ParadoxesandRevisionTheoryofTruthwangyanjing.com/wp-content/uploads/2019/08/ParadoxesRTT.pdfAbriefintroductiontoRTT...
Transcript of ParadoxesandRevisionTheoryofTruthwangyanjing.com/wp-content/uploads/2019/08/ParadoxesRTT.pdfAbriefintroductiontoRTT...
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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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A brief introduction to RTT An approach to designing paradoxes
Main Target
A brief introduction to revision theory of truth
A general approach to designing paradoxesvia revision theory of truth
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A brief introduction to RTT An approach to designing paradoxes
Main Target
A brief introduction to revision theory of truth
A general approach to designing paradoxesvia revision theory of truth
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A brief introduction to RTT An approach to designing paradoxes
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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A brief introduction to RTT An approach to designing paradoxes
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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A brief introduction to RTT An approach to designing paradoxes
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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A brief introduction to RTT An approach to designing paradoxes
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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A brief introduction to RTT An approach to designing paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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.
12 / 92
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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
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.
12 / 92
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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
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.
12 / 92
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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
⌜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⌝
).
13 / 92
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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
⌜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⌝
).
13 / 92
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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
⌜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⌝
).
13 / 92
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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
We can construct a sentence λ of LT such that
λ ≡ ¬T ⌜λ⌝.
The liar sentence
Sentence (λ) is untrue (λ)
14 / 92
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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
We can construct a sentence λ of LT such that
λ ≡ ¬T ⌜λ⌝.
The liar sentence
Sentence (λ) is untrue (λ)
14 / 92
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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
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⌝
15 / 92
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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
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 .
16 / 92
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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
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}
17 / 92
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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
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}
17 / 92
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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
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}
17 / 92
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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
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}
17 / 92
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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
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}
17 / 92
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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
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}
17 / 92
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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
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}
17 / 92
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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
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}
17 / 92
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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
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}
17 / 92
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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
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}
17 / 92
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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
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}
17 / 92
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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
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}
17 / 92
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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
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}h2 = V⟨N,Γ1⟩ Γ2 = {A|h2(A) = 1}. . . . . . . . . . . . . . . . . .
18 / 92
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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
Starting from the initial hypothesis h0, we obtain thefollowing two sequences:
h0 Γ0 = {A|h0(A) = 1}h1 = V⟨N,Γ0⟩ Γ1 = {A|h1(A) = 1}h2 = V⟨N,Γ1⟩ Γ2 = {A|h2(A) = 1}. . . . . . . . . . . . . . . . . .
What about hω?
19 / 92
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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
Starting from the initial hypothesis h0, we obtain thefollowing two sequences:
h0 Γ0 = {A|h0(A) = 1}h1 = V⟨N,Γ0⟩ Γ1 = {A|h1(A) = 1}h2 = V⟨N,Γ1⟩ Γ2 = {A|h2(A) = 1}. . . . . . . . . . . . . . . . . .
What about hω?
19 / 92
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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
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.
20 / 92
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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
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 … ?
21 / 92
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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
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 … ?
21 / 92
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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
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 … ?
21 / 92
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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
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 … ?
21 / 92
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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
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 … ?
21 / 92
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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
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 … ?
21 / 92
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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
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 … ?
21 / 92
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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
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 … ?
21 / 92
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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
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 … ?
21 / 92
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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
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 … ?
21 / 92
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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
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 … ?
21 / 92
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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
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 … ?
21 / 92
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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
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 … ?
22 / 92
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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
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 … ?
22 / 92
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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
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 … ?
22 / 92
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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
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 … ?
22 / 92
![Page 64: ParadoxesandRevisionTheoryofTruthwangyanjing.com/wp-content/uploads/2019/08/ParadoxesRTT.pdfAbriefintroductiontoRTT Anapproachtodesigningparadoxes Inthistalk,weonlyassume somebasicsyntaxandsemanticsoffirst-orderlogic(for](https://reader033.fdocuments.in/reader033/viewer/2022050310/5f72a64f6dd53a28841f81e1/html5/thumbnails/64.jpg)
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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
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 … ?
22 / 92
![Page 65: ParadoxesandRevisionTheoryofTruthwangyanjing.com/wp-content/uploads/2019/08/ParadoxesRTT.pdfAbriefintroductiontoRTT Anapproachtodesigningparadoxes Inthistalk,weonlyassume somebasicsyntaxandsemanticsoffirst-orderlogic(for](https://reader033.fdocuments.in/reader033/viewer/2022050310/5f72a64f6dd53a28841f81e1/html5/thumbnails/65.jpg)
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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
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 … ?
