deductive system

deductive system

Introduction

A

deductive system

is a formal (mathematical) setup of reasoning. In order to describe a deductive system, a (formal)

language

system must first be in place, consisting of (well-formed)

formulas

, strings of symbols constructed according to some prescribed syntax. With the language in place, reasoning, from a formal point of view, is just

derivation

of a formula, called a

conclusion

, from a given set of formulas, called

assumptions

, via a set of formulas, called axioms, and rules, called

rules of inference

.

More specifically, given a language

L

of

well-formed formulas

, a

deductive system

D

consists of

1.

a set of formulas in

L

called

axioms

, and

2.

a set of

binary relations

on subsets of

L

; each

relation

is called a

rule of inference

, or simply

rule

; if

R

is a rule, and

(

Δ

,

Γ

)

∈

R

, we say that from

Δ

we infer

Γ

via

R

; elements of

Δ

are called

premises

, and elements of

Γ

are

conclusions

. Typically,

Δ

is assumed finite (and maybe empty), and

Γ

a

singleton

.

There are also

variations

to the setup above. Sometimes the formulas are replaced by other

expressions

(sequents, etc…), sometimes the rules are not relations, but other constructs (trees, etc…)

A deduction system is also called a

deduction system

or

proof system

.

The

central

task of a deduction system is the construction of

deductions

, which, informally, is the application of inference rules to assumptions (or axioms, if any) to arrive at a conclusion, in a finite number of steps. For more detail, please see this entry (

http://planetmath.org/Deduction

).

A deductive system together with the underlying language is called a

formal system

.

Some Major Formulations of Deductive Systems in Logic

Let us fix a language

L

(of well-formed formulas). There are four main formulations of deductive systems:

•

Hilbert system

, or axiom system: in this formulation, axioms are the main ingredient, and there are only one or two rules of inference (

modus ponens

is usually one of them).

Theorems

in a Hilbert system are those formulas which are conclusions of deductions from axioms.

•

natural deduction

: as opposed to a Hilbert system, a natural deduction system consists of all rules and no axioms, so that the focus is on deductions from assumptions. One special aspect about some of the inference rules is the allowance to cancel assumptions, so that theorems are conclusions from those deductions where all of the premises may be cancelled.

•

Gentzen system

, or sequent system: unlike the last two systems, where axioms and inference rules are based on formulas from

L

, in a Gentzen system, the building blocks of axioms and inference rules are sequents, which are expressions of the form

Δ

⇒

Γ

,

where

Δ

and

Γ

are

finite sequences

of formulas (in

L

, and

⇒

is a symbol not in

L

. In a Gentzen system, all axioms are of the form

A

⇒

A

, for each formula

A

in

L

. Theorems in a Gentzen system are those formulas

B

(in

L

) such that

⇒

B

is the conclusion of a deduction.

•

tableau system: in a tableau system, like natural deduction, there are only inference rules and no axioms. However, unlike all of the systems above, the rules are not relations, but

labeled trees

, which are used to construct larger labeled trees called tableaux from given formulas. Theorems are those formulas corresponding to tableaux satisfying a certain

property

(namely, being closed).

Of the four formulations above, Hilbert systems are most widely used in logic, and are applicable to many types of logics. Natural deduction and Gentzen systems are more amenable to

analysis

of deductions, and therefore are mainly used in structural proof theory.

Given a language

L

, two deductive systems

D

1

and

D

2

are

deductively equivalent

if any theorem

deducible

from one system is deducible from another. In other words,

⊢

D

1

A

iff

⊢

D

2

A

.

There is also a stronger notion of deductive equivalence:

D

1

is (strongly) deductively equivalent to

D

2

exactly when

Δ

⊢

D

1

A

iff

Δ

⊢

D

2

A

,

where

Δ

is a set of formulas, and

A

is a formula, in

L

. Note that strong deductive equivalence implies weak deductive equivalence. On the other hand, if the

deduction theorem

and its

converse

hold in both

D

1

and

D

2

, then weak also implies strong.

In classical

propositional logic

(as well as predicate logic), all four formulations of deductive systems which are pairwise deductively equivalent exist. In

addition

, each can be shown to be sound and

complete

with respect to the usual truth-valuation

semantics

.

References

1

A. S. Troelstra, H. Schwichtenberg,

Basic

Proof Theory

, 2nd Edition, Cambridge University Press, 2000

2

D. Dalen,

Logic and

Structure

, 4th Edition, Springer, 2008

3

G. Priest,

Introduction

to Non-Classical Logic: From If to Is

, 2nd Edition, Cambridge University Press, 2008

Title

deductive system

Canonical

name

DeductiveSystem

Date of creation

2013-03-22 19:12:47

Last modified on

2013-03-22 19:12:47

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

25

Author

CWoo (3771)

Entry type

Topic

Classification

msc 03F03

Classification

msc 03B99

Classification

msc 03B22

Synonym

deduction system

Synonym

proof system

Related topic

FormalSystem

Related topic

InferenceRule

Related topic

LogicalAxiom

Defines

deductively equivalent

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