Arrows and Boxes Tadeusz Litak (based on a joint work with Albert - PowerPoint PPT Presentation

• Now consider another language: L i 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

• Now consider another language: L i 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

• Now consider another language: L i 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

• Now consider another language: L i 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

Historical aside • Lewis himself wanted to have involutive negation 8

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

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

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

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

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

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

• 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

[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 co¨ ordinates, are themselves principles of mathematics, or the reason for the radix 10 is of the essence of number. 11

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

• 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

Returning to our semantics . . . • . . . is Bo˘ zi´ c and Do˘ sen enough for persistence? 13

Returning to our semantics . . . • . . . is Bo˘ zi´ c 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

Returning to our semantics . . . • . . . is Bo˘ zi´ c 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

Returning to our semantics . . . • . . . is Bo˘ zi´ c 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

• We’re not done with the relationship between φ � ψ and � ( φ → ψ ) = ⊤ � ( φ → ψ ) 14

• We’re not done with the relationship between φ � ψ and � ( φ → ψ ) = ⊤ � ( φ → ψ ) • Which of them is stronger? 14

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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 φ � ψ → � ( φ → ψ ) is equivalent to postfixing if ℓ ⊏ m � n , then ℓ ⊏ n i.e., ⊏ ; � being contained in ⊏ 14

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

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

� here is our � ENTCS 2011, proceedings of MSFP 2008 16

• I’d suggest calling FP arrows “strong arrows” 17

• I’d suggest calling FP arrows “strong arrows” • They satisfy in addition the axiom ( φ → ψ ) → φ � ψ 17

• 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

• 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

• 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

• 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

• 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

• 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

• This recaps most of my previous talk 18

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

• 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

• 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

• 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

• 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

L i � φ, ψ ::= ⊤ | ⊥ | p | φ → ψ | φ ∨ ψ | φ ∧ ψ | � φ L i φ, ψ ::= ⊤ | ⊥ | p | φ → ψ | φ ∨ ψ | φ ∧ ψ L i − φ, ψ ::= ⊤ | ⊥ | p | φ → ψ | φ ∧ ψ L i 0 φ, ψ ::= ⊤ | ⊥ | p | φ → ψ L i a φ, ψ ::= ⊤ | ⊥ | p | φ → ψ | φ ∨ ψ | φ ∧ ψ | φ � ψ L i − φ, ψ ::= ⊤ | ⊥ | p | φ → ψ | φ ∧ ψ | φ � ψ a L i 0 φ, ψ ::= ⊤ | ⊥ | p | φ → ψ | φ � ψ a L w a φ, ψ ::= ⊤ | ⊥ | p | φ ∨ ψ | φ ∧ ψ | φ � ψ 19

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

1. IPC 0 , i.e., intuitionism in just L i 0 : 2. ⊢ α → ( β → α ) 20

1. IPC 0 , i.e., intuitionism in just L i 0 : 2. ⊢ α → ( β → α ) 3. ⊢ ( α → ( β → γ )) → (( α → β ) → ( α → γ )) 20

1. IPC 0 , i.e., intuitionism in just L i 0 : 2. ⊢ α → ( β → α ) 3. ⊢ ( α → ( β → γ )) → (( α → β ) → ( α → γ )) 4. ⊢ ⊥ → α 20

1. IPC 0 , i.e., intuitionism in just L i 0 : 2. ⊢ α → ( β → α ) 3. ⊢ ( α → ( β → γ )) → (( α → β ) → ( α → γ )) 4. ⊢ ⊥ → α 5. For IPC − , add: 20

1. IPC 0 , i.e., intuitionism in just L i 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

1. IPC 0 , i.e., intuitionism in just L i 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

1. IPC 0 , i.e., intuitionism in just L i 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

1. IPC 0 , i.e., intuitionism in just L i 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

1. IPC 0 , i.e., intuitionism in just L i 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

1. IPC 0 , i.e., intuitionism in just L i 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

1. IPC 0 , i.e., intuitionism in just L i 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

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 ) φ → ψ N a φ � ψ. 21

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 ) φ → ψ N a 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

Derivation exercises A generalization of K a : φ � ( ψ → χ ) ⊢ ( φ ∧ ψ ) � ( ψ ∧ ( ψ → χ )) by N a and K a ⊢ ( φ ∧ ψ ) � χ by monotonicity of � 22

Derivation exercises A generalization of K a : φ � ( ψ → χ ) ⊢ ( φ ∧ ψ ) � ( ψ ∧ ( ψ → χ )) by N a and K a ⊢ ( φ ∧ ψ ) � χ by monotonicity of � Another curious one: ψ � χ ⊢ ψ � ( ψ → χ ) ∧ ¬ ψ � ( ψ → χ ) by Tra and N a ⊢ ( ψ ∨ ¬ ψ ) � ( ψ → χ ) by Di 22

Derivation exercises A generalization of K a : φ � ( ψ → χ ) ⊢ ( φ ∧ ψ ) � ( ψ ∧ ( ψ → χ )) by N a and K a ⊢ ( φ ∧ ψ ) � χ by monotonicity of � Another curious one: ψ � χ ⊢ ψ � ( ψ → χ ) ∧ ¬ ψ � ( ψ → χ ) by Tra and N a ⊢ ( ψ ∨ ¬ ψ ) � ( ψ → χ ) by Di We thus get ψ � χ ⊣⊢ ( ψ ∨ ¬ ψ ) � ( ψ → χ ) 22

• The validity of p � q ⊣⊢ ( p ∨ ¬ p ) � ( p → q ) implies that Col is valid over classical logic 23

• 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

• 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

• 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 24