Arrows and Boxes. Tadeusz Litak (based on a joint work with Albert Visser) November 15, 2016

Transcription

1 Arrows and Boxes Tadeusz Litak (based on a joint work with Albert Visser) November 15,

2 Reminder from the last week... (slides of Miriam and Ulrich, with some corrections) 2

3 Intuitionistic Modal Logic i Z Li φ, ψ ::= p φ ψ φ ψ φ ψ φ p V ars The intuitionistic modal logic i Z: All intuitionistic tautologies plus whichever (set of) axiom(s) Z you want Closure under MP and substitution Closure under Gödel rule: φ/ ψ. Contains (φ ψ) ( φ ψ) 3

4 Intuitionistic Modal Logic i Z Li φ, ψ ::= p φ ψ φ ψ φ ψ φ p V ars Kripke semantics for : Nonempty set of worlds Two relations: Intuitionistic relation, preorder Modal relation Semantics: w φ if for any v w, v φ Minimal frame condition (Bo zić and Do sen, 1984) w 1 w 2, ( w 3, w 1 w 3 w 3 w 2 ) ( w 3, w 1 w 3 w 3 w 2) 3

5 Recall: the Bo zić and Do sen condition equivalent to 4

6 Recall: the Bo zić and Do sen condition equivalent to for any -up-set A, A is -upward-closed too 4

7 Recall: the Bo zić and Do sen condition equivalent to for any -up-set A, A is -upward-closed too Canonical extension/duality theory lead to a stronger condition 4

8 Recall: the Bo zić and Do sen condition equivalent to for any -up-set A, A is -upward-closed too Canonical extension/duality theory lead to a stronger condition Consider a prime theory/prime filter Γ... 4

9 Recall: the Bo zić and Do sen condition equivalent to for any -up-set A, A is -upward-closed too Canonical extension/duality theory lead to a stronger condition Consider a prime theory/prime filter Γ i.e., a set of formulas/elements of algebra... 4

10 Recall: the Bo zić and Do sen condition equivalent to for any -up-set A, A is -upward-closed too Canonical extension/duality theory lead to a stronger condition Consider a prime theory/prime filter Γ i.e., a set of formulas/elements of algebra closed under MP (but not Gödel!)... 4

11 Recall: the Bo zić and Do sen condition equivalent to for any -up-set A, A is -upward-closed too Canonical extension/duality theory lead to a stronger condition Consider a prime theory/prime filter Γ i.e., a set of formulas/elements of algebra closed under MP (but not Gödel!) plus φ ψ Γ implies either φ or ψ belongs too 4

12 Recall: the Bo zić and Do sen condition equivalent to for any -up-set A, A is -upward-closed too Canonical extension/duality theory lead to a stronger condition Consider a prime theory/prime filter Γ i.e., a set of formulas/elements of algebra closed under MP (but not Gödel!) plus φ ψ Γ implies either φ or ψ belongs too on theories/filters: ordinary inclusion 4

13 Recall: the Bo zić and Do sen condition equivalent to for any -up-set A, A is -upward-closed too Canonical extension/duality theory lead to a stronger condition Consider a prime theory/prime filter Γ i.e., a set of formulas/elements of algebra closed under MP (but not Gödel!) plus φ ψ Γ implies either φ or ψ belongs too on theories/filters: ordinary inclusion Γ : α whenever α Γ 4

14 Consider now Γ Γ 5

15 Consider now Γ Γ Assume α Γ 5

16 Consider now Γ Γ Assume α Γ Then α Γ... 5

17 Consider now Γ Γ Assume α Γ Then α Γ then α... 5

18 Consider now Γ Γ Assume α Γ Then α Γ then α then α 5

19 Consider now Γ Γ Assume α Γ Then α Γ then α then α We have shown Γ 5

20 Consider now Γ Γ Assume α Γ Then α Γ then α then α We have shown Γ That is, ; ; is contained in 5

21 Consider now Γ Γ Assume α Γ Then α Γ then α then α We have shown Γ That is, ; ; is contained in contained in same as! 5

22 Consider now Γ Γ Assume α Γ Then α Γ then α then α We have shown Γ That is, ; ; is contained in contained in same as! is closed under pre- and postfixing with 5

23 Consider now Γ Γ Assume α Γ Then α Γ then α then α We have shown Γ That is, ; ; is contained in contained in same as! is closed under pre- and postfixing with This is obviously a much stronger condition! 5

24 Consider now Γ Γ Assume α Γ Then α Γ then α then α We have shown Γ That is, ; ; is contained in contained in same as! is closed under pre- and postfixing with This is obviously a much stronger condition! Following some authors (e.g., Wolter & Zakharyaschev) let s call it Mix 5

25 How do you Mix Bo zić and Do sen? 6

26 How do you Mix Bo zić and Do sen?... extend it with postfixing if l m n, then l n i.e., ; is contained in Iemhoff and coauthors call it brilliancy. In the Dec 14 talk, I called it Static. Another name, as we will soon see, is -collapse. 6

27 How do you Mix Bo zić and Do sen?... extend it with postfixing i.e., ; is contained in if l m n, then l n Iemhoff and coauthors call it brilliancy. In the Dec 14 talk, I called it Static. Another name, as we will soon see, is -collapse. Many authors (say, Goldblatt 81 Grothendieck topology as geometric modality... or Bo zić and Do sen)... 6

28 How do you Mix Bo zić and Do sen?... extend it with postfixing i.e., ; is contained in if l m n, then l n Iemhoff and coauthors call it brilliancy. In the Dec 14 talk, I called it Static. Another name, as we will soon see, is -collapse. Many authors (say, Goldblatt 81 Grothendieck topology as geometric modality... or Bo zić and Do sen) noted you can impose this condition on any Bo zić-do sen frame... 6

