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Square of oppositionF W,

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Contents

1 S

2 T

3 M

4 L -

5 S 6 R

7 E

Summar

I , (L: popoiio)

(oaio ennciaia), ,

. A caegoical popoiion , ,

.

E

. T :

T - 'A' ,

(nieali affimaia), L ' S

P', ' S P'.

T 'E' , (niealinegaia), L ' S P', '

S P'.

T 'I' , ( paiclai affimaia), L ' S P',

' S P'.

T 'O' , ( paiclai negaia), L ' S P',

' S P'.

I :

T F A P





Name Smbol Latin English SSF

Universal affirmative A Omne S est P Every S is P All S is P

Universal negative E Nullum S est P No S is P All S is not P

Particular affirmative I Quoddam S est P Some S is P Some S is P

Particular negative O Quoddam S non est P Some S is not P Some S is not P

Aristotle states (in chapters six and seven of the Pei hemaneia (Πε Ἑμηνεία, Latin De Inepeaione,

English 'On Exposition'), that there are certain logical relationships between these four kinds of proposition. He say

that to every affirmation there corresponds exactly one negation, and that every affirmation and its negation are

'opposed' such that always one of them must be true, and the other false. A pair of affirmative and negative

statements he calls a 'contradiction' (in medieval Latin, conadicio). Examples of contradictories are 'every man

white' and 'not every man is white', 'no man is white' and 'some man is white'.

'Contrary' (medieval: conaiae) statements, are such that both cannot at the same time be true. Examples of thes

are the universal affirmative 'every man is white', and the universal negative 'no man is white'. These cannot be true

at the same time. However, these are not contradictories because both of them may be false. For example, it isfalse that every man is white, since some men are not white. Yet it is also false that no man is white, since there are

some white men.

Since every statement has a contradictory opposite, and since a contradictory is true when its opposite is false, it

follows that the opposites of contraries (which the medievals called subcontraries, bconaiae) can both be tru

but they cannot both be false. Since subcontraries are negations of universal statements, they were called 'particula

statements by the medieval logicians.

A further logical relationship implied by this, though not mentioned explicitly by Aristotle, is subalternation

( balenaio). This is a relation between a particular statement and a universal statement such that the particular

implied by the other. For example, if 'every man is white' is true, its contrary 'no man is white' is false. Therefore th

contradictory 'some man is white' is true. Similarly the universal 'no man is white' implies the particular 'not every

man is white'.

In summary:

Universal statements are contraries: 'every man is just' and 'no man is just' cannot be true together,

although one may be true and the other false, and also both may be false (if at least one man is just, and

at least one man is not just).

Particular statements are subcontraries. 'Some man is just' and 'some man is not just' cannot be false

together The universal affirmative and the particular affirmative are subalternates, because in Aristotelian

semantics 'every A is B' implies 'some A is B'. Note that modern formal interpretations of English

sentences interpret 'every A is B' as 'for any x, x is A implies x is B', which does no imply 'some x is A

This is a matter of semantic interpretation, however, and does not mean, as is sometimes claimed, that

Aristotelian logic is 'wrong'.

The universal affirmative and the particular negative are contradictories. If some A is not B, not every A

is B. Conversely, though this is not the case in modern semantics, it was thought that if every A is not B

some A is not B. This interpretation has caused difficulties (see below). While Aristotle's Greek does no





F'

T contrr :

I subcontrr

' A B', ' A B', B[citation needed

Peri hermaneias, ' A B',

' A B',

.

T B

. T ,

, 'T S O'.

The problem of eistential import

S, ' A B' ( A B)

' A B' ( A B) ,

( A B / A B) . T P

A. 'S A B' ' A'. F 'S '

, ' ' . B '

' , ' '

. B A . B . T( ) , .. . B (

A , D) ?[1]

F , ' ' '

'.[2]

A , '

.

I ' - ' , per accidens (' - ').B -, . . B '

' , . [3]

[ : M , S' , P' ]

Modern squares of opposition

I 19 , G B

(I O),

(A E) . T V . T ,

B , S

. I , A O

, E I,

; , , . T,

, ""

,

, .





Frege's Begriffsschrift also presents a square of oppositions, organised in

an almost identical manner to the classical square, showing the contradictories, subalternates and contraries

between four formulae constructed from universal quantification, negation and implication.

Logical heagons and other bi-simplees

Main article: Logical heagon

The square of opposition has been extended to a logical hexagon which includes the relationships of six statements

It was discovered independently by both Augustin Sesmat and Robert Blanché.[4] It has been proven that both the

square and the hexagon, followed by a logical cube, belong to a regular series of n-dimensional objects called

logical bi-simplexes of dimension n. The pattern also goes even beyond this.[5]

See also

Boole's syllogistic

Free logic

Semiotic square

References

1. ^ In his Dialectica, and in his commentary on the Perihermaneias

2. ^ Re enim hominis prorsus non eistente neque ea vera est quae ait: omnis homo est homo, nec ea quae proponit:

quidam homo non est homo

3. ^ Si enim vera est: Omnis homo qui lapis est, est lapis, et eius conversa per accidens vera est: Quidam lapis est

homo qui est lapis. Sed nullus lapis est homo qui est lapis, quia neque hic neque ille etc. Sed et illam: Quidam

homo qui est lapis, non est lapis, falsam esse necesse est, cum impossibile ponat

4. ^ N-Opposition Theory Logical heagon (http://alessiomoretti.perso.sfr. fr/NOTLogicalHexagon.html)

5. ^ Moretti, Pellissier

Eternal links

The traditional Square of Opposition (http://plato.stanford.edu/entries/square) entry by Terence Parson

in the Stanford Encclopedia of Philosoph

International Congress on the Square of Opposition (http://www.square-of-opposition.org/)

Special Issue of Logica Universalis Vol2 N1 (2008) on the Square of Opposition

(http://www.springerlink.com/content/l8rj3631747w/)Color-coded traditional square illustrating the various inferences

(http://bearspace.baylor.edu/M_Boone/www/color%20coded%20traditional%20square.JPG)

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