A stratified pointfree definition of probability via constructive - PowerPoint PPT Presentation

Probability on fuzzy subsets Give a probability space ( Ω , E , ̃ P ) 1 Fuzzy subsets of Ω are f ∈ [ 0 , 1 ] Ω , 2 they form a Heyting algebra with inf, sup, pointwise ≤ and ⇒ : ⎧ ⎪ ⎪ ( f ⇒ g )( ω ) = 1 if f ( ω ) ≤ g ( ω ) ⎨ ⎪ ( f ⇒ g )( ω ) = g ( ω ) otherwise ⎪ ⎩ They form a de Morgan algebra with ( ¬ f )( ω ) ∶ = 1 − f ( ω ) . If ∫ Ω f d ̃ P is defined, we say that f is a measurable fuzzy set. 3

Probability on fuzzy subsets Give a probability space ( Ω , E , ̃ P ) 1 Fuzzy subsets of Ω are f ∈ [ 0 , 1 ] Ω , 2 they form a Heyting algebra with inf, sup, pointwise ≤ and ⇒ : ⎧ ⎪ ⎪ ( f ⇒ g )( ω ) = 1 if f ( ω ) ≤ g ( ω ) ⎨ ⎪ ( f ⇒ g )( ω ) = g ( ω ) otherwise ⎪ ⎩ They form a de Morgan algebra with ( ¬ f )( ω ) ∶ = 1 − f ( ω ) . If ∫ Ω f d ̃ P is defined, we say that f is a measurable fuzzy set. 3 We call that integral P ( f ) .

Probability on fuzzy subsets Give a probability space ( Ω , E , ̃ P ) 1 Fuzzy subsets of Ω are f ∈ [ 0 , 1 ] Ω , 2 they form a Heyting algebra with inf, sup, pointwise ≤ and ⇒ : ⎧ ⎪ ⎪ ( f ⇒ g )( ω ) = 1 if f ( ω ) ≤ g ( ω ) ⎨ ⎪ ( f ⇒ g )( ω ) = g ( ω ) otherwise ⎪ ⎩ They form a de Morgan algebra with ( ¬ f )( ω ) ∶ = 1 − f ( ω ) . If ∫ Ω f d ̃ P is defined, we say that f is a measurable fuzzy set. 3 We call that integral P ( f ) . Subsets in E are identified with their characteristic functions: 4 ̃ P ( E ) = P ( χ E ) for every E ∈ R .

Probability on fuzzy subsets Give a probability space ( Ω , E , ̃ P ) 1 Fuzzy subsets of Ω are f ∈ [ 0 , 1 ] Ω , 2 they form a Heyting algebra with inf, sup, pointwise ≤ and ⇒ : ⎧ ⎪ ⎪ ( f ⇒ g )( ω ) = 1 if f ( ω ) ≤ g ( ω ) ⎨ ⎪ ( f ⇒ g )( ω ) = g ( ω ) otherwise ⎪ ⎩ They form a de Morgan algebra with ( ¬ f )( ω ) ∶ = 1 − f ( ω ) . If ∫ Ω f d ̃ P is defined, we say that f is a measurable fuzzy set. 3 We call that integral P ( f ) . Subsets in E are identified with their characteristic functions: 4 ̃ P ( E ) = P ( χ E ) for every E ∈ R . These are exactly those measurable fuzzy sets for which f ⇒ � and ¬ f 5 coincide.

A common shape

A common shape a collection of potential events P . 1

A common shape a collection of potential events P . 1 P ( R ) , P ([ 0 , 1 ]) , [ 0 , 1 ] Ω

A common shape a collection of potential events P . 1 P ( R ) , P ([ 0 , 1 ]) , [ 0 , 1 ] Ω a collection of easily-valuable events, regular events R . 2

A common shape a collection of potential events P . 1 P ( R ) , P ([ 0 , 1 ]) , [ 0 , 1 ] Ω a collection of easily-valuable events, regular events R . 2 finite unions of left-open right-closed intervals, Borel subsets, Measurable subsets.

