[PPT] - Validity of bilateral classical logic and Verificationist semantics PowerPoint Presentation

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Validity of bilateral classical logic and its application

Yoriyuki Yamagata July 6, 2017 Kyoto University

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Verificationist semantics

Theory of meaning should be molecular
Otherwise the theory is not learnable
Meaning of statements are their verification
Truth is not decidable
Thus grasp of truth cannot be manifested
Logical Inferences are constructions of verification

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Proof theoretical semantics

Verification = direct proofs
Direct proofs = proofs by introduction rules
Direct proofs are molecular
Introduction build proofs from simpler formulas
(Notion of harmony)
Inferences are valid = verification can extracted from

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Principle of excluded middle

A ∨ ¬A There is only limited case in which the principle of excluded middle is valid as long as disjunction is interpreted constructively

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Bilateral classical logic

Classical logic comes with two linguistic forces
Affirmation
Denial
Contradiction is a punctuation symbol, not sentence
Logical rules + coordination rules between two

linguistic forces

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

BCL: Language

Definition (Proposition)

A := a | A → A.

Definition (Statement)

α := +A | −A.

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

BCL: Logical inferences

[+A] . . . . +B +A → B + → I +A → B +A +B + → E +A −B −A → B − → I −A → B +A − → E1 −A → B −B − → E1

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

BCL: Coordination rules

+A −A ⊥ ⊥ [α] . . . . ⊥ α∗ RAA

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Normalization of BCL

[+A] . . . . +B +A → B . . . . +A +B = ⇒ . . . . +A . . . . +B

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Normalization of BCL

+A −B −A → B +A = ⇒ +A +A −B −A → B −B = ⇒ −B

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Normalization of BCL

[+A] . . . . +B +A → B . . . . +A −B −A → B ⊥ = ⇒ . . . . +A . . . . +B −B ⊥

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Normalization of BCL

[α] . . . . ⊥ α∗ . . . . α ⊥ = ⇒ . . . . α . . . . ⊥

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Normalization of BCL

[−A → B] . . . . ⊥ +A → B . . . . +A +B = ⇒ . . . . +A [−B] −A → B . . . . ⊥ +B

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Normalization of BCL

[+A → B] . . . . ⊥ −A → B +A = ⇒ [+A] [−A] ⊥ +B +A → B . . . . ⊥ +A

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Normalization of BCL

[+A → B] . . . . ⊥ −A → B −B = ⇒ [+B] +A → B . . . . ⊥ −B

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Our claim

Introduction Introduction of logical symbols and RAA Elimination Elimination of logical symbols and the contradiction rule

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Evidence for ⊥

α ∈ S α α∗ ∈ S α∗ ⊥

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Evidence for an atom a

+a(S) := Ax(+a)(S) ∪ −a∗(S) −a(S) := Ax(−a)(S) ∪ +a∗(S) We define +a by the smallest solution of this equation Ax(α) is the set of axioms which derives α in BCL(S)

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

m∗(S)

[α] . . . . π ⊥ α∗ ∈ m∗(S) if for any σ ∈ m(S′), S′ ⊇ S, . . . . σ α . . . . π ⊥ always reduces an evidence of ⊥ in BCL(S′).

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Evidence for +A → B, −A → B

+A → B(S) := → (+A, +B)(S) ∪ −A → B∗(S) −A → B(S) := •(+A, −B)(S) ∪ +A → B∗(S) We define +A → B by the smallest solution of this equation

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

(+A, −B)(S)

. . . . σA +A . . . . σB −B −A → B where σA ∈ +A(S) σB ∈ −B(S)

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

→ (+A, +B)(S)

[+A] . . . . π +B +A → B if for any σ ∈ +A(S′), S′ ⊇ S . . . . σ +A . . . . π +B always reduces an evidence of +B in BCL(S′).

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Semantics space M(α)

Let A be the set of atomic sentences Let D(α) be the set of closed derivations of α M(α) = {m: 2A → 2D(α)} ∴ α ∈ M(α) M(α) is a complete lattice by point-wise ordering

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Increasing (monotone)

perator

F is increasing (monotone) if m1 ≤ m2 ∈ M(α) = ⇒ F(m1) ≤ F(m2) ∈ M(α) F(m)(S) =→ (+A, +B)(S)∪ (•(+A, −B)(S) ∪ m∗(S))∗(S) In particular, this F is monotone.

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Knaster and Tarski’s theorem

L : be a complete lattice f : L → L : an increasing function F := {x ∈ L | f (x) = x} Then, F forms a complete lattice. In particular, F is not empty.

