Are we ready yet? Friday, February 18,
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1 Are we ready yet? 1
2 2
3 Tableau/Sequent Duality Pushed a Bit Further Melvin Fitting CUNY 2
4 It is common knowledge that tableau proofs and sequent proofs are the same thing. 3
5 It is common knowledge that tableau proofs and sequent proofs are the same thing. One is the other, upside down. 3
6 A Simple Tableau Proof F (P Q) ( P Q) 4
7 A Simple Tableau Proof F (P Q) ( P Q) TP Q F P Q 4
8 A Simple Tableau Proof F (P Q) ( P Q) TP Q F P Q FP TQ 4
9 A Simple Tableau Proof F (P Q) ( P Q) TP Q F P Q FP F P FQ TQ 4
10 A Simple Tableau Proof F (P Q) ( P Q) TP Q F P Q FP F P FQ TP TQ 4
11 A Simple Tableau Proof F (P Q) ( P Q) TP Q F P Q FP F P FQ TP TQ F P FQ 4
12 Turn the tree over TP FQ F P FP FQ F P TQ F P Q TP Q F (P Q) ( P Q) 5
13 Now turn branches into sequents, T s on the left, F s on the right, and undo the tree construction. 6
14 TP FQ F P FP FQ F P TQ F P Q TP Q F (P Q) ( P Q) 7
15 P Q, P TP FQ F P FP FQ F P TQ F P Q TP Q F (P Q) ( P Q) 7
16 P Q, P FQ F P FP FQ F P TQ F P Q TP Q F (P Q) ( P Q) 7
17 P Q, P FQ F P FP FQ F P TQ F P Q TP Q F (P Q) ( P Q) 7
18 P Q, P P, Q, P FQ F P FP FQ F P TQ F P Q TP Q F (P Q) ( P Q) 7
19 P Q, P P, Q, P FQ F P F P FP TQ F P Q TP Q F (P Q) ( P Q) 7
20 P Q, P P, Q, P FP FQ F P TQ F P Q TP Q F (P Q) ( P Q) 7
21 P Q, P P, Q, P FP FQ F P TQ F P Q TP Q F (P Q) ( P Q) 7
22 P Q, P P, Q, P P Q, P FP FQ F P TQ F P Q TP Q F (P Q) ( P Q) 7
23 P Q, P P, Q, P P Q, P FP Q P, Q FQ F P TQ F P Q TP Q F (P Q) ( P Q) 7
24 P Q, P P, Q, P Q P, Q P Q, P FP F P TQ F P Q TP Q F (P Q) ( P Q) 7
25 P Q, P P, Q, P Q P, Q P Q, P FP TQ F P Q TP Q F (P Q) ( P Q) 7
26 P Q, P P, Q, P Q P, Q P Q, P FP TQ F P Q TP Q F (P Q) ( P Q) 7
27 P Q, P P, Q, P P Q, P FP Q P, Q Q P Q TQ F P Q TP Q F (P Q) ( P Q) 7
28 P Q, P P, Q, P P Q, P Q P, Q Q P Q TQ F P Q TP Q F (P Q) ( P Q) 7
29 P Q, P P, Q, P P Q, P Q P, Q Q P Q F P Q TP Q F (P Q) ( P Q) 7
30 P Q, P P, Q, P P Q, P Q P, Q Q P Q F P Q TP Q F (P Q) ( P Q) 7
31 P Q, P P, Q, P Q P, Q P Q, P Q P Q F P Q TP Q F (P Q) ( P Q) P Q P Q 7
32 P Q, P P, Q, P Q P, Q P Q, P Q P Q TP Q F (P Q) ( P Q) P Q P Q 7
33 P Q, P P, Q, P Q P, Q P Q, P Q P Q F (P Q) ( P Q) P Q P Q 7
34 P Q, P P, Q, P Q P, Q P Q, P Q P Q F (P Q) ( P Q) P Q P Q 7
35 P Q, P P, Q, P Q P, Q P Q, P Q P Q F (P Q) ( P Q) P Q P Q (P Q) ( P Q) 7
36 P Q, P P, Q, P Q P, Q P Q, P Q P Q P Q P Q (P Q) ( P Q) 7
37 Now, prefixed tableaus Fitting 1972, Massacci
38 A prefix is a sequence, say , of positive integers. 9
39 A prefix is a sequence, say , of positive integers. Think of a prefix as naming a possible world. 9
40 A prefix is a sequence, say , of positive integers. Think of a prefix as naming a possible world. Think of 1.2.1, 1.2.2, etc. as possible worlds accessible from 1.2 9
41 A prefixed formula is σϕ prefix formula 10
42 A prefixed formula is σϕ prefix formula Intuitively: formula is true at prefix world. 10
43 A prefixed tableau has prefixed formulas on its branches. 11
44 Uniform Notation α α 1 α 2 X Y X Y (X Y ) X Y (X Y ) X Y β β 1 β 2 X Y X Y (X Y ) X Y X Y X Y 12
45 Modal Uniform Notation ν ν 0 X X X X π π 0 X X X X 13