29 How do you Mix Bo zić and Do sen?... extend it with postfixing i.e., ; is contained in if l m n, then l n Iemhoff and coauthors call it brilliancy. In the Dec 14 talk, I called it Static. Another name, as we will soon see, is -collapse. Many authors (say, Goldblatt 81 Grothendieck topology as geometric modality... or Bo zić and Do sen) noted you can impose this condition on any Bo zić-do sen frame just replace with ; 6

30 How do you Mix Bo zić and Do sen?... extend it with postfixing i.e., ; is contained in if l m n, then l n Iemhoff and coauthors call it brilliancy. In the Dec 14 talk, I called it Static. Another name, as we will soon see, is -collapse. Many authors (say, Goldblatt 81 Grothendieck topology as geometric modality... or Bo zić and Do sen) noted you can impose this condition on any Bo zić-do sen frame just replace with ; this does not change the validity of formulas in Li because all extensions are upward closed!... and because of the satisfaction clause for 6

31 Now consider another language: Li a φ, ψ ::= p φ ψ φ ψ φ ψ φ ψ 7

32 Now consider another language: Li a φ, ψ ::= p φ ψ φ ψ φ ψ φ ψ is the strict implication of C. I. Lewis who isn t C.S. Lewis, D. Lewis or Lewis Carroll 7

33 Now consider another language: Li a φ, ψ ::= p φ ψ φ ψ φ ψ φ ψ is the strict implication of C. I. Lewis who isn t C.S. Lewis, D. Lewis or Lewis Carroll Semantics: w φ ψ if for any v w, v φ implies v ψ 7

34 Now consider another language: Li a φ, ψ ::= p φ ψ φ ψ φ ψ φ ψ is the strict implication of C. I. Lewis who isn t C.S. Lewis, D. Lewis or Lewis Carroll Semantics: w φ ψ if for any v w, v φ implies v ψ φ is clearly definable... 7

35 Now consider another language: Li a φ, ψ ::= p φ ψ φ ψ φ ψ φ ψ is the strict implication of C. I. Lewis who isn t C.S. Lewis, D. Lewis or Lewis Carroll Semantics: w φ ψ if for any v w, v φ implies v ψ φ is clearly definable as φ. If discrete, i.e., model of classical propositional calculus, converse holds too... 7

36 Now consider another language: Li a φ, ψ ::= p φ ψ φ ψ φ ψ φ ψ is the strict implication of C. I. Lewis who isn t C.S. Lewis, D. Lewis or Lewis Carroll Semantics: w φ ψ if for any v w, v φ implies v ψ φ is clearly definable as φ. If discrete, i.e., model of classical propositional calculus, converse holds too i.e., φ ψ is same as (φ ψ), i.e., (φ ψ) on discrete/classical models 7

37 Historical aside Lewis himself wanted to have involutive negation 8

38 Historical aside Lewis himself wanted to have involutive negation In fact, he introduced as defined using somehow did not explicitly work with in the signature 8

39 Historical aside Lewis himself wanted to have involutive negation In fact, he introduced as defined using somehow did not explicitly work with in the signature But perhaps this is why slid into irrelevance... 8

40 Historical aside Lewis himself wanted to have involutive negation In fact, he introduced as defined using somehow did not explicitly work with in the signature But perhaps this is why slid into irrelevance which did not seem to make him happy 8

41 Historical aside Lewis himself wanted to have involutive negation In fact, he introduced as defined using somehow did not explicitly work with in the signature But perhaps this is why slid into irrelevance which did not seem to make him happy He didn t even like the name modal logic... 8

42 There is a logic restricted to indicatives; the truth-value logic most impressively developed in Principia Mathematica. But those who adhere to it usually have thought of it so far as they understood what they were doing as being the universal logic of propositions which is independent of mode. And when that universal logic was first formulated in exact terms, they failed to recognize it as the only logic which is independent of the mode in which propositions are entertained and dubbed it modal logic. 9

43 Curiously, Lewis was opened towards non-classical systems (mostly MV of Lukasiewicz) 10

44 Curiously, Lewis was opened towards non-classical systems (mostly MV of Lukasiewicz) I found just one reference where he mentions (rather favourably) Brouwer and intuitionism... 10

45 [T]he mathematical logician Brouwer has maintained that the law of the Excluded Middle is not a valid principle at all. The issues of so difficult a question could not be discussed here; but let us suggest a point of view at least something like his.... The law of the Excluded Middle is not writ in the heavens: it but reflects our rather stubborn adherence to the simplest of all possible modes of division, and our predominant interest in concrete objects as opposed to abstract concepts. The reasons for the choice of our logical categories are not themselves reasons of logic any more than the reasons for choosing Cartesian, as against polar or Gaussian coördinates, are themselves principles of mathematics, or the reason for the radix 10 is of the essence of number. 11

46 As we will see, maybe he should ve followed up on that 12

47 As we will see, maybe he should ve followed up on that... especially that there were more analogies between him and Brouwer almost perfectly parallel life dates wrote his 1910 PhD on The Place of Intuition in Knowledge a solid background in/influence of idealism and Kant... 12

48 Returning to our semantics is Bo zić and Do sen enough for persistence? 13

49 Returning to our semantics is Bo zić and Do sen enough for persistence? That is, given A, B upward closed, is A B = {w for any v w, v A implies v B} upward closed? 13

