Square of opposition

In term logic, the square of opposition is a diagram representing the relations between the four basic categorical propositions. The origin of the square can be traced back to Aristotle's tractate On Interpretation and its distinction between two oppositions: contradiction and contrariety. However, Aristotle did not draw any diagram; this was done several centuries later.

Summary

In traditional logic, a proposition is a spoken assertion, not the meaning of an assertion, as in modern philosophy of language and logic. A categorical proposition is a simple proposition containing two terms, subject and predicate, in which the predicate is either asserted or denied of the subject.
Every categorical proposition can be reduced to one of four logical forms, named,,, and based on the Latin ', for the affirmative propositions and, and ', for the negative propositions and. These are:

The proposition, the universal affirmative, whose form in Latin is 'omne est ', usually translated as 'every is a '.
The proposition, the universal negative, Latin form 'nullum est ', usually translated as 'no are '.
The proposition, the particular affirmative, Latin 'quoddam est ', usually translated as 'some are '.
The proposition, the particular negative, Latin 'quoddam nōn est ', usually translated as 'some are not '.

In tabular form:

Name	Symbol	Latin	English*	Mnemonic	Modern form
Universal affirmative		Omne est.	Every is.	'
Universal negative		Nullum est.	No is.	'
Particular affirmative		Quoddam est.	Some is.	'
Particular negative		Quoddam nōn est.	Some is not.	'

*Proposition may be stated as "All is." However, Proposition when stated correspondingly as "All is not." is ambiguous because it can be either an or proposition, thus requiring a context to determine the form; the standard form "No is " is unambiguous, so it is preferred. Proposition also takes the forms "Some is not." and "A certain is not."
** in the modern forms means that a statement applies on an object. It may be simply interpreted as " is " in many cases. can be also written as.
Aristotle states, that there are certain logical relationships between these four kinds of proposition. He says 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 an affirmative statement and its negation is, he calls, a 'contradiction'. Examples of contradictories are 'every man is white' and 'not every man is white', 'no man is white' and 'some man is white'.
The below relations, contrary, subcontrary, subalternation, and superalternation, do hold based on the traditional logic assumption that things stated as exist. If this assumption is taken out, then these relations do not hold.
'Contrary' statements, are such that both statements cannot be true at the same time. Examples of these 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 is false 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 the contradictory opposite, and since a contradicting statement is true when its opposite is false, it follows that the opposites of contraries can both be true, but they cannot both be false. Since subcontraries are negations of universal statements, they were called 'particular' statements by the medieval logicians.
Another logical relation implied by this, though not mentioned explicitly by Aristotle, is 'alternation', consisting of 'subalternation' and 'superalternation'. Subalternation is a relation between the particular statement and the universal statement of the same quality such that the particular is implied by the universal, while superalternation is a relation between them such that the falsity of the universal is implied by the falsity of the particular. In these relations, the particular is the subaltern of the universal, which is the particular's superaltern. For example, if 'every man is white' is true, its contrary 'no man is white' is false. Therefore, the 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.
Particular statements are subcontraries. 'Some man is just' and 'some man is not just' cannot be false together.
The particular statement of one quality is the subaltern of the universal statement of that same quality, which is the superaltern of the particular statement because in Aristotelian semantics 'every is ' implies 'some is ' and 'no is ' implies 'some is not '. Note that modern formal interpretations of English sentences interpret 'every is ' as 'for any, a statement that is implies a statement that is ', which does not imply 'some is. 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 is not, then not every is. Conversely, though this is not the case in modern semantics, it was thought that if every is not, some is not. This interpretation has caused difficulties. While Aristotle's Greek does not represent the particular negative as 'some is not, but as 'not every is ', someone in his commentary on the Peri Hermaneias, renders the particular negative as 'quoddam A nōn est ', literally 'a certain is not a ', and in all medieval writing on logic it is customary to represent the particular proposition in this way.

