Square of opposition

Square of opposition
In the Venn diagrams, black areas are empty and red areas are nonempty.
The faded arrows and faded red areas apply in traditional logic.

Depiction from the 15th century

In the system of Aristotelian logic, the square of opposition is a diagram representing the different ways in which each of the four propositions of the system is logically related ('opposed') to each of the others. The system is also useful in the analysis of syllogistic logic, serving to identify the allowed logical conversions from one type to another.

Summary

In traditional logic, a proposition (Latin: propositio) is a spoken assertion (oratio enunciativa), 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. These are:

The so-called 'A' proposition, the universal affirmative (universalis affirmativa), whose form in Latin is 'omne S est P', usually translated as 'every S is a P'.
The 'E' proposition, the universal negative (universalis negativa), Latin form 'nullum S est P', usually translated as 'no S are P'.
The 'I' proposition, the particular affirmative (particularis affirmativa), Latin 'quoddam S est P', usually translated as 'some S are P'.
The 'O' proposition, the particular negative (particularis negativa), Latin 'quoddam S non est P', usually translated as 'some S are not P'.

In tabular form:

The Four Aristotelian Propositions
Name	Symbol	Latin	English
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.
Particular negative	O	Quoddam S non est P.	Some S is not P.

Aristotle states (in chapters six and seven of the Peri hermaneias (Περὶ Ἑρμηνείας, Latin De 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 affirmative and negative statements he calls a 'contradiction' (in medieval Latin, contradictio). Examples of contradictories are 'every man is white' and 'not every man is white', 'no man is white' and 'some man is white'.

'Contrary' (medieval: contrariae) statements, are such that both cannot at the same time be true. 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 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, 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 opposition implied by this, though not mentioned explicitly by Aristotle, is 'alternation' (alternatio), consisting of 'subalternation' and 'superalternation'. Alternation is a relation between a particular statement and a universal statement of the same quality such that the particular is implied by the other. 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'.^[1]^[2]

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 'every A is B' implies 'some A is B' and 'no A is B' implies 'some A is not 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 not 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 not represent the particular negative as 'some A is not B', but as 'not every A is B', someone in his commentary on the Peri hermaneias, renders the particular negative as 'quoddam A non est B', literally 'a certain A is not a B', 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 originating with 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 'The Square of Opposition'.

The problem of existential import

Subcontraries, which medieval logicians represented in the form 'quoddam A est B' (some particular A is B) and 'quoddam A non est B' (some particular A is not B) cannot both be false, since their universal contradictory statements (every A is B / no A is B) cannot both be true. This leads to a difficulty that was first identified by Peter Abelard. 'Some A is B' seems to imply 'something is A'. For example 'Some man is white' seems to imply that at least one thing 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 seems to imply that something is a man, namely the man who is not white if 'some man is not white' is true. But Aristotelian logic requires that necessarily one of these statements is true. Both cannot be false. Therefore (since both imply that something is a man) it follows that necessarily something is a man, i.e. men exist. But (as Abelard points out, in the Dialectica) surely men might not exist?^[3]

For with absolutely no man existing, neither the proposition 'every man is a man' is true nor 'some man is not a man'.^[4]

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 conversion per 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.^[5]

Terence Parsons argues that ancient philosophers did not experience the problem of existential import as only the A and I forms had existential import.

Affirmatives have existential import, and negatives do not. The ancients thus did not see the incoherence of the square as formulated by Aristotle because there was no incoherence to see.^[6]

He goes on to cite medieval philosopher William of Ockham

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.^[7]

And points to Boethius' translation of Aristotle's work as giving rise to the mistaken notion that the O form has existential import.

But when Boethius 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.^[8]

Modern squares of opposition

Frege's square of opposition
The conträr below is an erratum:
It should read subconträr

In the 19th century, George Boole argued for requiring existential import on both terms in particular claims (I and O), but allowing all terms of universal claims (A and E) 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, A and O claims are contradictories, as are E and I, but all other forms of opposition cease to hold; there are no contraries, subcontraries, or subalterns. 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.

Logical hexagons and other bi-simplexes

Main article: Logical hexagon

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.^[9] 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.^[10]

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 white, the universal affirmative and its negation Not every man is white (or Some men are not white), the particular negative on the one hand, Some men are white, the particular affirmative and its negation No man is white, the universal negative on the other. 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.

References

↑ Parry & Hacker, Aristotelian Logic (SUNY Press, 1990), p. 158.
↑ Cohen & Nagel, Introduction to Logic Second Edition (Hackett Publishing, 1993), p. 55.
↑ In his Dialectica, and in his commentary on the Perihermaneias
↑ Re enim hominis prorsus non existente neque ea vera est quae ait: omnis homo est homo, nec ea quae proponit: quidam homo non est homo
↑ 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
↑ in The Traditional Square of Opposition in the Stanford Encyclopedia of Philosophy
↑ (SL I.72) Loux 1974, 206
↑ The Traditional Square of Opposition
↑ N-Opposition Theory Logical hexagon
↑ Moretti, Pellissier

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Mathematical logic

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Traditional logic	Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram

Propositional calculus Boolean logic	Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables

Predicate logic	First-order Quantifiers Predicate Second-order Monadic predicate calculus

Naive set theory	Set Empty set Enumeration Extensionality Finite set Infinite set Subset Power set Countable set Uncountable set Recursive set Domain Range Map Function Relation Ordered pair

Set theory	Foundations of mathematics Zermelo–Fraenkel set theory Axiom of choice General set theory Kripke–Platek set theory Von Neumann–Bernays–Gödel set theory Morse–Kelley set theory Tarski–Grothendieck set theory

Model theory	Model Interpretation Non-standard model Finite model theory Truth value Validity

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