This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Square of opposition" – news ·newspapers ·books ·scholar ·JSTOR(February 2015) (Learn how and when to remove this message)

Interm logic (a branch ofphilosophical logic), thesquare of opposition is adiagramrepresenting therelations between the four basiccategorical propositions. The origin of the square can be traced back toAristotle's tractateOn Interpretation and its distinction between two oppositions:contradiction andcontrariety. However, Aristotle did not draw any diagram; this was done several centuries later.

Intraditional logic, aproposition (Latin:propositio) is aspokenassertion (oratio enunciativa), not the meaning of an assertion, as inmodern philosophy oflanguage andlogic. Acategorical proposition is a simple proposition containing two terms, subject (S) and predicate (P), in which the predicate is either asserted or denied of the subject.

Every categorical proposition can be reduced to one of fourlogical forms, namedA,E,I, andO based on the Latinaffirmo (I affirm), for the affirmative propositionsA andI, andnego (I deny), for the negative propositionsE andO. These are:

• TheA proposition, the universal affirmative (universalis affirmativa), whose form in Latin is 'omneS estP', usually translated as 'everyS is aP'.
• TheE proposition, the universal negative (universalis negativa), Latin form 'nullumS estP', usually translated as 'noS areP'.
• TheI proposition, the particular affirmative (particularis affirmativa), Latin 'quoddamS estP', usually translated as 'someS areP'.
• TheO proposition, the particular negative (particularis negativa), Latin 'quoddamS nōn estP', usually translated as 'someS are notP'.

In tabular form:

The four Aristotelian propositions

Name	Symbol	Latin	English*	Mnemonic	Modern form[1]
Universal affirmative	A	OmneS estP.	EveryS isP. (S is alwaysP.)	affirmo (I affirm)	${\displaystyle \mathrm{\forall }x(Sx\to Px)}$
Universal negative	E	NullumS estP.	NoS isP. (S is neverP.)	nego (I deny)	${\displaystyle \mathrm{\forall }x(Sx\to \mathrm{\neg }Px)}$
Particular affirmative	I	QuoddamS estP.	SomeS isP. (S is sometimesP.)	affirmo (I affirm)	${\displaystyle \mathrm{\exists }x(Sx\wedge Px)}$
Particular negative	O	QuoddamS nōn estP.	SomeS is notP. (S is not alwaysP.)	nego (I deny)	${\displaystyle \mathrm{\exists }x(Sx\wedge \mathrm{\neg }Px)}$

*PropositionA may be stated as "AllS isP." However, PropositionE when stated correspondingly as "AllS is notP." is ambiguous^[2] because it can be either anE orO proposition, thus requiring a context to determine the form; the standard form "NoS isP" is unambiguous, so it is preferred. PropositionO also takes the forms "SomeS is notP." and "A certainS is notP." (Latin 'QuoddamS nōn estP.')

**${\displaystyle Sx}$ in the modern forms means that a statement${\displaystyle S}$ applies on an object${\displaystyle x}$. It may be simply interpreted as "${\displaystyle x}$ is${\displaystyle S}$" in many cases.${\displaystyle Sx}$ can be also written as${\displaystyle S(x)}$.

Aristotle states (in chapters six and seven of thePeri Hermēneias (Περὶ Ἑρμηνείας, LatinDe Interpretatione, English 'On Interpretation')), 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' (in medieval Latin,contradictio). Examples of contradictories are 'every man is white' and 'not every man is white' (also read as 'some men are not 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 asS (or things satisfying a statementS in modern logic) exist. If this assumption is taken out, then these relations do not hold.