22 / 92
![Page 66: ParadoxesandRevisionTheoryofTruthwangyanjing.com/wp-content/uploads/2019/08/ParadoxesRTT.pdfAbriefintroductiontoRTT Anapproachtodesigningparadoxes Inthistalk,weonlyassume somebasicsyntaxandsemanticsoffirst-orderlogic(for](https://reader033.fdocuments.in/reader033/viewer/2022050310/5f72a64f6dd53a28841f81e1/html5/thumbnails/66.jpg)
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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
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 … ?
22 / 92
![Page 67: ParadoxesandRevisionTheoryofTruthwangyanjing.com/wp-content/uploads/2019/08/ParadoxesRTT.pdfAbriefintroductiontoRTT Anapproachtodesigningparadoxes Inthistalk,weonlyassume somebasicsyntaxandsemanticsoffirst-orderlogic(for](https://reader033.fdocuments.in/reader033/viewer/2022050310/5f72a64f6dd53a28841f81e1/html5/thumbnails/67.jpg)
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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
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 … ?
22 / 92
![Page 68: ParadoxesandRevisionTheoryofTruthwangyanjing.com/wp-content/uploads/2019/08/ParadoxesRTT.pdfAbriefintroductiontoRTT Anapproachtodesigningparadoxes Inthistalk,weonlyassume somebasicsyntaxandsemanticsoffirst-orderlogic(for](https://reader033.fdocuments.in/reader033/viewer/2022050310/5f72a64f6dd53a28841f81e1/html5/thumbnails/68.jpg)
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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
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 … ?
22 / 92
![Page 69: ParadoxesandRevisionTheoryofTruthwangyanjing.com/wp-content/uploads/2019/08/ParadoxesRTT.pdfAbriefintroductiontoRTT Anapproachtodesigningparadoxes Inthistalk,weonlyassume somebasicsyntaxandsemanticsoffirst-orderlogic(for](https://reader033.fdocuments.in/reader033/viewer/2022050310/5f72a64f6dd53a28841f81e1/html5/thumbnails/69.jpg)
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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
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 … ?
22 / 92
![Page 70: ParadoxesandRevisionTheoryofTruthwangyanjing.com/wp-content/uploads/2019/08/ParadoxesRTT.pdfAbriefintroductiontoRTT Anapproachtodesigningparadoxes Inthistalk,weonlyassume somebasicsyntaxandsemanticsoffirst-orderlogic(for](https://reader033.fdocuments.in/reader033/viewer/2022050310/5f72a64f6dd53a28841f81e1/html5/thumbnails/70.jpg)
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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
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 … ?
22 / 92
![Page 71: ParadoxesandRevisionTheoryofTruthwangyanjing.com/wp-content/uploads/2019/08/ParadoxesRTT.pdfAbriefintroductiontoRTT Anapproachtodesigningparadoxes Inthistalk,weonlyassume somebasicsyntaxandsemanticsoffirst-orderlogic(for](https://reader033.fdocuments.in/reader033/viewer/2022050310/5f72a64f6dd53a28841f81e1/html5/thumbnails/71.jpg)
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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
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 … ?
22 / 92
![Page 72: ParadoxesandRevisionTheoryofTruthwangyanjing.com/wp-content/uploads/2019/08/ParadoxesRTT.pdfAbriefintroductiontoRTT Anapproachtodesigningparadoxes Inthistalk,weonlyassume somebasicsyntaxandsemanticsoffirst-orderlogic(for](https://reader033.fdocuments.in/reader033/viewer/2022050310/5f72a64f6dd53a28841f81e1/html5/thumbnails/72.jpg)
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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
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 … ?
23 / 92
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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
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 … ?
23 / 92
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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
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.
24 / 92
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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
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.
24 / 92
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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
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.
24 / 92
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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
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.
24 / 92
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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
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.
25 / 92
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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
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 … ?
26 / 92
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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
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 … ?
26 / 92
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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
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 … ?
26 / 92
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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
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 … ?
26 / 92
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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
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 … ?
26 / 92
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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
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 … ?
26 / 92
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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
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 … ?
26 / 92
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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
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 … ?
26 / 92
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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
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 … ?
26 / 92
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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
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 … ?
26 / 92
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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
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 … ?
26 / 92
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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
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 … ?
26 / 92
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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
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?
27 / 92
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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
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
The limit rule for unstable case: choice 2
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
The limit rule for unstable case: choice 3
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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.
