propositional calculus

propositional calculus

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A

propositional calculus

is a

formal system

whose expressions

represent

formal objects

known as

propositions

and whose distinguished

relations

among expressions represent existing relations among propositions. Many different propositional calculi represent what is recognizably the same subject matter of propositions and their relations, which more

generic

subject matter is conveniently described as

propositional logic

. For the purposes of mathematical discussion, and especially in computational applications, it is

sufficient

to identify a proposition with a boolean-valued function, that is, a

mapping

of the type

X

→

𝔹

, where

X

is some set and

𝔹

=

{

0

,

1

}

.

As a general consideration, a calculus is a formal system that consists of a set of syntactic expressions, a distinguished subset of these expressions, plus a set of transformation rules that define a

binary relation

on the space of expressions.

When the expressions are interpreted for mathematical purposes, the transformation rules are typically intended to preserve some type of semantic equivalence relation among the expressions. In particular, when the expressions are intepreted as a logical system, the semantic equivalence is typically intended to be

logical equivalence

. In this setting, the transformation rules can be used to derive logically equivalent expressions from any given expression. These

derivations

include as special cases (1) the problem of

simplifying

expressions and (2) the problem of deciding whether a given expression is

equivalent

to an expression in the distinguished subset, typically interpreted as the subset of logical

axioms

.

The

formal language

component of a propositional calculus consists of (1) a set of

primitive

symbols, variously referred to as

atomic formulas

,

placeholders

,

proposition letters

, or

variables

, and (2) a set of

operator symbols

, variously interpreted as

logical operators

or

logical connectives

. A

well-formed formula

(

wff

) is any atomic formula or any

formula

that can be built up from atomic formulas by means of operator symbols.

The following outlines a standard propositional calculus. Many different formulations exist which are all more or less equivalent but differ in (1) their

language

, that is, the particular

collection

of primitive symbols and operator symbols, (2) the set of axioms, or distingushed formulas, and (3) the set of transformation rules that are available.

Contents:

1

Abstraction and application

2

Generic description of a propositional calculus

3

Example 1. Simple axiom system

4

Example 2. Natural deduction system

5

Document history

1

Abstraction and application

Although it is possible to

construct

an abstract formal calculus that has no immediate practical use and next to nothing in the way of obvious applications, the very name

calculus

indicates that this species of formal system owes its origin to the utility of its prototypical members in practical calculation. Generally speaking, any mathematical calculus is designed with the intention of representing a given domain of formal objects, and typically with the aim of facilitating the computations and inferences that need to be carried out in this

representation

. Thus some idea of the intended

denotation

, the formal objects that the formulas of the calculus are intended to denote, is given in advance of developing the calculus itself.

Viewed over the course of its historical

development

, a formal calculus for any given subject matter normally arises through a process of gradual abstraction, stepwise

refinement

, and trial-and-error synthesis from the array of informal notational systems that inform prior use, each of which covers the object domain only in part or from a particular angle.

2

Generic description of a propositional calculus

A

propositional calculus

is a formal system

ℒ

=

ℒ

⁢

(

A

,

Ω

,

Z

,

I

)

, whose formulas are constructed in the following manner:

•

The

alpha set

A

is a

finite set

of

elements

called

proposition symbols

or

propositional variables

. Syntactically speaking, these are the most basic elements of the formal language

ℒ

, otherwise referred to as

atomic formulas

or

terminal elements

. In the examples to follow the elements of

A

are typically the letters

p

,

q

,

r

, and so on.

•

The

omega set

Ω

is a finite set of elements called

operator symbols

or

logical connectives

. The set

Ω

is partitioned into disjoint subsets as follows:

Ω

=

⋃

j

=

0

m

Ω

j

.

In this

partition

,

Ω

j

is the set of operator symbols of arity

j

.

In the more familiar propositional calculi,

Ω

is typically partitioned as follows:

Ω

1

=

{

¬

}

,

Ω

2

⊆

{

∧

,

∨

,

⇒

,

⇔

}

.

A frequently adopted option treats the constant logical values as

operators

of arity zero, thus:

Ω

0

=

{

0

,

1

}

.

Some writers use the tilde

(

~

)

instead of

(

¬

)

and some use the ampersand

(

&

)

instead of

(

∧

)

. Notation varies even more for the set of logical values, with symbols like

{

false

,

true

}

,

{

F

,

T

}

,

{

0

,

1

}

, and

{

⊥

,

⊤

}

all being seen in various

contexts

.

