CHAPTER 13
PREDICATE LANGUAGES
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Predicate Languages
Propositional Languages are also called Zero Order Languages, as opposed to
Predicate Languages that are called First Order Languages. The same applies
to the use of terms Propositional and Predicate Logic; they are often called zero
Order and First Order Logics and we will use both terms equally.
We will work with several different predicate languages, depending on what ap
plications we have in mind. All of those languages have some common features,
and we begin with these.
Propositional connectives
We define the set of propositional connectives
CON
in the same way as in the case of the propositional languages. It means
that we assume the following.
1. The set of connectives is nonempty and finite, i.e.
0
< cardCON <
ℵ
0
.
2. We consider only the connectives with one or two arguments.
Quantifiers
We adopt two quantifiers;
∀
(for all, the universal quantifier) and
∃
(there exists, the existential quantifier), i.e. we have the following set of
quantifiers
Q
=
{∀
,
∃}
.
In a case of the classical logic and the logics that extend it, it is possible
to adopt only one quantifier and to define the other in terms of it and
propositional connectives. It is impossible in a case of some nonclassical
logics, for example the intuitionistic logic. But even in the case of classical
logic two quantifiers express better the common intuition, so we assume
that we have two of them.
Parenthesis.
As in the propositional case, we adopt the signs ( and ) for our
parenthesis., i.e. we define a set
PAR
as
PAR
=
{
(
,
)
}
.
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Variables
We assume that we always have a countably infinite set
V AR
of
variables, i.e. we assume that
cardV AR
=
ℵ
0
.
We denote variables by
x, y, z, ...
, with indices, if necessary, what we often
express by writing
V AR
=
{
x
1
, x
2
,
....
}
.
The set of propositional connectives
CON
defines a propositional part of the
predicate logic language.
What really differ one predicate language from the
other is the choice of additional symbols to the symbols described above. These
are called predicate symbols, function symbols, and constant symbols.
I.e.
a
particular predicate language is determined by specifying the following sets of
symbols.
Predicate symbols
Predicate symbols represent relations. We assume that
we have an non empty, finite or countably infinite set
P
of predicate, or relation symbols. I.e. we assume that
0
< card
P
≤ ℵ
0
.
We denote predicate symbols by
P, Q, R, ...
, with indices, if necessary.
Each predicate symbol
P
∈
P
has a positive integer #
P
assigned to it;
if #
P
=
n
then say
P
is called an nary (n  place) predicate (relation)
symbol.
Function symbols
We assume that we have a finite (may be empty) or count
ably infinite set
F
of function symbols. I.e. we assume that
0
≤
card
F
≤ ℵ
0
.
When the set
F
is empty we say that we deal with a language without
functional symbols.
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 Spring '08
 Bachmair,L
 Computer Science, Logic, Tn, Predicate logic, predicate language

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