50 Returning to our semantics is Bo zić and Do sen enough for persistence? That is, given A, B upward closed, is A B = {w for any v w, v A implies v B} upward closed? Clearly no. So what is the minimal condition now? 13

51 Returning to our semantics is Bo zić and Do sen enough for persistence? That is, given A, B upward closed, is A B = {w for any v w, v A implies v B} upward closed? Clearly no. So what is the minimal condition now? People in the Netherlands found out: prefixing if k l m, then k m i.e., ; is contained in (same as) Curiously, this condition already considered in Goldblatt 81 Grothendieck topology as geometric modality 13

52 We re not done with the relationship between φ ψ and (φ ψ) = (φ ψ) 14

53 We re not done with the relationship between φ ψ and (φ ψ) = (φ ψ) Which of them is stronger? 14

54 We re not done with the relationship between φ ψ and (φ ψ) = (φ ψ) Which of them is stronger?... it is (φ ψ) 14

55 We re not done with the relationship between φ ψ and (φ ψ) = (φ ψ) Which of them is stronger?... it is (φ ψ) (φ ψ) φ ψ is valid... because is reflexive. Btw, binding priorities are as follows: 14

56 We re not done with the relationship between φ ψ and (φ ψ) = (φ ψ) Which of them is stronger?... it is (φ ψ) (φ ψ) φ ψ is valid... because is reflexive. Btw, binding priorities are as follows: and bind strongest 14

57 We re not done with the relationship between φ ψ and (φ ψ) = (φ ψ) Which of them is stronger?... it is (φ ψ) (φ ψ) φ ψ is valid... because is reflexive. Btw, binding priorities are as follows: and bind strongest next comes 14

58 We re not done with the relationship between φ ψ and (φ ψ) = (φ ψ) Which of them is stronger?... it is (φ ψ) (φ ψ) φ ψ is valid... because is reflexive. Btw, binding priorities are as follows: and bind strongest next comes then and (associative) 14

59 We re not done with the relationship between φ ψ and (φ ψ) = (φ ψ) Which of them is stronger?... it is (φ ψ) (φ ψ) φ ψ is valid... because is reflexive. Btw, binding priorities are as follows: and bind strongest next comes then and (associative) and finally (which like associates to the right) 14

60 We re not done with the relationship between φ ψ and (φ ψ) = (φ ψ) Which of them is stronger?... it is (φ ψ) (φ ψ) φ ψ is valid... because is reflexive. Btw, binding priorities are as follows: and bind strongest next comes then and (associative) and finally (which like associates to the right) How about the converse? 14

61 We re not done with the relationship between φ ψ and (φ ψ) = (φ ψ) Which of them is stronger?... it is (φ ψ) (φ ψ) φ ψ is valid... because is reflexive. Btw, binding priorities are as follows: and bind strongest next comes then and (associative) and finally (which like associates to the right) How about the converse? The validity of Col postfixing i.e., ; being contained in φ ψ (φ ψ) is equivalent to if l m n, then l n 14

62 As you told you two years ago... 15

63 As you told you two years ago after a while, the TCS/CT/FP crowd caught up with this distinction 15

64 here is our ENTCS 2011, proceedings of MSFP

65 I d suggest calling FP arrows strong arrows 17

66 I d suggest calling FP arrows strong arrows They satisfy in addition the axiom (φ ψ) φ ψ 17

67 I d suggest calling FP arrows strong arrows They satisfy in addition the axiom (φ ψ) φ ψ... or, equivalently, S a φ φ 17

68 I d suggest calling FP arrows strong arrows They satisfy in addition the axiom (φ ψ) φ ψ... or, equivalently, S a φ φ Why equivalently? φ ψ (φ ψ) φ ψ 17

69 I d suggest calling FP arrows strong arrows They satisfy in addition the axiom (φ ψ) φ ψ... or, equivalently, S a φ φ Why equivalently? φ ψ (φ ψ) φ ψ Recall: this forces contained in 17

70 I d suggest calling FP arrows strong arrows They satisfy in addition the axiom (φ ψ) φ ψ... or, equivalently, S a φ φ Why equivalently? φ ψ (φ ψ) φ ψ Recall: this forces contained in... rather degenerate in the classical case... 17

71 I d suggest calling FP arrows strong arrows They satisfy in addition the axiom (φ ψ) φ ψ... or, equivalently, S a φ φ Why equivalently? φ ψ (φ ψ) φ ψ Recall: this forces contained in... rather degenerate in the classical case only three consistent logics of (disjoint unions of) singleton(s)... 17

72 I d suggest calling FP arrows strong arrows They satisfy in addition the axiom (φ ψ) φ ψ... or, equivalently, S a φ φ Why equivalently? φ ψ (φ ψ) φ ψ Recall: this forces contained in... rather degenerate in the classical case only three consistent logics of (disjoint unions of) singleton(s) and yet intuitionistically you have a whole zoo: logics of (type inhabitation of) idioms, arrows, strong monads/pll with superintuitionistic logics as a degenerate case also recent attempts at intuitionistic epistemic logics, esp. Artemov and Protopopescu 17

73 This recaps most of my previous talk 18

74 This recaps most of my previous talk Now it s time to proceed in an orderly fashion... 18

75 This recaps most of my previous talk Now it s time to proceed in an orderly fashion starting from the axiomatization of the minimal logic of all frames (W,, ) where 18

76 This recaps most of my previous talk Now it s time to proceed in an orderly fashion starting from the axiomatization of the minimal logic of all frames (W,, ) where is a partial order 18