•

Depending on the precise formal grammar or the

grammar

formalism that is being used, syntactic auxiliaries like the left parenthesis,

`

`

(

"

, and the right parentheses,

`

`

)

"

, may be necessary to complete the construction of formulas.

•

The

zeta set

Z

is a finite set of

transformation rules

that are called

inference rules

when they acquire logical applications.

•

The

iota set

I

is a finite set of

initial points

that are called

axioms

when they receive logical

interpretations

.

3

Example 1. Simple axiom system

Let

ℒ

1

=

ℒ

⁢

(

A

,

Ω

,

Z

,

I

)

, where

A

,

Ω

,

Z

,

I

are defined as follows:

The alpha set

A

is a finite set of symbols that is large enough to supply the needs of a given discussion, for example:

A

=

{

p

,

q

,

r

,

s

,

t

,

u

}

.

The omega set

Ω

is a set of two symbols

{

¬

,

⇒

}

that partitions as follows:

Ω

1

=

{

¬

}

,

Ω

2

=

{

⇒

}

.

Under the usual interpretation of these symbols, this choice amounts to adopting

negation

and

implication

as the primitive

operations

of the propositional calculus

ℒ

1

.

Of the three

connectives

for

conjunction

(

∧

)

,

disjunction

(

∨

)

, and implication

(

⇒

)

, one can be taken as primitive and the other two can be defined in terms of it and negation

(

¬

)

. The equivalence

(

⇔

)

is defined in terms of conjunction and implication, with

p

⇔

q

defined as

(

p

⇒

q

)

∧

(

q

⇒

p

)

.

The zeta set

Z

formalizes the inference rule known as

modus ponens

(

http://planetmath.org/ModusPonens

)

:

From

p

,

(

p

⇒

q

)

, infer

q

.

The iota set

I

abstracts an

axiom system for propositional logic

that was discovered by Jan Łukasiewicz:

p

⇒

(

q

⇒

p

)

(

p

⇒

(

q

⇒

r

)

)

⇒

(

(

p

⇒

q

)

⇒

(

p

⇒

r

)

)

(

¬

p

⇒

¬

q

)

⇒

(

q

⇒

p

)

We have the following auxiliary definitions:

p

∨

q

:=

¬

⁢

p

⇒

q

.

p

∧

q

:=

¬

⁡

(

p

⇒

¬

⁢

q

)

.

4

Example 2. Natural deduction system

Let

ℒ

2

=

ℒ

⁢

(

A

,

Ω

,

Z

,

I

)

, where

A

,

Ω

,

Z

,

I

are defined as follows:

The alpha set

A

is a finite set of symbols that is large enough to supply the needs of a given discussion, for example:

A

=

{

p

,

q

,

r

,

s

,

t

,

u

}

.

The omega set

Ω

=

Ω

1

∪

Ω

2

partitions as follows:

Ω

1

=

{

¬

}

,

Ω

2

=

{

∧

,

∨

,

⇒

,

⇔

}

.

In the following example of a propositional calculus, the transformation rules are intended to be interpreted as the inference rules of a so-called

natural deduction

system

. The particular system presented here has no initial points, which means that its interpretation for logical applications derives its

theorems

from an empty axiom set.

To be continued

…

5

Document history

Portions of the above article are adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.

•

http://www.mywikibiz.com/Propositional_calculus

Propositional calculus,

http://www.mywikibiz.com/

MyWikiBiz.

•

http://www.getwiki.net/-Propositional_Calculus

Propositional calculus,

http://www.getwiki.net/

GetWiki.

•

http://www.wikinfo.org/index.php/Propositional_calculus

Propositional calculus,

http://www.wikinfo.org/

Wikinfo.

•

http://en.wikipedia.org/w/index.php?title=Propositional_calculus&oldid=74069381

Propositional calculus,

http://en.wikipedia.org/

Wikipedia.

Title

propositional calculus

Canonical name

PropositionalCalculus

Date of creation

2013-03-22 17:54:40

Last modified on

2013-03-22 17:54:40

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

22

Author

Jon Awbrey (15246)

Entry type

Definition

Classification

msc 03F03

Classification

msc 03C05

Classification

msc 03B22

Classification

msc 03B05

Synonym

propositional logic

Synonym

sentential calculus

Synonym

sentential logic

Related topic

ZerothOrderLogic

Related topic

MinimalNegationOperator

Related topic

DifferentialLogic

Related topic

DifferentialPropositionalCalculus

Related topic

DifferentialPropositionalCalculusExamples

Related topic

DifferentialPropositionalCalculusAppendices

Related topic

DifferentialPropositionalCalculusAppendix2

Related topic

DifferentialPropositionalC

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