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

Arrows and Boxes

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

1

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

2

Intuitionistic Modal Logic iZ

Li φ, ψ ::= ⊤ | ⊥ | p | φ → ψ | φ∨ψ | φ∧ψ | φ p ∈ V ars The intuitionistic modal logic iZ:

• All intuitionistic tautologies plus whichever (set of)

axiom(s) Z you want

• Closure under MP and substitution
• Closure under G¨
• del rule: φ/ψ.
• Contains (φ → ψ) → (φ → ψ)

3

Intuitionistic Modal Logic iZ

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˘

zi´ c and Do˘ sen, 1984) ∀w1w2, (∃w3, w1 w3 ∧ w3 ⊏ w2) ⇒ (∃w′

3, w1 ⊏ w′ 3 ∧ w′ 3 w2)

3

• Recall: the Bo˘

zi´ c and Do˘ sen condition equivalent to

4

• Recall: the Bo˘

zi´ c and Do˘ sen condition equivalent to

• for any -up-set A, A is -upward-closed too

4

• Recall: the Bo˘

zi´ c 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

• Recall: the Bo˘

zi´ c 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

• Recall: the Bo˘

zi´ c 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

• Recall: the Bo˘

zi´ c 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 G¨
• del!) . . .

4

• Recall: the Bo˘

zi´ c 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 G¨
• del!) . . .
• . . . plus φ ∨ ψ ∈ Γ implies either φ or ψ belongs too

4

• Recall: the Bo˘

zi´ c 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 G¨
• del!) . . .
• . . . plus φ ∨ ψ ∈ Γ implies either φ or ψ belongs too
• on theories/filters: ordinary inclusion

4

• Recall: the Bo˘

zi´ c 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 G¨
• del!) . . .
• . . . plus φ ∨ ψ ∈ Γ implies either φ or ψ belongs too
• on theories/filters: ordinary inclusion
• Γ ⊏ ∆: α ∈ ∆ whenever α ∈ Γ

4

• Consider now Γ Γ′ ⊏ ∆′ ∆

5

• Consider now Γ Γ′ ⊏ ∆′ ∆
• Assume α ∈ Γ

5

• Consider now Γ Γ′ ⊏ ∆′ ∆
• Assume α ∈ Γ
• Then α ∈ Γ′ . . .

5

• Consider now Γ Γ′ ⊏ ∆′ ∆
• Assume α ∈ Γ
• Then α ∈ Γ′ . . .
• . . . then α ∈ ∆′ . . .

5

• Consider now Γ Γ′ ⊏ ∆′ ∆
• Assume α ∈ Γ
• Then α ∈ Γ′ . . .
• . . . then α ∈ ∆′ . . .
• . . . then α ∈ ∆

5

• Consider now Γ Γ′ ⊏ ∆′ ∆
• Assume α ∈ Γ
• Then α ∈ Γ′ . . .
• . . . then α ∈ ∆′ . . .
• . . . then α ∈ ∆
• We have shown Γ ⊏ ∆

5

• Consider now Γ Γ′ ⊏ ∆′ ∆
• Assume α ∈ Γ
• Then α ∈ Γ′ . . .
• . . . then α ∈ ∆′ . . .
• . . . then α ∈ ∆
• We have shown Γ ⊏ ∆
• That is, ; ⊏; is contained in ⊏

5

• 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

• 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

• 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

• 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

• How do you Mix Bo˘

zi´ c and Do˘ sen?

6

• How do you Mix Bo˘

zi´ c 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

• How do you Mix Bo˘

zi´ c 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”.

• Many authors (say, Goldblatt’81 Grothendieck topology as

geometric modality . . . or Bo˘ zi´ c and Do˘ sen) . . .

6

• How do you Mix Bo˘

zi´ c 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”.

• Many authors (say, Goldblatt’81 Grothendieck topology as

geometric modality . . . or Bo˘ zi´ c and Do˘ sen) . . .

• . . . noted you can impose this condition on any Bo˘

zi´ c-Do˘ sen frame . . .

6

• How do you Mix Bo˘

zi´ c 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”.

• Many authors (say, Goldblatt’81 Grothendieck topology as

geometric modality . . . or Bo˘ zi´ c and Do˘ sen) . . .