77 This recaps most of my previous talk Now it s time to proceed in an orderly fashion starting from the axiomatization of the minimal logic of all frames (W,, ) where is a partial order prefixing holds: if k l m, then k m 18

78 This recaps most of my previous talk Now it s time to proceed in an orderly fashion starting from the axiomatization of the minimal logic of all frames (W,, ) where is a partial order prefixing holds: if k l m, then k m Let us begin by introducing several language fragments... 18

79 Li φ, ψ ::= p φ ψ φ ψ φ ψ φ Li φ, ψ ::= p φ ψ φ ψ φ ψ Li φ, ψ ::= p φ ψ φ ψ Li 0 φ, ψ ::= p φ ψ Li a φ, ψ ::= p φ ψ φ ψ φ ψ φ ψ Li a φ, ψ ::= p φ ψ φ ψ φ ψ Li 0 a φ, ψ ::= p φ ψ φ ψ Lw a φ, ψ ::= p φ ψ φ ψ φ ψ 19

80 1. IPC 0, i.e., intuitionism in just Li 0 : 20

81 1. IPC 0, i.e., intuitionism in just Li 0 : 2. α (β α) 20

82 1. IPC 0, i.e., intuitionism in just Li 0 : 2. α (β α) 3. (α (β γ)) ((α β) (α γ)) 20

83 1. IPC 0, i.e., intuitionism in just Li 0 : 2. α (β α) 3. (α (β γ)) ((α β) (α γ)) 4. α 20

84 1. IPC 0, i.e., intuitionism in just Li 0 : 2. α (β α) 3. (α (β γ)) ((α β) (α γ)) 4. α 5. For IPC, add: 20

85 1. IPC 0, i.e., intuitionism in just Li 0 : 2. α (β α) 3. (α (β γ)) ((α β) (α γ)) 4. α 5. For IPC, add: 6. (α β) ((α γ) (α (β γ))) 20

86 1. IPC 0, i.e., intuitionism in just Li 0 : 2. α (β α) 3. (α (β γ)) ((α β) (α γ)) 4. α 5. For IPC, add: 6. (α β) ((α γ) (α (β γ))) 7. α β α 20

87 1. IPC 0, i.e., intuitionism in just Li 0 : 2. α (β α) 3. (α (β γ)) ((α β) (α γ)) 4. α 5. For IPC, add: 6. (α β) ((α γ) (α (β γ))) 7. α β α 8. α β β 20

88 1. IPC 0, i.e., intuitionism in just Li 0 : 2. α (β α) 3. (α (β γ)) ((α β) (α γ)) 4. α 5. For IPC, add: 6. (α β) ((α γ) (α (β γ))) 7. α β α 8. α β β 9. For full IPC, add moreover: 20

89 1. IPC 0, i.e., intuitionism in just Li 0 : 2. α (β α) 3. (α (β γ)) ((α β) (α γ)) 4. α 5. For IPC, add: 6. (α β) ((α γ) (α (β γ))) 7. α β α 8. α β β 9. For full IPC, add moreover: 10. α (α β) 20

90 1. IPC 0, i.e., intuitionism in just Li 0 : 2. α (β α) 3. (α (β γ)) ((α β) (α γ)) 4. α 5. For IPC, add: 6. (α β) ((α γ) (α (β γ))) 7. α β α 8. α β β 9. For full IPC, add moreover: 10. α (α β) 11. β (α β) 20

91 1. IPC 0, i.e., intuitionism in just Li 0 : 2. α (β α) 3. (α (β γ)) ((α β) (α γ)) 4. α 5. For IPC, add: 6. (α β) ((α γ) (α (β γ))) 7. α β α 8. α β β 9. For full IPC, add moreover: 10. α (α β) 11. β (α β) 12. (α γ) ((β γ) ((α β) γ)) 20

92 Axioms and rules of the disjunction-free fragment ia : Those of IPC plus: Tra φ ψ ψ χ φ χ K a φ ψ φ χ φ (ψ χ) N a φ ψ φ ψ. 21

93 Axioms and rules of the disjunction-free fragment ia : Those of IPC plus: Tra φ ψ ψ χ φ χ K a φ ψ φ χ φ (ψ χ) N a φ ψ φ ψ. (I believe Tra and N a are enough for ia 0, but haven t verified this yet) Axioms and rules of the full minimal system ia: All the axioms and rules of IPC and ia and Di φ χ ψ χ (φ ψ) χ. 21

94 Derivation exercises A generalization of K a : φ (ψ χ) (φ ψ) (ψ (ψ χ)) (φ ψ) χ by N a and K a by monotonicity of 22

95 Derivation exercises A generalization of K a : φ (ψ χ) (φ ψ) (ψ (ψ χ)) (φ ψ) χ by N a and K a by monotonicity of Another curious one: ψ χ ψ (ψ χ) ψ (ψ χ) (ψ ψ) (ψ χ) by Tra and N a by Di 22

96 Derivation exercises A generalization of K a : φ (ψ χ) (φ ψ) (ψ (ψ χ)) (φ ψ) χ by N a and K a by monotonicity of Another curious one: ψ χ ψ (ψ χ) ψ (ψ χ) (ψ ψ) (ψ χ) by Tra and N a by Di We thus get ψ χ (ψ ψ) (ψ χ) 22

97 The validity of p q (p p) (p q) implies that Col is valid over classical logic 23

98 The validity of p q (p p) (p q) implies that Col is valid over classical logic We derived syntactically why you need IPC to get to work 23

99 The validity of p q (p p) (p q) implies that Col is valid over classical logic We derived syntactically why you need IPC to get to work Note no other classical tautology in one variable would do: p q ( p p) (p q) 23