A common shape a collection of potential events P . 1 P ( R ) , P ([ 0 , 1 ]) , [ 0 , 1 ] Ω a collection of easily-valuable events, regular events R . 2 finite unions of left-open right-closed intervals, Borel subsets, Measurable subsets. a collection of actual events A with R ⊆ A ⊆ P 3

A common shape a collection of potential events P . 1 P ( R ) , P ([ 0 , 1 ]) , [ 0 , 1 ] Ω a collection of easily-valuable events, regular events R . 2 finite unions of left-open right-closed intervals, Borel subsets, Measurable subsets. a collection of actual events A with R ⊆ A ⊆ P 3 on which probability P is defined.

A common shape a collection of potential events P . 1 P ( R ) , P ([ 0 , 1 ]) , [ 0 , 1 ] Ω a collection of easily-valuable events, regular events R . 2 finite unions of left-open right-closed intervals, Borel subsets, Measurable subsets. a collection of actual events A with R ⊆ A ⊆ P 3 on which probability P is defined. Borel subsets of R , Lebesgue measurable subsets of [ 0 , 1 ] , Measurable fuzzy subsets.

A common shape a collection of potential events P . 1 P ( R ) , P ([ 0 , 1 ]) , [ 0 , 1 ] Ω a collection of easily-valuable events, regular events R . 2 finite unions of left-open right-closed intervals, Borel subsets, Measurable subsets. a collection of actual events A with R ⊆ A ⊆ P 3 on which probability P is defined. Borel subsets of R , Lebesgue measurable subsets of [ 0 , 1 ] , Measurable fuzzy subsets.

Constructive frequentist probability Classically natural density for A ⊆ N + : ∣ A ∩ { 1 ,..., n }∣ δ ( A ) ∶ = lim n →∞ (if this limit exists ) n

Constructive frequentist probability Classically natural density for A ⊆ N + : ∣ A ∩ { 1 ,..., n }∣ δ ( A ) ∶= lim n →∞ (if this limit exists ) n Subsets of natural numbers can be seen classically as sequences in { 0 , 1 }

Constructive frequentist probability Classically natural density for A ⊆ N + : ∣ A ∩ { 1 ,..., n }∣ δ ( A ) ∶= lim n →∞ (if this limit exists ) n Subsets of natural numbers can be seen classically as sequences in { 0 , 1 } We can think of them as sequences of trials in which the success of some event is recorded.

Constructive frequentist probability Classically natural density for A ⊆ N + : ∣ A ∩ { 1 ,..., n }∣ δ ( A ) ∶= lim n →∞ (if this limit exists ) n Subsets of natural numbers can be seen classically as sequences in { 0 , 1 } We can think of them as sequences of trials in which the success of some event is recorded. With this interpretation: ∣ A ∩ { 1 ,..., n }∣ n is the rate of success in the first n trials.

Constructive frequentist probability Classically natural density for A ⊆ N + : ∣ A ∩ { 1 ,..., n }∣ δ ( A ) ∶= lim n →∞ (if this limit exists ) n Subsets of natural numbers can be seen classically as sequences in { 0 , 1 } We can think of them as sequences of trials in which the success of some event is recorded. With this interpretation: ∣ A ∩ { 1 ,..., n }∣ n is the rate of success in the first n trials. We try to develop this idea in a constructive framework Minimalist Foundation (+ AC!)

Potential events A potential event e is a sequence of { 0 , 1 } : 1 e ( n ) ∈ { 0 , 1 } [ n ∈ N + ]

Potential events A potential event e is a sequence of { 0 , 1 } : 1 e ( n ) ∈ { 0 , 1 } [ n ∈ N + ] To every potential event is associated a sequence of rates of success Φ 2 i = 1 e ( n ) Φ ( e )( n ) ∶= ∑ n ∈ Q [ n ∈ N + ] n

