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Admissible Rules and Beyond George Metcalfe Mathematics Institute University of Bern WARU II, Les Diablerets, February 2015 George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 1 / 28

A First Question What is an admissible rule ? George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 2 / 28

Two Informal Answers (A) “A rule is admissible in a system if the set of theorems does not change when the rule is added to the system.” (B) “A rule is admissible in a system if any substitution sending its premises to theorems, sends its conclusion to a theorem.” George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 3 / 28

Two Informal Answers (A) “A rule is admissible in a system if the set of theorems does not change when the rule is added to the system.” (B) “A rule is admissible in a system if any substitution sending its premises to theorems, sends its conclusion to a theorem.” George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 3 / 28

Admissibility in Intuitionistic Logic The “independence of premises” rule {¬ p → ( q ∨ r ) } ⇒ ( ¬ p → q ) ∨ ( ¬ p → r ) is not derivable in intuitionistic logic, but it is admissible because. . . (A) adding it to an axiomatization gives no new theorems (B) if ¬ ϕ → ( ψ ∨ χ ) is a theorem, so is ( ¬ ϕ → ψ ) ∨ ( ¬ ϕ → χ ) . George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 4 / 28

Admissibility in Intuitionistic Logic The “independence of premises” rule {¬ p → ( q ∨ r ) } ⇒ ( ¬ p → q ) ∨ ( ¬ p → r ) is not derivable in intuitionistic logic, but it is admissible because. . . (A) adding it to an axiomatization gives no new theorems (B) if ¬ ϕ → ( ψ ∨ χ ) is a theorem, so is ( ¬ ϕ → ψ ) ∨ ( ¬ ϕ → χ ) . George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 4 / 28

Admissibility in Intuitionistic Logic The “independence of premises” rule {¬ p → ( q ∨ r ) } ⇒ ( ¬ p → q ) ∨ ( ¬ p → r ) is not derivable in intuitionistic logic, but it is admissible because. . . (A) adding it to an axiomatization gives no new theorems (B) if ¬ ϕ → ( ψ ∨ χ ) is a theorem, so is ( ¬ ϕ → ψ ) ∨ ( ¬ ϕ → χ ) . George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 4 / 28

Multiple-Conclusion Rules The “disjunction property” { p ∨ q } ⇒ { p , q } is admissible in intuitionistic logic because. . . (A) adding it to an axiomatization gives no new theorems (B) if ϕ ∨ ψ is a theorem, either ϕ or ψ is a theorem. George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 5 / 28

Multiple-Conclusion Rules The “disjunction property” { p ∨ q } ⇒ { p , q } is admissible in intuitionistic logic because. . . (A) adding it to an axiomatization gives no new theorems (B) if ϕ ∨ ψ is a theorem, either ϕ or ψ is a theorem. George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 5 / 28

Multiple-Conclusion Rules The “disjunction property” { p ∨ q } ⇒ { p , q } is admissible in intuitionistic logic because. . . (A) adding it to an axiomatization gives no new theorems (B) if ϕ ∨ ψ is a theorem, either ϕ or ψ is a theorem. George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 5 / 28

A Splitting of the Notions The “linearity property” ⇒ { p → q , q → p } is admissible in Gödel logic according to. . . (A) because adding it to an axiomatization gives no new theorems but not according to. . . (B) because it may be that neither ϕ → ψ nor ψ → ϕ is a theorem. George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 6 / 28

A Splitting of the Notions The “linearity property” ⇒ { p → q , q → p } is admissible in Gödel logic according to. . . (A) because adding it to an axiomatization gives no new theorems but not according to. . . (B) because it may be that neither ϕ → ψ nor ψ → ϕ is a theorem. George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 6 / 28

A Splitting of the Notions The “linearity property” ⇒ { p → q , q → p } is admissible in Gödel logic according to. . . (A) because adding it to an axiomatization gives no new theorems but not according to. . . (B) because it may be that neither ϕ → ψ nor ψ → ϕ is a theorem. George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 6 / 28

A More Exotic Example The “Takeuti-Titani density rule” { (( ϕ → p ) ∨ ( p → ψ )) ∨ χ } ⇒ ( ϕ → ψ ) ∨ χ where p does not occur in ϕ , ψ , or χ is admissible in Gödel logic because. . . (A) adding it to an axiomatization gives no new theorems (B) if (( ϕ → p ) ∨ ( p → ψ )) ∨ χ is a theorem, ( ϕ → ψ ) ∨ χ is a theorem. George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 7 / 28

