Square of opposition

In term logic (a branch of philosophical 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 by Apuleius and Boethius.

The $A$ 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 $S$ nōn est $P$ ', usually translated as 'some $S$ are not $P$ '.

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 ( $S$ ) and predicate ( $P$ ), in which the predicate is either asserted or denied of the subject.

Every categorical proposition can be reduced to one of four logical forms, named $A$ , $E$ , $I$ , and $O$ based on the Latin affirmo (I affirm), for the affirmative propositions $A$ and $I$ , and nego (I deny), for the negative propositions $E$ and $O$ . These are:

In tabular form:

*Proposition $A$ may be stated as "All $S$ is $P$ ." However, Proposition $E$ when stated correspondingly as "All $S$ is not $P$ ." is ambiguous^[2] because it can be either an $E$ or $O$ proposition, thus requiring a context to determine the form; the standard form "No $S$ is $P$ " is unambiguous, so it is preferred. Proposition $O$ also takes the forms "Sometimes $S$ is not $P$ ." and "A certain $S$ is not $P$ ." (literally the Latin 'Quoddam $S$ nōn est $P$ .')

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

Aristotle states (in chapters six and seven of the Peri hermēneias (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 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 as $S$ (or things satisfying a statement $S$ 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 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 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 the contrapositive 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:

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'. Therefore, the following cases can be made:^[6]

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

It affirms that $A$ and $E$ are not neither both true nor both false in each of the above cases. The same applies to $I$ and $O$ . While the first two are universal statements, the couple $I$ / $O$ refers to particular ones.

The Square of Oppositions was used for the categorical inferences described by the Greek philosopher Aristotle: conversion, obversion and contraposition. Each of those three types of categorical inference was applied to the four Boethian logical forms: $A$ , $E$ , $I$ , and $O$ .

Square of opposition (or logical square) and modal logic[edit]

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" (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 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.

"All $A$ is $B$ " (AaB) is equivalent to " $s(A)$ is a of $s(B)$ ", or $s(A)\subseteq s(B)$ .

subset

"No $A$ is $B$ " (AeB) is equivalent to "The of $s(A)$ and $s(B)$ is empty", or $s(A)\cap s(B)=\emptyset$ .

intersection

"Some $A$ is $B$ " (AiB) is equivalent to "The intersection of $s(A)$ and $s(B)$ is not empty", or $s(A)\cap s(B)\neq \emptyset$ .

"Some $A$ is not $B$ " (AoB) is equivalent to " $s(A)$ is not a subset of $s(B)$ ", or $s(A)\nsubseteq s(B)$ .

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 $A$ 's is labeled as $s(A)$ and the set of all $B$ 's as $s(B)$ , then:

By definition, the empty set $\emptyset$ is a subset of all sets. From this fact it follows that, according to this mathematical convention, if there are no $A$ 's, then the statements "All $A$ is $B$ " and "No $A$ is $B$ " are always true whereas the statements "Some $A$ is $B$ " and "Some $A$ is not $B$ " 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 $A$ 's ( $s(A)=\emptyset$ ).

Boole's syllogistic

Free logic

Logical cube

Logical hexagon

Octagon of Prophecies

Triangle of opposition

Parsons, Terence. . In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.

"The Traditional Square of Opposition"

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

with links to video and digitized manuscript LJS 101 of Boethius' Latin translation of Aristotle's De interpretatione, which contains 9th/11th century copies (some with color) of Aristotle's square of opposition on leaves 36r and 36v.