• . . . noted you can impose this condition on any Bo˘

zi´ c-Do˘ sen frame . . .

• . . . just replace ⊏ with ⊏;

6

• How do you Mix Bo˘

zi´ c 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”.

• Many authors (say, Goldblatt’81 Grothendieck topology as

geometric modality . . . or Bo˘ zi´ c and Do˘ sen) . . .

• . . . noted you can impose this condition on any Bo˘

zi´ c-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

• Now consider another language:

Lia φ, ψ ::= ⊤ | ⊥ | p | φ → ψ | φ ∨ ψ | φ ∧ ψ | φ ψ

7

• Now consider another language:

Lia 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

• Now consider another language:

Lia 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:

Lia 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:

Lia 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:

Lia 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

• f 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

• f 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¨

• rdinates, 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, Sa

φ → φ

17

• I’d suggest calling FP arrows “strong arrows”
• They satisfy in addition the axiom (φ → ψ) → φ ψ
• . . . or, equivalently, Sa

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, Sa

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, Sa

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, Sa

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, Sa

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
• f 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
• f 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
• f 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
• f 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

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 Li0 φ, ψ ::= ⊤ | ⊥ | p | φ → ψ Lia φ, ψ ::= ⊤ | ⊥ | p | φ → ψ | φ ∨ ψ | φ ∧ ψ | φ ψ Li−

a

φ, ψ ::= ⊤ | ⊥ | p | φ → ψ | φ ∧ ψ | φ ψ Li0

a

φ, ψ ::= ⊤ | ⊥ | p | φ → ψ | φ ψ Lwa φ, ψ ::= ⊤ | ⊥ | p | φ ∨ ψ | φ ∧ ψ | φ ψ

19

• 1. IPC0, i.e., intuitionism in just Li0:

20

• 1. IPC0, i.e., intuitionism in just Li0:
• 2. ⊢ α → (β → α)

20

• 1. IPC0, i.e., intuitionism in just Li0:
• 2. ⊢ α → (β → α)
• 3. ⊢ (α → (β → γ)) → ((α → β) → (α → γ))

20

• 1. IPC0, i.e., intuitionism in just Li0:
• 2. ⊢ α → (β → α)
• 3. ⊢ (α → (β → γ)) → ((α → β) → (α → γ))
• 4. ⊢ ⊥ → α

20

• 1. IPC0, i.e., intuitionism in just Li0:
• 2. ⊢ α → (β → α)
• 3. ⊢ (α → (β → γ)) → ((α → β) → (α → γ))
• 4. ⊢ ⊥ → α
• 5. For IPC−, add:

20

• 1. IPC0, i.e., intuitionism in just Li0:
• 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. IPC0, i.e., intuitionism in just Li0:
• 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. IPC0, i.e., intuitionism in just Li0:
• 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. IPC0, i.e., intuitionism in just Li0:
• 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. IPC0, i.e., intuitionism in just Li0:
• 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. IPC0, i.e., intuitionism in just Li0:
• 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. IPC0, i.e., intuitionism in just Li0:
• 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 Ka φ ψ → φ χ → φ (ψ ∧ χ)

Na

φ → ψ φ ψ.

21

Axioms and rules of the disjunction-free fragment iA−: Those of IPC− plus: Tra φ ψ → ψ χ → φ χ Ka φ ψ → φ χ → φ (ψ ∧ χ)

Na

φ → ψ φ ψ.

(I believe Tra and Na are enough for iA0, 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 Ka: φ (ψ → χ) ⊢ (φ ∧ ψ) (ψ ∧ (ψ → χ)) by Na and Ka ⊢ (φ ∧ ψ) χ by monotonicity of

22

Derivation exercises

A generalization of Ka: φ (ψ → χ) ⊢ (φ ∧ ψ) (ψ ∧ (ψ → χ)) by Na and Ka ⊢ (φ ∧ ψ) χ by monotonicity of Another curious one: ψ χ ⊢ ψ (ψ → χ) ∧ ¬ψ (ψ → χ) by Tra and Na ⊢ (ψ ∨ ¬ψ) (ψ → χ) by Di

22

Derivation exercises

A generalization of Ka: φ (ψ → χ) ⊢ (φ ∧ ψ) (ψ ∧ (ψ → χ)) by Na and Ka ⊢ (φ ∧ ψ) χ by monotonicity of Another curious one: ψ χ ⊢ ψ (ψ → χ) ∧ ¬ψ (ψ → χ) by Tra and Na ⊢ (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