Potential events A potential event e is a sequence of { 0 , 1 } : 1 e ( n ) ∈ { 0 , 1 } [ n ∈ N + ] To every potential event is associated a sequence of rates of success Φ 2 i = 1 e ( n ) Φ ( e )( n ) ∶= ∑ n ∈ Q [ n ∈ N + ] n � ∶= λ n . 0 and ⊺ ∶= λ n . 1 3

Potential events A potential event e is a sequence of { 0 , 1 } : 1 e ( n ) ∈ { 0 , 1 } [ n ∈ N + ] To every potential event is associated a sequence of rates of success Φ 2 i = 1 e ( n ) Φ ( e )( n ) ∶= ∑ n ∈ Q [ n ∈ N + ] n � ∶= λ n . 0 and ⊺ ∶= λ n . 1 3 e ∧ e ′ and e ∨ e ′ are the pointwise inf and sup .

Potential events A potential event e is a sequence of { 0 , 1 } : 1 e ( n ) ∈ { 0 , 1 } [ n ∈ N + ] To every potential event is associated a sequence of rates of success Φ 2 i = 1 e ( n ) Φ ( e )( n ) ∶= ∑ n ∈ Q [ n ∈ N + ] n � ∶= λ n . 0 and ⊺ ∶= λ n . 1 3 e ∧ e ′ and e ∨ e ′ are the pointwise inf and sup . ¬ e ∶= λ n . ( 1 − e ( n ))

Potential events A potential event e is a sequence of { 0 , 1 } : 1 e ( n ) ∈ { 0 , 1 } [ n ∈ N + ] To every potential event is associated a sequence of rates of success Φ 2 i = 1 e ( n ) Φ ( e )( n ) ∶= ∑ n ∈ Q [ n ∈ N + ] n � ∶= λ n . 0 and ⊺ ∶= λ n . 1 3 e ∧ e ′ and e ∨ e ′ are the pointwise inf and sup . ¬ e ∶= λ n . ( 1 − e ( n )) e ≤ e ′ is pointwise ≤ .

Potential events A potential event e is a sequence of { 0 , 1 } : 1 e ( n ) ∈ { 0 , 1 } [ n ∈ N + ] To every potential event is associated a sequence of rates of success Φ 2 i = 1 e ( n ) Φ ( e )( n ) ∶= ∑ n ∈ Q [ n ∈ N + ] n � ∶= λ n . 0 and ⊺ ∶= λ n . 1 3 e ∧ e ′ and e ∨ e ′ are the pointwise inf and sup . ¬ e ∶= λ n . ( 1 − e ( n )) e ≤ e ′ is pointwise ≤ .

Actual events An actual event is a pair ( e ,γ ) where: 1

Actual events An actual event is a pair ( e ,γ ) where: 1 e is a potential event.

Actual events An actual event is a pair ( e ,γ ) where: 1 e is a potential event. γ ∈ N + → N + is an increasing sequence.

Actual events An actual event is a pair ( e ,γ ) where: 1 e is a potential event. γ ∈ N + → N + is an increasing sequence. ∣ Φ ( e ,γ ( n ) + i ) − Φ ( e ,γ ( n ) + j )∣ < 1 n [ n ∈ N + , i ∈ N , j ∈ N ]

Actual events An actual event is a pair ( e ,γ ) where: 1 e is a potential event. γ ∈ N + → N + is an increasing sequence. ∣ Φ ( e ,γ ( n ) + i ) − Φ ( e ,γ ( n ) + j )∣ < 1 n [ n ∈ N + , i ∈ N , j ∈ N ] ( e ,γ ) and ( e ′ ,γ ′ ) are equal iff e = e ′ ∈ { 0 , 1 } N + 2

Actual events An actual event is a pair ( e ,γ ) where: 1 e is a potential event. γ ∈ N + → N + is an increasing sequence. ∣ Φ ( e ,γ ( n ) + i ) − Φ ( e ,γ ( n ) + j )∣ < 1 n [ n ∈ N + , i ∈ N , j ∈ N ] ( e ,γ ) and ( e ′ ,γ ′ ) are equal iff e = e ′ ∈ { 0 , 1 } N + 2

