 0 1

h1(λ) = V⟨N,Γ0⟩(¬T ⌜λ⌝)= 0 (∵ λ ∈ Γ0) 1 0

h2(λ) = V⟨N,Γ1⟩(¬T ⌜λ⌝)= 1 (∵ λ /∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

λ 1 0 1 0 1 … ?

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Example

stage λ

h0(λ) = 1 0 1

h1(λ) = V⟨N,Γ0⟩(¬T ⌜λ⌝)= 0 (∵ λ ∈ Γ0) 1 0

h2(λ) = V⟨N,Γ1⟩(¬T ⌜λ⌝)= 1 (∵ λ /∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

λ 1 0 1 0 1 … ?

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Example

stage λ

h0(λ) = 1 0 1

h1(λ) = V⟨N,Γ0⟩(¬T ⌜λ⌝)= 0 (∵ λ ∈ Γ0) 1 0

h2(λ) = V⟨N,Γ1⟩(¬T ⌜λ⌝)= 1 (∵ λ /∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

λ 1 0 1 0 1 … ?

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Example

stage λ

h0(λ) = 1 0 1

h1(λ) = V⟨N,Γ0⟩(¬T ⌜λ⌝)= 0 (∵ λ ∈ Γ0) 1 0

h2(λ) = V⟨N,Γ1⟩(¬T ⌜λ⌝)= 1 (∵ λ /∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

λ 1 0 1 0 1 … ?

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Example

stage λ

h0(λ) = 1 0 1

h1(λ) = V⟨N,Γ0⟩(¬T ⌜λ⌝)= 0 (∵ λ ∈ Γ0) 1 0

h2(λ) = V⟨N,Γ1⟩(¬T ⌜λ⌝)= 1 (∵ λ /∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

λ 1 0 1 0 1 … ?

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Example

stage λ

h0(λ) = 1 0 1

h1(λ) = V⟨N,Γ0⟩(¬T ⌜λ⌝)= 0 (∵ λ ∈ Γ0) 1 0

h2(λ) = V⟨N,Γ1⟩(¬T ⌜λ⌝)= 1 (∵ λ /∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

λ 1 0 1 0 1 … ?

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Example

stage λ

h0(λ) = 1 0 1

h1(λ) = V⟨N,Γ0⟩(¬T ⌜λ⌝)= 0 (∵ λ ∈ Γ0) 1 0

h2(λ) = V⟨N,Γ1⟩(¬T ⌜λ⌝)= 1 (∵ λ /∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

λ 1 0 1 0 1 … ?

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Example

stage λ

h0(λ) = 1 0 1

h1(λ) = V⟨N,Γ0⟩(¬T ⌜λ⌝)= 0 (∵ λ ∈ Γ0) 1 0

h2(λ) = V⟨N,Γ1⟩(¬T ⌜λ⌝)= 1 (∵ λ /∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

λ 1 0 1 0 1 … ?

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Example

stage λ

h0(λ) = 1 0 1

h1(λ) = V⟨N,Γ0⟩(¬T ⌜λ⌝)= 0 (∵ λ ∈ Γ0) 1 0

h2(λ) = V⟨N,Γ1⟩(¬T ⌜λ⌝)= 1 (∵ λ /∈ Γ1) 2 1

. . . . . . . . .

0 1 2 3 4 … ω

λ 1 0 1 0 1 … ?

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Example

0 1 2 3 4 … ω

λ 1 0 1 0 1 … ?T ⌜λ⌝ 0 1 0 1 0 … ?

Seeing that both of λ and T ⌜λ⌝ are unstable before stageω, we ask: what shall be their truth values at stage ω?

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The limit rule for unstable case: choice 1

0 1 2 3 4 5 6 … ω

λ 1 0 1 0 1 0 1 … 0T ⌜λ⌝ 0 1 0 1 0 1 0 … 0

Herzberger 1982a sentence will always be false at a limit stage γ, if thatsentence is unstable before that limit stage.

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0 1 2 3 4 5 6 … ω

λ 1 0 1 0 1 0 1 … 1T ⌜λ⌝ 0 1 0 1 0 1 0 … 0

Gupta 1982the truth-value of a sentence at a limit stage will always bethe same as that at the initial stage, if that sentence isunstable before that limit stage.

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0 1 2 3 4 5 6 … ω

λ 1 0 1 0 1 0 1 … *T ⌜λ⌝ 0 1 0 1 0 1 0 … *

Belnap 1982The truth value of a sentence at a limit stage can bechosen as you like, if that sentence is unstable before thatlimit stage.

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There are other limit rules for unstable case.

See Yaqūb, Aladdin Mahmūd (1993). The Liar Speaks the Truth:A Defense of the Revision Theory of Truth. Oxford University Press.

I give an improved limit rule (an improvement of Gupta limit rule)in my paper.

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The definition of hω

If applying Gupta limit rule, we can define

hω(A) =

1 if A is stably true before stage ω,0 if A is stably false before stage ω,

h0(A) if A is unstable before stage ω.

