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Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

A

formal model for explicit knowledge as

awareness

of

plus awareness

that

Claudia Fern´andez-Fern´andez

1

, Fernando R. Vel´azquez-Quesada

2

Model-based Reasoning in Science and Technology

(2)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

E

xplicit knowledge as awareness

of

+ awareness

that

1

I

ntroduction

2

S

ystem of

E

xplicit

K

nowledge

The model

The concepts

3

P

roperties and relationships

Awareness-of and Awareness-that

Effects of the closure operation

Moorean Phenomena

Other Alternatives for the Concepts

(3)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

I

n a nutshell

Context:

logical omniscience vs agents with limited abilities.

Purpose:

reconsider what constitutes

explicit knowledge

.

(4)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

I

n a nutshell

Context:

logical omniscience vs agents with limited abilities.

Purpose:

reconsider what constitutes

explicit knowledge

.

(5)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

I

n a nutshell

Context:

logical omniscience vs agents with limited abilities.

Purpose:

reconsider what constitutes

explicit knowledge

.

(6)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

T

ypes of knowledge

Di

ff

erent types of knowledge:

explicit

and

implicit

w.r.t.

deduction

(e.g.,

Konolige 1984,

Levesque 1984);

explicit

and

implicit

w.r.t.

awareness

(Fagin and Halpern 1988);

Note.

Explicit knowledge:

what the agent

actually has

.

(7)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

T

ypes of knowledge

Di

ff

erent types of knowledge:

explicit

and

implicit

w.r.t.

deduction

(e.g.,

Konolige 1984,

Levesque 1984);

explicit

and

implicit

w.r.t.

awareness

(Fagin and Halpern 1988);

Note.

Explicit knowledge:

what the agent

actually has

.

(8)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

T

ypes of knowledge

Di

ff

erent types of knowledge:

explicit

and

implicit

w.r.t.

deduction

(e.g.,

Konolige 1984,

Levesque 1984);

explicit

and

implicit

w.r.t.

awareness

(Fagin and Halpern 1988);

Note.

Explicit knowledge:

what the agent

actually has

.

(9)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

T

ypes of knowledge

Di

ff

erent types of knowledge:

explicit

and

implicit

w.r.t.

deduction

(e.g.,

Konolige 1984,

Levesque 1984);

explicit

and

implicit

w.r.t.

awareness

(Fagin and Halpern 1988);

Note.

Explicit knowledge:

what the agent

actually has

.

(10)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

T

he ingredients

Awareness-of

and

awareness-that

.

Fagin and Halpern (1988): awareness has

di

ff

erent interpretations

.

Dretske (1993):

Awareness of things

vs

awareness of facts

.

Here:

Awareness-of

as

entertaining

(‘

working memory

’), not implying

any attitude in favour or against.

(11)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

T

he ingredients

Awareness-of

and

awareness-that

.

Fagin and Halpern (1988): awareness has

di

ff

erent interpretations

.

Dretske (1993):

Awareness of things

vs

awareness of facts

.

Here:

Awareness-of

as

entertaining

(‘

working memory

’), not implying

any attitude in favour or against.

(12)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

T

he ingredients

Awareness-of

and

awareness-that

.

Fagin and Halpern (1988): awareness has

di

ff

erent interpretations

.

Dretske (1993):

Awareness of things

vs

awareness of facts

.

Here:

Awareness-of

as

entertaining

(‘

working memory

’), not implying

any attitude in favour or against.

(13)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

T

he ingredients

Awareness-of

and

awareness-that

.

Fagin and Halpern (1988): awareness has

di

ff

erent interpretations

.

Dretske (1993):

Awareness of things

vs

awareness of facts

.

Here:

Awareness-of

as

entertaining

(‘

working memory

’), not implying

any attitude in favour or against.

