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 

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

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 

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 

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 

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 

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. 

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 

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 

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 

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 

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 ⊥ 

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 . . . . ⊥ 

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 

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 

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 

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 

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∗ ⊥ 

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) 

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 ). 

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 

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) 

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 ). 

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 

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) operator 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. 

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. 

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) 

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 

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 ) 

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 

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 

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 

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 

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) 

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 

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 

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 

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 

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 • The notion of decidability depends on an underlining theory T • Constructivist requires to explicit construction of a decision procedure when claiming decidability • Impredicativity is considered problematic by constructivists • No apparent reason to deny impredicativity in the theory of meaning 

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 

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 of evidences 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 

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