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
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
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
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
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
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.
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
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)
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
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
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.
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)
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
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
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
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
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
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
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
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
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