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Revision sequence

Starting from the initial hypothesis h0, we obtain thefollowing two arbitrarily long sequences:

hω Γω = {A|hω(A) = 1}hω+1 = V⟨N,Γω⟩ Γω+1 = {A|hω+1(A) = 1}. . . . . . . . . . . . . . . . . .

According to the limit rule we apply, the sequence h0, h1,…, can be called the Gupta (Herzberger) revisionsequence starting from h0.

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Revision sequence

Starting from the initial hypothesis h0, we obtain thefollowing two arbitrarily long sequences:

hω Γω = {A|hω(A) = 1}hω+1 = V⟨N,Γω⟩ Γω+1 = {A|hω+1(A) = 1}. . . . . . . . . . . . . . . . . .

According to the limit rule we apply, the sequence h0, h1,…, can be called the Gupta (Herzberger) revisionsequence starting from h0.

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Two applications of revision sequence

Apply to the study of the truth predicate

Apply to the study of paradoxes

I will focus on the second application.

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Revision period

(Herzberger JPL-1982) Let ∆ be a set of sentences, andlet ⟨h0, h1, . . .⟩ be a revision sequence. An ordinal π ≥ 1 isa (revision) period of ∆ in the sequence ⟨h0, h1, . . .⟩, ifthere exists a non-zero ordinal π and an ordinal θ such thatfor all A ∈ ∆, whenever α ≥ β, we always havehα+π(A) = hα(A).

(Herzberger JPL-1982) fundamental period(icity): ......

(Herzberger JPL-1982) stabilization point: ......

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Example

0 1 2 3 4 5 6 … ω …λ 1 0 1 0 1 0 1 … 1 …

In the above revision sequence, we have:

2 is a period of λ: hα+2(λ) = hα(λ) for all α.

Any positive multiple of 2 is also a period of λ.

2 is the smallest one among all the periods of λ, which iscalled the fundamental period(icity).

stage 0 is the stabilization point of λ, that is, the smalleststage in which λ becomes periodic.

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Example

0 1 2 3 4 5 6 … ω …λ 1 0 1 0 1 0 1 … 1 …

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Example

0 1 2 3 4 5 6 … ω …λ 1 0 1 0 1 0 1 … 1 …

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Example

0 1 2 3 4 5 6 … ω …λ 1 0 1 0 1 0 1 … 1 …

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Example

0 1 2 3 4 5 6 … ω …λ 1 0 1 0 1 0 1 … 1 …

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Example

0 1 2 3 4 5 6 … ω …¬T ⌜0 = 0⌝ 0 1 0 0 0 0 0 … 0 …

1 is a period of ¬T ⌜0 = 0⌝:hα+1(¬T ⌜0 = 0⌝) = hα(¬T ⌜0 = 0⌝) for all α ≥ 2.

In other words, the truth value of ¬T ⌜0 = 0⌝ becomesfixed after some stage.

2 is the stabilization point of ¬T ⌜0 = 0⌝.

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Example

0 1 2 3 4 5 6 … ω …¬T ⌜0 = 0⌝ 0 1 0 0 0 0 0 … 0 …

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Example

0 1 2 3 4 5 6 … ω …¬T ⌜0 = 0⌝ 0 1 0 0 0 0 0 … 0 …

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Paradoxicality

A sentence is paradoxical, if 1 is never aperiod of it in any revision sequence. (Gupta1982; Herzberger 1982)

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The 2-cycle liar paradox

John Buridan, 1300–1362

sentence (λ1) is false (λ0)

sentence (λ0) is true (λ1)

{λ0 ≡ ¬T ⌜λ1⌝λ1 ≡ T ⌜λ0⌝

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John Buridan, 1300–1362

sentence (λ1) is false (λ0)

{λ0 ≡ ¬T ⌜λ1⌝λ1 ≡ T ⌜λ0⌝

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Here is a quick way to compute the truth values of λ0 andλ1 at any stage:{

λ0 ≡ ¬T ⌜λ1⌝λ1 ≡ T ⌜λ0⌝

{hn+1(λ0) = ¬hn(λ1)

hn+1(λ1) = hn(λ0)

Just kick out the truth predicate!

the truth value of λ0 at a stage is the inverse value of λ1 atthe preceding stage;

the truth value of λ1 at a stage is the same as that of λ0 atthe preceding stage.

We will appeal to similar facts again and again. See(Hsiung 2017) for the proof of ‘⇒’ (for the general case).

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{hn+1(λ0) = ¬hn(λ1)

hn+1(λ1) = hn(λ0)

0 1 2 3 4 5 … ω ω + 1 …λ0 0 1 1 0 0 1 … 0 1 …λ1 0 0 1 1 0 0 … 0 0 …

In the above revision sequence,

4 is the fundamental period of the 2-cycle liar paradox.

0 is the stabilization point of the 2-cycle liar paradox.