These relationships became the basis of a diagram drawn by Boethius and used by medieval logicians to classify the logical relationships. The propositions are placed in the four corners of a square, and the relations represented as lines drawn between them, whence the name 'Square of Opposition'. Therefore, the following cases can be made:

If is true, then is false, is true, is false;
If is true, then is false, is false, is true;
If is true, then is false, and are indeterminate;
If is true, then is false, and are indeterminate;
If is false, then is true, and are indeterminate;
If is false, then is true, and are indeterminate;
If is false, then is false, is true, is true;
If is false, then is true, is false, is true.

To memorise them, the medievals invented the following Latin rhyme:
It affirms that and are not neither both true nor both false in each of the above cases. The same applies to and. While the first two are universal statements, the couple / refers to particular ones.
The Square of Oppositions was used for the categorical inferences described by medieval logicians: conversion and obversion and contraposition. Each of those three types of categorical inference was applied to the four logical forms:,,, and.

The problem of existential import

Subcontraries, which medieval logicians represented in the form 'quoddam est ' and 'quoddam non est ' cannot both be false, since their universal contradictory statements cannot both be true. This leads to a difficulty firstly identified by Peter Abelard. 'Some is ' seems to imply 'something is ', in other words, there exists something that is. For example, 'Some man is white' seems to imply that at least one thing that exists is a man, namely the man who has to be white, if 'some man is white' is true. But, 'some man is not white' also implies that something as a man exists, namely the man who is not white, if the statement 'some man is not white' is true. But Aristotelian logic requires that, necessarily, one of these statements is true, i.e., they cannot both be false. Therefore, since both statements imply the presence of at least one thing that is a man, the presence of a man or men is followed. But, as Abelard points out in the Dialectica, surely men might not exist?
Abelard also points out that subcontraries containing subject terms denoting nothing, such as 'a man who is a stone', are both false.
Terence Parsons argues that ancient philosophers did not experience the problem of existential import as only the A and I forms had existential import.
He goes on to cite a medieval philosopher William of Ockham,
And points to Boethius' commentaryof Aristotle's work as giving rise to the mistaken notion that the form has existential import.

Modern squares of opposition

In the 19th century, George Boole argued for requiring existential import on both terms in particular claims, but allowing all terms of universal claims to lack existential import. This decision made Venn diagrams particularly easy to use for term logic. The square of opposition, under this Boolean set of assumptions, is often called the modern square of opposition. In the modern square of opposition, and claims are contradictories, as are and, but all other forms of opposition cease to hold; there are no contraries, subcontraries, subalternations, and superalternations. Thus, from a modern point of view, it often makes sense to talk about 'the' opposition of a claim, rather than insisting, as older logicians did, that a claim has several different opposites, which are in different kinds of opposition with the claim.
Gottlob 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.
Algirdas Julien Greimas ' semiotic square was derived from Aristotle's work.
The traditional square of opposition is now often compared with squares based on inner- and outer-negation.

Logical hexagons and other bi-simplexes

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. 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 ". The pattern also goes even beyond this.

Square of opposition (or logical square) and modal logic

The logical square, also called square of opposition or square of Apuleius, has its origin in the four marked sentences to be employed in syllogistic reasoning: "Every man is bad," the universal affirmative – The negation of the universal affirmative "Not every man is bad" – "Some men are bad," the particular affirmative – and finally, the negation of the particular affirmative "No man is bad". Robert Blanché published with Vrin his Structures intellectuelles in 1966 and since then many scholars think that the logical square or square of opposition representing four values should be replaced by the logical hexagon which by representing six values is a more potent figure because it has the power to explain more things about logic and natural language.

Set-theoretical interpretation of categorical statements

In modern mathematical logic, statements containing words "all", "some" and "no", can be stated in terms of set theory if we assume a set-like domain of discourse. If the set of all 's is labeled as and the set of all 's as, then:

"All is " is equivalent to " is a subset of ", or.
"No is " is equivalent to "The intersection of and is empty", or.
"Some is " is equivalent to "The intersection of and is not empty", or.
"Some is not " is equivalent to " is not a subset of ", or.

By definition, the empty set is a subset of all sets. From this fact it follows that, according to this mathematical convention, if there are no 's, then the statements "All is " and "No is " are always true whereas the statements "Some is " and "Some is not " are always false. This also implies that AaB does not entail AiB, and some of the syllogisms mentioned above are not valid when there are no 's.