'Contrary' (medieval:contrariae) 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 (its negation), and since a contradicting statement is true when its opposite is false, it follows that the opposites of contraries (which the medievals calledsubcontraries,subcontrariae) 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' (alternatio), consisting of 'subalternation' and 'superalternation'. Subalternation is a relation between the particular statement and the universal statement of the same quality (affirmative or negative) such that the particular is implied by the universal, while superalternation is a relation between them such that the falsity of the universal (equivalently the negation of the universal) is implied by the falsity of the particular (equivalently the negation of the particular).^[3] (The superalternation is thecontrapositive of the subalternation.) 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'.^[4][5]

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 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 'everyA isB' implies 'someA isB' and 'noA isB' implies 'someA is notB'. Note that modern formal interpretations of English sentences interpret 'everyA isB' as 'for anyx, a statement thatx isA implies a statement thatx isB', which doesnot imply 'somex isA'. This is a matter of semantic interpretation, however, and does not mean, as is sometimes claimed, that Aristotelian logic is 'wrong'.
• The universal affirmative (A) and the particular negative (O) are contradictories. If someA is notB, then not everyA isB. Conversely, though this is not the case in modern semantics, it was thought that if everyA is notB, someA is notB. This interpretation has caused difficulties (see below). While Aristotle's Greek does not represent the particular negative as 'someA is notB, but as 'not everyA isB', someone in his commentary on thePeri Hermaneias, renders the particular negative as 'quoddam A nōn estB', literally 'a certainA is not aB', 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 byBoethius 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:^[6]

1. IfA is true, thenE is false,I is true,O is false;
2. IfE is true, thenA is false,I is false,O is true;
3. IfI is true, thenE is false,A andO are indeterminate;
4. IfO is true, thenA is false,E andI are indeterminate;
5. IfA is false, thenO is true,E andI are indeterminate;
6. IfE is false, thenI is true,A andO are indeterminate;
7. IfI is false, thenA is false,E is true,O is true;
8. IfO is false, thenA is true,E is false,I is true.

To memorise them, the medievals invented the following Latin rhyme:^[7]

A adfirmat, negatE, sed universaliter ambae;
I firmat, negatO, sed particulariter ambae.

It affirms thatA andE are not neither both true nor both false in each of the above cases. The same applies toI andO. While the first two are universal statements, the coupleI /O refers to particular ones.

The Square of Oppositions was used for the categorical inferences described by medieval logicians:conversion andobversion andcontraposition. Each of those three types of categorical inference was applied to the four logical forms:A,E,I, andO.

Subcontraries (I andO), which medieval logicians represented in the form 'quoddamA estB' (some particularA isB) and 'quoddamA non estB' (some particularA is notB) cannot both be false, since their universal contradictory statements (noA isB / everyA isB) cannot both be true. This leads to a difficulty firstly identified byPeter Abelard (12 February 1079 – 21 April 1142). 'SomeA isB' seems to imply 'something isA', in other words, there exists something that isA. 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 (more generally 'some particularA isB' and 'some particularA is notB') 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 theDialectica, surely men might not exist?^[8]

For with absolutely no man existing, neither the proposition 'every man is a man' is true nor 'some man is not a man'.^[9]

Abelard also points out that subcontraries containing subject terms denoting nothing, such as 'a man who is a stone', are both false.

If 'every stone-man is a stone' is true, also its conversionper accidens is true ('some stones are stone-men'). But no stone is a stone-man, because neither this man nor that man etc. is a stone. But also this 'a certain stone-man is not a stone' is false by necessity, since it is impossible to suppose it is true.^[10]

Terence Parsons (1939 – 2022) argues that ancient philosophers did not experience the problem ofexistential import as only the A (universal affirmative) and I (particular affirmative) forms had existential import. (If a statement includes a term such that the statement is false if the term has no instances, i.e., no thing associated with the term exists, then the statement is said to haveexistential import with respect to that term.)

Affirmatives have existential import, and negatives do not. The ancients thus did not see the incoherence of the square as formulated byAristotle because there was no incoherence to see.^[11]

He goes on to cite a medieval philosopherWilliam of Ockham (c. 1287 – 9/10 April 1347 ),

In affirmative propositions a term is always asserted to supposit for something. Thus, if it supposits for nothing the proposition is false. However, in negative propositions the assertion is either that the term does not supposit for something or that it supposits for something of which the predicate is truly denied. Thus a negative proposition has two causes of truth.[12]

And points toBoethius' commentaryof Aristotle's work as giving rise to the mistaken notion that theO form has existential import.

