logical implication

logical implication

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1

Short version

Logical implication

is an

operation

on two logical values, typically the values of two

propositions

(

http://planetmath.org/PropositionalCalculus

), that produces a value of false just in case the first operand is true and the second operand is false.

The

truth table

for the logical implication operation that is written as

p

⇒

q

and read as

`

⁢

`

⁢

p

⁢

implies

⁡

q

⁢

"

,

also written as

p

→

q

and read as

`

⁢

`

⁢

if

⁡

p

⁢

then

⁡

q

⁢

"

,

is as follows:

Logical Implication

p

q

p

⇒

q

F

F

T

F

T

T

T

F

F

T

T

T

2

Long version

The mathematical objects that inform our capacity for logical reasoning are easier to describe in a straightforward way than it is to reconcile their traditional accounts. Still, some discussion of the

language

that

occurs in

the literature cannot be avoided.

The

concept

of

logical implication

encompasses a specific logical

function

, a specific logical relation, and the various symbols that are used to denote this function and this

relation

. In order to define the specific function, relation, and symbols in question it is first necessary to establish a few ideas about the connections among them.

Close approximations to the concept of logical implication are expressed in ordinary language by means of linguistic forms like the following:

•

p

implies

q

.

•

If

p

then

q

.

Here

p

and

q

are

propositional variables

that stand for any propositions in a given language. In a statement of the form “if

p

then

q

”, the first term,

p

, is called the

antecedent

and the second term,

q

, is called the

consequent

, while the statement as a whole is called either the

conditional

or the

consequence

. Assuming that the conditional statement is true, then the truth of the antecedent is a

sufficient

condition

for the truth of the consequent, while the truth of the consequent is a

necessary condition

for the truth of the antecedent.

Many writers draw a technical distinction between the form “

p

implies

q

” and the form “if

p

then

q

”. In this view, writing “

p

implies

q

” asserts the

existence

of a certain relation between the logical value of

p

and the logical value of

q

while writing “if

p

then

q

” simply forms a compound

sentence

whose logical value is a function of the logical values of

p

and

q

. Notice that a relation is a mathematical object while a sentence, whether open or closed, is a syntactic form that exists in the domain of signs (

http://planetmath.org/SignRelation

).

Two factors enter at cross-purposes to our understanding at this point, and they have historically been the source of many divergent thinkers talking skew to each other on this score.

•

There is the contrast between “object” and “sign”. Objects include among their number abstract, formal, and mathematical objects. Signs include the full

variety

of syntactic forms in general. For some thinkers, objects are the primary concerns and language merely accessory to them. For other thinkers, signs are the primary business and objects always mediated by them, with

formal objects

being perhaps the mere illusions that are produced in our minds by forms of words.

•

There is the contrast between the relational aspect and the

functional

aspect of the mathematical objects in question. This issue is less problematic within mathematics itself, it being axiomatic that a

binary operation

is a

ternary relation

and because there are standard ways of relating

characteristic functions

with the sets and relations that are their fibers.

2.1

Definition

The concept of logical implication is associated with an operation on two logical values, typically the values of two propositions, that produces a value of false just in case the first operand is true and the second operand is false.

In the interpretation where

0

=

false

and

1

=

true

, the truth table associated with the statement “

p

implies

q

”, symbolized as

p

⇒

q

, is as follows:

Logical Implication

p

q

p

⇒

q

0

0

1

0

1

1

1

0

0

1

1

1

2.2

Discussion

The usage of the terms

logical implication

and

material conditional

varies from field to field and even across different contexts of discussion. One way to minimize the potential confusion is to begin with a focus on the various types of formal objects that are being discussed, of which there are only a few, taking up the

variations

in language as a secondary matter.

The main formal object under discussion is a logical operation on two logical values, typically the values of two propositions, that produces a value of false just in case the first operand is true and the second operand is false. By way of a temporary name, the logical operation in question may be written as

cond

⁡

(

p

,

q

)

, where

p

and

q

are logical values. The truth table associated with this operation is as follows:

cond

:

𝔹

×

𝔹

→

𝔹

p

q

cond

⁡

(

p

,

q

)

0

0

1

0

1

1

1

0

0

1

1

1

Some writers draw a firm distinction between the

conditional

connective

(the syntactic sign

`

⁢

`

→

"

) and the

implication

relation (the formal object denoted by the sign

`

⁢

`

⇒

"

). These writers use the phrase “if–then” for the conditional connective and the term “implies” for the implication relation. The

difference

is sometimes explained by saying that the conditional is the “contemplated” relation while the implication is the “asserted” relation. In most areas of mathematics, the distinction is treated as a variation in the usage of the single sign

`

⁢

`

⇒

"

, not requiring two separate signs. Not all of those who use the sign

`

⁢

`

→

"

for the conditional connective regard it as a sign that denotes any kind of formal object, but treat it as a so-called

syncategorematic sign

, that is, a sign with a purely syntactic function. For the sake of clarity and simplicity in the present

introduction

, it is convenient to use the two-sign notation, but allow the sign

`

⁢

`

→

"

to denote the

boolean function

that is associated with the truth table of the material conditional. These considerations result in the following scheme of notation.

p

→

q

p

⇒

q

if

⁢

p

⁢

then

⁢

q

p

⁢

implies

⁢

q

Let

𝔹

=

{

0

,

1

}

, where 0 is interpreted as the logical value

false

and 1 is interpreted as the logical value

true

. The truth table shows the ordered triples of a triadic relation

L

⊆

𝔹

×

𝔹

×

𝔹

that is defined as follows:

L

=

{

(

p

,

q

,

r

)

∈

𝔹

×

𝔹

×

𝔹

:

cond

⁡

(

p

,

q

)

=

r

}

.

