Square of opposition

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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 are logically related ('opposed') to each of the others.

1 Summary
2 The problems of existential import
3 The modern square of opposition
4 See also
5 External links

[edit] 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.

Every 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 'omnis S est P', usually translated as 'every S is P'.

The 'E' proposition, the universal negative (universalis negativa), Latin form 'nullus S est P', usually translated as 'no S is P'.

The 'I' proposition, the particular affirmative (particularis affirmativa), Latin 'quidam S est P', usually translated as 'some S is P'.

The 'O' proposition, the particular negative (particularis negativa), Latin 'quidam S non est P', usually translated as 'some S is not P'.

In tabular form:

The Four Aristotelian Propositions
Name	Symbol	Latin	English
Universal affirmative	A	Omnis S est P	Every S is P
Universal negative	E	Nullus S est P	No S is P
Particular affirmative	I	Quidam S est P	Some S is P
Particular negative	O	Quidam S non est P	Some S is not P

Aristotle states (in chapters six and seven of the Perihermaneias (Latin De Interpretatione, English 'On Exposition'), that there are certain logical relationships between these four kinds of proposition.

According to Aristotle there are four distinct kinds of opposition between pairs of statements. Pairs of propositions are called contradictories (contradictoriae) when they cannot at the same time both be true or both be false, contraries (contrariae) when both cannot at the same time be true, subcontraries (subcontrariae) when both cannot at the same time be false, and subalternates (subalternae) when the truth of the one proposition implies the truth of the other, but not conversely. The corresponding relations are known as contradiction (contradictio), contrariety (contrarietas), subcontrariety (subcontrarietas) and subalternation (subalternatio). Thus:

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 not imply 'some x is A'. This is a matter of semantics, however.
The universal affirmative and the particular negative are contradictories. Clearly 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 diffculties (see below). Note that Aristotle's Greek does not represent the particular negative as 'some A is not B', but as 'not every A is B', and Boethius' Latin faithfully follows him. However, in Boethius' commentary on the Perihermaneias, he renders the particular negative as 'quidam 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 these possible logical relations. 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'.

[edit] The problems of existential import

Aristotle assumes that all terms used in a term logic will have non-empty references, that we won't even attempt to reason formally about non-existent things. Thus in a strongly Aristotelian version of term logic, you can infer from the claim "No dogs are tygers" to the claim "Tygers exist." A term is said to have "existential import" if use of the term implies the existence of the thing that the term refers to. For early Aristotelian logics, all terms have existential import in all formal uses. However medieval Christians often wanted to be able to talk formally about something without covertly assuming that it already existed. Consider the theological claim "God knew that mankind would fall from grace prior to creating mankind." If we used an early Aristotelian logic to think about this we would get the absurd result that mankind existed before mankind was created. But as soon as one loosens the restriction that all terms have existential import, some of the inferences embodied in the traditional square of opposition cease to hold. "No unicorns are mammals" and "All unicorns are mammals" are not contraries, because they might both be true if no unicorns exist at all. Logicians working in the term logic tradition tried a lot of different compromises for how to get existential import to work. Perhaps we require existential import on all terms in particular claims, but not terms in universal claims. Perhaps we require subject terms to have existential import, but not predicate terms. Perhaps we require that the subject terms of particular claims have existential import but no others. Perhaps we require no terms to have existential import (now called a free logic). Each decision about which terms are required to have existential import will generate slightly different versions of the square of opposition. Note that if existential import holds on the A and I propositions but not the E and O propositions, the traditional square and all its inferences hold without inconsistency, although then obversion and controposition fail in some cases. The following table lists some approaches to existential import and their consequences:

Approaches to Existential Import
Name	Existential Symbols	Empty terms	Traditional Inferences	Contraposition / Obversion
Aristotlean	AI	Yes	Yes	No
Boolean	IO	Yes	No	Yes
Medieval?	AIEO	No	Yes	Yes
Free Logic	none	Yes	No	Yes

[edit] The modern square of opposition

In the 19th century, George Boole argues 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 winds up making Venn diagrams particularly easy to use for term logic, and becomes quite popular. 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.