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# 証明論的意味論の具体例

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2010-04-24 (土) 応用哲学会＠札幌
ワークショップ「証明論的意味論の可能性」（大西琢朗、伊藤遼、久木田水生、丸山善宏、矢田部俊介）での発表

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### 証明論的意味論の具体例

1. 1. 1 2010 5 22 1
2. 2. • Hilbert (H) A HA (N) A NA • H N H N 2 2010 5 22 2
3. 3. PTS • • H N etc. • • • H N 3 2010 5 22 3
4. 4. ML-ITT PTS (ML-ITT) • • BHK • Propositions-as-types λ • Curry-Howard • 4 2010 5 22 4
5. 5. BHK A x x:A • a: A • g(a): B h B x g λxg(x):A→B g(x):B +v λx.g(x):A B 5 2010 5 22 5
6. 6. • Propositions as Types • λ • = • A a:A a • λ • λ ML-ITT T • • To know the meaning of A is to know the conditions for asserting A. The condition for asserting A is to know a cannonical proof of A (Rathjen). 6 2010 5 22 6
7. 7. • BHK • P(x) x • x P P D!Sets • Π Σ • ∀xP(x) =Π(x: D)P(x) = {λx.t(x): t(x) P(x) } • ∃xP(x) =Σ(x: D)P(x) = {(a,t(a)): a:D t(x) P(x) } 7 2010 5 22 7
8. 8. H,N ML-ITT • H N ML-ITT • • • H,N • 8 2010 5 22 8
9. 9. H,N • H A HA N A NA • A HA β σ • A NA λ τ • • σ τ • σ τ β- β σ τ 9 2010 5 22 9
10. 10. (ML-ITT ) • • Hilbert (H) A HA (N) A NA λ • H N ML- ITT 10 2010 5 22 10
11. 11. • ML-ITT • ML-ITT ZF Universe [Werner] • • • Curry-Howard [Schwichtenberg] • λ • ! λμ [Parigot] • ! λ [Avron] 11 2010 5 22 11
12. 12. • • λ ML-ITT 12 2010 5 22 12