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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes

Contents

1 A brief introduction to RTT

2 An approach to designing paradoxesPrimary period and critical pointDesigner paradoxes

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From now on, when we say a revision sequence, wealways mean a revision sequence whose limit rule isGupta limit rule.

We actually need to employ a stronger limit rule forthe present purpose (see my paper).

Let ∆ be a set of sentences. By a period of ∆, wemean a period of ∆ in some revision sequence.Similarly, by a stabilization point of ∆, we mean astabilization point of ∆ in some revision sequence.

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Primary period and Critical point

(Hsiung 2017) π is a primary period of ∆, if π is a periodof ∆, and it is not a multiple of any other period of ∆.(Roughly, primary period = ‘independent’ period)

The critical point of ∆ is the supremum (the least upperbound) of all its stabilization points.(the critical point of a sentence is the smallest stage atwhich that sentence must become periodic)

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Primary period and Critical point

(Hsiung 2017) π is a primary period of ∆, if π is a periodof ∆, and it is not a multiple of any other period of ∆.(Roughly, primary period = ‘independent’ period)

The critical point of ∆ is the supremum (the least upperbound) of all its stabilization points.(the critical point of a sentence is the smallest stage atwhich that sentence must become periodic)

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The liar paradox

Starting from the hypothesis h with h(λ) = 0,

0 1 2 3 …λ 0 1 0 1 …

0 1 2 3 …λ 1 0 1 0 …

The primary period of the liar paradox: 2

The critical point of the liar paradox: 0

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The liar paradox

0 1 2 3 …λ 0 1 0 1 …

0 1 2 3 …λ 1 0 1 0 …

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The liar paradox

0 1 2 3 …λ 0 1 0 1 …

0 1 2 3 …λ 1 0 1 0 …

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0 1 2 3 4 …λ0 0 0 1 1 0 …λ1 0 1 1 0 0 …

0 1 2 3 4 …λ0 0 1 1 0 0 …λ1 1 1 0 0 1 …

……

The primary period of the 2-cycle liar paradox: 4

The critical point of the 2-cycle liar paradox: 0

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0 1 2 3 4 …λ0 0 0 1 1 0 …λ1 0 1 1 0 0 …

0 1 2 3 4 …λ0 0 1 1 0 0 …λ1 1 1 0 0 1 …

……

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0 1 2 3 4 …λ0 0 0 1 1 0 …λ1 0 1 1 0 0 …

0 1 2 3 4 …λ0 0 1 1 0 0 …λ1 1 1 0 0 1 …

……

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0 1 2 3 4 …λ0 0 0 1 1 0 …λ1 0 1 1 0 0 …

0 1 2 3 4 …λ0 0 1 1 0 0 …λ1 1 1 0 0 1 …

……

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The n-cycle liar paradox

sentence (λn) is false (λ1)

… …

sentence (λn−1) is true (λn)

For any n ≥ 1, let n = 2i(2j + 1), then

The primary period of the n-cycle liar paradox: 2i+1

The critical point of the n-cycle liar paradox: 0

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The n-cycle liar paradox

sentence (λn) is false (λ1)

… …

sentence (λn−1) is true (λn)

For any n ≥ 1, let n = 2i(2j + 1), then

The primary period of the n-cycle liar paradox: 2i+1

The critical point of the n-cycle liar paradox: 0

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Wen’s paradox

sentence (δ2) is true, but sentence (δ3) is false, (δ1)

either sentence (δ1) is false, or sentence (δ3) is true, (δ2)

both (δ1) and (δ2) are true. (δ3)

The primary period of Wen’s paradox: 3

The critical point of Wen’s paradox: 2

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The ω-cycle liar paradox (Hertzbeger 1982, Yablo 1985)

sentence (λω) is false (λ0)

… …

for i ≥ 0, sentence (λi) is true (λω)

λ0 ≡ ¬T ⌜λω⌝

λ1 ≡ T ⌜λ0⌝

λ2 ≡ T ⌜λ1⌝

… …

λω ≡ ∀xT ⌜λx⌝

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λ0 ≡ ¬T ⌜λω⌝λ1 ≡ T ⌜λ0⌝. . . . . . . . .

λω ≡ T ⌜λ0⌝ ∧ T ⌜λ1⌝ ∧ . . .

Our ground model is based upon the standard structure ofnatural numbers! We are thus allowed to formulate theω-cycle liar paradox in infinitary language.

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λ0 ≡ ¬T ⌜λω⌝λ1 ≡ T ⌜λ0⌝. . . . . . . . .

λω ≡ T ⌜λ0⌝ ∧ T ⌜λ1⌝ ∧ . . .

hn+1(λ0) = ¬hn(λω)

hn+1(λ1) = hn(λ0)

. . . . . . . . .

hn+1(λω) = hn(λ0) ∧ hn(λ1) ∧ . . .

Again just kick out the truth predicate!

