Proof Theory: Logical and Philosophical Aspects Class 1: Foundations Greg Restall and Shawn Standefer nasslli · july 2016 · rutgers
Our Aim To introduce proof theory , with a focus on its applications in philosophy, linguistics and computer science. Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 2 of 66
Our Aim for Today Introduce the basics of sequent systems and Gentzen’s Cut Elimination Theorem . Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 3 of 66
Today's Plan Sequents Left and Right Rules Structural Rules Cut Elimination Consequences Onward to Classical Logic Another approach to Cut Elimination Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 4 of 66
sequents
Gerhard Gentzen
Natural deduction to sequents Sequents record consequences of premises Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer Lay out relations explicitly 7 of 66 ▶ A → ( B → C ) , A ⊢ B → C A [ 1 ] A → ( B → C ) ▶ A → ( B → C ) , A, B ⊢ C [ → E ] B → C B [ → E ] ▶ A → ( B → C ) , B ⊢ A → C C [ → I] 1 A → C
Natural deduction to sequents Sequents record consequences of premises Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer Lay out relations explicitly 7 of 66 ▶ A → ( B → C ) , A ⊢ B → C A [ 1 ] A → ( B → C ) ▶ A → ( B → C ) , A, B ⊢ C [ → E ] B → C B [ → E ] ▶ A → ( B → C ) , B ⊢ A → C C [ → I] 1 A → C
Natural deduction to sequents Sequents record consequences of premises Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer Lay out relations explicitly 7 of 66 ▶ A → ( B → C ) , A ⊢ B → C A [ 1 ] A → ( B → C ) ▶ A → ( B → C ) , A, B ⊢ C [ → E ] B → C B [ → E ] ▶ A → ( B → C ) , B ⊢ A → C C [ → I] 1 A → C
Natural deduction to sequents Sequents record consequences of premises Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer Lay out relations explicitly 7 of 66 ▶ A → ( B → C ) , A ⊢ B → C A [ 1 ] A → ( B → C ) ▶ A → ( B → C ) , A, B ⊢ C [ → E ] B → C B [ → E ] ▶ A → ( B → C ) , B ⊢ A → C C [ → I] 1 A → C
Sequents Could also use sets , multisets , or more general structures Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 8 of 66 X ⊢ A X is a sequence
Sequent proofs Rather than introduction and elimination rules, sequent systems use left and right introduction rules Proofs are trees built up by rules. There are two sorts of rules: Connective rules and structural rules Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 9 of 66
left and right rules
Left and right rules Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 11 of 66 X, A, Y ⊢ C [ ∧ L 1 ] X, A, Y ⊢ C U, B, V ⊢ C [ ∨ L ] X, A ∧ B, Y ⊢ C X, U, A ∨ B, Y, V ⊢ C X, B, Y ⊢ C [ ∧ L 2 ] X ⊢ A [ ∨ R 1 ] X, A ∧ B, Y ⊢ C X ⊢ A ∨ B X ⊢ A Y ⊢ B X ⊢ B [ ∨ R 2 ] [ ∧ R ] X, Y ⊢ A ∧ B X ⊢ A ∨ B
Left and right rules Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 12 of 66 X ⊢ A X ⊢ A Y, B, Z ⊢ C [ ¬ L ] [ → L ] X, ¬ A ⊢ Y, X, A → B, Z ⊢ C X, A ⊢ X, A ⊢ B [ ¬ R ] [ → R ] X ⊢ ¬ A X ⊢ A → B
Sequent Calculus Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 13 of 66 p ⊢ p [ ∧ L 1 ] p ∧ r ⊢ p q ⊢ q [ ∨ R 1 ] [ ∨ R 2 ] p ∧ r ⊢ p ∨ q q ⊢ p ∨ q ( p ∧ r ) ∨ q ⊢ p ∨ q s ⊢ s ( p ∧ r ) ∨ q, s ⊢ ( p ∨ q ) ∧ s p ⊢ p [ ¬ L ] p, ¬ p ⊢ [ ¬ R ] p ⊢ ¬¬ p [ → R ] ⊢ p → ¬¬ p
structural rules
Identity axiom What about arbitrary formulas in the axioms? Either prove a theorem or take generalizations as axioms Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 15 of 66 p ⊢ p A ⊢ A
Weakening Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 16 of 66 X, Y ⊢ C X ⊢ [ KL ] [ KR ] X, A, Y ⊢ C X ⊢ A
Contraction Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 17 of 66 X, A, A, Z ⊢ C [ WL ] X, A, Z ⊢ C
Permutation Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 18 of 66 X, A, B, Z ⊢ C [ CL ] X, B, A, Z ⊢ C
Cut Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 19 of 66 X ⊢ A Y, A, Z ⊢ B [ Cut ] Y, X, Z ⊢ B