Probability Bishop real numbers: sequences x ( n ) ∈ Q [ n ∈ N + ] such that 1 ∣ x ( n ) − x ( m )∣ < 1 n + 1 m [ n ∈ N + , m ∈ N + ]

Probability Bishop real numbers: sequences x ( n ) ∈ Q [ n ∈ N + ] such that 1 ∣ x ( n ) − x ( m )∣ < 1 n + 1 m [ n ∈ N + , m ∈ N + ] and x and y are equal Bishop reals iff ∣ x ( n ) − y ( n )∣ < 2 n [ n ∈ N + ]

Probability Bishop real numbers: sequences x ( n ) ∈ Q [ n ∈ N + ] such that 1 ∣ x ( n ) − x ( m )∣ < 1 n + 1 m [ n ∈ N + , m ∈ N + ] and x and y are equal Bishop reals iff ∣ x ( n ) − y ( n )∣ < 2 n [ n ∈ N + ] If ( e ,γ ) is an actual event, then Φ ( e ) ○ γ is a Bishop real number. 2

Probability Bishop real numbers: sequences x ( n ) ∈ Q [ n ∈ N + ] such that 1 ∣ x ( n ) − x ( m )∣ < 1 n + 1 m [ n ∈ N + , m ∈ N + ] and x and y are equal Bishop reals iff ∣ x ( n ) − y ( n )∣ < 2 n [ n ∈ N + ] If ( e ,γ ) is an actual event, then Φ ( e ) ○ γ is a Bishop real number. 2 If ( e ,γ ) and ( e ′ ,γ ′ ) are equal actual events, then Φ ( e ) ○ γ and Φ ( e ′ ) ○ γ ′ are equal Bishop reals.

Probability Bishop real numbers: sequences x ( n ) ∈ Q [ n ∈ N + ] such that 1 ∣ x ( n ) − x ( m )∣ < 1 n + 1 m [ n ∈ N + , m ∈ N + ] and x and y are equal Bishop reals iff ∣ x ( n ) − y ( n )∣ < 2 n [ n ∈ N + ] If ( e ,γ ) is an actual event, then Φ ( e ) ○ γ is a Bishop real number. 2 If ( e ,γ ) and ( e ′ ,γ ′ ) are equal actual events, then Φ ( e ) ○ γ and Φ ( e ′ ) ○ γ ′ are equal Bishop reals. P ( e ,γ ) ∶= Φ ( e ) ○ γ is a well-defined operation 3 from actual events to Bishop reals.

Properties of probability (1) We denote with R the set of Bishop reals, with P the set of potential events and with A the set of actual events.

Properties of probability (1) We denote with R the set of Bishop reals, with P the set of potential events and with A the set of actual events. (Strictness) (� ,λ n . n ) ∈ A and P (� ,λ n . n ) = R 0 1

Properties of probability (1) We denote with R the set of Bishop reals, with P the set of potential events and with A the set of actual events. (Strictness) (� ,λ n . n ) ∈ A and P (� ,λ n . n ) = R 0 1 (Involution) if ( e ,γ ) ∈ A , then (¬ e ,γ ) ∈ A and P (¬ e ,γ ) = R 1 − P ( e ,γ ) . 2

Properties of probability (1) We denote with R the set of Bishop reals, with P the set of potential events and with A the set of actual events. (Strictness) (� ,λ n . n ) ∈ A and P (� ,λ n . n ) = R 0 1 (Involution) if ( e ,γ ) ∈ A , then (¬ e ,γ ) ∈ A and P (¬ e ,γ ) = R 1 − P ( e ,γ ) . 2 (Monotonicity) if ( e ,γ ) ∈ A , ( e ′ ,γ ′ ) ∈ A and e ≤ e ′ , then P ( e ,γ ) ≤ R P ( e ′ ,γ ′ ) . 3

