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S
I V
.
T
.
D 15
Square of oppositionF W,
I A , square of opposition
('') . T
,
.
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
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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
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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,
, ""
,
, .
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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)
Retrieved from "http://en.wikipedia.org/w/index.php?title=Square_of_opposition&oldid=463175140"
Categories: Conceptual models Traditional logic Inference
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