100 Completeness results for many such systems published by Iemhoff et al in early naughties Her 2001 PhD, 2003 MLQ, 2005 SL with de Jongh and Zhou Also Zhou s ILLC MSc in

101 Completeness results for many such systems published by Iemhoff et al in early naughties Her 2001 PhD, 2003 MLQ, 2005 SL with de Jongh and Zhou Also Zhou s ILLC MSc in 2003 She was continuing research of Visser on preservativity logic of HA we ll get there soon 24

102 Completeness results for many such systems published by Iemhoff et al in early naughties Her 2001 PhD, 2003 MLQ, 2005 SL with de Jongh and Zhou Also Zhou s ILLC MSc in 2003 She was continuing research of Visser on preservativity logic of HA we ll get there soon Two things to be discussed earlier: 24

103 Completeness results for many such systems published by Iemhoff et al in early naughties Her 2001 PhD, 2003 MLQ, 2005 SL with de Jongh and Zhou Also Zhou s ILLC MSc in 2003 She was continuing research of Visser on preservativity logic of HA we ll get there soon Two things to be discussed earlier: the connection with weak logics with strict implication and weak Heyting algebras (and perhaps also BI)? 24

104 Completeness results for many such systems published by Iemhoff et al in early naughties Her 2001 PhD, 2003 MLQ, 2005 SL with de Jongh and Zhou Also Zhou s ILLC MSc in 2003 She was continuing research of Visser on preservativity logic of HA we ll get there soon Two things to be discussed earlier: the connection with weak logics with strict implication and weak Heyting algebras (and perhaps also BI)? the possibility of obtaining such results in a systematic manner? 24

105 Weak logics with strict implication and weak Heyting algebras Corsi 1987, Došen 1993, Celani and Jansana 2001, 2005 Related system: Visser 1981, Epstein and Horn

106 Weak logics with strict implication and weak Heyting algebras Corsi 1987, Došen 1993, Celani and Jansana 2001, 2005 Related system: Visser 1981, Epstein and Horn 1976 Briefly discussing this was in fact the only reason to introduce Lw a 25

107 Weak logics with strict implication and weak Heyting algebras Corsi 1987, Došen 1993, Celani and Jansana 2001, 2005 Related system: Visser 1981, Epstein and Horn 1976 Briefly discussing this was in fact the only reason to introduce Lw a Ordinary Kripke frames with discrete... 25

108 Weak logics with strict implication and weak Heyting algebras Corsi 1987, Došen 1993, Celani and Jansana 2001, 2005 Related system: Visser 1981, Epstein and Horn 1976 Briefly discussing this was in fact the only reason to introduce Lw a Ordinary Kripke frames with discrete but in the absence of, how would you know the difference? 25

109 Weak logics with strict implication and weak Heyting algebras Corsi 1987, Došen 1993, Celani and Jansana 2001, 2005 Related system: Visser 1981, Epstein and Horn 1976 Briefly discussing this was in fact the only reason to introduce Lw a Ordinary Kripke frames with discrete but in the absence of, how would you know the difference? Problematic even from the point of view of algebraizability 25

110 Weak logics with strict implication and weak Heyting algebras Corsi 1987, Došen 1993, Celani and Jansana 2001, 2005 Related system: Visser 1981, Epstein and Horn 1976 Briefly discussing this was in fact the only reason to introduce Lw a Ordinary Kripke frames with discrete but in the absence of, how would you know the difference? Problematic even from the point of view of algebraizability Došen proposed a Hilbert-style system, but it does not capture local consequence... 25

111 Weak logics with strict implication and weak Heyting algebras Corsi 1987, Došen 1993, Celani and Jansana 2001, 2005 Related system: Visser 1981, Epstein and Horn 1976 Briefly discussing this was in fact the only reason to introduce Lw a Ordinary Kripke frames with discrete but in the absence of, how would you know the difference? Problematic even from the point of view of algebraizability Došen proposed a Hilbert-style system, but it does not capture local consequence and the deductive systems capturing either relation in Lw a are not even protoalgebraic 25

112 But of course things get fixed if both implications are combined 26

113 But of course things get fixed if both implications are combined ia has proper algebraic semantics: 26

114 But of course things get fixed if both implications are combined ia has proper algebraic semantics: Definition A constructive Lewis algebra or ia-algebra is a tuple of the form where A := (A,,,,,, ), 26

115 But of course things get fixed if both implications are combined ia has proper algebraic semantics: Definition A constructive Lewis algebra or ia-algebra is a tuple of the form A := (A,,,,,, ), where (A,,,,, ) is a Heyting algebra and 26

116 But of course things get fixed if both implications are combined ia has proper algebraic semantics: Definition A constructive Lewis algebra or ia-algebra is a tuple of the form where A := (A,,,,,, ), (A,,,,, ) is a Heyting algebra and (A,,,,, ) is a weakly Heyting algebra (Celani, Jansana), i.e., 26

117 But of course things get fixed if both implications are combined ia has proper algebraic semantics: Definition A constructive Lewis algebra or ia-algebra is a tuple of the form where A := (A,,,,,, ), (A,,,,, ) is a Heyting algebra and (A,,,,, ) is a weakly Heyting algebra (Celani, Jansana), i.e., C1 a b a c = a (b c), 26

118 But of course things get fixed if both implications are combined ia has proper algebraic semantics: Definition A constructive Lewis algebra or ia-algebra is a tuple of the form where A := (A,,,,,, ), (A,,,,, ) is a Heyting algebra and (A,,,,, ) is a weakly Heyting algebra (Celani, Jansana), i.e., C1 a b a c = a (b c), C2 a c b c = (a b) c, 26