Properties of probability (1) We denote with R the set of Bishop reals, with P the set of potential events and with A the set of actual events. (Strictness) (� ,λ n . n ) ∈ A and P (� ,λ n . n ) = R 0 1 (Involution) if ( e ,γ ) ∈ A , then (¬ e ,γ ) ∈ A and P (¬ e ,γ ) = R 1 − P ( e ,γ ) . 2 (Monotonicity) if ( e ,γ ) ∈ A , ( e ′ ,γ ′ ) ∈ A and e ≤ e ′ , then P ( e ,γ ) ≤ R P ( e ′ ,γ ′ ) . 3 (Null events) if ( e ,γ ) ∈ A , P ( e ,γ ) = R 0 and e ′ ≤ e, then ( e ′ ,λ n .γ ( 6 n )) ∈ A 4

Properties of probability (2) (Incompatible events) If ( e ,γ ) ∈ A , ( e ′ ,γ ′ ) ∈ A and e ∧ e ′ = P � , 1 then ( e ∧ e ′ ,λ n . ( γ ( 2 n ) + γ ′ ( 2 n ))) ∈ A .

Properties of probability (2) (Incompatible events) If ( e ,γ ) ∈ A , ( e ′ ,γ ′ ) ∈ A and e ∧ e ′ = P � , 1 then ( e ∧ e ′ ,λ n . ( γ ( 2 n ) + γ ′ ( 2 n ))) ∈ A . (Modularity) If ( e ,γ ) , ( e ′ ,γ ′ ) , ( e ∨ e ′ ,γ ′′ ) , ( e ∧ e ′ ,γ ′′′ ) ∈ A , 2 then P ( e ,γ ) + P ( e ′ ,γ ′ ) = P ( e ∨ e ′ ,γ ′′ ) + P ( e ∧ e ′ ,γ ′′′ )

Regular events For a = [ a 1 ,..., a n ] , p = [ p 1 ,..., p m ] finite lists of 0s and 1s,

Regular events For a = [ a 1 ,..., a n ] , p = [ p 1 ,..., p m ] finite lists of 0s and 1s, ∥ a , p ∥ ∈ P is [ a 1 ,..., a n , p 1 ,..., p m , p 1 ,..., p m .... ]

Regular events For a = [ a 1 ,..., a n ] , p = [ p 1 ,..., p m ] finite lists of 0s and 1s, ∥ a , p ∥ ∈ P is [ a 1 ,..., a n , p 1 ,..., p m , p 1 ,..., p m .... ] It is sort of deterministic event (up to a finite number of errors).

Regular events For a = [ a 1 ,..., a n ] , p = [ p 1 ,..., p m ] finite lists of 0s and 1s, ∥ a , p ∥ ∈ P is [ a 1 ,..., a n , p 1 ,..., p m , p 1 ,..., p m .... ] It is sort of deterministic event (up to a finite number of errors). (∥ a , [ 0 ]∥ ,λ i . 2 in ) ∈ A and P (∥ a , [ 0 ]∥ ,λ i . 2 in )) = R 0. 1

Regular events For a = [ a 1 ,..., a n ] , p = [ p 1 ,..., p m ] finite lists of 0s and 1s, ∥ a , p ∥ ∈ P is [ a 1 ,..., a n , p 1 ,..., p m , p 1 ,..., p m .... ] It is sort of deterministic event (up to a finite number of errors). (∥ a , [ 0 ]∥ ,λ i . 2 in ) ∈ A and P (∥ a , [ 0 ]∥ ,λ i . 2 in )) = R 0. 1 ∑ m i = 1 p i (∥[ ] , p ∥ ,λ i . 4 im ) ∈ A and P (∥[ ] , p ∥ ,λ i . 4 im )) = R . 2 m

