An Introduction to Proof Theory

Transcription

1 An Introduction to Proof Theory Class 1: Foundations Agata Ciabattoni and Shawn Standefer anu lss december 2016 anu

2 Our Aim To introduce proof theory, with a focus on its applications in philosophy, linguistics and computer science. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 2 of 76

3 Our Aim for Today Introduce the basics of sequent systems and Gentzen s Cut Elimination Theorem. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 3 of 76

4 Today's Plan Sequents Left and Right Rules Structural Rules Cut Elimination Consequences Onward to Classical Logic Another approach to Cut Elimination Proofs and Models Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 4 of 76

5 sequents

6 Gerhard Gentzen

7 Natural deduction to sequents A (B C) A [1] [ E] B C B [ E] C [ I] 1 A C A (B C), A B C A (B C), A, B C A (B C), B A C Sequents record consequences of premises Lay out relations explicitly Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 7 of 76

8 Natural deduction to sequents A (B C) A [1] [ E] B C B [ E] C [ I] 1 A C A (B C), A B C A (B C), A, B C A (B C), B A C Sequents record consequences of premises Lay out relations explicitly Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 7 of 76

9 Natural deduction to sequents A (B C) A [1] [ E] B C B [ E] C [ I] 1 A C A (B C), A B C A (B C), A, B C A (B C), B A C Sequents record consequences of premises Lay out relations explicitly Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 7 of 76

10 Natural deduction to sequents A (B C) A [1] [ E] B C B [ E] C [ I] 1 A C A (B C), A B C A (B C), A, B C A (B C), B A C Sequents record consequences of premises Lay out relations explicitly Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 7 of 76

11 Sequents X A X is a sequence Could also use sets, multisets, or more general structures Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 8 of 76

12 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 Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 9 of 76

13 left and right rules

14 Left and right rules X, A, Y C [ L1 ] X, A B, Y C X, B, Y C [ L2 ] X, A B, Y C X A Y B [ R] X, Y A B X, A, Y C U, B, V C [ L] X, U, A B, Y, V C X A X A B X B X A B [ R 1 ] [ R 2 ] Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 11 of 76

15 Left and right rules X A [ L] X, A X, A X A [ R] X A Y, B, Z C [ L] Y, X, A B, Z C X, A B X A B [ R] Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 12 of 76

16 Sequent Calculus p p p r p [ L 1 ] [ R 1 ] q q [ R 2 ] p r p q q p q [ L] (p r) q p q s s [ R] (p r) q, s (p q) s p p p, p [ L] p p [ R] p p [ R] Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 13 of 76

17 structural rules

18 Identity axiom p p What about arbitrary formulas in the axioms? Either prove a theorem or take generalizations as axioms A A Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 15 of 76

19 Weakening X, Y C [KL] X, A, Y C X X A [KR] Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 16 of 76

20 Contraction X, A, A, Z C [WL] X, A, Z C Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 17 of 76

21 Permutation X, A, B, Z C [CL] X, B, A, Z C Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 18 of 76

22 Cut X A Y, A, Z B [Cut] Y, X, Z B Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 19 of 76

23 Sequent system The system with all the connective rules, the axiom rule, and the structural rules [KL], [KR], [CL], [WL] will be LJ LJ+Cut will be LJ with the addition of [Cut] Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 20 of 76

24 Sequent Proof p p q, p p p q, p p p, p q p p q, p q p p q p p p p, p p, p q [ L] [KR] [KL] [ L 2 ] [CL] [ L 1 ] [WL] Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 21 of 76

25 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 Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 22 of 76

26 cut elimination

27 Hauptsatz Gentzen called his Elimination Theorem the Hauptsatz He showed that for sequent derivable with a Cut, there is a Cut-free derivation Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 24 of 76

28 Admissibility and derivability 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 Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 25 of 76

29 Admissibility and derivability Consider the system LJ. The rule X, A, B C [ L3 ] X, A B C is derivable in LJ The Elimination Theorem shows that Cut is admissible in LJ, even though it is not derivable Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 26 of 76

