formal system

formal system

In mathematical logic, a

formal system

is a “set-up” to study the syntactic structures of statements that we see in everyday mathematics. Thus, instead of looking at the meaning of the statement

1

+

1

=

2

, one looks at the expression made up of symbols

1

,

+

,

1

,

=

, and

2

.

Formally, a formal system

S

consists of the following:

1.

the

language

of

S

, which is formed by

(a)

a set

Σ

of symbols;

finite sequences

of these symbols are called

words

or

expressions

over

Σ

;

(b)

a set of expressions called

well-formed formulas

, or simply

formulas

;

(c)

a set of rules that specify how these words are generated; the set is called the

syntax of the language

; typically, the rules are

*

certain expressions are singled out, which are called atoms, or

atomic formulas

*

a set of rules (inductive in nature) showing how formulas can be formed from other formulas that have already been formed

2.

a

deductive system

(or

proof system

) associated with

S

, which consists of

(a)

a set of axioms; each axiom is usually (but not always) a formula;

(b)

a set of rules called

rules of inference

on

S

; the rules of inferences are used to generate formulas, which are known as

theorems

of

S

; typically,

*

an axiom is a theorem,

*

a formula that can be formed (or deduced) from other theorems by a rule of inference is a theorem.

Thus, in a formal system

S

, one starts with a set

Σ

of symbol. Three sets are associated:

Σ

*

⊇

Fmla

⁡

(

S

)

⊇

Thm

⁡

(

S

)

where

Σ

*

is the set of all expressions over

Σ

,

Fmla

⁡

(

S

)

is the set of all (well-formed) formulas obtained by a set of formula formation rules, and

Thm

⁡

(

S

)

is the set of all theorems obtained from the formulas by a set of rules of inference.

Here’s an example of a formal system: the (classical)

propositional logic

P

⁢

R

⁢

O

⁢

P

. The symbols of

P

⁢

R

⁢

O

⁢

P

consists of

→

,

¬

,

(

,

)

,

(not including

,

the symbol for commas, and a set of

elements

called

propositional variables

. Formulas of

P

⁢

R

⁢

O

⁢

P

are generated as follows:

•

propositional variables are formulas; and

•

if

α

,

β

are formulas, so are

(

¬

⁢

α

)

and

(

α

∨

β

)

.

A deduction system for

P

⁢

R

⁢

O

⁢

P

may consist the following:

•

all formulas of the forms

(

α

→

(

α

→

β

)

)

,

(

(

α

→

(

β

→

γ

)

)

→

(

(

α

→

β

)

→

(

α

→

γ

)

)

)

,

(

(

¬

(

¬

α

)

)

→

α

)

for all formulas

α

,

β

, and

•

γ

, and the (only) rule of inference is

modus ponens

: from

α

and

(

α

→

β

)

, we may infer (or deduce)

β

.

The (classical)

first-order logic

is another example of a formal system.

Group theory

, number theory (a la

Peano arithmetic

), theory of

partially ordered sets

, and

set theory

(a la ZFC) are other examples of formal systems that are extensively studied.

Remark

. A formal system is sometimes called a

logical system

, although the latter term usually either refers to a formal system where the

deducibility relation

⊢

arising out of the associated deductive system

satisfies

certain properties (for example,

A

⊢

A

for all wffs, etc…), or a formal system together with a semantical

structure

.

References

1

R. M. Smullyan,

Theory of Formal Systems

, Princeton University Press, 1961

2

J. R. Shoenfield,

Mathematical Logic

, AK Peters, 2001

Title

formal system

Canonical name

FormalSystem

Date of creation

2013-03-22 19:12:25

Last modified on

2013-03-22 19:12:25

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Topic

Classification

msc 03B99

Synonym

logical system

Related topic

DeductiveSystem

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