Regular events For a = [ a 1 ,..., a n ] , p = [ p 1 ,..., p m ] finite lists of 0s and 1s, ∥ a , p ∥ ∈ P is [ a 1 ,..., a n , p 1 ,..., p m , p 1 ,..., p m .... ] It is sort of deterministic event (up to a finite number of errors). (∥ a , [ 0 ]∥ ,λ i . 2 in ) ∈ A and P (∥ a , [ 0 ]∥ ,λ i . 2 in )) = R 0. 1 ∑ m i = 1 p i (∥[ ] , p ∥ ,λ i . 4 im ) ∈ A and P (∥[ ] , p ∥ ,λ i . 4 im )) = R . 2 m If ( e ,γ ) ∈ A , then ( e + ∶= [ 0 , e ] ,λ i .γ ( 3 i ) + 1 ) ∈ A and 3 P ( e + ,λ i .γ ( 3 i ) + 1 ) = R P ( e ,γ )

Regular events For a = [ a 1 ,..., a n ] , p = [ p 1 ,..., p m ] finite lists of 0s and 1s, ∥ a , p ∥ ∈ P is [ a 1 ,..., a n , p 1 ,..., p m , p 1 ,..., p m .... ] It is sort of deterministic event (up to a finite number of errors). (∥ a , [ 0 ]∥ ,λ i . 2 in ) ∈ A and P (∥ a , [ 0 ]∥ ,λ i . 2 in )) = R 0. 1 ∑ m i = 1 p i (∥[ ] , p ∥ ,λ i . 4 im ) ∈ A and P (∥[ ] , p ∥ ,λ i . 4 im )) = R . 2 m If ( e ,γ ) ∈ A , then ( e + ∶= [ 0 , e ] ,λ i .γ ( 3 i ) + 1 ) ∈ A and 3 P ( e + ,λ i .γ ( 3 i ) + 1 ) = R P ( e ,γ ) Hence for every a and p , there is γ such that ∑ m i = 1 p i (∥ a , p ∥ ,γ ) ∈ A and P (∥ a , p ∥ ,γ ) = R m

...the same shape a collection of potential events P . 1

...the same shape a collection of potential events P . 1 a collection of easily-valuable events, regular events R . 2

...the same shape a collection of potential events P . 1 a collection of easily-valuable events, regular events R . 2 they can be seen here as events coming from a classical probability

...the same shape a collection of potential events P . 1 a collection of easily-valuable events, regular events R . 2 they can be seen here as events coming from a classical probability going from R to P ≡ passing from classical to frequentist approach:

...the same shape a collection of potential events P . 1 a collection of easily-valuable events, regular events R . 2 they can be seen here as events coming from a classical probability going from R to P ≡ passing from classical to frequentist approach: equally possible elementary cases ↝ identically independently distributed.

...the same shape a collection of potential events P . 1 a collection of easily-valuable events, regular events R . 2 they can be seen here as events coming from a classical probability going from R to P ≡ passing from classical to frequentist approach: equally possible elementary cases ↝ identically independently distributed. a collection of actual events A with R ⊆ A ⊆ P 3

...the same shape a collection of potential events P . 1 a collection of easily-valuable events, regular events R . 2 they can be seen here as events coming from a classical probability going from R to P ≡ passing from classical to frequentist approach: equally possible elementary cases ↝ identically independently distributed. a collection of actual events A with R ⊆ A ⊆ P 3 on which probability P is defined.

...the same shape a collection of potential events P . 1 a collection of easily-valuable events, regular events R . 2 they can be seen here as events coming from a classical probability going from R to P ≡ passing from classical to frequentist approach: equally possible elementary cases ↝ identically independently distributed. a collection of actual events A with R ⊆ A ⊆ P 3 on which probability P is defined.

...the same shape a collection of potential events P . 1 a collection of easily-valuable events, regular events R . 2 they can be seen here as events coming from a classical probability going from R to P ≡ passing from classical to frequentist approach: equally possible elementary cases ↝ identically independently distributed. a collection of actual events A with R ⊆ A ⊆ P 3 on which probability P is defined.