30 Theorem If there is a derivation of X A in LJ + Cut, then there is a Cut-free derivation of X A Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 27 of 76

31 Auxiliary concepts (L) X A In the Cut rule, Y, A, Z B (R) (C) Y, X, Z B [Cut] the displayed A is the cut formula There are two ways of measuring the complexity of a Cut: grade and rank Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 28 of 76

32 Auxiliary concepts The grade, γ, of an application of [Cut] is the number of logical symbols in the cut formula A. The left rank, ρ L, of an application of [Cut] is the length of the longest path starting with (L) containing A in the succeedent The right rank, ρ R, of an application of [Cut] is the length of the longest path starting with (R) containing A in the antecedent The rank, ρ, of an application of [Cut] is ρ L + ρ R Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 29 of 76

33 Proof setup Double induction on grade and rank of a Cut Outer induction is on grade, inner induction is on rank Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 30 of 76

34 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 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 Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 31 of 76

35 Eliminating Cuts: Parametric. π 1 X A. π 2 [#] X A A, Y C [Cut] X, Y C. π 2. π 1 A, Y C [ ] X A A, Y C [Cut] X, Y C. π 1. π 2. π 1. π 2 X A A, Y C [Cut] X A A, Y C [Cut] X, Y C [#] X, Y C X, Y C [ ] X, Y C Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 32 of 76

36 Eliminating Cuts: Parametric. π 1. π 2 X, A C Y, B C. π 3 [ L] X, Y, A B C C, Z D [Cut] X, Y, A B, Z D. π 1. π 3 X, A C C, Z D [Cut] X, A, Z D. π 2. π 3 Y, B C C, Z D [Cut] Y, B, Z D [ L] X, Y, A B, Z, Z D [WL] X, Y, A B, Z D Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 33 of 76

37 Eliminating Cuts: Principal. π 1 X A X A B [ R]. π 2 A, Y C. π 3 B, Z C [ L] A B, Y, Z C [Cut] X, Y, Z C. π 1 X A. π 2 A, Y C [Cut] X, Y C [KL] X, Y, Z C Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 34 of 76

38 Eliminating Cuts: Principal. π 1 X, A B X A B [ R]. π 2 U A. π 3 Y, B, Z C [ L] Y, U, A B, Z C [Cut] Y, U, X, Z C. π 2. π 1 U A X, A B [Cut] X, U B. π 3 Y, B, Z C [Cut] Y, X, U, Z C [CL] Y, U, X, Z C Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 35 of 76

39 Eliminating Cuts: Special Cases. π 1 X p p p X p [Cut]. π 2. π 1 Y C [KL] X A A, Y C [Cut] X, Y C. π 1 X p. π 2 Y C [KL] X, Y C Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 36 of 76

40 Contraction Contraction causes some problems for this proof Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 37 of 76

41 Contraction. π 2. π 1 A, A, Y C [WL] X A A, Y C [Cut] X, Y C. π 1 X A. π 1 X A. π 2 A, A, Y C [Cut] X, A, Y C [Cut] X, X, Y C Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 38 of 76

42 Solution Use a stronger rule that removes all copies of the formula in one go X A Y B [Mix] X, Y A B Y is required to contain at least one copy of A We can extend the proof to cover contraction by proving that Mix is admissible The admissibility of Mix has the admissibility of Cut as a corollary Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 39 of 76

43 Mix cases. π 2. π 1 A, A, Y C [WL] X A A, Y C [Mix] X, Y A C. π 1. π 2 X A A, A, Y C [Mix] X, Y A C Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 40 of 76

44 Eliminating Mix: Complications with rank. π 1 X, A X A [ R] X, Y A. π 2 A, Y A [ L] A, Y, A [Mix]. π 1 X, A X A [ R]. π 1 X, A X A [ R]. π 2 A, Y A [Mix] X, Y A A [ L] X, Y A, A [Mix] X, X A, Y A [WL] X, Y A Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 41 of 76

45 Eliminating Mix: Complications with grade. π 1 X, A X A. π 2 Y A [ R] [ L] Y, A [Mix] X, Y. π 2 Y A. π 1 X, A [Mix] Y, X A [KL] Y, X [CL] X, Y Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 42 of 76

