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

Validity of bilateral classical logic and its application Yoriyuki Yamagata Validity of bilateral classical logic and Verificationist semantics its application Bilateral classical logic Proof theoretical semantics of BCL Evidences and Yoriyuki Yamagata verificationist semantics Summary July 6, 2017 Kyoto University

Validity of bilateral classical Table of contents logic and its application Yoriyuki Yamagata Verificationist 1 Verificationist semantics semantics Bilateral classical logic 2 Bilateral classical logic Proof theoretical semantics of BCL Evidences and verificationist semantics 3 Proof theoretical semantics of BCL Summary 4 Evidences and verificationist semantics 5 Summary

Validity of bilateral classical Verificationist semantics logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic • Theory of meaning should be molecular Proof theoretical • Otherwise the theory is not learnable semantics of BCL Evidences and • Meaning of statements are their verification verificationist semantics • Truth is not decidable Summary • Thus grasp of truth cannot be manifested • Logical Inferences are constructions of verification

Validity of bilateral classical Proof theoretical semantics logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical • Verification = direct proofs logic Proof theoretical • Direct proofs = proofs by introduction rules semantics of BCL Evidences and • Direct proofs are molecular verificationist semantics • Introduction build proofs from simpler formulas Summary • (Notion of harmony) • Inferences are valid = verification can extracted from

Validity of bilateral classical Principle of excluded middle logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL A ∨ ¬ A Evidences and verificationist There is only limited case in which the principle of semantics excluded middle is valid Summary as long as disjunction is interpreted constructively

Validity of bilateral classical Bilateral classical logic logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic • Classical logic comes with two linguistic forces Proof theoretical • Affirmation semantics of BCL • Denial Evidences and verificationist semantics • Contradiction is a punctuation symbol, not sentence Summary • Logical rules + coordination rules between two linguistic forces

Validity of bilateral classical BCL: Language logic and its application Yoriyuki Yamagata Verificationist semantics Definition (Proposition) Bilateral classical logic Proof theoretical semantics of BCL A := a | A → A . Evidences and verificationist semantics Definition (Statement) Summary α := + A | − A .

Validity of bilateral classical BCL: Logical inferences logic and its application Yoriyuki Yamagata Verificationist semantics [+ A ] Bilateral classical . logic . . . Proof theoretical semantics of BCL + B + A → B + A + A → B + → I + → E Evidences and + B verificationist semantics Summary + A − B − A → B − A → B − → I − → E 1 + A − A → B − → E 1 − B

Validity of bilateral classical BCL: Coordination rules logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL [ α ] . Evidences and . . verificationist . semantics + A − A ⊥ ⊥ Summary α ∗ RAA ⊥

Validity of bilateral classical Normalization of BCL logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical [+ A ] . semantics of BCL . . . . . . Evidences and . . verificationist + A . . . semantics + B = ⇒ . . . . Summary + A → B + A + B + B

Validity of bilateral classical Normalization of BCL logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic + A − B Proof theoretical − A → B = ⇒ + A semantics of BCL + A Evidences and verificationist semantics Summary + A − B − A → B = ⇒ − B − B

Validity of bilateral classical Normalization of BCL logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic . Proof theoretical . [+ A ] . semantics of BCL . . . . . . . Evidences and + A . . . verificationist . . semantics + B + A − B = ⇒ . Summary + A → B − A → B + B − B ⊥ ⊥

Validity of bilateral classical Normalization of BCL logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical [ α ] semantics of BCL . . . . . Evidences and . . . verificationist α . . . semantics = ⇒ . ⊥ . . . . α ∗ Summary α ⊥ ⊥

Validity of bilateral classical Normalization of BCL logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic . . Proof theoretical . . semantics of BCL [ − A → B ] . + A [ − B ] . Evidences and . . . verificationist . − A → B semantics . = ⇒ . ⊥ . . . Summary . + A → B + A ⊥ + B + B

Validity of bilateral classical Normalization of BCL logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic [+ A ] [ − A ] Proof theoretical semantics of BCL [+ A → B ] ⊥ . . Evidences and . + B . verificationist semantics = ⇒ ⊥ + A → B . . Summary − A → B . . + A ⊥ + A

Validity of bilateral classical Normalization of BCL logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical [+ A → B ] [+ B ] semantics of BCL . . Evidences and . + A → B . . verificationist . . semantics ⊥ = ⇒ . Summary − A → B ⊥ − B − B

Validity of bilateral classical Our claim logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Introduction Introduction of logical symbols and RAA Evidences and verificationist Elimination Elimination of logical symbols and the semantics contradiction rule Summary

Validity of bilateral classical Evidence for ⊥ logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL α ∗ ∈ S Evidences and α ∈ S verificationist α ∗ α semantics ⊥ Summary

Validity of bilateral classical Evidence for an atom a logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL � + a � ( S ) := Ax (+ a )( S ) ∪ � − a � ∗ ( S ) Evidences and verificationist � − a � ( S ) := Ax ( − a )( S ) ∪ � + a � ∗ ( S ) semantics Summary 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 m ∗ ( S ) logic and its application Yoriyuki Yamagata Verificationist [ α ] semantics . . . . π Bilateral classical ∈ m ∗ ( S ) logic ⊥ Proof theoretical α ∗ semantics of BCL Evidences and if for any σ ∈ m ( S ′ ) , S ′ ⊇ S , verificationist semantics . Summary . . . σ α . . . . π ⊥ always reduces an evidence of ⊥ in BCL ( S ′ ).

Validity of bilateral classical Evidence for logic and its application + A → B , − A → B Yoriyuki Yamagata Verificationist semantics Bilateral classical logic Proof theoretical semantics of BCL Evidences and � + A → B � ( S ) := → ( � + A � , � + B � )( S ) ∪ � − A → B � ∗ ( S ) verificationist semantics � − A → B � ( S ) := • ( � + A � , � − B � )( S ) ∪ � + A → B � ∗ ( S ) Summary We define � + A → B � by the smallest solution of this equation

Validity of bilateral classical • ( � + A � , � − B � )( S ) logic and its application Yoriyuki Yamagata Verificationist semantics . . Bilateral classical . . . . σ A . σ B . logic Proof theoretical + A − B semantics of BCL − A → B Evidences and verificationist semantics where Summary σ A ∈ � + A � ( S ) σ B ∈ � − B � ( S )

Validity of bilateral classical → ( � + A � , � + B � )( S ) logic and its application Yoriyuki Yamagata Verificationist [+ A ] semantics . . . π . Bilateral classical logic + B Proof theoretical + A → B semantics of BCL Evidences and if for any σ ∈ � + A � ( S ′ ) , S ′ ⊇ S verificationist semantics . Summary . . . σ + A . . . . π + B always reduces an evidence of + B in BCL ( S ′ ).

Validity of bilateral classical Semantics space M ( α ) logic and its application Yoriyuki Yamagata Verificationist semantics Bilateral classical Let A be the set of atomic sentences logic Proof theoretical Let D ( α ) be the set of closed derivations of α semantics of BCL Evidences and M ( α ) = { m : 2 A → 2 D ( α ) } verificationist semantics Summary ∴ � α � ∈ M ( α ) M ( α ) is a complete lattice by point-wise ordering