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
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
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
.
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
.
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
.
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
.
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
.
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
.
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
.
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.
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.
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.
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.
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
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
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
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
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
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
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
.
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
.
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
.
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
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
∈
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
∈
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
∗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
∗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
∗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
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
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
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
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
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
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
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
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
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
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
(ϕ
∧
ψ)
1
A
t
(ϕ
∧
ψ)
→
A
t
ϕ
and
1
A
t
(ϕ
∧
ψ)
→
A
t
ψ
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
(ϕ
∧
ψ)
1
A
t
(ϕ
∧
ψ)
→
A
t
ϕ
and
1
A
t
(ϕ
∧
ψ)
→
A
t
ψ
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
(ϕ
∧
ψ)
1
A
t
(ϕ
∧
ψ)
→
A
t
ϕ
and
1
A
t
(ϕ
∧
ψ)
→
A
t
ψ
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
Aϕ
→
Aϕ
, 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:
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
Aϕ
→
Aϕ
, 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:
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
(ϕ
→
ψ)
→
(K
Ex
ϕ
→
K
Ex
ψ)
1
K
Ex
(ϕ
→
ψ)
→
(K
Ex
ϕ
∧
A
o
ψ)
→
K
Ex
ψ
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
(ϕ
→
ψ)
→
(K
Ex
ϕ
→
K
Ex
ψ)
1
K
Ex
(ϕ
→
ψ)
→
(K
Ex
ϕ
∧
A
o
ψ)
→
K
Ex
ψ
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
(ϕ
→
ψ)
→
(K
Ex
ϕ
→
K
Ex
ψ)
1
K
Ex
(ϕ
→
ψ)
→
(K
Ex
ϕ
∧
A
o
ψ)
→
K
Ex
ψ
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
(ϕ
→
ψ)
→
(K
Ex
ϕ
→
K
Ex
ψ)
1
K
Ex
(ϕ
→
ψ)
→
(K
Ex
ϕ
∧
A
o
ψ)
→
K
Ex
ψ
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
(ϕ
→
ψ)
→
(K
Ex
ϕ
→
K
Ex
ψ)
1
K
Ex
(ϕ
→
ψ)
→
(K
Ex
ϕ
∧
A
o
ψ)
→
K
Ex
ψ
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
(ϕ
→
ψ)
→
(K
Ex
ϕ
→
K
Ex
ψ)
1
K
Ex
(ϕ
→
ψ)
→
(K
Ex
ϕ
∧
A
o
ψ)
→
K
Ex
ψ
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
(ϕ
→
ψ)
→
(K
Ex
ϕ
→
K
Ex
ψ)
1
K
Ex
(ϕ
→
ψ)
→
(K
Ex
ϕ
∧
A
o
ψ)
→
K
Ex
ψ
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
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:
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:
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:
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:
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:
Introduction System ofExplicitKnowledge Properties and relationships EpistemicActions Closing References