46 Propositional Tableau Rules Prefix Tableau Double Negation Rule Prefix Tableau α Rule Prefix Tableau β Rule σ X σx σ α σα 1 σα 2 σβ σβ 1 σβ 2 14
47 Modal K Rules Prefix Tableau ν Rule Prefix Tableau π Rule σν σ.n ν 0 σ.n not new σπ σ.n π 0 σ.n new 15
48 K Tableau Example Here is a K tableau proof of ( P Q) ( (P R) (Q S)) 16
49 1 (P R) 8. 1 (Q S) (P R) P R (( P Q) ( (P R) (Q S))) 1. 1 P Q 2. 1 ( (P R) (Q S)) 3. 1 P 4. 1 Q P Q (Q S) Q S
50 Next, nested sequents Brünnler
51 We now use one-sided (or Tait) sequences. 19
52 We now use one-sided (or Tait) sequences. Instead of A, B C, D use A, B,C,D and think of it disjunctively. 19
53 The Definition A nested sequent is a non-empty finite set of formulas and nested sequents. Think of a sequent as a disjunction and nesting as necessitation. 20
54 {A, B, {C, {D, E}, {F, G}}} is a nested sequent We generally write it as A, B, [C, [D, E], [F, G]] Think of it as A B (C (D E) (F G)) 21
55 We use the following notation. It is introduced here informally, but a formal definition is easy. 22
56 We use the following notation. It is introduced here informally, but a formal definition is easy. Γ( ) is a (nested) sequent with a hole and Γ(A, B,...) fills that hole with what you see. 22
57 Propositional Nested Sequent Rules Axioms Γ(A, A), A a propositional letter Double Negation Γ(X) Γ( X) α Rule Γ(α 1 ) Γ(α 2 ) Γ(α) Γ(β 1,β 2 ) β Rule Γ(β) 23
58 Nested K Rules ν Rule π Rule Γ([ν 0 ]) Γ(ν) Γ(π, [π 0,X,...]) Γ(π, [X,...]) 24
59 K Sequent Example Here is a nested sequent proof of ( P Q) ( (P R) (Q S)) 25
60 (P R), P, P, R, Q 11, 12 (P R), P, P R, Q (Q S), P, Q, Q, S 10 (P R), P, Q (Q S), P, Q, Q S 8 (Q S), P, Q 9 (P R) (Q S), P, Q 7 (P R) (Q S), Q, P 6 (P R) (Q S), P, Q 4, 5 ( P Q), (P R) (Q S) 2, 3 ( P Q) ( (P R) (Q S)) 1 14,
61 27
62 This is just the earlier prefixed tableau proof turned over, with nesting replacing prefixes, and formulas dualized. 27
63 The idea is, tableau branches turn into nested sequents. Formulas with the same prefix all go into the same nested subsequent, and prefix structure corresponds to nesting levels. 28
64 For example, {1 A, 1.1 B,1.1.1 C, D, 1.2 E,1.2.1 F } translates to {A, {B,{C}, {D}}, {E,{F }}} or A, [B,[C], [D]], [E,[F ]] 29
65 Nested sequent proofs convert to prefixed tableau proofs too. This is a bit harder, and I ll skip the details. 30
66 Other Tableau Rules Prefixed Tableau T Rule Prefixed Tableau B Rule Prefixed Tableau 4 Rule Prefixed Tableau 4r Rule σν σν 0 σ.n ν σν 0 σν σ.n ν σ.n ν σν In the 4 rule, the prefix introduced must not be new. 31
67 And Sequent Rules Nested Sequent t Rule Nested Sequent b Rule Nested Sequent 4 Rule Nested Sequent 4r Rule Γ(π, π 0 ) Γ(π) Γ(π 0, [π,...]) Γ([π,...]) Γ(π, [π, X,... ]) Γ(π, [X,...]) Γ(π, [π,...]) Γ([π,...]) 32
68 Quantification Possibilist quantification (constant domain) is the simplest. We ll begin with it. 33
69 Uniform Notation γ γ(a) δ δ(a) ( x)ϕ(x) ϕ(a) ( x)ϕ(x) ϕ(a) ( x)ϕ(x) ϕ(a) ( x)ϕ(x) ϕ(a) 34
70 Possibilist Tableau Rules γ Rule σγ σγ(a) any parameter a δ Rule σδ σδ(a) new parameter a 35
71 Possibilist Sequent Rules γ Rule δ Rule Γ(γ(a)) Γ(γ) Γ(δ, δ(a)) Γ(δ) In the γ rule, a must not occur in the conclusion. 36