A More Exotic Example The “Takeuti-Titani density rule” { (( ϕ → p ) ∨ ( p → ψ )) ∨ χ } ⇒ ( ϕ → ψ ) ∨ χ where p does not occur in ϕ , ψ , or χ is admissible in Gödel logic because. . . (A) adding it to an axiomatization gives no new theorems (B) if (( ϕ → p ) ∨ ( p → ψ )) ∨ χ is a theorem, ( ϕ → ψ ) ∨ χ is a theorem. George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 7 / 28

A More Exotic Example The “Takeuti-Titani density rule” { (( ϕ → p ) ∨ ( p → ψ )) ∨ χ } ⇒ ( ϕ → ψ ) ∨ χ where p does not occur in ϕ , ψ , or χ is admissible in Gödel logic because. . . (A) adding it to an axiomatization gives no new theorems (B) if (( ϕ → p ) ∨ ( p → ψ )) ∨ χ is a theorem, ( ϕ → ψ ) ∨ χ is a theorem. George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 7 / 28

More Generally. . . What does it mean for a first-order sentence such as or ( ∃ x )( ∀ y )( x ≤ y ) ( ∀ x )( ∃ y ) ¬ ( x ≤ y ) to be admissible in a logic or class of algebras? George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 8 / 28

The Main Question How can these notions of admissibility be characterized ? George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 9 / 28

References We take a “first-order” approach as described in G. Metcalfe. Admissible Rules: From Characterizations to Applications. Proceedings of WoLLIC 2012 , LNCS 7456, Springer (2012), 56–69. A “consequence relations” approach is described in R. Iemhoff. A Note on Consequence. Submitted. George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 10 / 28

References We take a “first-order” approach as described in G. Metcalfe. Admissible Rules: From Characterizations to Applications. Proceedings of WoLLIC 2012 , LNCS 7456, Springer (2012), 56–69. A “consequence relations” approach is described in R. Iemhoff. A Note on Consequence. Submitted. George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 10 / 28

First-Order Logic Assume the usual terminology of first-order logic with equality , using the symbols ∀ , ∃ , ⊓ , ⊔ , ⇒ , ∼ , 0, 1, and ≈ . Fix an algebraic language L with terms Tm ( L ) and sentences Sen ( L ) . For sets of L -equations Γ and ∆ , denote by Γ ⇒ ∆ the L -clause ( ∀ ¯ x )( ⊓ Γ ⇒ ⊔ ∆) called an L -quasiequation if | ∆ | = 1 and a positive L -clause if Γ = ∅ . George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 11 / 28

First-Order Logic Assume the usual terminology of first-order logic with equality , using the symbols ∀ , ∃ , ⊓ , ⊔ , ⇒ , ∼ , 0, 1, and ≈ . Fix an algebraic language L with terms Tm ( L ) and sentences Sen ( L ) . For sets of L -equations Γ and ∆ , denote by Γ ⇒ ∆ the L -clause ( ∀ ¯ x )( ⊓ Γ ⇒ ⊔ ∆) called an L -quasiequation if | ∆ | = 1 and a positive L -clause if Γ = ∅ . George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 11 / 28

First-Order Logic Assume the usual terminology of first-order logic with equality , using the symbols ∀ , ∃ , ⊓ , ⊔ , ⇒ , ∼ , 0, 1, and ≈ . Fix an algebraic language L with terms Tm ( L ) and sentences Sen ( L ) . For sets of L -equations Γ and ∆ , denote by Γ ⇒ ∆ the L -clause ( ∀ ¯ x )( ⊓ Γ ⇒ ⊔ ∆) called an L -quasiequation if | ∆ | = 1 and a positive L -clause if Γ = ∅ . George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 11 / 28

Admissibility Algebraically Let Tm ( L ) denote the term algebra of L , and consider a class of L -algebras K and a set of L -equations Γ . A K -unifier of Γ is a homomorphism σ : Tm ( L ) → Tm ( L ) such that K | = σ ( s ) ≈ σ ( t ) for all s ≈ t ∈ Γ . We say that an L -clause Γ ⇒ ∆ is K -admissible if ⇒ σ is a K-unifier of some s ≈ t ∈ ∆ . σ is a K-unifier of Γ = George Metcalfe (University of Bern) Admissible Rules and Beyond WARU II, Les Diablerets 12 / 28