• 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
• f HA—we’ll get there soon

24

• 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
• f HA—we’ll get there soon
• Two things to be discussed earlier:

24

• 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
• f 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

• 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
• f 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

• “Weak logics with strict implication” and “weak Heyting

algebras”

Corsi 1987, Doˇ sen 1993, Celani and Jansana 2001, 2005 Related system: Visser 1981, Epstein and Horn 1976

25

• “Weak logics with strict implication” and “weak Heyting

algebras”

Corsi 1987, Doˇ sen 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 Lwa

25

• “Weak logics with strict implication” and “weak Heyting

algebras”

Corsi 1987, Doˇ sen 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 Lwa

• Ordinary Kripke frames with discrete . . .

25

• “Weak logics with strict implication” and “weak Heyting

algebras”

Corsi 1987, Doˇ sen 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 Lwa

• Ordinary Kripke frames with discrete . . .
• . . . but in the absence of →, how would you know the

difference?

25

• “Weak logics with strict implication” and “weak Heyting

algebras”

Corsi 1987, Doˇ sen 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 Lwa

• 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

• “Weak logics with strict implication” and “weak Heyting

algebras”

Corsi 1987, Doˇ sen 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 Lwa

• 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
• Doˇ

sen proposed a Hilbert-style system, but it does not capture local consequence . . .

25

• “Weak logics with strict implication” and “weak Heyting

algebras”

Corsi 1987, Doˇ sen 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 Lwa

• 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
• Doˇ

sen proposed a Hilbert-style system, but it does not capture local consequence . . .

• . . . and the deductive systems capturing either relation in

Lwa are not even protoalgebraic

25

• But of course things get fixed if both implications are

combined

26

• But of course things get fixed if both implications are

combined

• iA has proper algebraic semantics:

26

• 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

26

• 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

• 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
• (A, ∧, ∨, , ⊥, ⊤) is a weakly Heyting algebra (Celani,

Jansana), i.e.,

26

• 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
• (A, ∧, ∨, , ⊥, ⊤) is a weakly Heyting algebra (Celani,

Jansana), i.e., C1 a b ∧ a c = a (b ∧ c),

26

• 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
• (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

• 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
• (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

• 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
• (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

• An interesting exercise, which I guess should be automatic:

27

• 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

• 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

• 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

• 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

• 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

• Recall the rule

Na

φ → ψ φ ψ

28

• Recall the rule

Na

φ → ψ φ ψ

• If is now taken to satisfy BI axioms and, in particular,

have a residual

28

• Recall the rule

Na

φ → ψ φ ψ

• If is now taken to satisfy BI axioms and, in particular,

have a residual

• . . . many additional laws will follow

28

• Recall the rule

Na

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: Sa

φ → φ and the law of weakening would hold

28

• Recall the rule

Na

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: Sa

φ → φ and the law of weakening would hold

• Semantical considerations show that among BI axioms,

commutativity is tough

28

• Recall the rule

Na

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: Sa

φ → φ and the law of weakening would hold

• Semantical considerations show that among BI axioms,

commutativity is tough

• something akin to monadicity would hold, don’t have the

full picture yet

28

Systematic completeness/correspondence results . . .

• . . . by reducing to a classical (multi-)modal language?

29

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

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

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

• G¨
• del-(McKinsey-Tarski) translation for Li:

t(φ) := [ i ][m](tφ) and [ i ] in front of every subformula

29

• t embeds faithfully every i-logic into a whole cluster of

extensions of BM . . .

30

• t embeds faithfully every i-logic into a whole cluster of

extensions of BM . . .

• . . . the latter being the logic with S4 axioms for

30

• 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

• 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

• 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

• 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

• Extending the G¨
• del-(McKinsey-Tarski) translation to Lia

t(φ ψ) := [ i ][m](tφ → tψ)

31

• Extending the G¨
• del-(McKinsey-Tarski) translation to Lia

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

• Extending the G¨
• del-(McKinsey-Tarski) translation to Lia

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

• The missing bit of the story: why researchers like Visser

and Iemhoff were /are interested in the first place

32

• 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

• 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

• 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

• 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 Σ0

1-preservativity

32

• 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 Σ0

1-preservativity

• Let us first recall the simpler idea of the logic of provability

. . .

