CHAPTER 4
CLASSICAL PROPOSITIONAL SEMANTICS
1
Language
There are several propositional languages that are routinely called classical
propositional logic languages. It is due to the functional dependency of classi
cal connectives discussed briefly in chapter 2. They all share the same logical
meaning, called semantics.
They also define the same set of universally true
formulas called tautologies.
We adopt here as classical propositional language the language
L
with the full
set of connectives
CON
=
{¬
,
∪
,
∩
,
⇒
,
⇔
,
⇔}
i.e. the language
L
{¬
,
∪
,
∩
,
⇒
,
⇔
,
⇔}
.
As the choice of the set of connectives is now fixed, we will use the symbol
L
to denote the language
L
{¬
,
∪
,
∩
,
⇒
,
}
.
In previous chapters we have already established how we
read
and what are the
natural language
names
of the connectives of
L
.
For example, we
read
the symbol
¬
as
”not”, ”not true”
and its
name
is
negation
.
We
read
the formula (
A
⇒
B
) as
” if A, then B”, ”A implies B”,
or
”from
the fact that A we deduce B”
, and the
name
of the connective
⇒
is
implication
.
We refer to the established reading of propositional connectives and formulas
involving them as
a natural language meaning
.
Propositional logic defines and studies their
a logical meaning
called
a se
mantics
of the language
L
. We define here a
classical semantics
, some other
semantics are defined in next chapter (Chapter 5).
1
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2
Classical Semantics, Satisfaction
We based the classical logic, i.e.
classical semantics
on the following two
assumptions.
TWO VALUES:
there are only two logical values.
We denote them T (for
true) and F (for false). Other common notations are 1
,
>
for true and 0
,
⊥
for false.
EXTENSIONALITY:
the logical value of a formula depends only on a main
connective and logical values of its subformulas.
We define a
classical semantics
for
L
in terms of two factors: classical truth
tables (reflexes the extensionality of connectives) and a truth assignment.
In
Chapter 2 we provided a motivation for the notion of classical logical connectives
and introduced their informal definitions. We summarize here the truth tables
for propositional connectives defined in chapter 2 in the following one table.
A
B
¬
A
(
A
∩
B
)
(
A
∪
B
)
(
A
⇒
B
)
(
A
⇔
B
)
T
T
F
T
T
T
T
T
F
F
F
T
F
F
F
T
T
F
T
T
F
F
F
T
F
F
T
T
(1)
The first truth values row of the above table 2 reads:
For any formulas
A, B
, if the logical value of
A
=
T
and
B
=
T
, then logical
values of
¬
A
=
T
, (
A
∩
B
) =
T
, (
A
∪
B
) =
T
and (
A
⇒
B
) =
T
.
We read and write the other rows in a similar manner.
The table
2 indicates that the logical value of of propositional formulas de
pends on the logical values of its factors; i.e. fulfils the condition of extension
ality. Moreover, it shows that the logical value of of propositional connectives
depends only on the logical values of its factors; i.e. it is
independent of the
formulas
A, B
. It gives us the following important property of our proposi
tional connectives.
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 Spring '08
 Bachmair,L
 Computer Science, Logic, Logical connective

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