Regarded as a set, this triadic relation is the same thing as the binary operation:

cond

:

𝔹

×

𝔹

→

𝔹

.

The relationship between

cond

and

L

exemplifies the standard association that exists between any binary operation and its corresponding triadic relation.

The conditional sign

`

⁢

`

→

"

denotes the same formal object as the function name

cond

, the only difference being that the first is written infix while the second is written prefix. Thus we have the following

equation

:

(

p

→

q

)

=

cond

(

p

,

q

)

.

Consider once again the triadic relation

L

⊆

𝔹

×

𝔹

×

𝔹

that is defined in the following

equivalent

fashion:

L

=

{

(

p

,

q

,

cond

⁡

(

p

,

q

)

)

:

(

p

,

q

)

∈

𝔹

×

𝔹

}

.

Associated with the 3-adic relation

L

is a 2-adic relation

L

..1

⊆

𝔹

×

𝔹

that is called the

fiber

of

L

with 1 in the third place. This object is defined as follows:

L

..1

:=

{

(

p

,

q

)

∈

𝔹

×

𝔹

:

(

p

,

q

,

1

)

∈

L

}

.

The same object is achieved in the following way. Begin with the 2-adic operation:

cond

:

𝔹

×

𝔹

→

𝔹

.

Form the 2-adic relation that is called the

fiber

of

cond

at 1, notated as follows:

cond

-

1

⁡

(

1

)

⊆

𝔹

×

𝔹

.

This object is defined as follows:

cond

-

1

⁡

(

1

)

=

{

(

p

,

q

)

∈

𝔹

×

𝔹

:

cond

⁡

(

p

,

q

)

=

1

}

.

The implication sign

`

⁢

`

⇒

"

denotes the same formal object as the relation names

`

⁢

`

⁢

L

..1

⁢

"

and

`

⁢

`

⁢

cond

-

1

⁡

(

1

)

⁢

"

, the only differences being purely syntactic. Thus we have the following

logical equivalence

:

(

p

⇒

q

)

⇔

(

p

,

q

)

∈

L

..1

⇔

(

p

,

q

)

∈

cond

-

1

(

1

)

.

This completes the derivation of the mathematical objects that are denoted by the signs

`

⁢

`

→

"

and

`

⁢

`

⇒

"

in this discussion. It needs to be remembered, though, that not all writers observe this distinction in every context. Especially in mathematics, where the sign

`

⁢

`

→

"

is reserved for function notation, it is common to see the sign

`

⁢

`

⇒

"

being used for both concepts.

3

Bibliography

•

Brown, Frank Markham (2003),

Boolean

Reasoning : The Logic of Boolean Equations

, 1st edition, Kluwer Academic Publishers, Norwell, MA, 1990. 2nd edition, Dover Publications, Mineola, NY.

•

Church, Alonzo (1962), “Logic, formal”, pp. 170–181 in Runes (1962).

•

Church, Alonzo (1996),

Introduction to Mathematical Logic

. Originally published, Annals of Mathemtics Studies, 1944. Revised and enlarged edition, Princeton Mathematical Series, 1956. 10th printing, Princeton Landmarks in Mathematics and Physics, Princeton University Press, Princeton, NJ.

•

van Heijenoort, Jean (1967, ed.),

From Frege To Gödel : A Source Book in Mathematical Logic, 1879–1931

, Harvard University Press, Cambridge, MA.

•

Quine, Willard Van Orman (1982),

Methods of Logic

, 1st ed. 1950, 2nd ed. 1959, 3rd ed. 1972. 4th edition, Harvard University Press, Cambridge, MA.

•

Runes, Dagobert D. (1962, ed.),

Dictionary of Philosophy

, Littlefield, Adams, and Company, Totowa, NJ.

•

Styazhkin, N.I. (1969),

History of Mathematical Logic from Leibniz to Peano

, MIT Press, Cambridge, MA.

•

Tarski, Alfred (1983), Logic,

Semantics

,

Metamathematics

: Papers from 1923 to 1938, J.H. Woodger (trans.), Oxford University Press, Oxford, UK, 1956. 2nd edition, John Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983.

4

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/Logical_implication

Logical implication,

http://www.mywikibiz.com/

MyWikiBiz.

•

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

Logical implication,

http://www.getwiki.net/

GetWiki.

•

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

Logical implication,

http://www.wikinfo.org/

Wikinfo.

•

http://en.wikipedia.org/w/index.php?title=Logical_implication&oldid=77109738

Logical implication,

http://en.wikipedia.org/

Wikipedia.

Title

logical implication

Canonical

name

LogicalImplication

Date of creation

2013-03-22 17:55:15

Last modified on

2013-03-22 17:55:15

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

24

Author

Jon Awbrey (15246)

Entry type

Definition

Classification

msc 03F03

Classification

msc 03C05

Classification

msc 03B22

Classification

msc 03B05

Synonym

material implication

Related topic

Implication

Related topic

LogicalConnective

Defines

material conditional

Defines

antecedent

Defines

consequent

Defines

conditional

Defines

consequence

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