119 But of course things get fixed if both implications are combined ia has proper algebraic semantics: Definition A constructive Lewis algebra or ia-algebra is a tuple of the form where A := (A,,,,,, ), (A,,,,, ) is a Heyting algebra and (A,,,,, ) is a weakly Heyting algebra (Celani, Jansana), i.e., C1 a b a c = a (b c), C2 a c b c = (a b) c, C3 a b b c a c, 26

120 But of course things get fixed if both implications are combined ia has proper algebraic semantics: Definition A constructive Lewis algebra or ia-algebra is a tuple of the form where A := (A,,,,,, ), (A,,,,, ) is a Heyting algebra and (A,,,,, ) is a weakly Heyting algebra (Celani, Jansana), i.e., C1 a b a c = a (b c), C2 a c b c = (a b) c, C3 a b b c a c, C4 a a =. 26

121 An interesting exercise, which I guess should be automatic: 27

122 An interesting exercise, which I guess should be automatic: Could one obtain ia by fibring or dovetailing IPC with the minimal weak logic with strict implication? 27

123 An interesting exercise, which I guess should be automatic: Could one obtain ia by fibring or dovetailing IPC with the minimal weak logic with strict implication? This brings us to extension of intuitionism with an additional implication... 27

124 An interesting exercise, which I guess should be automatic: Could one obtain ia by fibring or dovetailing IPC with the minimal weak logic with strict implication? This brings us to extension of intuitionism with an additional implication BI with its and 27

125 An interesting exercise, which I guess should be automatic: Could one obtain ia by fibring or dovetailing IPC with the minimal weak logic with strict implication? This brings us to extension of intuitionism with an additional implication BI with its and How closely these two are related? 27

126 An interesting exercise, which I guess should be automatic: Could one obtain ia by fibring or dovetailing IPC with the minimal weak logic with strict implication? This brings us to extension of intuitionism with an additional implication BI with its and How closely these two are related? As it turns out, not quite, unless you want to add some powerful axioms 27

127 Recall the rule N a φ ψ φ ψ 28

128 Recall the rule N a φ ψ φ ψ If is now taken to satisfy BI axioms and, in particular, have a residual 28

129 Recall the rule N a φ ψ φ ψ If is now taken to satisfy BI axioms and, in particular, have a residual... many additional laws will follow 28

130 Recall the rule N a φ ψ φ ψ If is now taken to satisfy BI axioms and, in particular, have a residual... many additional laws will follow One example: S a would hold φ φ and the law of weakening 28

131 Recall the rule N a φ ψ φ ψ If is now taken to satisfy BI axioms and, in particular, have a residual... many additional laws will follow One example: S a would hold φ φ and the law of weakening Semantical considerations show that among BI axioms, commutativity is tough 28

132 Recall the rule N a φ ψ φ ψ If is now taken to satisfy BI axioms and, in particular, have a residual... many additional laws will follow One example: S a would hold φ φ and the law of weakening Semantical considerations show that among BI axioms, commutativity is tough something akin to monadicity would hold, don t have the full picture yet 28

133 Systematic completeness/correspondence results by reducing to a classical (multi-)modal language? 29

134 Systematic completeness/correspondence results by reducing to a classical (multi-)modal language? For Li, methodology developed by Wolter & Zakharyashev in the late 1990 s 29

135 Systematic completeness/correspondence results by reducing to a classical (multi-)modal language? For Li, methodology developed by Wolter & Zakharyashev in the late 1990 s Correspondence language: Li [i m] φ, ψ ::= p φ ψ φ ψ φ ψ [ i ]φ [m]φ 29

136 Systematic completeness/correspondence results by reducing to a classical (multi-)modal language? For Li, methodology developed by Wolter & Zakharyashev in the late 1990 s Correspondence language: Li [i m] φ, ψ ::= p φ ψ φ ψ φ ψ [ i ]φ [m]φ Gödel-(McKinsey-Tarski) translation for Li : t ( φ) := [ i ][m](tφ) and [ i ] in front of every subformula 29

137 t embeds faithfully every i -logic into a whole cluster of extensions of BM... 30

138 t embeds faithfully every i -logic into a whole cluster of extensions of BM the latter being the logic with S4 axioms for 30

139 t embeds faithfully every i -logic into a whole cluster of extensions of BM the latter being the logic with S4 axioms for Each such cluster has a maximal element, obtained with the help of the Grzegorczyk axiom and mix [m]φ [ i ][m][ i ]φ 30

140 t embeds faithfully every i -logic into a whole cluster of extensions of BM the latter being the logic with S4 axioms for Each such cluster has a maximal element, obtained with the help of the Grzegorczyk axiom and mix [m]φ [ i ][m][ i ]φ The translation reflects decidability, completeness, fmp. Above mix, it also reflects canonicity 30

141 t embeds faithfully every i -logic into a whole cluster of extensions of BM the latter being the logic with S4 axioms for Each such cluster has a maximal element, obtained with the help of the Grzegorczyk axiom and mix [m]φ [ i ][m][ i ]φ The translation reflects decidability, completeness, fmp. Above mix, it also reflects canonicity... enough to find one Li [i m] -counterpart with the desired property! 30

142 t embeds faithfully every i -logic into a whole cluster of extensions of BM the latter being the logic with S4 axioms for Each such cluster has a maximal element, obtained with the help of the Grzegorczyk axiom and mix [m]φ [ i ][m][ i ]φ The translation reflects decidability, completeness, fmp. Above mix, it also reflects canonicity... enough to find one Li [i m] -counterpart with the desired property!... can use the Sahlqvist algorithm, SQEMA... 30

