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Monotone Calculi.
The inference opeartion
M
defined by a logical matrix
M
satisfies not only (c1)-(c3) but also (c4) and (c5). Furthermore, for every
propositional calculus
L,
there exists a class
K
of logical matrices for
L
such
=
{
M
:
that
M∈K}
.
Beyond Structurality: Admissible Valuations.
One way of extending ma-
trix semantics to cover non-structural inference systems is to define the semantic
entailment in terms of 'admissible interpretations', i.e., to consider
generalized
matrices
of the form
A,d,H
A
and
d
are as above, and
H
,where
is a subset
L
A
of the set of all interpretations of
. In this semantic framework, every
inference operation that satisfies (c1)-(c4) can be defined by a class of gener-
alized matrices. A similar approach of admitting only some interpretations to
model non-structural nonmonotonic inference systems has been also developed
for preferential model semantics (cf. [7]).
into
Beyond Monotonicity: Preferential Matrices
The notion of cumulativ-
ity arose as a result of the search for desired and natural formal properties of
nonmonotonic inference systems. A desired 'degree' of nonmonotonicity can be
semantically modeled in terms of logical matrices of the form
M
=
A, D,H, ≺,
where
A
and
H
are as in a generalized matrix,
D
is a family of subsets of the uni-
verse of
A
,and
≺
is a binary (preference) relation on
D
. The inference operation
M
is defined as follows:
X
M
α
iff
for every
h ∈H
and every
d ∈D
,if
d
is a minimal element of
D
(with respect to
≺
) such that
h
(
X
)
⊆ d
,then
h
(
α
)
∈ d
.
Preferential matrices have the same semantic scope as preferential model struc-
tures (cf. [8,14]).
Logical Matrices with Completion.
It is not essential to interpret the under-
lying algebra
A
of a logical matrix
M
for a language
L
as a space of truth-values
for the formulas of
can be interpreted as propositions,
events, and even infons of the Situation Theory of Barwise and Perry. If one
views subsets of the universe of
L
. The elements of
A
as situations (partial or complete), then pref-
erential matrices can be replaced by structures of the form
A
M
=
A,H,
called
matrices with completion
,where
A
,and
H
are as above and
is a function that
B
. In the language of
,
B ⊆ B
=
maps 2
|A|
into 2
|A|
such that for every
B ⊆|A|
universal algebra,
. This operation can be thought
of as a
completion function
that assigns an actual and complete situation
B
to
a (possibly partial) situation
B
which is a part of
B
. The inference operation
M
associated with such a matrix is defined as follows: for every set
X ∪{α}
is a closure operator on
A
of
formulas,
h
(
X
).
X
M
α
iff
for every
h ∈H,h
(
α
)
∈
Matrices with completion can be used to semantically model cumulativity with-
out any explicit reference to preference.
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