32

• 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 Σ0

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

• Extend Li to L⊚0,...,⊚k. with operators ⊚0, . . . , ⊚k

where ⊚i has arity ni

• F assigns to every ⊚i an arithmetical formula

A(v0, . . . , vni−1)

where all free variables are among the variables shown

• We write ⊚i,F (B0, . . . , Bni−1) for F(⊚i)(B0, . . . , Bni−1)

Here C is the numeral of the G¨

• del 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(φ0, . . . , φni−1))f

F := ⊚F ((φ0)f F , . . . , (φni−1)f F )

33

• Let T be an arithmetical theory

An extension of i-EA, the intuitionistic version of Elementary Arithmetic, in the arithmetical language

34

• 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

• 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 ΛT,F for the set of L⊚0,...,⊚k-formulas that are

T-valid w.r.t. F.

34

• 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 ΛT,F for the set of L⊚0,...,⊚k-formulas that are

T-valid w.r.t. F.

• Of course, ΛT,F interesting only for well-chosen F

34

• Consider first empty set of ⊚’s

35

• Consider first empty set of ⊚’s
• . . . i.e., our language is pure Li

35

• Consider first empty set of ⊚’s
• . . . i.e., our language is pure Li
• If T is Heyting Arithmetic (HA), then ΛT is precisely IPC

35

• Consider first empty set of ⊚’s
• . . . i.e., our language is pure Li
• If T is Heyting Arithmetic (HA), then ΛT is precisely IPC
• The property of theories that ΛT = IPC is called the de

Jongh property.

35

• Consider first empty set of ⊚’s
• . . . i.e., our language is pure Li
• If T is Heyting Arithmetic (HA), then ΛT is precisely IPC
• The property of theories that ΛT = IPC is called the de

Jongh property.

• There are theories for which ΛT is an intermediate logic

strictly between IPC and CPC.

35

• Consider first empty set of ⊚’s
• . . . i.e., our language is pure Li
• If T is Heyting Arithmetic (HA), then ΛT is precisely IPC
• The property of theories that ΛT = IPC is called the de

Jongh property.

• There are theories for which ΛT 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

• Now consider a single unary ⊚ = and any arithmetical

theory T . . .

36

• Now consider a single unary ⊚ = and any arithmetical

theory T . . .

• . . . which comes equipped with a ∆0(exp)-predicate αT

encoding its axiom set.

36

• Now consider a single unary ⊚ = and any arithmetical

theory T . . .

• . . . which comes equipped with a ∆0(exp)-predicate αT

encoding its axiom set.

• Let provability in T be arithmetised by provT .

36

• Now consider a single unary ⊚ = and any arithmetical

theory T . . .

• . . . which comes equipped with a ∆0(exp)-predicate αT

encoding its axiom set.

• Let provability in T be arithmetised by provT .
• Set F0,T () := provT(v0). Let Λ∗

T := ΛT,F0,T .

36

• Now consider a single unary ⊚ = and any arithmetical

theory T . . .

• . . . which comes equipped with a ∆0(exp)-predicate αT

encoding its axiom set.

• Let provability in T be arithmetised by provT .
• Set F0,T () := provT(v0). Let Λ∗

T := ΛT,F0,T .

• Intuitionistic L¨
• b’s logic i-GL is given by the following

axioms over IPC.

36

• Now consider a single unary ⊚ = and any arithmetical

theory T . . .

• . . . which comes equipped with a ∆0(exp)-predicate αT

encoding its axiom set.

• Let provability in T be arithmetised by provT .
• Set F0,T () := provT(v0). Let Λ∗

T := ΛT,F0,T .

• Intuitionistic L¨
• b’s logic i-GL is given by the following

axioms over IPC.

N ⊢ φ ⇒ ⊢ φ

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• Now consider a single unary ⊚ = and any arithmetical

theory T . . .

• . . . which comes equipped with a ∆0(exp)-predicate αT

encoding its axiom set.

• Let provability in T be arithmetised by provT .
• Set F0,T () := provT(v0). Let Λ∗

T := ΛT,F0,T .

• Intuitionistic L¨
• b’s logic i-GL is given by the following

axioms over IPC.