143 Extending the Gödel-(McKinsey-Tarski) translation to Li a t(φ ψ) := [ i ][m](tφ tψ) 31

144 Extending the Gödel-(McKinsey-Tarski) translation to Li a t(φ ψ) := [ i ][m](tφ tψ) The only change one seems to need in the preceding slide is (obviously) replacing mix with lewis [m]φ [ i ][m]φ 31

145 Extending the Gödel-(McKinsey-Tarski) translation to Li a t(φ ψ) := [ i ][m](tφ tψ) The only change one seems to need in the preceding slide is (obviously) replacing mix with lewis [m]φ [ i ][m]φ Apart from this, everything seems to work (still verifying details) 31

146 The missing bit of the story: why researchers like Visser and Iemhoff were /are interested in the first place 32

147 The missing bit of the story: why researchers like Visser and Iemhoff were /are interested in the first place (nothing to do with, e.g., idioms/arrow/monads or guarded recursion... 32

148 The missing bit of the story: why researchers like Visser and Iemhoff were /are interested in the first place (nothing to do with, e.g., idioms/arrow/monads or guarded recursion at least not on the surface... and not now) 32

149 The missing bit of the story: why researchers like Visser and Iemhoff were /are interested in the first place (nothing to do with, e.g., idioms/arrow/monads or guarded recursion at least not on the surface... and not now) I mentioned it has to do with metatheory of intuitionistic arithmetic... 32

150 The missing bit of the story: why researchers like Visser and Iemhoff were /are interested in the first place (nothing to do with, e.g., idioms/arrow/monads or guarded recursion at least not on the surface... and not now) I mentioned it has to do with metatheory of intuitionistic arithmetic more specifically, with the logic of Σ 0 1 -preservativity 32

151 The missing bit of the story: why researchers like Visser and Iemhoff were /are interested in the first place (nothing to do with, e.g., idioms/arrow/monads or guarded recursion at least not on the surface... and not now) I mentioned it has to do with metatheory of intuitionistic arithmetic more specifically, with the logic of Σ 0 1 -preservativity Let us first recall the simpler idea of the logic of provability... 32

152 The missing bit of the story: why researchers like Visser and Iemhoff were /are interested in the first place (nothing to do with, e.g., idioms/arrow/monads or guarded recursion at least not on the surface... and not now) I mentioned it has to do with metatheory of intuitionistic arithmetic more specifically, with the logic of Σ 0 1 -preservativity Let us first recall the simpler idea of the logic of provability or even more generally, that of arithmetical interpretation of a propositional logic 32

153 Extend Li to L 0,..., k. with operators 0,..., k where i has arity n i F assigns to every i an arithmetical formula A(v 0,..., v ni 1) where all free variables are among the variables shown We write i,f (B 0,..., B ni 1) for F ( i )( B 0,..., B ni 1 ) Here C is the numeral of the Gödel number of C f maps V ars to arithmetical sentences. Define (φ) f F : (p) f F := f(p) ( ) f F commutes with the propositional connectives ( i (φ 0,..., φ ni 1)) f F := F ((φ 0 ) f F,..., (φ n i 1) f F ) 33

154 Let T be an arithmetical theory An extension of i-ea, the intuitionistic version of Elementary Arithmetic, in the arithmetical language 34

155 Let T be an arithmetical theory An extension of i-ea, the intuitionistic version of Elementary Arithmetic, in the arithmetical language A modal formula in L 0,..., k is T -valid w.r.t. F iff, for all assignments f of arithmetical sentences to V ars, we have T (φ) f F. 34

156 Let T be an arithmetical theory An extension of i-ea, the intuitionistic version of Elementary Arithmetic, in the arithmetical language A modal formula in L 0,..., k is T -valid w.r.t. F iff, for all assignments f of arithmetical sentences to V ars, we have T (φ) f F. Write Λ T,F for the set of L 0,..., k -formulas that are T -valid w.r.t. F. 34

157 Let T be an arithmetical theory An extension of i-ea, the intuitionistic version of Elementary Arithmetic, in the arithmetical language A modal formula in L 0,..., k is T -valid w.r.t. F iff, for all assignments f of arithmetical sentences to V ars, we have T (φ) f F. Write Λ T,F for the set of L 0,..., k -formulas that are T -valid w.r.t. F. Of course, Λ T,F interesting only for well-chosen F 34

158 Consider first empty set of s 35

159 Consider first empty set of s... i.e., our language is pure Li 35

160 Consider first empty set of s... i.e., our language is pure Li If T is Heyting Arithmetic (HA), then Λ T is precisely IPC 35

161 Consider first empty set of s... i.e., our language is pure Li If T is Heyting Arithmetic (HA), then Λ T is precisely IPC The property of theories that Λ T = IPC is called the de Jongh property. 35

162 Consider first empty set of s... i.e., our language is pure Li If T is Heyting Arithmetic (HA), then Λ T is precisely IPC The property of theories that Λ T = IPC is called the de Jongh property. There are theories for which Λ T is an intermediate logic strictly between IPC and CPC. 35

163 Consider first empty set of s... i.e., our language is pure Li If T is Heyting Arithmetic (HA), then Λ T is precisely IPC The property of theories that Λ T = IPC is called the de Jongh property. There are theories for which Λ T is an intermediate logic strictly between IPC and CPC. In fact, if you pick any intermediate logic Θ with the fmp, just extend HA with the corresponding scheme 35