(14)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

C

ombined

D

iagram

3

2

4

1

5

(5) Awareness-that

(not in

working memory)

(4) ‘Implicit’

Awareness-that

(not in working

mem-ory)

(3) Awareness-of

(2) Aware-of not

aware-that, but deducible

(15)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

C

ombined

D

iagram

3

2

4

1

5

(5) Awareness-that

(not in

working memory)

(4) ‘Implicit’

Awareness-that

(not in working

mem-ory)

(3) Awareness-of

(2) Aware-of not

aware-that, but deducible

(16)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

C

ombined

D

iagram

3

2

4

1

5

(5) Awareness-that

(not in

working memory)

(4) ‘Implicit’

Awareness-that

(not in working

mem-ory)

(3) Awareness-of

(2) Aware-of not

aware-that, but deducible

(17)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

C

ombined

D

iagram

3

2

4

1

5

(5) Awareness-that

(not in

working memory)

(4) ‘Implicit’

Awareness-that (not in working

mem-ory)

(3) Awareness-of

(2) Aware-of not

aware-that, but deducible

(18)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

C

ombined

D

iagram

3

2

4

1

5

(5) Awareness-that (not in

working memory)

(4) ‘Implicit’

Awareness-that (not in working

mem-ory)

(3) Awareness-of

(2) Aware-of not

aware-that, but deducible

(19)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

E

xplicit knowledge as awareness

of

+ awareness

that

1

I

ntroduction

2

S

ystem of

E

xplicit

K

nowledge

The model

The concepts

3

P

roperties and relationships

Awareness-of and Awareness-that

Effects of the closure operation

Moorean Phenomena

Other Alternatives for the Concepts

(20)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

A

wareness neighbourhood model

(ANM)

D

efinition

(A

wareness neighbourhood model

(ANM))

Let

P

be a set of atoms. An

ANM

is a tuple

M

=

h

W

,

N

,

V

,

A

i

where

W

,

N

:

W

(

(

W

))

V

:

P

(

W

)

A

P

Awareness-that

: (local) neighbourhood function

N

.

(21)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

A

wareness neighbourhood model

(ANM)

D

efinition

(A

wareness neighbourhood model

(ANM))

Let

P

be a set of atoms. An

ANM

is a tuple

M

=

h

W

,

N

,

V

,

A

i

where

W

,

N

:

W

(

(

W

))

V

:

P

(

W

)

A

P

Awareness-that

: (local) neighbourhood function

N

.

(22)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

A

wareness neighbourhood model

(ANM)

D

efinition

(A

wareness neighbourhood model

(ANM))

Let

P

be a set of atoms. An

ANM

is a tuple

M

=

h

W

,

N

,

V

,

A

i

where

W

,

N

:

W

(

(

W

))

V

:

P

(

W

)

A

P

Awareness-that

: (local) neighbourhood function

N

.

(23)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

L

anguage and semantic interpretation

(1)

D

efinition

(L

anguage

L

)

ϕ

,

ψ

::

=

>

|

p

|

¬

ϕ

|

ϕ

ψ

|

A

o

ϕ

|

A

t

ϕ

|

[

]

ϕ

J

>

K

M

:

=

W

,

J

p

K

M

:

=

V

(

p

),

J

¬

ϕ

K

M

:

=

W

\

J

ϕ

K

M

,

J

ϕ

ψ

K

M

:

=

J

ϕ

K

M

J

ψ

K

M

(24)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

L

anguage and semantic interpretation

(2)

D

efinition

(L

anguage

L

)

ϕ

,

ψ

::

=

>

|

p

|

¬

ϕ

|

ϕ

ψ

|

A

o

ϕ

|

A

t

ϕ

|

[

]

ϕ

J

A

o

ϕ

K

M

:

=

W

if atm(

ϕ

)

A

otherwise

,

J

A

t

ϕ

K

M

:

=

n

w

W

|

J

ϕ

K

M

(25)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

L

anguage and semantic interpretation

(2)

D

efinition

(L

anguage

L

)

ϕ

,

ψ

::

=

>

|

p

|

¬

ϕ

|

ϕ

ψ

|

A

o

ϕ

|

A

t

ϕ

|

[

]