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Proof of Knaster and Tarski’s theorem

l :=

{x ∈ L | x ≥ f (x)}

(1) x ≥ f (x) (assumption) (2) x ≥ l (1) (3) f (x) ≥ f (l) (monotonicity) (4) x ≥ f (l) (2) & (4) (5) l ≥ f (l) (x arbitrary) (6) f (l) ≥ f (f (l)) (monotonicity) (7) f (l) ∈ {x ∈ L | x ≥ f (x)} (8) f (l) ≥ l (1) (9) f (l) = l (5) & (9) (10)

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Induction on the least fixed point

P ⊆ L

i∈I

xi ∈ P if ∀i ∈ I, xi ∈ P x ∈ P = ⇒ f (x) ∈ P implies l ∈ P

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Validity

A closed derivation π is valid in BCL(S) if π always reduces an evidence in BCL(S) π : a derivation with assumptions α1, . . . , αn is valid in BCL(S) if ∀S′ ⊇ S, ∀valid closed derivations σ1, . . . , σn in BCL(S′), π[σ1/α1, . . . , σn/αn] is valid in BCL(S′)

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

All derivations are valid

Theorem

All derivations in BCL(S) are valid

Lemma

If all one-step reducta of π are valid, π is valid

Lemma

σ is evidence, its one-step reducta are also evidences

Corollary

Evidences are valid

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Validity of the contradiction rule

[α] . . . . π1 ⊥ α∗ . . . . π2 α ⊥ = ⇒ . . . . π2 α . . . . π1 ⊥ ∵ by induction hypothesis, π1 and π2 are valid

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Validity of + →-elimination

[−A → B] . . . . π1 ⊥ +A → B . . . . π2 +A +B = ⇒ . . . . π2 +A [−B] −A → B . . . . π1 ⊥ +B

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Validity of + →-elimination

For any σ ∈ −B(S′), S′ ⊃ S, . . . . π2 +A . . . . σ −B −A → B is valid Therefore, by induction hypothesis, . . . . π2 +A . . . . σ −B −A → B . . . . π1 ⊥ is valid

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Validity of + →-elimination

Being σ taken arbitrary, . . . . π2 +A [−B] −A → B . . . . π1 ⊥ +B ∈ −B∗(S) ⊆ +B(S)

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Application: Strong Normalization

π is strongly normalizable if always reduces a normal form

Theorem

Any derivation π in BCL(S) is strongly normalizable

Proof.

1 Evidences are strongly normalizable 2 All assumptions have derivations for some S′ ⊇ S′ 3 π is valid

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Meaning, evidence and decidability

Understanding of meaning must be manifested in

the speaker

Ability to affirm/deny a statement must be

manifested in the speaker

What is counted as an evidence for a statement,

must be decidable

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Interpretation of decidability

Realist
Relativist
(Constructivist)

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Realist view to decidability

A property is decidable or not, independent of our

knowledge

Decidability is proven by a classical mathematics
but never be completely described by a particular

theory

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Relativist view to decidability

Decidability, if claimed, must be proven by some

theory

Constructivist requires to construct a decision

procedure

Impredicativity is considered problematic by

constructivists

No apparent reason to deny impredicativity in the

theory of meaning

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Characterization of evidences

Our set of evidences are decidable
We can give a concrete decision procedure

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Theorem: Characterization

f evidence

π is an evidence if and only if either

π is an axiom
π ends with an introduction rule for a logical symbol
π ends with RAA

holds

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Validity of bilateral classical logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics Summary

Summary

We claim that in BCL

Introduction Introduction of logical symbols and RAA Elimination Elimination of logical symbols and the contradiction rule

We define evidences and validity
We show that the set of evidences is decidable

Validity of bilateral classical logic and its application

Yoriyuki Yamagata July 6, 2017 Kyoto University

Table of contents

1 Verificationist semantics 2 Bilateral classical logic 3 Proof theoretical semantics of BCL 4 Evidences and verificationist semantics 5 Summary

Verificationist semantics

Proof theoretical semantics

Principle of excluded middle

A ∨ ¬A There is only limited case in which the principle of excluded middle is valid as long as disjunction is interpreted constructively

Bilateral classical logic

linguistic forces

BCL: Language

Definition (Proposition)

A := a | A → A.

Definition (Statement)

α := +A | −A.

BCL: Logical inferences

[+A] . . . . +B +A → B + → I +A → B +A +B + → E +A −B −A → B − → I −A → B +A − → E1 −A → B −B − → E1

BCL: Coordination rules

+A −A ⊥ ⊥ [α] . . . . ⊥ α∗ RAA

Normalization of BCL

[+A] . . . . +B +A → B . . . . +A +B = ⇒ . . . . +A . . . . +B

Normalization of BCL

+A −B −A → B +A = ⇒ +A +A −B −A → B −B = ⇒ −B

Normalization of BCL

[+A] . . . . +B +A → B . . . . +A −B −A → B ⊥ = ⇒ . . . . +A . . . . +B −B ⊥

Normalization of BCL

[α] . . . . ⊥ α∗ . . . . α ⊥ = ⇒ . . . . α . . . . ⊥

Normalization of BCL

[−A → B] . . . . ⊥ +A → B . . . . +A +B = ⇒ . . . . +A [−B] −A → B . . . . ⊥ +B

Normalization of BCL

[+A → B] . . . . ⊥ −A → B +A = ⇒ [+A] [−A] ⊥ +B +A → B . . . . ⊥ +A

Normalization of BCL

[+A → B] . . . . ⊥ −A → B −B = ⇒ [+B] +A → B . . . . ⊥ −B

Our claim

Introduction Introduction of logical symbols and RAA Elimination Elimination of logical symbols and the contradiction rule