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hn+1(λ0) = ¬hn(λω)

hn+1(λ1) = hn(λ0)

. . . . . . . . .

hn+1(λω) = hn(λ0) ∧ hn(λ1) ∧ . . .

0 1 2 3 4 5 … ω ω + 1 …λ0 1 1 0 0 1 1 … 1 1 …λ1 1 1 1 0 0 1 … 1 1 …λ2 1 1 1 1 0 0 … 1 1 …λ3 1 1 1 1 1 0 … 1 1 …… … …λω 0 1 1 0 0 0 … 0 1 …

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hn+1(λ0) = ¬hn(λω)

hn+1(λ1) = hn(λ0)

. . . . . . . . .

hn+1(λω) = hn(λ0) ∧ hn(λ1) ∧ . . .

Case 1:

0 1 2 3 4 5 … ω ω + 1 …λ0 1 * 0 * 1 1 … 1 0 …λ1 1 1 * 0 * 1 … 1 1 …λ2 1 1 1 * 0 * … 1 1 …λ3 1 1 1 1 * 0 … 1 1 …… … …λω * 1 * 0 0 0 … 0 1 …

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hn+1(λ0) = ¬hn(λω)

hn+1(λ1) = hn(λ0)

. . . . . . . . .

hn+1(λω) = hn(λ0) ∧ hn(λ1) ∧ . . .

Case 2:

0 1 2 3 4 5 … ω ω + 1 …λ0 * 1 1 1 1 1 … 1 0 …λ1 * * 1 1 1 1 … 1 1 …λ2 0 * * 1 1 1 … 1 1 …λ3 * 0 * * 1 1 … 1 1 …… … …λω * 0 0 0 0 0 … 0 1 …

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λ0 ≡ ¬T ⌜λω⌝λ1 ≡ T ⌜λ0⌝. . . . . . . . .

The primary period of the ω-cycle liar paradox: ω

The critical point of the ω-cycle liar paradox: ω

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Yablo’s paradox (1985)

for any k greater than 0, sentence (Yk) is untrue (Y0)

for any k greater than 1, sentence (Yk) is untrue (Y1)

… …

for any k greater than n, sentence (Yk) is untrue (Yn)

… …

The primary period of Yablo’s paradox: 2

The critical point of Yablo’s paradox: 2

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paradox primary period critical pointthe liar 2 0

the 2i(2j + 1)-cycle liar 2i+1 0Wen’s paradox 3 2Yablo’s paradox 2 2the ω-cycle liar ω ω

All the known paradoxes have a unique primaryperiod.Question: is there a paradox with two or more primaryperiods?

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In principle, no matter what paradox youwant (but you should describe it by an effectiveprocedure), then we can always ‘design’ aparadox as you like!

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Three typical designer examples

A paradox with primary periods 2 and 3.

A paradox with primary periods 2 and ω.

A paradox with all prime numbers as itsprimary periods.

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We can always push the critical point ofthese paradoxes as far as you like. But we willnot do this here.

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Design a paradox with primary periods 2 and 3

The designer paradox

⇑ translation (翻译)

Truth table

⇑ transcription (转录)

Revision sequencs

⇑ customization (定制)δ1 ≡ A(T ⌜δ1⌝, T ⌜δ2⌝, T ⌜δ3⌝)δ2 ≡ B(T ⌜δ1⌝, T ⌜δ2⌝, T ⌜δ3⌝)δ3 ≡ C(T ⌜δ1⌝, T ⌜δ2⌝, T ⌜δ3⌝)

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The equivalences

δ1 ≡ A(T ⌜δ1⌝, T ⌜δ2⌝, T ⌜δ3⌝)δ2 ≡ B(T ⌜δ1⌝, T ⌜δ2⌝, T ⌜δ3⌝)δ3 ≡ C(T ⌜δ1⌝, T ⌜δ2⌝, T ⌜δ3⌝)

A(T ⌜δ1⌝, T ⌜δ2⌝, T ⌜δ3⌝) is obtained from a propositionalformula A(p1, p2, p3) by simultaneously substituting p1, p2,p3 with T ⌜δ1⌝, T ⌜δ2⌝, T ⌜δ3⌝.

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Customization

0 1 2 3 … 0 1 2 3 4 5 …δ1 1 1 1 1 … 1 0 0 0 0 0 …δ2 1 1 0 1 … 0 1 1 0 0 1 …δ3 1 0 1 0 … 0 1 0 1 0 0 …

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Transcription

0 1 2 3 … 0 1 2 3 4 5 …δ1 1 1 1 1 … 1 0 0 0 0 0 …δ2 1 1 0 1 … 0 1 1 0 0 1 …δ3 1 0 1 0 … 0 1 0 1 0 0 …

∵ h1(δ1) = A(h0(δ1), h0(δ2), h0(δ3)))

(recall: kick out the truth predicate!)