ϕ

J

A

o

ϕ

K

M

:

=

W

if atm(

ϕ

)

A

otherwise

,

JA

t

ϕ

K

M

:

=

n

w

W

|

J

ϕ

K

M

(26)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

L

anguage and semantic interpretation

(3)

D

efinition

(L

anguage

L

)

ϕ

,

ψ

::

=

>

|

p

|

¬

ϕ

|

ϕ

ψ

|

A

o

ϕ

|

A

t

ϕ

|

[

]

ϕ

Given

M

=

h

W

,

N

,

V

,

A

i

, define

M

=

h

W

,

N

,

V

,

A

i

with

N

(

w

) :

=

n

U

W

|

\

N

(

w

)

U

o

Then

J[

]

ϕ

K

M

:

=

J

ϕ

K

M

(27)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

L

anguage and semantic interpretation

(3)

D

efinition

(L

anguage

L

)

ϕ

,

ψ

::

=

>

|

p

|

¬

ϕ

|

ϕ

ψ

|

A

o

ϕ

|

A

t

ϕ

|

[

]

ϕ

Given

M

=

h

W

,

N

,

V

,

A

i

, define

M

=

h

W

,

N

,

V

,

A

i

with

N

(

w

) :

=

n

U

W

|

\

N

(

w

)

U

o

Then

J[

]

ϕ

K

M

:

=

J

ϕ

K

M

(28)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

L

anguage and semantic interpretation

(3)

D

efinition

(L

anguage

L

)

ϕ

,

ψ

::

=

>

|

p

|

¬

ϕ

|

ϕ

ψ

|

A

o

ϕ

|

A

t

ϕ

|

[

]

ϕ

Given

M

=

h

W

,

N

,

V

,

A

i

, define

M

=

h

W

,

N

,

V

,

A

i

with

N

(

w

) :

=

n

U

W

|

\

N

(

w

)

U

o

Then

J[

]

ϕ

K

M

:

=

J

ϕ

K

M

(29)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

E

xplicit knowledge as awareness

of

+ awareness

that

1

I

ntroduction

2

S

ystem of

E

xplicit

K

nowledge

The model

The concepts

3

P

roperties and relationships

Awareness-of and Awareness-that

Effects of the closure operation

Moorean Phenomena

Other Alternatives for the Concepts

(30)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

T

he concepts of

K

nowledge

Aware knowledge

Explicit knowledge:

K

Ex

ϕ

:

=

A

o

ϕ

A

t

ϕ

Implicit knowledge:

K

Im

ϕ

:

=

A

o

ϕ

[

] A

t

ϕ

Unaware knowledge

(31)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

T

he concepts of

K

nowledge

Aware knowledge

Explicit knowledge:

K

Ex

ϕ

:

=

A

o

ϕ

A

t

ϕ

Implicit knowledge:

K

Im

ϕ

:

=

A

o

ϕ

[

] A

t

ϕ

Unaware knowledge

(32)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

T

he concepts of

K

nowledge

Aware knowledge

Explicit knowledge:

K

Ex

ϕ

:

=

A

o

ϕ

A

t

ϕ

Implicit knowledge:

K

Im

ϕ

:

=

A

o

ϕ

[

] A

t

ϕ

Unaware knowledge

‘Disassociated’ knowledge:

K

Ex

o

ϕ

:

=

¬

A

o

ϕ

A

t

ϕ

(33)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

T

he concepts of

K

nowledge

Aware knowledge

Explicit knowledge:

K

Ex

ϕ

:

=

A

o

ϕ

A

t

ϕ

Implicit knowledge:

K

Im

ϕ

:

=

A

o

ϕ

[

] A

t

ϕ

Unaware knowledge

(34)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

E

xplicit knowledge as awareness

of

+ awareness

that

1

I

ntroduction

2

S

ystem of

E

xplicit

K

nowledge

The model

The concepts

3

P

roperties and relationships

Awareness-of and Awareness-that

E

ff

ects of the closure operation

Moorean Phenomena

Other Alternatives for the Concepts

(35)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

P

roperties of

A

wareness

-

of

(A

o

)