46 consequences

47 Subformula property In rules besides Cut, all formulas appearing in the premises appear in the conclusion This is the Subformula Property In Cut-free derivations, formulas not appearing in the end sequent don t appear in the rest of the proof, which makes proof search easier Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 44 of 76

48 Conservative extension One sequent system T + is a conservative extension of another sequent system T, just in case the language of T + extends that of T, and if X A is derivable in T + then X A is derivable in T, when X, A are in the language of T. The Elimination Theorem yields conservative extension results via the Subformula Property. If X and A are all in the base language, then the Subformula Property guarantees that a proof of X A in T + will not use any of the rules not available in T. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 45 of 76

49 Consistency In the presence of [KL] and [KR], says everything implies everything. The Elimination Theorem implies that that is not provable Suppose that it is. There is then a Cut-free derivation. All the axioms have formulas on both sides, and no rules delete formulas. So there is no derivation of. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 46 of 76

50 Unprovability results Similar arguments can be used to show that p p isn t derivable. How would a Cut-free derivation go? The last rule would have to be [ R], applied to either p or p, neither of which is provable Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 47 of 76

51 Disjunction property Suppose that A B is derivable There is a Cut-free derivation, so the last rule has to be [ R]. So either A or B is derivable. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 48 of 76

52 onward to classical logic

53 A seemingly magical fact LJ is complete for intuitionistic logic A sequent system for classical logic, LK, can be obtained by allowing the succedent to contain more than one formula A 1,..., A k B 1,..., B n says that if all the A i s hold, then one of the B j s does too. Ian Hacking remarked that this seemed magical, and it was explored in Peter Milne s paper Harmony, Purity, Simplicity, and a Seemingly Magical Fact Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 50 of 76

54 Left and right rules X, A, Y Z [ L1 ] X, A B, Y Z X, B, Y Z [ L2 ] X, A B, Y Z X Y, A, Z U V, B, W [ R] X, U Y, V, A B, Z, W X, A, Y Z U, B, V W [ L] X, U, A B, Y, V Z, W X Y, A, Z X Y, A B, Z X Y, B, Z X Y, A B, Z [ R] [ R] Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 51 of 76

55 Left and right rules X A, Y [ L] X, A Y X Y, A, Z U, B, V W [ L] U, X, A B, V Y, Z, W X, A Y X A, Y [ R] X, A B, Y X A B, Y [ R] Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 52 of 76

56 Weakening X Y [KL] A, X Y X Y X Y, A [KR] Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 53 of 76

57 Contraction X, A, A, Z Y [WL] X, A, Z Y X Y, A, A, Z [WR] X Y, A, Z Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 54 of 76

58 Permutation X, A, B, Z Y [CL] X, B, A, Z Y X Y, A, B, Z [CR] X Y, B, A, Z Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 55 of 76

59 Classical proofs p p p, p [ R] p p, p [ R 1 ] p p, p p p p [ R 2 ] [WR] q q q, q q q, q q q, q q q q [ L] [ L 1 ] [ L 2 ] [WL] Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 56 of 76

60 Some features An Elimination Theorem is provable for LK Since LK can have multiple formulas on the right, one can apply [WR] as well as the connective rules as the final rule in a proof of A Consequently, LK does not have the Disjunction Property Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 57 of 76

61 another approach to cut elimination

62 Alternatives Different ways of setting up a sequent system may lead to different ways to prove the Elimination Theorem One way, explored by Dyckhoff, Negri and von Plato, originally due to Dragalin, is to absorb the structural rules into the connective rules There are no structural rules in this system, but their effects are implicit in the connective rules Instead of sequences in the sequents, we will use multisets Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 59 of 76

63 Rules Identity axiom: X, p p, Y A, B, X Y [ L] A B, X Y X Y, A X Y, B [ R] X Y, A B X Y, A, B X Y, A B [ R] A, X Y B, X Y [ L] A B, X Y Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 60 of 76