∴ 1 = A(1,1,1)

δ1 δ2 δ3 A

1 1 1 1

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Transcription

0 1 2 3 … 0 1 2 3 4 5 …δ1 1 1 1 1 … 1 0 0 0 0 0 …δ2 1 1 0 1 … 0 1 1 0 0 1 …δ3 1 0 1 0 … 0 1 0 1 0 0 …

Similarly,

∵ h2(δ1) = A(h1(δ1), h1(δ2), h1(δ3)))

∴ 1 = A(1,1,0)

δ1 δ2 δ3 A

1 1 1 11 1 0 1

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Transcription

0 1 2 3 … 0 1 2 3 4 5 …δ1 1 1 1 1 … 1 0 0 0 0 0 …δ2 1 1 0 1 … 0 1 1 0 0 1 …δ3 1 0 1 0 … 0 1 0 1 0 0 …

In some sense, the truth table can be taken as the‘transversion’ of the revision sequence table.

δ1 δ2 δ3 A B C

1 1 1 1 1 01 1 0 1 0 1

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Transcription result

δ1 δ2 δ3 A B C

1 1 1 1 1 01 1 0 1 0 11 0 1 1 1 01 0 0 0 1 10 1 1 0 1 00 1 0 0 0 10 0 1 0 0 00 0 0 0 1 0

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Translation

δ1 δ2 δ3 A B C

1 1 1 1 1 01 1 0 1 0 11 0 1 1 1 01 0 0 0 1 10 1 1 0 1 00 1 0 0 0 10 0 1 0 0 00 0 0 0 1 0

A(δ1, δ2, δ3) ⇔ (δ1 ∧ δ2 ∧ δ3) ∨ (δ1 ∧ δ2 ∧ ¬δ3)

∨(δ1 ∧ ¬δ2 ∧ δ3)

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Translation

δ1 δ2 δ3 A B C

1 1 1 1 1 01 1 0 1 0 11 0 1 1 1 01 0 0 0 1 10 1 1 0 1 00 1 0 0 0 10 0 1 0 0 00 0 0 0 1 0

A(δ1, δ2, δ3) ⇔ (δ1 ∧ δ2 ∧ δ3) ∨ (δ1 ∧ δ2 ∧ ¬δ3)

∨(δ1 ∧ ¬δ2 ∧ δ3)

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Translation

A(δ1, δ2, δ3) ⇔ (δ1 ∧ δ2 ∧ δ3) ∨ (δ1 ∧ δ2 ∧ ¬δ3)

∨(δ1 ∧ ¬δ2 ∧ δ3)

This can be simplified

A(δ1, δ2, δ3) ⇔ δ1 ∧ (δ2 ∨ δ3)

Then we obtain

δ1 ≡ T ⌜δ1⌝ ∧ (T ⌜δ2⌝) ∨ T ⌜δ3⌝)

(bring back the truth predicate you kicked out!)

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Translation

A(δ1, δ2, δ3) ⇔ (δ1 ∧ δ2 ∧ δ3) ∨ (δ1 ∧ δ2 ∧ ¬δ3)

∨(δ1 ∧ ¬δ2 ∧ δ3)

This can be simplified

A(δ1, δ2, δ3) ⇔ δ1 ∧ (δ2 ∨ δ3)

Then we obtain

δ1 ≡ T ⌜δ1⌝ ∧ (T ⌜δ2⌝) ∨ T ⌜δ3⌝)

(bring back the truth predicate you kicked out!)

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The designer paradox

δ1 ≡ T ⌜δ1⌝ ∧ (T ⌜δ2⌝) ∨ T ⌜δ3⌝)δ2 ≡ (T ⌜δ1⌝ ∧ ¬T ⌜δ2⌝) ∨ (T ⌜δ2⌝ ↔ T ⌜δ3⌝)δ3 ≡ (T ⌜δ1⌝ ∨ T ⌜δ2⌝) ∧ ¬T ⌜δ3⌝

(δ1) is true and at least one of (δ2) and (δ3) is true (δ1)

either (δ1) is true but δ2 is false, or δ2 and δ3 have the same truth value (δ2)

(δ3) is false but at least one of (δ1) and (δ2) is true (δ3)

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Design a paradox with primary periods 2 and ω

0 1 2 3 … ω 0 1 2δ0 1 0 1 1 … 1 0 1 0δ1 1 1 0 1 … 1 0 0 0δ2 1 1 1 0 … 1 0 0 0δ3 1 1 1 1 … 1 0 0 0… … … … … … … … …

Good design makes the form simple and beautiful!

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Design a paradox with primary periods 2 and ω

0 1 2 3 … ω 0 1 2 0 1δ0 1 0 1 1 … 1 0 1 0 * 0δ1 1 1 0 1 … 1 0 0 0 * 0δ2 1 1 1 0 … 1 0 0 0 * 0δ3 1 1 1 1 … 1 0 0 0 * 0… … … … … … … … … … … …

Make sure what you design is paradoxical: considerall revision sequences!