Toward a pointfree stratified definition of probability structure A probability structure is ( P , A , R , � , ∧ , ∨ , → , ¬ , ≤ , P ) ( P , ≤ , � , ∧ , ∨ , → ) is a Heyting algebra 1

Toward a pointfree stratified definition of probability structure A probability structure is ( P , A , R , � , ∧ , ∨ , → , ¬ , ≤ , P ) ( P , ≤ , � , ∧ , ∨ , → ) is a Heyting algebra 1 ( P , ≤ , � , ∧ , ∨ , ¬) is a de Morgan algebra 2

Toward a pointfree stratified definition of probability structure A probability structure is ( P , A , R , � , ∧ , ∨ , → , ¬ , ≤ , P ) ( P , ≤ , � , ∧ , ∨ , → ) is a Heyting algebra 1 ( P , ≤ , � , ∧ , ∨ , ¬) is a de Morgan algebra 2 R ⊆ A ⊆ P 3

Toward a pointfree stratified definition of probability structure A probability structure is ( P , A , R , � , ∧ , ∨ , → , ¬ , ≤ , P ) ( P , ≤ , � , ∧ , ∨ , → ) is a Heyting algebra 1 ( P , ≤ , � , ∧ , ∨ , ¬) is a de Morgan algebra 2 R ⊆ A ⊆ P 3 ( R , ≤ , � , ∧ , ∨ , ¬) is a Boolean algebra 4

Toward a pointfree stratified definition of probability structure A probability structure is ( P , A , R , � , ∧ , ∨ , → , ¬ , ≤ , P ) ( P , ≤ , � , ∧ , ∨ , → ) is a Heyting algebra 1 ( P , ≤ , � , ∧ , ∨ , ¬) is a de Morgan algebra 2 R ⊆ A ⊆ P 3 ( R , ≤ , � , ∧ , ∨ , ¬) is a Boolean algebra 4 P ∈ A → R 5

Toward a pointfree stratified definition of probability structure A probability structure is ( P , A , R , � , ∧ , ∨ , → , ¬ , ≤ , P ) ( P , ≤ , � , ∧ , ∨ , → ) is a Heyting algebra 1 ( P , ≤ , � , ∧ , ∨ , ¬) is a de Morgan algebra 2 R ⊆ A ⊆ P 3 ( R , ≤ , � , ∧ , ∨ , ¬) is a Boolean algebra 4 P ∈ A → R 5 e ∈ A e ∈ A e ≤ e ′ ¬ e ∈ A P (�) = 0 ∈ R P (¬ e ) = 1 − P ( e ) ∈ R P ( e ) ≤ R P ( e ′ )

Toward a pointfree stratified definition of probability structure A probability structure is ( P , A , R , � , ∧ , ∨ , → , ¬ , ≤ , P ) ( P , ≤ , � , ∧ , ∨ , → ) is a Heyting algebra 1 ( P , ≤ , � , ∧ , ∨ , ¬) is a de Morgan algebra 2 R ⊆ A ⊆ P 3 ( R , ≤ , � , ∧ , ∨ , ¬) is a Boolean algebra 4 P ∈ A → R 5 e ∈ A e ∈ A e ≤ e ′ ¬ e ∈ A P (�) = 0 ∈ R P (¬ e ) = 1 − P ( e ) ∈ R P ( e ) ≤ R P ( e ′ ) e ∈ A P ( e ) = 0 ∈ R e ′ ≤ e e ∈ A e ′ ∈ A e ∧ e ′ = � e ′ ∈ A e ∨ e ′ ∈ A

Toward a pointfree stratified definition of probability structure A probability structure is ( P , A , R , � , ∧ , ∨ , → , ¬ , ≤ , P ) ( P , ≤ , � , ∧ , ∨ , → ) is a Heyting algebra 1 ( P , ≤ , � , ∧ , ∨ , ¬) is a de Morgan algebra 2 R ⊆ A ⊆ P 3 ( R , ≤ , � , ∧ , ∨ , ¬) is a Boolean algebra 4 P ∈ A → R 5 e ∈ A e ∈ A e ≤ e ′ ¬ e ∈ A P (�) = 0 ∈ R P (¬ e ) = 1 − P ( e ) ∈ R P ( e ) ≤ R P ( e ′ ) e ∈ A P ( e ) = 0 ∈ R e ′ ≤ e e ∈ A e ′ ∈ A e ∧ e ′ = � e ′ ∈ A e ∨ e ′ ∈ A e ∈ A e ′ ∈ A e ∧ e ′ ∈ A e ∨ e ′ ∈ A P ( e ) + P ( e ′ ) = P ( e ∨ e ′ ) + P ( e ∧ e ′ ) ∈ R