Sequent system The system with all the connective rules, the axiom rule, LJ+Cut will be LJ with the addition of [Cut] Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 20 of 66 and the structural rules [ KL ] , [ KR ] , [ CL ] , [ WL ] will be LJ
Sequent Proof Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 21 of 66 p ⊢ p [ KL ] q, p ⊢ p [ ∧ L 2 ] p ∧ q, p ⊢ p [ CL ] p, p ∧ q ⊢ p [ ∧ L 1 ] p ∧ q, p ∧ q ⊢ p [ WL ] p ∧ q ⊢ p p ⊢ p [ ¬ L ] p, ¬ p ⊢ [ KR ] p, ¬ p ⊢ q
Cut Cut is the only rule in which formulas disappear going from premiss to conclusion A proof is Cut-free iff it does not contain an application of the Cut rule If you know there is a Cut-free derivation of a sequent, it can make finding a proof easier Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 22 of 66
cut elimination
Hauptsatz Gentzen called his Elimination Theorem the Hauptsatz He showed that for sequent derivable with a Cut, there is a Cut-free derivation Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 24 of 66
Greg Restall and Shawn Standefer Admissibility and derivability Proof Theory:, Logical and Philosophical Aspects 25 of 66 S 1 , . . . , S n [ R ] S A rule [ R ] is derivable iff given derivations of S 1 , . . . , S n , one can extend those derivations to obtain a derivation of S A rule [ R ] is admissible iff if there are derivations of S 1 , . . . , S n , then there is a derivation of S
Admissibility and derivability The rule is derivable The Elimination Theorem shows that Cut is admissible , even though it is not derivable Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 26 of 66 X, A, B ⊢ C [ ∧ L 3 ] X, A ∧ B ⊢ C
Theorem Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 27 of 66 If there is a derivation of X ⊢ A in LJ + Cut , then there is a Cut-free derivation of X ⊢ A
Auxiliary concepts In the Cut rule, There are two ways of measuring the complexity of a Cut: grade and rank of cut formula Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 28 of 66 ( L ) X ⊢ A Y, A, Z ⊢ B ( R ) [ Cut ] ( C ) Y, X, Z ⊢ B the displayed A is the cut formula
Auxiliary concepts Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 29 of 66 The grade , γ ( A ) , of A is the number of logical symbols in A . The left rank , ρ L ( A ) , of A is the length of the longest path starting with ( L ) containing A in the succeedent The right rank , ρ R ( A ) , is the length of the longest path starting with ( R ) containing A in the antecedent The rank , ρ ( A ) , is ρ L ( A ) + ρ R ( A )
Proof setup Double induction on grade and rank of a Cut Outer induction is on grade, inner induction is on rank Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 30 of 66
Proof strategy Show how to move Cuts above rules, lowering left rank, then right rank, then lowering grade Parametric Cuts are cuts in which the Cut formula is not the one displayed in a rule, and principal Cuts are ones in which the Cut formula is the one displayed in a rule If one premiss of a Cut comes via an axiom or a weakening step, then the Cut can be eliminated entirely Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 31 of 66
Eliminating Cuts: Parametric . Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer . . . . . . . . . . 32 of 66 . . . . . . π 1 . π 2 X ′ ⊢ A [#] A, Y ′ ⊢ C . π 2 . π 1 [ ♭ ] X ⊢ A A, Y ⊢ C X ⊢ A A, Y ⊢ C [ Cut ] [ Cut ] X, Y ⊢ C X, Y ⊢ C . π 1 . π 2 . π 2 . π 1 X ′ ⊢ A A, Y ′ ⊢ C A, Y ⊢ C X ⊢ A [ Cut ] [ Cut ] X, Y ′ ⊢ C X ′ , Y ⊢ C [#] [ ♭ ] X, Y ⊢ C X, Y ⊢ C
Eliminating Cuts: Parametric . Proof Theory:, Logical and Philosophical Aspects Greg Restall and Shawn Standefer . . . . . . . . . . . . . 33 of 66 . π 1 . π 2 . π 3 X, A ⊢ C Y, B ⊢ C [ ∨ L ] X, Y, A ∨ B ⊢ C C, Z ⊢ D [ Cut ] X, Y, A ∨ B, Z ⊢ D . π 1 . π 3 . π 2 . π 3 X, A ⊢ C C, Z ⊢ D Y, B ⊢ C C, Z ⊢ D [ Cut ] [ Cut ] X, A, Z ⊢ D Y, B, Z ⊢ D [ ∨ L ] X, Y, A ∨ B, Z, Z ⊢ D [ WL ] X, Y, A ∨ B, Z ⊢ D
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