But when Boethius (c. 480 – 524) comments on this text he illustrates Aristotle's doctrine with the now-famous diagram, and he uses the wording 'Some man is not just'. So this must have seemed to him to be a natural equivalent in Latin. It looks odd to us in English, but he wasn't bothered by it.[13]

In the 19th century,George Boole (November 1815 – 8 December 1864) argued for requiringexistential import on both terms in particular claims (I andO), but allowing all terms of universal claims (A andE) to lack existential import. This decision madeVenn diagrams particularly easy to use for term logic. The square of opposition, under this Boolean set of assumptions, is often called themodern square of opposition. In the modern square of opposition,A andO claims are contradictories, as areE andI, 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 (8 November 1848 – 26 July 1925)'sBegriffsschrift 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 (9 March 1917 – 27 February 1992)'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.^[14]

The square of opposition has been extended to a logical hexagon which includes the relationships of six statements. It was discovered independently by bothAugustin Sesmat (April 7, 1885 – December 12, 1957) andRobert Blanché (1898–1975).^[15] 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 dimensionn". The pattern also goes even beyond this.^[16]

The logical square, also called square of opposition or square ofApuleius, 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" (or "Some men are not 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 hisStructures intellectuelles in 1966 and since then many scholars think that the logical square or square of opposition representing four values should be replaced by thelogical 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.

In modernmathematical logic, statements containing words "all", "some" and "no", can be stated in terms ofset theory if we assume a set-like domain of discourse. If the set of allA's is labeled as${\displaystyle s(A)}$ and the set of allB's as${\displaystyle s(B)}$, then:

• "AllA isB" (AaB) is equivalent to "${\displaystyle s(A)}$ is asubset of${\displaystyle s(B)}$", or${\displaystyle s(A)\subseteq s(B)}$.
• "NoA isB" (AeB) is equivalent to "Theintersection of${\displaystyle s(A)}$ and${\displaystyle s(B)}$ isempty", or${\displaystyle s(A)\cap s(B)=\mathrm{\varnothing }}$.
• "SomeA isB" (AiB) is equivalent to "The intersection of${\displaystyle s(A)}$ and${\displaystyle s(B)}$ is not empty", or${\displaystyle s(A)\cap s(B)\ne \mathrm{\varnothing }}$.
• "SomeA is notB" (AoB) is equivalent to "${\displaystyle s(A)}$ is not a subset of${\displaystyle s(B)}$", or${\displaystyle s(A)⊈s(B)}$.

By definition, theempty set${\displaystyle \mathrm{\varnothing }}$ is a subset of all sets. From this fact it follows that, according to this mathematical convention, if there are noA's, then the statements "AllA isB" and "NoA isB" are always true whereas the statements "SomeA isB" and "SomeA is notB" 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 noA's (${\displaystyle s(A)=\mathrm{\varnothing }}$).

• Boole's syllogistic
• Free logic
• Lambda cube
• Logical cube
• Logical hexagon
• Triangle of opposition

Wikimedia Commons has media related toSquare of opposition.

• Parsons, Terence."The Traditional Square of Opposition". InZalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.
• International Congress on the Square of Opposition
• Special Issue of Logica Universalis Vol. 2 N. 1 (2008) on the Square of Opposition
• Catlogic: An open source computer script written in Ruby to construct, investigate, and compute categorical propositions and syllogisms
• Periermenias Aristotelis with links to video and digitised manuscript LJS 101 ofBoethius' Latin translation of Aristotle'sDe Interpretatione, which contains 9th/11th century copies (some with color) of Aristotle's square of opposition on leaves 36r and 36v.

• Conceptual models
• Term logic
• Inference
• Syllogism

• CS1 Italian-language sources (it)
• Articles with short description
• Short description matches Wikidata
• Articles needing additional references from February 2015
• All articles needing additional references
• Articles containing Latin-language text
• Commons category link is on Wikidata