logical connective

logical connective

In

propositional logic

, the usual way of forming a

well-formed formula

out of existing ones is by attaching a designated symbol to the existing wffs. This designated symbol is variously known as a

logical connective

,

logical symbol

, or simply a

connective

. Given a connective, the number of wffs needed to form a new wff is a fixed

integer

, and is called the

arity

of the connective. For example, if

#

is a connective of arity 3, and

p

,

q

,

r

are three existing wffs, then

#

⁢

p

⁢

q

⁢

r

is the wff formed by attaching

#

to the

p

,

q

and

r

in order

. Instead of prefixing to the string of wffs as above, the connective may also be attached at the end (as a suffix), or customarily infixed in case it has arity of 2 (binary).

0

-ary connectives are also allowed, in which case it is just a symbol with no wffs attached.

The common classical logical connectives are:

•

¬

:

logical not

(

http://planetmath.org/

Negation

);

•

∨

:

logical or

;

•

∧

:

logical and

;

•

→

or

⊃

:

material implication

; and

•

↔

or

≡

: material equivalence (

http://planetmath.org/

Biconditional

).

The symbols

⊃

and

≡

are due to Russell.

Remark

. “

Logical implication

”

→

and “

logical is

equivalent

to

”

↔

symbols are typically used for logicians. Nevertheless, the symbols

⇒

for material implication and

⇔

for material equivalence are commonly used in the literature. In particular,

→

is usually reserved for the

concept

of limit.

Usually, given a logical connective

#

, a

truth function

is associated. The arity of the truth function is defined to be the arity of the connective. When there is no confusion, the symbol for the associated truth function is the same as the symbol of the connective. A truth function of small arity can be easily represented by a table, called the

truth table

of the truth function. The truth functions and truth tables associated with the connectives listed above are

•

¬

:

𝟐

→

𝟐

given by

¬

⁡

(

x

)

=

1

-

x

;

P

¬

⁢

P

F

T

T

F

•

∨

:

𝟐

2

→

𝟐

given by

∨

(

x

,

y

)

=

max

⁡

(

x

,

y

)

;

P

Q

P

∨

Q

F

F

F

F

T

T

T

F

T

T

T

T

•

∧

:

𝟐

2

→

𝟐

given by

∧

(

x

,

y

)

=

min

⁡

(

x

,

y

)

;

P

Q

P

∧

Q

F

F

F

F

T

F

T

F

F

T

T

T

•

→

:

𝟐

2

→

𝟐

given by

→

(

x

,

y

)

=

max

⁡

(

1

-

x

,

y

)

; and

P

Q

P

→

Q

F

F

T

F

T

T

T

F

F

T

T

T

•

↔

:

𝟐

2

→

𝟐

given by

↔

(

x

,

y

)

=

(

x

+

y

+

1

)

(

mod

2

)

.

P

Q

P

↔

Q

F

F

T

F

T

F

T

F

F

T

T

T

where

𝟐

=

{

0

,

1

}

. Note that

0

and

1

have been converted to

F

and

T

in the tables above.

Any truth function of any finite arity can be written as a finite

combination

of these connectives (see the entry on

functional completeness

). However, the

collection

is redundant; the final three symbols,

∧

,

→

, and

↔

, can be defined in terms of prior ones. By DeMorgan’s law, we can define logical and by

P

∧

Q

:=

¬

⁢

P

∨

¬

⁢

Q

.

Material implication can be defined by

P

→

Q

:=

¬

⁢

P

∨

Q

.

Finally, material equivalence can be defined by

P

↔

Q

:=

(

P

→

Q

)

∧

(

Q

→

P

)

=

¬

⁡

(

¬

⁢

P

∨

Q

)

∨

¬

⁡

(

¬

⁢

Q

∨

P

)

.

Hence

¬

and

∨

suffice to define all other connectives.

Title

logical connective

Canonical name

LogicalConnective

Date of creation

2013-03-22 16:26:58

Last modified on

2013-03-22 16:26:58

Owner

mps (409)

Last modified by

mps (409)

Numerical id

19

Author

mps (409)

Entry type

Definition

Classification

msc 03B05

Synonym

logical symbol

Synonym

connective

Synonym

conjunctive connective

Synonym

disjunctive connective

Related topic

Ampheck

Related topic

ContradictoryStatement

Related topic

LogicalAxiom

Related topic

SoleSufficientOperator

Related topic

PropositionalCalculus

Related topic

LogicalImplication

Related topic

ZerothOrderLogic

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