Toward a pointfree stratified definition of probability structure A probability structure is ( P , A , R , � , ∧ , ∨ , → , ¬ , ≤ , P ) ( P , ≤ , � , ∧ , ∨ , → ) is a Heyting algebra 1 ( P , ≤ , � , ∧ , ∨ , ¬) is a de Morgan algebra 2 R ⊆ A ⊆ P 3 ( R , ≤ , � , ∧ , ∨ , ¬) is a Boolean algebra 4 P ∈ A → R 5 e ∈ A e ∈ A e ≤ e ′ ¬ e ∈ A P (�) = 0 ∈ R P (¬ e ) = 1 − P ( e ) ∈ R P ( e ) ≤ R P ( e ′ ) e ∈ A P ( e ) = 0 ∈ R e ′ ≤ e e ∈ A e ′ ∈ A e ∧ e ′ = � e ′ ∈ A e ∨ e ′ ∈ A e ∈ A e ′ ∈ A e ∧ e ′ ∈ A e ∨ e ′ ∈ A P ( e ) + P ( e ′ ) = P ( e ∨ e ′ ) + P ( e ∧ e ′ ) ∈ R

or... A “censored” probability structure is ( P , A , R , � , ∧ , ∨ , → , , ≤ , P ) ( P , ≤ , � , ∧ , ∨ , → ) is a Heyting algebra 1 2 R ⊆ A ⊆ P 3 ( R , ≤ , � , ∧ , ∨ , ¬) is a Boolean algebra 4 P ∈ A → R 5 e ≤ e ′ P (�) = 0 ∈ R P ( e ) ≤ R P ( e ′ ) e ∈ A e ′ ∈ A e ∧ e ′ = � e ∨ e ′ ∈ A e ∈ A e ′ ∈ A e ∧ e ′ ∈ A e ∨ e ′ ∈ A P ( e ) + P ( e ′ ) = P ( e ∨ e ′ ) + P ( e ∧ e ′ ) ∈ R

Future work Explore the relation with other pointfree approaches like that of valuations 1

Future work Explore the relation with other pointfree approaches like that of valuations 1 Describe subjective probability in terms of these structures. 2

Future work Explore the relation with other pointfree approaches like that of valuations 1 Describe subjective probability in terms of these structures. 2 Use constructive notion of natural density to study probabilistic versions of 3 principles of limited omniscience, e. g. LPO : ∀ n ( x ( n ) = 0 ) ∨ ∃ n ( x ( n ) = 1 ) for x ∶ N → { 0 , 1 }

Future work Explore the relation with other pointfree approaches like that of valuations 1 Describe subjective probability in terms of these structures. 2 Use constructive notion of natural density to study probabilistic versions of 3 principles of limited omniscience, e. g. LPO : ∀ n ( x ( n ) = 0 ) ∨ ∃ n ( x ( n ) = 1 ) for x ∶ N → { 0 , 1 } ⇊ P − LPO : P ( e ) = 0 ∨ ∃ n ( e ( n ) = 1 ) for e ∈ P , A ...

Future work Explore the relation with other pointfree approaches like that of valuations 1 Describe subjective probability in terms of these structures. 2 Use constructive notion of natural density to study probabilistic versions of 3 principles of limited omniscience, e. g. LPO : ∀ n ( x ( n ) = 0 ) ∨ ∃ n ( x ( n ) = 1 ) for x ∶ N → { 0 , 1 } ⇊ P − LPO : P ( e ) = 0 ∨ ∃ n ( e ( n ) = 1 ) for e ∈ P , A ...