31 / 92
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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
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
Revision sequence
Starting from the initial hypothesis h0, we obtain thefollowing two arbitrarily long sequences:
h0 Γ0 = {A|h0(A) = 1}h1 = V⟨N,Γ0⟩ Γ1 = {A|h1(A) = 1}h2 = V⟨N,Γ1⟩ Γ2 = {A|h2(A) = 1}. . . . . . . . . . . . . . . . . .
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
Revision sequence
Starting from the initial hypothesis h0, we obtain thefollowing two arbitrarily long sequences:
h0 Γ0 = {A|h0(A) = 1}h1 = V⟨N,Γ0⟩ Γ1 = {A|h1(A) = 1}h2 = V⟨N,Γ1⟩ Γ2 = {A|h2(A) = 1}. . . . . . . . . . . . . . . . . .
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
Example
0 1 2 3 4 5 6 … ω …¬T ⌜0 = 0⌝ 0 1 0 0 0 0 0 … 0 …
In the above revision sequence, we have:
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
Example
0 1 2 3 4 5 6 … ω …¬T ⌜0 = 0⌝ 0 1 0 0 0 0 0 … 0 …
In the above revision sequence, we have:
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
Example
0 1 2 3 4 5 6 … ω …¬T ⌜0 = 0⌝ 0 1 0 0 0 0 0 … 0 …
In the above revision sequence, we have:
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
Paradoxicality
A sentence is paradoxical, if 1 is never aperiod of it in any revision sequence. (Gupta1982; Herzberger 1982)
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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
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
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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A brief introduction to RTT An approach to designing paradoxes Arithmetic language with T Revision sequence An application of revision sequence
{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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
The liar paradox
Starting from the hypothesis h with h(λ) = 0,
0 1 2 3 …λ 0 1 0 1 …
Starting from the hypothesis h with h(λ) = 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
45 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
The liar paradox
Starting from the hypothesis h with h(λ) = 0,
0 1 2 3 …λ 0 1 0 1 …
Starting from the hypothesis h with h(λ) = 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
45 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
The liar paradox
Starting from the hypothesis h with h(λ) = 0,
0 1 2 3 …λ 0 1 0 1 …
Starting from the hypothesis h with h(λ) = 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
45 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
The 2-cycle liar paradox
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
46 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
The 2-cycle liar paradox
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
46 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
The 2-cycle liar paradox
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
46 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
The 2-cycle liar paradox
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
46 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
The n-cycle liar paradox
sentence (λn) is false (λ1)
sentence (λ1) is true (λ2)
… …
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
47 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
The n-cycle liar paradox
sentence (λn) is false (λ1)
sentence (λ1) is true (λ2)
… …
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
47 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
Wen’s paradox
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
48 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
The ω-cycle liar paradox (Hertzbeger 1982, Yablo 1985)
sentence (λω) is false (λ0)
sentence (λ0) is true (λ1)
sentence (λ1) is true (λ2)
… …
for i ≥ 0, sentence (λi) is true (λω)
λ0 ≡ ¬T ⌜λω⌝
λ1 ≡ T ⌜λ0⌝
λ2 ≡ T ⌜λ1⌝
… …
λω ≡ ∀xT ⌜λx⌝
49 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
λ0 ≡ ¬T ⌜λω⌝λ1 ≡ T ⌜λ0⌝. . . . . . . . .
λω ≡ ∀xT ⌜λx⌝
⇒
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.
50 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
λ0 ≡ ¬T ⌜λω⌝λ1 ≡ T ⌜λ0⌝. . . . . . . . .
λω ≡ ∀xT ⌜λx⌝
The primary period of the ω-cycle liar paradox: ω
The critical point of the ω-cycle liar paradox: ω
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
Yes!
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
We can always push the critical point ofthese paradoxes as far as you like. But we willnot do this here.
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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 …
74 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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 ∧ . . .) ∨ . . ..
75 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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 ∧ . . .) ∨ . . ..
75 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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 ∧ . . .) ∨ . . . .
76 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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 . . .).
79 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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⌝)
80 / 92
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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 brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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
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 brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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
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 brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
Another 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 ≡ T ⌜δk⌝, k ≥ 1.
See my paper for details.
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
It from bit.
J. A. Wheeler
We can create a paradox by ‘editing’certain ‘genes’ — 0-1 matrices!
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
It from bit.
J. A. Wheeler
We can create a paradox by ‘editing’certain ‘genes’ — 0-1 matrices!
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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
Some references
Semantic theories of truth
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
Some references
Revision theory of truth
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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A brief introduction to RTT An approach to designing paradoxes Primary period and critical point Designer paradoxes
Thanks for your attention!Q & A
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