72 Possibilist K Tableau Example 1 [( x)p (x) ( x)p (x)] 1. 1 ( x)p (x) 2. 1 ( x)p (x) ( x)p (x) P (a) 5. 1 P (a) P (a) 7. 37
73 Possibilist K Sequent Example ( x)p (x), P (a), [P (a), P (a)] 7 ( x)p (x), P (a), [P (a)] 6 ( x)p (x), [P (a)] 5 ( x)p (x), [( x)p (x)] ( x)p (x), ( x)p (x) ( x)p (x) ( x)p (x) 4 2,
74 Existence Predicate Add a special predicate symbol, E(x), and read it x actually exists. 39
75 Existence Predicate Add a special predicate symbol, E(x), and read it x actually exists. Let A E be formula A with quantifiers relativized to E. 39
76 Assume something exists Tableau σe(a) new parameter a Γ( E(a),X) Sequent Γ(X) a not in consequent 40
77 Adding these rules, and relativizing quantifiers, gives systems for actualist modal logics (varying domain). 41
78 Monotonic or Anti-monotonic Systems Also add: Tableau Monotonic σe(a) σ.n E(a) σ.n not new Tableau Anti-Monotonic σ.n E(a) σe(a) 42
79 or Sequent Monotonic Γ( E(a), [ E(a),X,...]) Γ( E(a), [X,...]) Sequent Anti-Monotonic Γ( E(a), [ E(a),...]) Γ([ E(a),...]) 43
80 Example ( x)p (x) ( x)p (x) is the Barcan formula. Here s a proof of the relativized formula, assuming anti-monotonicity. 44
81 Tableau Anti-Monotonic K Example 1 {( x)(e(x) P E (x)) ( x)(e(x) P E (x))} 1. 1 ( x)(e(x) P E (x)) 2. 1 ( x)(e(x) P E (x)) ( x)(e(x) P E (x)) (E(a) P E (a)) E(a) P E (a) 7. 1 E(a) P E (a) 8. 1 E(a) 9. 1 P E (a) E(a) P E (a)
82 Sequent Anti-Monotonic K Example ( x)(e(x) P E (x)),e(a), E(a), E(a),P E (a) 11 ( x)(e(x) P E (x)),e(a), E(a),P E (a) 9 ( x)(e(x) P E (x)), P E (a), P E (a), E(a),P E (a) 12 ( x)(e(x) P E (x)), P E (a), E(a),P E (a) , , 3 1 ( x)(e(x) P E (x)), (E(a) P E (a)), [ E(a),P E (a)] ( x)(e(x) P E (x)), [ E(a),P E (a)] ( x)(e(x) P E (x)), [E(a) P E (a)] ( x)(e(x) P E (x)), [( x)(e(x) P E (x))] ( x)(e(x) P E (x)), ( x)(e(x) P E (x)) ( x)(e(x) P E (x)) ( x)(e(x) P E (x)) 46
83 Justification Logics Natural systems exist, including for S4LP, and its relatives. 47
84 Justification Logics Natural systems exist, including for S4LP, and its relatives. I ll skip them here. 47
85 Intuitionistic Logic A long time ago, I sketched a prefixed tableau system for intuitionistic logic (1983). Here are the propositional rules. They use signed formulas. 48
86 σtx Y σtx σty σtx Y σtx σty σfx Y σfx σfy σfx Y σfx σfy σtx Y σfx σty σt X σfx σtx σ.n T X σ.n not new σfx Y σ.n T X σ.n F Y σ.n new σf X σ.n T X σ.n new 49
87 And Quantifier Rules σt( x)ϕ(x) σtϕ(a) a new σf( x)ϕ(x) σfϕ(a) any a σt( x)ϕ(x) σtϕ(a) any a σf( x)ϕ(x) σfϕ(a) a new 50
88 This is NOT first-order intuitionistic logic. 51
89 This is NOT first-order intuitionistic logic. It is constant-domain first-order intuitionistic logic. 51
90 This is NOT first-order intuitionistic logic. It is constant-domain first-order intuitionistic logic. This seems to have been introduced by Grzegorczyk. 51
91 It is axiomatized by adding ( x)(a B(x)) (A ( x)b(x)) to intuitionistic logic. 52
92 Tableau Example Here is a proof of ( x)(a B(x)) (A (C ( x)b(x))) And I should explain why this formula is interesting. 53
93 1 F ( x)(a B(x)) (A (C ( x)b(x))) T ( x)(a B(x)) FA (C ( x)b(x)) FA FC ( x)b(x) TC F ( x)b(x) FB(a) TA B(a) TA TB(a) TB(a)