The agent is aware-of the

concept of truth

:

A

o

>

But,

ϕ

does not imply

A

o

ϕ

Since A

o

is defined as a set of atomic propositions, it is

closed

under subformulas and superformulas

:

A

o

¬

ϕ

A

o

ϕ

A

o

(

ϕ

ψ

)

(A

o

ϕ

A

o

ψ

)

A

o

A

o

ϕ

A

o

ϕ

A

o

A

t

ϕ

A

o

ϕ

(36)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

P

roperties of

A

wareness

-

of

(A

o

)

The agent is aware-of the

concept of truth

:

A

o

>

But,

ϕ

does not imply

A

o

ϕ

Since A

o

is defined as a set of atomic propositions, it is

closed

under subformulas and superformulas

:

A

o

¬

ϕ

A

o

ϕ

A

o

(

ϕ

ψ

)

(A

o

ϕ

A

o

ψ

)

A

o

A

o

ϕ

A

o

ϕ

A

o

A

t

ϕ

A

o

ϕ

(37)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

P

roperties of

A

wareness

-

of

(A

o

)

The agent is aware-of the

concept of truth

:

A

o

>

But,

ϕ

does not imply

A

o

ϕ

Since A

o

is defined as a set of atomic propositions, it is

closed

under subformulas and superformulas

:

A

o

¬

ϕ

A

o

ϕ

A

o

(

ϕ

ψ

)

(A

o

ϕ

A

o

ψ

)

A

o

A

o

ϕ

A

o

ϕ

A

o

A

t

ϕ

A

o

ϕ

(38)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

P

roperties of

A

wareness

-

of

(A

o

)

The agent is aware-of the

concept of truth

:

A

o

>

But,

ϕ

does not imply

A

o

ϕ

Since A

o

is defined as a set of atomic propositions, it is

closed

under subformulas and superformulas

:

A

o

¬

ϕ

A

o

ϕ

A

o

(

ϕ

ψ

)

(A

o

ϕ

A

o

ψ

)

A

o

A

o

ϕ

A

o

ϕ

A

o

A

t

ϕ

A

o

ϕ

(39)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

P

roperties of

A

wareness

-

that

(A

t

)

A

t

is what appears in

N

(

w

). This pureley semantic concept is

closed under logical equivalence

(some kind of omniscience):

ϕ

ψ

implies

A

t

ϕ

A

t

ψ

But it is the only closure property, since

ϕ

does not imply

A

t

ϕ

1

(A

t

ϕ

A

t

ψ)

A

t

ψ)

1

A

t

ψ)

A

t

ϕ

and

1

A

t

ψ)

A

t

ψ

(40)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

P

roperties of

A

wareness

-

that

(A

t

)

A

t

is what appears in

N

(

w

). This pureley semantic concept is

closed under logical equivalence

(some kind of omniscience):

ϕ

ψ

implies

A

t

ϕ

A

t

ψ

But it is the only closure property, since

ϕ

does not imply

A

t

ϕ

1

(A

t

ϕ

A

t

ψ)

A

t

ψ)

1

A

t

ψ)

A

t

ϕ

and

1

A

t

ψ)

A

t

ψ

(41)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

P

roperties of

A

wareness

-

that

(A

t

)

A

t

is what appears in

N

(

w

). This pureley semantic concept is

closed under logical equivalence

(some kind of omniscience):

ϕ

ψ

implies

A

t

ϕ

A

t

ψ

But it is the only closure property, since

ϕ

does not imply

A

t

ϕ

1

(A

t

ϕ

A

t

ψ)

A

t

ψ)

1

A

t

ψ)

A

t

ϕ

and

1

A

t

ψ)

A

t

ψ

(42)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

A

wareness

-

of and

A

wareness

-

that

In contrast to what happens in

Awareness Logic

by

Fagin and

Halpern

, where

, with a global awareness set, we do

not obtain this result, thanks to the di

ff

erent concepts of awareness

we defined.