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δ0 ≡ A0(T ⌜δ0⌝, T ⌜δ1⌝, . . .)δ1 ≡ A1(T ⌜δ0⌝, T ⌜δ1⌝, . . .). . . . . . . . .

As before, we first determine the (possibly infinitary)propositional formulas Ak(δ0, δ1, . . .).

Remember: hα+1(δk) = Ak(hα(δ0), hα(δ1), hα(. . .))!

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δ0 ≡ A0(T ⌜δ0⌝, T ⌜δ1⌝, . . .)δ1 ≡ A1(T ⌜δ0⌝, T ⌜δ1⌝, . . .). . . . . . . . .

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δ0 ≡ A0(T ⌜δ0⌝, T ⌜δ1⌝, . . .)δ1 ≡ A1(T ⌜δ0⌝, T ⌜δ1⌝, . . .). . . . . . . . .

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0 1 2 3 … ω 0 1 2 0 1δ0 1 0 1 1 … 1 0 1 0 * 0δ1 1 1 0 1 … 1 0 0 0 * 0δ2 1 1 1 0 … 1 0 0 0 * 0δ3 1 1 1 1 … 1 0 0 0 * 0… … … … … … … … … … … …

δ0 δ1 δ2 δ3 δ4 … A0 A1 A2 A3 A4 …1 1 1 1 1 … 0 1 1 1 1 …0 1 1 1 1 … 1 0 1 1 1 …1 0 1 1 1 … 1 1 0 1 1 …… … … … … … … … … … … …0 0 0 0 0 … 1 0 0 0 0 …1 0 0 0 0 … 0 0 0 0 0 …* * * * * … 0 0 0 0 0 …

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0 1 2 3 … ω 0 1 2 0 1δ0 1 0 1 1 … 1 0 1 0 * 0δ1 1 1 0 1 … 1 0 0 0 * 0δ2 1 1 1 0 … 1 0 0 0 * 0δ3 1 1 1 1 … 1 0 0 0 * 0… … … … … … … … … … … …

δ0 δ1 δ2 δ3 δ4 … A0 A1 A2 A3 A4 …1 1 1 1 1 … 0 1 1 1 1 …0 1 1 1 1 … 1 0 1 1 1 …1 0 1 1 1 … 1 1 0 1 1 …… … … … … … … … … … … …0 0 0 0 0 … 1 0 0 0 0 …1 0 0 0 0 … 0 0 0 0 0 …* * * * * … 0 0 0 0 0 …

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δ0 δ1 δ2 δ3 δ4 … A0 A1 A2 A3 A4 …1 1 1 1 1 … 0 1 1 1 1 …0 1 1 1 1 … 1 0 1 1 1 …1 0 1 1 1 … 1 1 0 1 1 …… … … … … … … … … … … …0 0 0 0 0 … 1 0 0 0 0 …1 0 0 0 0 … 0 0 0 0 0 …* * * * * … 0 0 0 0 0 …

A0(δ0, δ1, . . .) = (¬δ0 ∧ ¬δ1 ∧ ¬δ2 ∧ . . .)

∨(¬δ0 ∧ δ1 ∧ δ2 ∧ δ3 ∧ . . .)

∨(δ0 ∧ ¬δ1 ∧ δ2 ∧ δ3 ∧ . . .) ∨ . . ..

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δ0 δ1 δ2 δ3 δ4 … A0 A1 A2 A3 A4 …1 1 1 1 1 … 0 1 1 1 1 …0 1 1 1 1 … 1 0 1 1 1 …1 0 1 1 1 … 1 1 0 1 1 …… … … … … … … … … … … …0 0 0 0 0 … 1 0 0 0 0 …1 0 0 0 0 … 0 0 0 0 0 …* * * * * … 0 0 0 0 0 …

A0(δ0, δ1, . . .) = (¬δ0 ∧ ¬δ1 ∧ ¬δ2 ∧ . . .)

∨(¬δ0 ∧ δ1 ∧ δ2 ∧ δ3 ∧ . . .)

∨(δ0 ∧ ¬δ1 ∧ δ2 ∧ δ3 ∧ . . .) ∨ . . ..

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δ0 δ1 δ2 δ3 δ4 … A0 A1 A2 A3 A4 …1 1 1 1 1 … 0 1 1 1 1 …0 1 1 1 1 … 1 0 1 1 1 …1 0 1 1 1 … 1 1 0 1 1 …… … … … … … … … … … … …0 0 0 0 0 … 1 0 0 0 0 …1 0 0 0 0 … 0 0 0 0 0 …* * * * * … 0 0 0 0 0 …

A1(δ0, δ1, . . .) = (δ0 ∧ δ1 ∧ δ2 ∧ . . .)

∨(δ0 ∧ ¬δ1 ∧ δ2 ∧ δ3 ∧ . . .)

∨(δ0 ∧ δ1 ∧ ¬δ2 ∧ δ3 ∧ . . .) ∨ . . . .

