logical language

logical language

In its most general form, a

logical language

is a set of rules for constructing

formulas

for some logic, which can then be assigned truth values based on the rules of that logic.

A logical language

ℒ

consists of:

•

A set

F

of

function symbols

(common examples include

+

and

⋅

)

•

A set

R

of relation symbols (common examples include

=

and

<

)

•

A set

C

of

logical connectives

(usually

¬

,

∧

,

∨

,

→

and

↔

)

•

A set

Q

of

quantifiers

(usuallly

∀

and

∃

)

•

A set

V

of

variables

Every function symbol, relation symbol, and

connective

is associated with an arity (the set of

n

-ary function symbols is denoted

F

n

, and similarly for relation symbols and connectives). Each quantifier is a

generalized quantifier

associated with a quantifier type

⟨

n

1

,

…

,

n

n

⟩

.

The underlying logic has a (possibly empty) set of types

T

. There is a

function

Type

:

F

∪

V

→

T

which assignes a type to each function and variable. For each arity

n

is a function

Inputs

n

:

F

n

∪

R

n

→

T

n

which gives the types of each of the

arguments

to a function symbol or

relation

. In

addition

, for each quantifier type

⟨

n

1

,

…

,

n

n

⟩

there is a function

Inputs

⟨

n

1

,

…

,

n

n

⟩

defined on

Q

⟨

n

1

,

…

,

n

n

⟩

(the set of quantifiers of that type) which gives an

n

-tuple of

n

i

-tuples of types of the arguments taken by formulas the quantifier applies to.

The

terms

of

ℒ

of type

t

∈

T

are built as follows:

1.

If

v

is a variable such that

Type

⁡

(

v

)

=

t

then

v

is a term of type

t

2.

If

f

is an

n

-ary function symbol such that

Type

⁡

(

f

)

=

t

and

t

1

,

…

,

t

n

are terms such that for each

i

<

n

Type

⁡

(

t

i

)

=

(

Inputs

n

⁡

(

f

)

)

i

then

f

⁢

t

1

,

…

,

t

n

is a term of type

t

The

formulas

of

ℒ

are built as follows:

1.

If

r

is an

n

-ary relation symbol and

t

1

,

…

,

t

n

are terms such that

Type

⁡

(

t

i

)

=

(

Inputs

n

⁡

(

r

)

)

i

then

r

⁢

t

1

,

…

,

t

n

is a formula

2.

If

c

is an

n

-ary connective and

f

1

,

…

,

f

n

are formulas then

c

⁢

f

1

,

…

,

f

n

is a formula

3.

If

q

is a quantifier of type

⟨

n

1

,

…

,

n

n

⟩

,

v

1

,

1

,

…

,

v

1

,

n

1

,

v

2

,

1

,

…

,

v

n

,

1

,

…

,

v

n

,

n

n

are a

sequence

of variables such that

Type

⁡

(

v

i

,

j

)

=

(

(

Inputs

⟨

n

1

,

…

,

n

n

⟩

⁡

(

q

)

)

j

)

i

and

f

1

,

…

,

f

n

are formulas then

q

⁢

v

1

,

1

,

…

,

v

1

,

n

1

,

v

2

,

1

,

…

,

v

n

,

1

,

…

,

v

n

,

n

n

⁢

f

1

,

…

,

f

n

is a formula

Generally the connectives, quantifiers, and variables are specified by the appropriate logic, while the function and relation symbols are specified for particular

languages

. Note that

0

-ary functions are usually called

constants

.

If there is only one type which is equated directly with truth values then this is essentially a

propositional logic

. If the standard quantifiers and connectives are used, there is only one type, and one of the relations is

=

(with its usual

semantics

), this produces

first order logic

. If the standard quantifiers and connectives are used, there are two types, and the relations include

=

and

∈

with appropriate semantics, this is

second order logic

(a slightly different formulation replaces

∈

with a

2

-ary function which represents function application; this views

second order

objects as functions rather than sets).

Note that often connectives are written with

infix notation

with parentheses used to control

order of operations

.

Title

logical language

Canonical

name

LogicalLanguage

Date of creation

2013-03-22 12:59:55

Last modified on

2013-03-22 12:59:55

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

14

Author

Henry (455)

Entry type

Definition

Classification

msc 03B15

Classification

msc 03B10

Related topic

QuantifierFree

Defines

term

Defines

formula

Defines

constant

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