Recall that

awareness-of

is a global notion and

awareness-that

is

locally defined.

Thus, analogous properties do not hold:

(43)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

A

wareness

-

of and

A

wareness

-

that

In contrast to what happens in

Awareness Logic

by

Fagin and

Halpern

, where

, with a global awareness set, we do

not obtain this result, thanks to the di

ff

erent concepts of awareness

we defined.

Recall that

awareness-of

is a global notion and

awareness-that

is

locally defined.

Thus, analogous properties do not hold:

(44)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

P

roperties of

E

xplicit

K

nowledge

(K

Ex

)

Recall that

K

Ex

ϕ

:

=

A

o

ϕ

A

t

ϕ

. This definition has the following

consequences

:

About

validities

:

ϕ

implies neither

K

Ex

ϕ;

nor

A

o

ϕ

K

Ex

ϕ;

nor

A

t

ϕ

K

Ex

ϕ

About

logical equivalence

:

ϕ

ψ

does not imply

K

Ex

ϕ

K

Ex

ψ

But,

ϕ

ψ

implies

(K

Ex

ϕ

A

o

ψ

)

K

Ex

ψ

About

Modus Ponens

:

1

K

Ex

ψ)

(K

Ex

ϕ

K

Ex

ψ)

1

K

Ex

ψ)

(K

Ex

ϕ

A

o

ψ)

K

Ex

ψ

(45)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

P

roperties of

E

xplicit

K

nowledge

(K

Ex

)

Recall that

K

Ex

ϕ

:

=

A

o

ϕ

A

t

ϕ

. This definition has the following

consequences

:

About

validities

:

ϕ

implies neither

K

Ex

ϕ;

nor

A

o

ϕ

K

Ex

ϕ;

nor

A

t

ϕ

K

Ex

ϕ

About

logical equivalence

:

ϕ

ψ

does not imply

K

Ex

ϕ

K

Ex

ψ

But,

ϕ

ψ

implies

(K

Ex

ϕ

A

o

ψ

)

K

Ex

ψ

About

Modus Ponens

:

1

K

Ex

ψ)

(K

Ex

ϕ

K

Ex

ψ)

1

K

Ex

ψ)

(K

Ex

ϕ

A

o

ψ)

K

Ex

ψ

(46)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

P

roperties of

E

xplicit

K

nowledge

(K

Ex

)

Recall that

K

Ex

ϕ

:

=

A

o

ϕ

A

t

ϕ

. This definition has the following

consequences

:

About

validities

:

ϕ

implies neither

K

Ex

ϕ;

nor

A

o

ϕ

K

Ex

ϕ;

nor

A

t

ϕ

K

Ex

ϕ

About

logical equivalence

:

ϕ

ψ

does not imply

K

Ex

ϕ

K

Ex

ψ

But,

ϕ

ψ

implies

(K

Ex

ϕ

A

o

ψ

)

K

Ex

ψ

About

Modus Ponens

:

1

K

Ex

ψ)

(K

Ex

ϕ

K

Ex

ψ)

1

K

Ex

ψ)

(K

Ex

ϕ

A

o

ψ)

K

Ex

ψ

(47)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

P

roperties of

E

xplicit

K

nowledge

(K

Ex

)

Recall that

K

Ex

ϕ

:

=

A

o

ϕ

A

t

ϕ

. This definition has the following

consequences

:

About

validities

:

ϕ

implies neither

K

Ex

ϕ;

nor

A

o

ϕ

K

Ex

ϕ;

nor

A

t

ϕ

K

Ex

ϕ

About

logical equivalence

:

ϕ

ψ

does not imply

K

Ex

ϕ

K

Ex

ψ

But,

ϕ

ψ

implies

(K

Ex

ϕ

A

o

ψ

)

K

Ex

ψ

About

Modus Ponens

:

1

K

Ex

ψ)

(K

Ex

ϕ

K

Ex

ψ)