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δ0 δ1 δ2 δ3 δ4 … A0 A1 A2 A3 A4 …1 1 1 1 1 … 0 1 1 1 1 …0 1 1 1 1 … 1 0 1 1 1 …1 0 1 1 1 … 1 1 0 1 1 …… … … … … … … … … … … …0 0 0 0 0 … 1 0 0 0 0 …1 0 0 0 0 … 0 0 0 0 0 …* * * * * … 0 0 0 0 0 …

A2(δ0, δ1, . . .) = (δ0 ∧ δ1 ∧ δ2 ∧ . . .)

∨(¬δ0 ∧ δ1 ∧ δ2 ∧ . . .).

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δ0 δ1 δ2 δ3 δ4 … A0 A1 A2 A3 A4 …1 1 1 1 1 … 0 1 1 1 1 …0 1 1 1 1 … 1 0 1 1 1 …1 0 1 1 1 … 1 1 0 1 1 …… … … … … … … … … … … …0 0 0 0 0 … 1 0 0 0 0 …1 0 0 0 0 … 0 0 0 0 0 …* * * * * … 0 0 0 0 0 …

A3(δ0, δ1, . . .) = (δ0 ∧ δ1 ∧ δ2 ∧ . . .)

∨(¬δ0 ∧ δ1 ∧ δ2 ∧ . . .)

∨(δ0 ∧ ¬δ1 ∧ δ2 ∧ . . .).

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δ0 δ1 δ2 δ3 δ4 … A0 A1 A2 A3 A4 …1 1 1 1 1 … 0 1 1 1 1 …0 1 1 1 1 … 1 0 1 1 1 …1 0 1 1 1 … 1 1 0 1 1 …… … … … … … … … … … … …0 0 0 0 0 … 1 0 0 0 0 …1 0 0 0 0 … 0 0 0 0 0 …* * * * * … 0 0 0 0 0 …

For all k ≥ 1,

Ak+1(δ0, δ1, . . .) = (δ0 ∧ δ1 ∧ δ2 ∧ . . .)

∨(¬δ0 ∧ δ1 ∧ δ2 ∧ . . .)

∨ . . .

∨(δ0 ∧ . . . ∧ ¬δk−1 ∧ δk . . .).

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A0(δ0, δ1, . . .) = (¬δ0 ∧ ¬δ1 ∧ ¬δ2 ∧ . . .)

∨(¬δ0 ∧ δ1 ∧ δ2 ∧ δ3 ∧ . . .)

∨(δ0 ∧ ¬δ1 ∧ δ2 ∧ δ3 ∧ . . .) ∨ . . . .

δ0 ≡ (¬T ⌜δ0⌝ ∧ ¬T ⌜δ1⌝ ∧ ¬T ⌜δ2⌝ ∧ . . .)

∨(¬T ⌜δ0⌝ ∧ T ⌜δ1⌝ ∧ T ⌜δ2⌝ ∧ T ⌜δ3⌝ ∧ . . .)

∨(T ⌜δ0⌝ ∧ ¬T ⌜δ1⌝ ∧ T ⌜δ2⌝ ∧ T ⌜δ3⌝ ∧ . . .) ∨ . . ..

(Again bring back the truth predicate you kicked out!)

δ0 ≡ ∀x¬T ⌜δx⌝ ∨ ∃x∀y (y = x ↔ ¬T ⌜δy⌝)

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A0(δ0, δ1, . . .) = (¬δ0 ∧ ¬δ1 ∧ ¬δ2 ∧ . . .)

∨(¬δ0 ∧ δ1 ∧ δ2 ∧ δ3 ∧ . . .)

∨(δ0 ∧ ¬δ1 ∧ δ2 ∧ δ3 ∧ . . .) ∨ . . . .

δ0 ≡ (¬T ⌜δ0⌝ ∧ ¬T ⌜δ1⌝ ∧ ¬T ⌜δ2⌝ ∧ . . .)

∨(¬T ⌜δ0⌝ ∧ T ⌜δ1⌝ ∧ T ⌜δ2⌝ ∧ T ⌜δ3⌝ ∧ . . .)

∨(T ⌜δ0⌝ ∧ ¬T ⌜δ1⌝ ∧ T ⌜δ2⌝ ∧ T ⌜δ3⌝ ∧ . . .) ∨ . . ..

δ0 ≡ ∀x¬T ⌜δx⌝ ∨ ∃x∀y (y = x ↔ ¬T ⌜δy⌝)

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A0(δ0, δ1, . . .) = (¬δ0 ∧ ¬δ1 ∧ ¬δ2 ∧ . . .)

∨(¬δ0 ∧ δ1 ∧ δ2 ∧ δ3 ∧ . . .)

∨(δ0 ∧ ¬δ1 ∧ δ2 ∧ δ3 ∧ . . .) ∨ . . . .

δ0 ≡ (¬T ⌜δ0⌝ ∧ ¬T ⌜δ1⌝ ∧ ¬T ⌜δ2⌝ ∧ . . .)