94 A Nested Sequent Version A sequent is Γ where Γ is a finite set of formulas and is a finite set of formulas and sequents. We ll write nested sequents using square brackets as before. 55
95 Informally, let us say we have a proof of a sequent Γ if we have an algorithm that takes as input proofs of all members of Γ and outputs a proof of some member of. 56
96 Informally, let us say we have a proof of a sequent Γ if we have an algorithm that takes as input proofs of all members of Γ and outputs a proof of some member of. For example, a proof of A, B C, [D, E F ] is an algorithm that converts proofs of A and B either into a proof of C, or into a proof of D, E F, which would be an algorithm that converts proofs of D and E into a proof of F. 56
97 Axioms are as follows. Γ 1,A Γ 2,A is an axiom, and if is an axiom, so is Γ 1 Γ 2, [ ] 57
98 Next, here are the sequent rules, beginning with what we might call the ordinary ones. 58
99 Next, here are the sequent rules, beginning with what we might call the ordinary ones. These apply not only at the top level, but at nested levels as well. 58
100 Γ 1, A, B Γ 2 Γ 1,A B Γ 2 Γ 1 Γ 2,A Γ 1 Γ 2,B Γ 1 Γ 2,A B Γ 1,A Γ 2 Γ 1,B Γ 2 Γ 1,A B Γ 2 Γ 1 Γ 2, A, B Γ 1 Γ 2,A B Γ 1 Γ 2,A Γ 1, A Γ 2 Γ 1 Γ 2,A Γ 1,B Γ 2 Γ 1,A B Γ 2 59
101 Next the special rules, the ones corresponding to prefix changes. 60
102 Γ 1 Γ 2, [A ] Γ 1 Γ 2, A Γ 1 Γ 2, [A B] Γ 1 Γ 2,A B Γ 1 Γ 2, [ 1,A 2 ] Γ 1,A Γ 2, [ 1 2 ] 61
103 And finally the constant domain quantifier rules. 62
104 Γ 1,ϕ(a) Γ 2 Γ 1, ( x)ϕ(x) Γ 2 a not in conclusion Γ 1,ϕ(a) Γ 2 Γ 1, ( x)ϕ(x) Γ 2 Γ 1 Γ 2,ϕ(a) Γ 1 Γ 2, ( x)ϕ(x) Γ 1 Γ Γ 2, [ ϕ(a)] 1 Γ 2,ϕ(a) Γ 1 Γ 2, ( x)ϕ(x) a not in conclusion not in conclusion 63
105 Here s the earlier tableau proof inverted to a sequent proof. 64
106 [A A, [C B(a)]] [ A, [B(a),C B(a)]] [B(a) A, [C B(a)]] [A B(a) A, [C B(a)]] [( x)(a B(x)) A, [C B(a)]] [( x)(a B(x)) A, [C ( x)b(x)]] [( x)(a B(x)) A, C ( x)b(x)] [( x)(a B(x)) (A (C ( x)b(x))] ( x)(a B(x)) (A (C ( x)b(x))) 65
107 One can introduce monotonicity by adding an E predicate (intuitionistically exists) and assumptions similar to the modal ones earlier. This gives real intuitionistic logic. 66
108 One can introduce monotonicity by adding an E predicate (intuitionistically exists) and assumptions similar to the modal ones earlier. This gives real intuitionistic logic. But, there are better systems for it. 66
109 Available to Read A paper, Prefixed Tableaus and Nested Sequents is on my web site, as are these slides. comet.lehman.cuny.edu/fitting 67
110 Conclusions We have a rich variety of forward reasoning systems, with the polarity preserving subformula property. 68
111 Nested sequent systems are under much development, notably by Kai Brünnler. 69
112 Nested sequent systems are under much development, notably by Kai Brünnler. The connection with prefixed tableau systems lets us make use of the years of work that went into tableaus. 69
113 Slogan: 70
114 Slogan: New is old, old is new. It s all a type of logic stew. 70
115 Thank You 71
116 71
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