1

K

Ex

ψ)

(K

Ex

ϕ

A

o

ψ)

K

Ex

ψ

(48)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

P

roperties of

E

xplicit

K

nowledge

(K

Ex

)

Recall that

K

Ex

ϕ

:

=

A

o

ϕ

A

t

ϕ

. This definition has the following

consequences

:

About

validities

:

ϕ

implies neither

K

Ex

ϕ;

nor

A

o

ϕ

K

Ex

ϕ;

nor

A

t

ϕ

K

Ex

ϕ

About

logical equivalence

:

ϕ

ψ

does not imply

K

Ex

ϕ

K

Ex

ψ

But,

ϕ

ψ

implies

(K

Ex

ϕ

A

o

ψ

)

K

Ex

ψ

About

Modus Ponens

:

1

K

Ex

ψ)

(K

Ex

ϕ

K

Ex

ψ)

1

K

Ex

ψ)

(K

Ex

ϕ

A

o

ψ)

K

Ex

ψ

(49)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

P

roperties of

E

xplicit

K

nowledge

(K

Ex

)

Recall that

K

Ex

ϕ

:

=

A

o

ϕ

A

t

ϕ

. This definition has the following

consequences

:

About

validities

:

ϕ

implies neither

K

Ex

ϕ;

nor

A

o

ϕ

K

Ex

ϕ;

nor

A

t

ϕ

K

Ex

ϕ

About

logical equivalence

:

ϕ

ψ

does not imply

K

Ex

ϕ

K

Ex

ψ

But,

ϕ

ψ

implies

(K

Ex

ϕ

A

o

ψ

)

K

Ex

ψ

About

Modus Ponens

:

1

K

Ex

ψ)

(K

Ex

ϕ

K

Ex

ψ)

1

K

Ex

ψ)

(K

Ex

ϕ

A

o

ψ)

K

Ex

ψ

(50)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

P

roperties of

E

xplicit

K

nowledge

(K

Ex

)

Recall that

K

Ex

ϕ

:

=

A

o

ϕ

A

t

ϕ

. This definition has the following

consequences

:

About

validities

:

ϕ

implies neither

K

Ex

ϕ;

nor

A

o

ϕ

K

Ex

ϕ;

nor

A

t

ϕ

K

Ex

ϕ

About

logical equivalence

:

ϕ

ψ

does not imply

K

Ex

ϕ

K

Ex

ψ

But,

ϕ

ψ

implies

(K

Ex

ϕ

A

o

ψ

)

K

Ex

ψ

About

Modus Ponens

:

1

K

Ex

ψ)

(K

Ex

ϕ

K

Ex

ψ)

1

K

Ex

ψ)

(K

Ex

ϕ

A

o

ψ)

K

Ex

ψ

(51)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

E

xplicit knowledge as awareness

of

+ awareness

that

1

I

ntroduction

2

S

ystem of

E

xplicit

K

nowledge

The model

The concepts

3

P

roperties and relationships

Awareness-of and Awareness-that

E

ff

ects of the closure operation

Moorean Phenomena

Other Alternatives for the Concepts

(52)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

A

wareness

-

that after full deductive inference

([

] A

t

)

The closure operation

[

]

makes

A

t

behave as

in relational models.

Some results on:

Validities:

ϕ

implies

[

] A

t

ϕ

Conjunction intr.:

[

] A

t

ϕ

[

] A

t

ψ

[

] A

t

(

ϕ

ψ

)

elim.:

[

] A

t

(

ϕ

ψ

)

[

] A

t

ϕ

and

[

] A

t

(

ϕ

ψ

)

[

] A

t

ψ

Closure under MP:

[

] A

t

(

ϕ

ψ

)

(A

t

ϕ

A

t

ψ

)

[

] A

t

(

ϕ

ψ

)

([

] A

t

ϕ

[

] A

t

ψ

)

Aware-that awareness-of is the case:

(53)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

A

wareness

-

that after full deductive inference

([

] A

t

)

The closure operation

[

]

makes

A

t

behave as

in relational models.

