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Syntax.
In the propositional case, a representational language
, defined by
a set of propositional variables
Var
and logical connectives
f
0
,...,f
n
,canbe
viewed as a term algebra (or
Lindenbaum's algebra of formulas
)
L
Terms
(
Var
)
,f
0
,...,f
n
,
generated by
Var
,where
Terms
(
Var
) denotes the set of all well-formed formu-
las of
. Syntactically richer languages can be adequately modeled using, for
instance, partial and many-sorted term algebras.
L
L
Inference Systems.
Given a propositional language
,arelation
between
sets of formulas of
L
and formulas of
L
is called an
inference operationon
on
L
,
if for every set
X
of formulas:
(c1)
X ⊆ C
(
X
)
(
inclusion
);
(c2)
C
(
C
(
X
))
⊆ C
(
X
)
(
idempotence
);
where
C
(
X
)=
{β
:
X β}
.An
inference system
on
L
is a pair
L,
,where
is an inference operation on
L
. Further conditions on
can be imposed: for
every
X, Y ⊆ Terms
(
Var
),
(c3)
X ⊆ Y ⊆ C
(
X
) implies
C
(
X
)=
C
(
Y
)
(
cumulativity
);
(c4)
X ⊆ Y
implies
C
(
X
)
⊆ C
(
Y
)
(
monotonicity
);
(c5)
for every endomorphism
e
of
L
,
e
(
C
(
X
))
⊆ C
(
e
(
X
))
(
structurality
).
Every inference system satisfying (c1)-(c5) is called a
propositional logic
.Since
Tarski's axiomatization of the concept of a consequence operation in formalized
languages, algebraic properties of monotonic and non-monotonic inference oper-
ations have been extensively studied in the literature. (cf. [1,10,13,16]).
Matrix Semantics.
The central idea behind classical matrix semantics is to
view algebras similar to a language
L
L
as models of
. Interpretations of formulas
of
L
in an algebra
A
similar to
L
are homomorphisms of
L
into
A
.When
A
is
augmented with a subset
d
of the universe of
A
, the resulting structure
M
=
A,d,
called a
logical matrix
for
L
, determines the inference operation
M
defined in
the following way: for every set
X ∪{α}
of formulas of
L
,
X
M
α
iff
for every homomorphism
h
of
L
into
A
,if
h
(
X
)
⊆ d
then
h
(
α
)
∈ d
.
The research on logical matrices has been strongly influenced by universal alge-
bra and model theory. Wojcicki's monograph [16] contains a detailed account of
the development of matrix semantics since its inception in the early 20th century.
In AI, matrix semantics (and a closely related discipline of many-valued logics)
has been successfully exploited in the areas of Automated Reasoning, KRR, and
Logic Programming (cf. [3,4,5,6,9,13,15]).
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