Disjunction Property . . . A derivable or B derivable A B - PowerPoint PPT Presentation

Utrecht University Jeroen Goudsmit The Admissible Rules of BD 2 August 8 th 2013

. Disjunction Property . . . A derivable or B derivable A ∨ B derivable

. . . ⊢ A ∨ B ⊢ A or ⊢ B

. . . . . semantics syntax ⊢ A ∨ B ⊢ A or ⊢ B

. . . . . semantics syntax ⊢ A ∨ B ⊢ A or ⊢ B

. . . . . semantics syntax ⊢ A ∨ B ⊢ A or ⊢ B

. . . . . semantics syntax ⊢ A ∨ B ⊢ A or ⊢ B

. Logic of Depth n BD n IPC bd n bd 0 = ⊥ bd n +1 = p n +1 ∨ ( p n +1 → bd n ) .

. Logic of Depth n bd 0 = ⊥ bd n +1 = p n +1 ∨ ( p n +1 → bd n ) . BD n = IPC + bd n

. Overview .

. Overview .

. Overview . . Axiomatising Admissibility in BD 2

. Overview . . . Admissible Approximation Axiomatising Admissibility in BD 2

. Overview . . . . Admissible Approximation Projectivity Axiomatising Admissibility in BD 2

. . A . . . / ∆ admissible

. . A . . . . σ A is derivable / ∆ admissible σ C is derivable for some C ∈ ∆

. . A . . . . . σ A is derivable ∆ admissible σ C is derivable for some C ∈ ∆

. . . ¬ C → A ∨ B ( ¬ C → A ) ∨ ( ¬ C → B )

. . . ¬ C → A ∨ B { ¬ C → A , ¬ C → B }

. . . . ¬¬ Disjunction Property ∨ ∆ {¬¬ C | C ∈ ∆ }

. An axiomatisation of admissibility is a set of rules R with ⊢ R = .

. . A ⊢ B

. . . A ⊢ B A . B

. Admissible Approximation A ⊢ B iff A . B

. If admissible approximations exists, and if A ⊢ R A then . ⊆ ⊢ R .

. If admissible approximations exists, and if A ⊢ R A and R ⊆ . then . = ⊢ R .

. Visser Rules . . ( ∨ ∆ → A ) → ∨ ∆ . ∨ { ( ∨ ∆ → A ) → C � } � C ∈ ∆

. Jankov–de Jongh formulae In suitable models have iff iff k ≤ l l ⊩ up k l ̸≤ k l ⊩ nd k

. .

. . k

. . k

. . k . up k

. . k

. . k . nd k

. .

. . . . w n w 1 . . .

. . . . w n . w 1 . . .

. . . . w n . w 1 . . .

. . . . w n . w 1 . . .

. . . . . w n . w 1 semantics . . .

. . . . . . w n . w 1 semantics . . . syntax

. . nd w i . n . w n n . . . . w 1 semantics . . . syntax ( n ) ∨ nd w i → ∨ n → ∨ i = 1 up w i i = 1 i = 1 ( n ) ∨ ∨ nd w i → ∨ n → nd w j i = 1 up w i j = 1 i = 1

. . . . w n . . . . w 1 semantics . . . syntax (∨ ∆ ) → ∨ ∆ → A (∨ ∆ ) ∨ → A → C C ∈ ∆

. A is projective when A A ⊢ σ A and A ⊢ σ B ≡ B for some σ .

. A is projective when ⊢ σ A and A ⊢ σ B ≡ B for some σ . A = A

. Ghilardi (1999) . .

. Ghilardi (1999) .

. Ghilardi (1999) . . A

. Ghilardi (1999) . . A .

. Ghilardi (1999) . . A .

. Iemhoff (2001) A formula is IPC-projective iff it admits DP and V n for n

. Goudsmit and Iemhoff (2012) it admits DP and V n A formula is T n -projective iff for n ≥ 2

. Visser Rules . . . ( ∨ ∆ → A ) → ∨ ∆ ( ∨ ∆ → A ) → C ∨ { � } � C ∈ ∆

. Skura (1992) . . ( ∨ ∆ → A ) → ∨ ∆ ¬¬ (( ∨ ∆ → A ) → C ) . { � } � C ∈ ∆

. it admits S A formula is BD 2 -projective iff

which shows A . . To each A there is set Γ of BD 2 -projectives with ∨ ∨ A ⊢ S Γ and Γ ⊢ A

. To each A there is set Γ of BD 2 -projectives with ∨ ∨ A ⊢ S Γ and Γ ⊢ A which shows A = ∨ Γ .

. Goudsmit (2013): S axiomatises admissibility of BD 2

.

. Preprint Series 297, pp. 1–18. : 0929-0710. : Intermediate Propositional Logics”. In: Notre Dame Journal of Skura, Tomasz F. (1992). “Refutation Calculi for Certain pp. 281–294. : 00224812. JSTOR: 2694922. Propositional Logic”. In: The Journal of Symbolic Logic 66.1, Iemhoff, Rosalie (2001). “On the Admissible Rules of Intuitionistic http://phil.uu.nl/preprints/lgps/number/297. admissible rules in Gabbay-de Jongh logics”. In: Logic Group References I Goudsmit, Jeroen P. and Rosalie Iemhoff (2012). “On unification and http://phil.uu.nl/preprints/lgps/number/313. In: Logic Group Preprint Series 313, pp. 1–23. : 2586506. Journal of Symbolic Logic 64.2, pp. 859–880. : 00224812. JSTOR: Ghilardi, Silvio (1999). “Unification in Intuitionistic Logic”. In: The Formal Logic 33.4, pp. 552–560. : 10.1305/ndjfl/1093634486. Goudsmit, Jeroen P. (2013). “The Admissible Rules of BD 2 and GSc”.