64 Three Lemmas Weakening Admissibility: If X Y is provable in n steps, then X Y is provable in at most n steps, where X X, Y Y Inversion Lemma: If the conclusion of a rule is provable in n steps, then the premiss of the rule is provable in at most n steps Contraction Admissibility: If A, A, X Y is provable in n steps, then A, X Y is; and if X Y, A, A is provable in at most n steps, then X Y, A is. These are height-preserving admissibility lemmas Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 61 of 76

65 Elimination Theorem One can show Cut is admissible Since there are no contraction rules, we do not have to use Mix Since there are fewer rules, there are fewer cases to check Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 62 of 76

66 proofs and models

67 Invalid sequents and positions Call an unprovable sequent X Y a position. Write as [X : Y]. Can use failed proof search to generate models. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 64 of 76

68 Models as Ideal Positions How does one get from proof to truth? Models are ways of systematically elaborating finite positions into ideal, infinite positions that settle every proposition In the propositional case, valuations are generated by ideal positions Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 65 of 76

69 Positions to models [X : Y] The members of X are true and the members of Y are false Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 66 of 76

70 Positions to models [X : Y] The members of X are true and the members of Y are false (relative to [X : Y]). Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 66 of 76

71 Example [p q, r : p] Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 67 of 76

72 Example [p q, r : p] p q, r Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 67 of 76

73 Example [p q, r : p] p q, r true Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 67 of 76

74 Example [p q, r : p] p q, r p true Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 67 of 76

75 Example [p q, r : p] p q, r true p false Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 67 of 76

76 Example [p q, r : p] p q, r true p false p Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 67 of 76

77 Example [p q, r : p] p q, r true p false p??? Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 67 of 76

78 Example [p q, r : p] p q, r true p false p??? definition: A is true at [X : Y] iff X A, Y. definition: A is false at [X : Y] iff X, A Y. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 67 of 76

79 Example [p q, r : p] p q, r true p false p true definition: A is true at [X : Y] iff X A, Y. definition: A is false at [X : Y] iff X, A Y. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 67 of 76

80 Example [p q, r : p] p q, r true p false p true p r definition: A is true at [X : Y] iff X A, Y. definition: A is false at [X : Y] iff X, A Y. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 67 of 76

81 Example [p q, r : p] p q, r true p false p true p r true definition: A is true at [X : Y] iff X A, Y. definition: A is false at [X : Y] iff X, A Y. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 67 of 76

82 Classical Logic A B is true at [X : Y] iff A and B are true at [X : Y]. A B is false at [X : Y] iff A and B are false at [X : Y]. A is true at [X : Y] iff A is false at [X : Y]. A is false at [X : Y] iff A is true at [X : Y]. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 68 of 76

83 Classical Logic A B is true at [X : Y] iff A and B are true at [X : Y]. A B is false at [X : Y] iff A and B are false at [X : Y]. A is true at [X : Y] iff A is false at [X : Y]. A is false at [X : Y] iff A is true at [X : Y]. However, p q is false at [ : p q] Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 68 of 76

84 Classical Logic A B is true at [X : Y] iff A and B are true at [X : Y]. A B is false at [X : Y] iff A and B are false at [X : Y]. A is true at [X : Y] iff A is false at [X : Y]. A is false at [X : Y] iff A is true at [X : Y]. However, p q is false at [ : p q] but neither p nor q is false at [ : p q] since neither p p q nor q p q. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 68 of 76

85 Classical Logic A B is true at [X : Y] iff A and B are true at [X : Y]. A B is false at [X : Y] iff A and B are false at [X : Y]. A is true at [X : Y] iff A is false at [X : Y]. A is false at [X : Y] iff A is true at [X : Y]. However, p q is false at [ : p q] but neither p nor q is false at [ : p q] since neither p p q nor q p q. Similarly, r is neither true nor false at [p : q]. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 68 of 76

86 Extensions fact: If A is neither true nor false in [X : Y] then both [X, A : Y] and [X : A, Y] is invalid, and each sequent settles A one as true and the other as false. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 69 of 76