N ⊢ φ ⇒ ⊢ φ K ⊢ (φ → ψ) → (φ → ψ)

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• Now consider a single unary ⊚ = and any arithmetical

theory T . . .

• . . . which comes equipped with a ∆0(exp)-predicate αT

encoding its axiom set.

• Let provability in T be arithmetised by provT .
• Set F0,T () := provT(v0). Let Λ∗

T := ΛT,F0,T .

• Intuitionistic L¨
• b’s logic i-GL is given by the following

axioms over IPC.

N ⊢ φ ⇒ ⊢ φ K ⊢ (φ → ψ) → (φ → ψ) L ⊢ (φ → φ) → φ

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The theory GL is obtained by extending i-GL with classical logic If T is a Σ1

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

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• Many possible interpretations of a binary connective

not all of them producing Lewis’ arrows!

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• Many possible interpretations of a binary connective

not all of them producing Lewis’ arrows!

• Interpretability

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• Many possible interpretations of a binary connective

not all of them producing Lewis’ arrows!

• Interpretability
• Π0

1-conservativity

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• Many possible interpretations of a binary connective

not all of them producing Lewis’ arrows!

• Interpretability
• Π0

1-conservativity

• Σ0

1-preservativity classically, the last two intertranslatable, like and ♦

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• The notion of Σ0

1-preservativity for a theory T (Visser

1985) is defined as follows:

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• The notion of Σ0

1-preservativity for a theory T (Visser

1985) is defined as follows:

• A T B iff, for all Σ0

1-sentences S, if T ⊢ S → A, then

T ⊢ S → B

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• The notion of Σ0

1-preservativity for a theory T (Visser

1985) is defined as follows:

• A T B iff, for all Σ0

1-sentences S, if T ⊢ S → A, then

T ⊢ S → B

• This actually yields Lewis’ arrow!

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• The notion of Σ0

1-preservativity for a theory T (Visser

1985) is defined as follows:

• A T B iff, for all Σ0

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 . . .

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Examples of valid principles

BL ⊢ (φ → ψ) → φ ψ

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Examples of valid principles

BL ⊢ (φ → ψ) → φ ψ Tr ⊢ (φ ψ ∧ ψ χ) → φ χ

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Examples of valid principles

BL ⊢ (φ → ψ) → φ ψ Tr ⊢ (φ ψ ∧ ψ χ) → φ χ Ka ⊢ (φ ψ ∧ φ χ) → φ (ψ ∧ χ)

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Examples of valid principles

BL ⊢ (φ → ψ) → φ ψ Tr ⊢ (φ ψ ∧ ψ χ) → φ χ Ka ⊢ (φ ψ ∧ φ χ) → φ (ψ ∧ χ) Di ⊢ (φ χ ∧ ψ χ) → (φ ∨ ψ) χ

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Examples of valid principles

BL ⊢ (φ → ψ) → φ ψ Tr ⊢ (φ ψ ∧ ψ χ) → φ χ Ka ⊢ (φ ψ ∧ φ χ) → φ (ψ ∧ χ) Di ⊢ (φ χ ∧ ψ χ) → (φ ∨ ψ) χ LB ⊢ φ ψ → (φ → ψ)

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Examples of valid principles

BL ⊢ (φ → ψ) → φ ψ Tr ⊢ (φ ψ ∧ ψ χ) → φ χ Ka ⊢ (φ ψ ∧ φ χ) → φ (ψ ∧ χ) Di ⊢ (φ χ ∧ ψ χ) → (φ ∨ ψ) χ LB ⊢ φ ψ → (φ → ψ) 4a ⊢ φ φ

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Examples of valid principles

BL ⊢ (φ → ψ) → φ ψ Tr ⊢ (φ ψ ∧ ψ χ) → φ χ Ka ⊢ (φ ψ ∧ φ χ) → φ (ψ ∧ χ) Di ⊢ (φ χ ∧ ψ χ) → (φ ∨ ψ) χ LB ⊢ φ ψ → (φ → ψ) 4a ⊢ φ φ Ma ⊢ (φ ψ) → (χ → φ) (χ → ψ)

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• Unfortunately, at this stage we’re running out of time

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• Unfortunately, at this stage we’re running out of time
• Just one comment:

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• 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)

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