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The Admissible Rules of BD2
Jeroen Goudsmit
Utrecht University
August 8th 2013
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Disjunction Property
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A ∨ B derivable
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Disjunction Property . . . A derivable or B derivable A B - - PowerPoint PPT Presentation
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
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⊢ A ∨ B
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⊢ A or ⊢ B
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semantics
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syntax
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⊢ A ∨ B
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⊢ A or ⊢ B
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semantics
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syntax
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⊢ A ∨ B
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⊢ A or ⊢ B
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semantics
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syntax
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⊢ A ∨ B
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⊢ A or ⊢ B
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semantics
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syntax
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⊢ A ∨ B
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⊢ A or ⊢ B
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Logic of Depth n bd0 = ⊥ bdn+1 = pn+1 ∨ (pn+1 → bdn). BDn IPC bdn
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Logic of Depth n bd0 = ⊥ bdn+1 = pn+1 ∨ (pn+1 → bdn). BDn = IPC + bdn
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Overview
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Overview
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Overview
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Axiomatising Admissibility in BD2
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Overview
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Admissible Approximation
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Axiomatising Admissibility in BD2
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Overview
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Admissible Approximation
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Projectivity
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Axiomatising Admissibility in BD2
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. A . / . ∆ admissible .
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. A . / . ∆ admissible .
σA is derivable
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σC is derivable for some C ∈ ∆
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. A . . . ∆ admissible .
σA is derivable
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σC is derivable for some C ∈ ∆
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¬C → A ∨ B
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(¬C → A ) ∨ ( ¬C → B)
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¬C → A ∨ B
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{ ¬C → A, ¬C → B }
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¬¬ Disjunction Property
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∨ ∆
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{¬¬C | C ∈ ∆}
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An axiomatisation of admissibility is a set of rules R with ⊢R = .
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A ⊢ B
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A ⊢ B
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A . B
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Admissible Approximation A ⊢ B iff A . B
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If admissible approximations exists, and if A ⊢R A then . ⊆ ⊢R.
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If admissible approximations exists, and if A ⊢R A and R ⊆ . then . = ⊢R.
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Visser Rules
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(∨ ∆ → A) → ∨ ∆
. ∨ {
(∨ ∆ → A) → C
- C ∈ ∆
}
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Jankov–de Jongh formulae In suitable models have l ⊩ upk iff k ≤ l l ⊩ ndk iff l ̸≤ k
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k
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k
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k
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upk
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k
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k
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ndk
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w1
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wn
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w1
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wn
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w1
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wn
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w1
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wn
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semantics
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w1
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wn
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semantics
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syntax
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w1
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wn
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semantics
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syntax
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w1
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wn
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( n ∨
i = 1
ndwi → ∨n
i = 1 upwi
) →
n
∨
i = 1
ndwi
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n
∨
j = 1
( n ∨
i = 1
ndwi → ∨n
i = 1 upwi
) → ndwj
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semantics
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syntax
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w1
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wn
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(∨ ∆ → A ) → ∨ ∆
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∨
C ∈ ∆
(∨ ∆ → A ) → C
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A is projective when ⊢ σA and A ⊢ σB ≡ B for some σ. A A
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A is projective when ⊢ σA and A ⊢ σB ≡ B for some σ. A = A
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Ghilardi (1999)
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Ghilardi (1999)
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Ghilardi (1999)
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A
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Ghilardi (1999)
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A
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Ghilardi (1999)
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A
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Iemhoff (2001) A formula is IPC-projective iff it admits DP and V
n
for n
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Goudsmit and Iemhoff (2012) A formula is Tn -projective iff it admits DP and Vn for n ≥ 2
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Visser Rules
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(∨ ∆ → A) → ∨ ∆
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∨ { (∨ ∆ → A) → C
- C ∈ ∆
}
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Skura (1992)
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(∨ ∆ → A) → ∨ ∆
. {
¬¬((∨ ∆ → A) → C )
- C ∈ ∆
}
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A formula is BD2-projective iff it admits S
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To each A there is set Γ of BD2-projectives with A ⊢S ∨ Γ and ∨ Γ ⊢ A which shows A .
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To each A there is set Γ of BD2-projectives with A ⊢S ∨ Γ and ∨ Γ ⊢ A which shows A = ∨ Γ.
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Goudsmit (2013): S axiomatises admissibility of BD2
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References I
Ghilardi, Silvio (1999). “Unification in Intuitionistic Logic”. In: The Journal of Symbolic Logic 64.2, pp. 859–880. : 00224812. JSTOR: 2586506. Goudsmit, Jeroen P. (2013). “The Admissible Rules of BD2 and GSc”. In: Logic Group Preprint Series 313, pp. 1–23. : http://phil.uu.nl/preprints/lgps/number/313. Goudsmit, Jeroen P. and Rosalie Iemhoff (2012). “On unification and admissible rules in Gabbay-de Jongh logics”. In: Logic Group Preprint Series 297, pp. 1–18. : 0929-0710. : http://phil.uu.nl/preprints/lgps/number/297. Iemhoff, Rosalie (2001). “On the Admissible Rules of Intuitionistic Propositional Logic”. In: The Journal of Symbolic Logic 66.1,
- pp. 281–294. : 00224812. JSTOR: 2694922.
Skura, Tomasz F. (1992). “Refutation Calculi for Certain Intermediate Propositional Logics”. In: Notre Dame Journal of Formal Logic 33.4, pp. 552–560. : 10.1305/ndjfl/1093634486.