Some results on:

Validities:

ϕ

implies

[

] A

t

ϕ

Conjunction intr.:

[

] A

t

ϕ

[

] A

t

ψ

[

] A

t

(

ϕ

ψ

)

elim.:

[

] A

t

(

ϕ

ψ

)

[

] A

t

ϕ

and

[

] A

t

(

ϕ

ψ

)

[

] A

t

ψ

Closure under MP:

[

] A

t

(

ϕ

ψ

)

(A

t

ϕ

A

t

ψ

)

[

] A

t

(

ϕ

ψ

)

([

] A

t

ϕ

[

] A

t

ψ

)

Aware-that awareness-of is the case:

(54)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

A

wareness

-

that after full deductive inference

([

] A

t

)

The closure operation

[

]

makes

A

t

behave as

in relational models.

Some results on:

Validities:

ϕ

implies

[

] A

t

ϕ

Conjunction intr.:

[

] A

t

ϕ

[

] A

t

ψ

[

] A

t

(

ϕ

ψ

)

elim.:

[

] A

t

(

ϕ

ψ

)

[

] A

t

ϕ

and

[

] A

t

(

ϕ

ψ

)

[

] A

t

ψ

Closure under MP:

[

] A

t

(

ϕ

ψ

)

(A

t

ϕ

A

t

ψ

)

[

] A

t

(

ϕ

ψ

)

([

] A

t

ϕ

[

] A

t

ψ

)

Aware-that awareness-of is the case:

(55)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

A

wareness

-

that after full deductive inference

([

] A

t

)

The closure operation

[

]

makes

A

t

behave as

in relational models.

Some results on:

Validities:

ϕ

implies

[

] A

t

ϕ

Conjunction intr.:

[

] A

t

ϕ

[

] A

t

ψ

[

] A

t

(

ϕ

ψ

)

elim.:

[

] A

t

(

ϕ

ψ

)

[

] A

t

ϕ

and

[

] A

t

(

ϕ

ψ

)

[

] A

t

ψ

Closure under MP:

[

] A

t

(

ϕ

ψ

)

(A

t

ϕ

A

t

ψ

)

[

] A

t

(

ϕ

ψ

)

([

] A

t

ϕ

[

] A

t

ψ

)

Aware-that awareness-of is the case:

(56)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

A

wareness

-

that after full deductive inference

([

] A

t

)

The closure operation

[

]

makes

A

t

behave as

in relational models.

Some results on:

Validities:

ϕ

implies

[

] A

t

ϕ

Conjunction intr.:

[

] A

t

ϕ

[

] A

t

ψ

[

] A

t

(

ϕ

ψ

)

elim.:

[

] A

t

(

ϕ

ψ

)

[

] A

t

ϕ

and

[

] A

t

(

ϕ

ψ

)

[

] A

t

ψ

Closure under MP:

[

] A

t

(

ϕ

ψ

)

(A

t

ϕ

A

t

ψ

)

[

] A

t

(

ϕ

ψ

)

([

] A

t

ϕ

[

] A

t

ψ

)

Aware-that awareness-of is the case:

(57)

Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References

A

wareness

-

that after full deductive inference

([

] A

t

)

The closure operation

[

]

makes

A

t

behave as

in relational models.

Some results on:

Validities:

ϕ

implies

[

] A

t

ϕ

Conjunction intr.:

[

] A

t

ϕ

[

] A

t

ψ

[

] A

t

(

ϕ

ψ

)

elim.:

[

] A

t

(

ϕ

ψ

)

[

] A

t

ϕ

and

[

] A

t

(

ϕ

ψ

)

[

] A

t

ψ

Closure under MP:

[

] A

t

(

ϕ

ψ

)

(A

t

ϕ

A

t

ψ

)

[

] A

t

(

ϕ

ψ

)

([

] A

t

ϕ

[

] A

t

ψ

)

Aware-that awareness-of is the case:

Referencias

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