Evidence for ⊥

α ∈ S α α∗ ∈ S α∗ ⊥

Evidence for an atom a

+a(S) := Ax(+a)(S) ∪ −a∗(S) −a(S) := Ax(−a)(S) ∪ +a∗(S) We define +a by the smallest solution of this equation Ax(α) is the set of axioms which derives α in BCL(S)

m∗(S)

[α] . . . . π ⊥ α∗ ∈ m∗(S) if for any σ ∈ m(S′), S′ ⊇ S, . . . . σ α . . . . π ⊥ always reduces an evidence of ⊥ in BCL(S′).

Evidence for +A → B, −A → B

+A → B(S) := → (+A, +B)(S) ∪ −A → B∗(S) −A → B(S) := •(+A, −B)(S) ∪ +A → B∗(S) We define +A → B by the smallest solution of this equation

. . . . σA +A . . . . σB −B −A → B where σA ∈ +A(S) σB ∈ −B(S)

→ (+A, +B)(S)

[+A] . . . . π +B +A → B if for any σ ∈ +A(S′), S′ ⊇ S . . . . σ +A . . . . π +B always reduces an evidence of +B in BCL(S′).

Semantics space M(α)

Let A be the set of atomic sentences Let D(α) be the set of closed derivations of α M(α) = {m: 2A → 2D(α)} ∴ α ∈ M(α) M(α) is a complete lattice by point-wise ordering

Increasing (monotone)

F is increasing (monotone) if m1 ≤ m2 ∈ M(α) = ⇒ F(m1) ≤ F(m2) ∈ M(α) F(m)(S) =→ (+A, +B)(S)∪ (•(+A, −B)(S) ∪ m∗(S))∗(S) In particular, this F is monotone.

Knaster and Tarski’s theorem

L : be a complete lattice f : L → L : an increasing function F := {x ∈ L | f (x) = x} Then, F forms a complete lattice. In particular, F is not empty.

Proof of Knaster and Tarski’s theorem

l :=

(1) x ≥ f (x) (assumption) (2) x ≥ l (1) (3) f (x) ≥ f (l) (monotonicity) (4) x ≥ f (l) (2) & (4) (5) l ≥ f (l) (x arbitrary) (6) f (l) ≥ f (f (l)) (monotonicity) (7) f (l) ∈ {x ∈ L | x ≥ f (x)} (8) f (l) ≥ l (1) (9) f (l) = l (5) & (9) (10)

Induction on the least fixed point

P ⊆ L

xi ∈ P if ∀i ∈ I, xi ∈ P x ∈ P = ⇒ f (x) ∈ P implies l ∈ P

Validity

A closed derivation π is valid in BCL(S) if π always reduces an evidence in BCL(S) π : a derivation with assumptions α1, . . . , αn is valid in BCL(S) if ∀S′ ⊇ S, ∀valid closed derivations σ1, . . . , σn in BCL(S′), π[σ1/α1, . . . , σn/αn] is valid in BCL(S′)

All derivations are valid

Theorem

All derivations in BCL(S) are valid

Lemma

If all one-step reducta of π are valid, π is valid

Lemma

σ is evidence, its one-step reducta are also evidences

Corollary

Evidences are valid

Validity of the contradiction rule

[α] . . . . π1 ⊥ α∗ . . . . π2 α ⊥ = ⇒ . . . . π2 α . . . . π1 ⊥ ∵ by induction hypothesis, π1 and π2 are valid

Validity of + →-elimination

[−A → B] . . . . π1 ⊥ +A → B . . . . π2 +A +B = ⇒ . . . . π2 +A [−B] −A → B . . . . π1 ⊥ +B

Validity of + →-elimination

For any σ ∈ −B(S′), S′ ⊃ S, . . . . π2 +A . . . . σ −B −A → B is valid Therefore, by induction hypothesis, . . . . π2 +A . . . . σ −B −A → B . . . . π1 ⊥ is valid

Validity of + →-elimination

Being σ taken arbitrary, . . . . π2 +A [−B] −A → B . . . . π1 ⊥ +B ∈ −B∗(S) ⊆ +B(S)

Application: Strong Normalization

π is strongly normalizable if always reduces a normal form

Theorem