∨(¬T ⌜δ0⌝ ∧ T ⌜δ1⌝ ∧ T ⌜δ2⌝ ∧ T ⌜δ3⌝ ∧ . . .)

∨(T ⌜δ0⌝ ∧ ¬T ⌜δ1⌝ ∧ T ⌜δ2⌝ ∧ T ⌜δ3⌝ ∧ . . .) ∨ . . ..

δ0 ≡ ∀x¬T ⌜δx⌝ ∨ ∃x∀y (y = x ↔ ¬T ⌜δy⌝)

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A paradox with primary periods 2 and ω

δ0 ≡ ∀x¬T ⌜δx⌝ ∨ ∃x∀y (y = x ↔ ¬T ⌜δy⌝) ,δ1 ≡ ∀xT ⌜δx⌝ ∨ ∃x > 0∀y (y = x ↔ ¬T ⌜δy⌝) ,

δk+1 ≡ ∀xT ⌜δx⌝ ∨ ∃x < k∀y (y = x ↔ ¬T ⌜δy⌝) , k ≥ 1.

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A summary

Design a 0-1 matrix

0 1 2 3 … ω 0 1 2 0 1δ0 1 0 1 1 … 1 0 1 0 * 0δ1 1 1 0 1 … 1 0 0 0 * 0δ2 1 1 1 0 … 1 0 0 0 * 0δ3 1 1 1 1 … 1 0 0 0 * 0… … … … … … … … … … … …

get a paradox

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A summary

Design a 0-1 matrix

0 1 2 3 … ω 0 1 2 0 1δ0 1 0 1 1 … 1 0 1 0 * 0δ1 1 1 0 1 … 1 0 0 0 * 0δ2 1 1 1 0 … 1 0 0 0 * 0δ3 1 1 1 1 … 1 0 0 0 * 0… … … … … … … … … … … …

get a paradox

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Another paradox with primary periods 2 and ω

δk+1 ≡ T ⌜δk⌝, k ≥ 1.

See my paper for details.

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The last example

Design a 0-1 matrix

0 1 2 0 1 2 3 0 1 2 3 4 …δ0 0 1 0 1 1 1 1 1 1 1 1 1 …δ1 0 0 0 1 1 1 1 1 1 1 1 1 …δ2 0 0 0 0 1 1 0 1 1 1 1 1 …δ3 0 0 0 0 0 1 0 1 1 1 1 1 …δ4 0 0 0 0 0 0 0 1 1 1 1 1 …δ5 0 0 0 0 0 0 0 0 1 1 1 0 …δ6 0 0 0 0 0 0 0 0 0 1 1 0 …δ7 0 0 0 0 0 0 0 0 0 0 1 0 …δ8 0 0 0 0 0 0 0 0 0 0 0 0 …… … … … … … … … … … … … … …

period 2 3 4 …

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Get a paradox with all prime numbers as its primaryperiods

{δ0 ≡ ∀x¬T ⌜δx⌝ ∨ ∃x > 1∀y (y ≤ x ↔ T ⌜δy⌝) ;δk ≡ ∃x > k ∀y (y ≤ x ↔ T ⌜δy⌝) , k > 0.

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It from bit.

J. A. Wheeler

We can create a paradox by ‘editing’certain ‘genes’ — 0-1 matrices!

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It from bit.

J. A. Wheeler

We can create a paradox by ‘editing’certain ‘genes’ — 0-1 matrices!

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A full view of paradoxes

classfinite primary infinite primary

critical point exampleperiods periods

(1) finitely many none finite the liar, example 1(2) infinitely many none finite example 3(3) none finitely many the ω-cycle liar(4) at least one at least one example 2(5) finitely many none infinite see my paper… … … … …

(1) (3)

(2) (4)

(5) …

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Some references

McGee, V. (1991). Truth, Vagueness and Paradox: an Essay on the Logicof Truth. Indianapolis: Hackett.

Axiomatic theories of truth

Halbach, V. (2014). Axiomatic theories of truth (revised edition).Cambridge: Cambridge University Press.

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Some references

Belnap, N. (1982). Gupta’s Rule of Revision Theory of Truth. Journal ofPhilosophical Logic, 11(1), 103-116.Gupta, A. (1982). Truth and paradox. Journal of Philosophical Logic,11(1), 1–60.Gupta, A., & Belnap, N. (1993). The Revision Theory of Truth. Cambridge:MIT Press.Herzberger, H. G. (1982-JP). Naive semantics and the Liar paradox.Journal of Philosophy, 79(9), 479–497.Herzberger, H. G. (1982-JPL). Notes on naive semantics. Journal ofPhilosophical Logic, 11(1), 61–102.

My paper

Hsiung, M. (2017). Boolean paradoxes and revision periods. StudiaLogica, 105(5), 881–914.Hsiung, M. (20xx). So many truth paradoxes: a revision-theoreticconstruction. Submitted.

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Thanks for your attention!Q & A

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