87 Extensions fact: If A is neither true nor false in [X : Y] then both [X, A : Y] and [X : A, Y] is invalid, and each sequent settles A one as true and the other as false. So, if [X : Y] doesn t settle the truth of a statement A, then we can throw A in on either side, to get a more comprehensive sequent which does settle it. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 69 of 76

88 Extensions fact: If A is neither true nor false in [X : Y] then both [X, A : Y] and [X : A, Y] is invalid, and each sequent settles A one as true and the other as false. So, if [X : Y] doesn t settle the truth of a statement A, then we can throw A in on either side, to get a more comprehensive sequent which does settle it. In general, if X Y then either X, A Y or X A, Y. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 69 of 76

89 Extensions fact: If A is neither true nor false in [X : Y] then both [X, A : Y] and [X : A, Y] is invalid, and each sequent settles A one as true and the other as false. So, if [X : Y] doesn t settle the truth of a statement A, then we can throw A in on either side, to get a more comprehensive sequent which does settle it. In general, if X Y then either X, A Y or X A, Y. X Y, A A, X Y [Cut] X Y Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 69 of 76

90 Maximal Sequents Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 70 of 76

91 Maximal Sequents [X : Y] is finitary, where X and Y are sets (or multisets or lists ). Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 70 of 76

92 Maximal Sequents [X : Y] is finitary, where X and Y are sets (or multisets or lists ). A maximal sequent is the limit of the process of throwing in each sentence in either the left or the right hand side. You can think of it as: Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 70 of 76

93 Maximal Sequents [X : Y] is finitary, where X and Y are sets (or multisets or lists ). A maximal sequent is the limit of the process of throwing in each sentence in either the left or the right hand side. You can think of it as: A pair [X : Y] of infinite sets, such that X Y and X Y is the whole language. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 70 of 76

94 Maximal Sequents [X : Y] is finitary, where X and Y are sets (or multisets or lists ). A maximal sequent is the limit of the process of throwing in each sentence in either the left or the right hand side. You can think of it as: A pair [X : Y] of infinite sets, such that X Y and X Y is the whole language. fact: Every maximal sequent makes each sentence either true or false. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 70 of 76

95 Maximal Sequents [X : Y] is finitary, where X and Y are sets (or multisets or lists ). A maximal sequent is the limit of the process of throwing in each sentence in either the left or the right hand side. You can think of it as: A pair [X : Y] of infinite sets, such that X Y and X Y is the whole language. fact: Every maximal sequent makes each sentence either true or false. fact: If X Y, there s a maximal [X : Y] extending [X : Y]. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 70 of 76

96 Models Assign truth values relative to maximal positions. Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 71 of 76

97 Models Assign truth values relative to maximal positions. In a slogan, truth value = location in a maximal position, Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 71 of 76

98 Variations The ideal position construction handles classical logic With some small adjustments, it can be used to provide models for intuitionistic logic The system LJ is single-conclusion, but there is a intuitionistic sequent system that has multiple conclusions The construction with these two systems yield Kripke models and Beth models Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 72 of 76

99 Classics gerhard gentzen Untersuchungen über das logische Schließen I Mathematische Zeitschrift, 39(1): , gerhard gentzen The Collected Papers of Gerhard Gentzen Translated and Edited by M. E. Szabo, North Holland, albert grigorevich dragalin Mathematical Intuitionism: Introduction to Proof Theory American Mathematical Society, Translations of Mathematical Monographs, Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 73 of 76

100 References roy dyckhoff Contraction-Free Sequent Calculi for Intuitionistic Logic Journal of Symbolic Logic, 57: , sara negri and jan von plato Structural Proof Theory Cambridge University Press, peter milne Harmony, Purity, Simplicity and a Seemingly Magical Fact The Monist, 85(4): , 2002 greg restall Truth Values and Proof Theory. Studia Logica, 92(2): , Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 74 of 76

101 Next Class Substructural Logics and their Proof Theory Agata Ciabattoni and Shawn Standefer An Introduction to Proof Theory 75 of 76

102 thank you! on Twitter Based on NASSLLI 2016 slides by Greg Restall and Shawn Standefer.