164 Now consider a single unary = and any arithmetical theory T... 36

165 Now consider a single unary = and any arithmetical theory T which comes equipped with a 0 (exp)-predicate α T encoding its axiom set. 36

166 Now consider a single unary = and any arithmetical theory T which comes equipped with a 0 (exp)-predicate α T encoding its axiom set. Let provability in T be arithmetised by prov T. 36

167 Now consider a single unary = and any arithmetical theory T which comes equipped with a 0 (exp)-predicate α T encoding its axiom set. Let provability in T be arithmetised by prov T. Set F 0,T ( ) := prov T (v 0 ). Let Λ T := Λ T,F 0,T. 36

168 Now consider a single unary = and any arithmetical theory T which comes equipped with a 0 (exp)-predicate α T encoding its axiom set. Let provability in T be arithmetised by prov T. Set F 0,T ( ) := prov T (v 0 ). Let Λ T := Λ T,F 0,T. Intuitionistic Löb s logic i-gl is given by the following axioms over IPC. 36

169 Now consider a single unary = and any arithmetical theory T which comes equipped with a 0 (exp)-predicate α T encoding its axiom set. Let provability in T be arithmetised by prov T. Set F 0,T ( ) := prov T (v 0 ). Let Λ T := Λ T,F 0,T. Intuitionistic Löb s logic i-gl is given by the following axioms over IPC. N φ φ 36

170 Now consider a single unary = and any arithmetical theory T which comes equipped with a 0 (exp)-predicate α T encoding its axiom set. Let provability in T be arithmetised by prov T. Set F 0,T ( ) := prov T (v 0 ). Let Λ T := Λ T,F 0,T. Intuitionistic Löb s logic i-gl is given by the following axioms over IPC. N φ φ K (φ ψ) ( φ ψ) 36

171 Now consider a single unary = and any arithmetical theory T which comes equipped with a 0 (exp)-predicate α T encoding its axiom set. Let provability in T be arithmetised by prov T. Set F 0,T ( ) := prov T (v 0 ). Let Λ T := Λ T,F 0,T. Intuitionistic Löb s logic i-gl is given by the following axioms over IPC. N φ φ K (φ ψ) ( φ ψ) L ( φ φ) φ 36

172 The theory GL is obtained by extending i-gl with classical logic If T is a Σ 1 0 -sound classical theory, then Λ T = GL (Solovay) In contrast, the logic i-gl is not complete for HA. For example, the following principles are valid for HA but not derivable in i-gl. φ φ. ( φ φ) φ (φ ψ) (φ ψ). Still unknown what the ultimate axiomatization is 37

173 Many possible interpretations of a binary connective not all of them producing Lewis arrows! 38

174 Many possible interpretations of a binary connective not all of them producing Lewis arrows! Interpretability 38

175 Many possible interpretations of a binary connective not all of them producing Lewis arrows! Interpretability Π 0 1 -conservativity 38

176 Many possible interpretations of a binary connective not all of them producing Lewis arrows! Interpretability Π 0 1 -conservativity Σ 0 1 -preservativity classically, the last two intertranslatable, like and 38

177 The notion of Σ 0 1-preservativity for a theory T (Visser 1985) is defined as follows: 39

178 The notion of Σ 0 1-preservativity for a theory T (Visser 1985) is defined as follows: A T B iff, for all Σ 0 1-sentences S, if T S A, then T S B 39

179 The notion of Σ 0 1-preservativity for a theory T (Visser 1985) is defined as follows: A T B iff, for all Σ 0 1-sentences S, if T S A, then T S B This actually yields Lewis arrow! 39

180 The notion of Σ 0 1-preservativity for a theory T (Visser 1985) is defined as follows: A T B iff, for all Σ 0 1-sentences S, if T S A, then T S B This actually yields Lewis arrow! And more curious axioms are valid under this interpretation too... 39

181 Examples of valid principles BL (φ ψ) φ ψ 40

182 Examples of valid principles BL (φ ψ) φ ψ Tr (φ ψ ψ χ) φ χ 40

183 Examples of valid principles BL (φ ψ) φ ψ Tr (φ ψ ψ χ) φ χ K a (φ ψ φ χ) φ (ψ χ) 40

184 Examples of valid principles BL (φ ψ) φ ψ Tr (φ ψ ψ χ) φ χ K a (φ ψ φ χ) φ (ψ χ) Di (φ χ ψ χ) (φ ψ) χ 40

185 Examples of valid principles BL (φ ψ) φ ψ Tr (φ ψ ψ χ) φ χ K a (φ ψ φ χ) φ (ψ χ) Di (φ χ ψ χ) (φ ψ) χ LB φ ψ ( φ ψ) 40

186 Examples of valid principles BL (φ ψ) φ ψ Tr (φ ψ ψ χ) φ χ K a (φ ψ φ χ) φ (ψ χ) Di (φ χ ψ χ) (φ ψ) χ LB φ ψ ( φ ψ) 4 a φ φ 40

187 Examples of valid principles BL (φ ψ) φ ψ Tr (φ ψ ψ χ) φ χ K a (φ ψ φ χ) φ (ψ χ) Di (φ χ ψ χ) (φ ψ) χ LB φ ψ ( φ ψ) 4 a φ φ M a (φ ψ) ( χ φ) ( χ ψ) 40

188 Unfortunately, at this stage we re running out of time 41

189 Unfortunately, at this stage we re running out of time Just one comment: 41

190 Unfortunately, at this stage we re running out of time Just one comment: would be nice to relate the preservativity interpretation to type-theoretical /FP work, esp. setups for guarded (co-)recursion (languages in question typically contain fragments of arithmetic) 41