# Who first studied asymmetric relations qua relation?

Who first asymmetric relations qua relation, viz., the fact that

A relates to B by some relation

does not always/necessarily imply that

B relates to A by the same relation.

?

• I think the term you're looking for is 'symmetry', not 'commutativity'. – Eliran Jul 5 '19 at 19:35
• @Eliran Thanks. I changed it to "asymmetric relations". – Geremia Jul 5 '19 at 20:10
• This is too vague, I am afraid. Euclid's Elements are filled with propositions about "greater", lines, angles, triangles, and "greater" is obviously asymmetric. Relations as a thing were only introduced into logic by de Morgan, Peirce studied them extensively, see Whelden, The Origins and Use of the Theory of Relations. Most of their examples are asymmetric, but it is not like symmetric ones were singled out and studied before. – Conifold Jul 5 '19 at 21:03
• @Conifold Those are specific asymmetric relations, but I'm interested in who first studied asymmetric relations qua relation. – Geremia Jul 5 '19 at 22:35
• Unfortunately "qua relation" does not really help, but look through Wikipedia's Relation (history of concept), see what fits. One does not really see much done with this sort of abstracting until mid-19th century. – Conifold Jul 5 '19 at 22:53

The Categories, chap.7 /On Relatives/ contains a remarkable discussion in just a few pages and its author, supposedly Aristotle, might well be the first to have 'studied' asymmetric relations. So Aristotle distinguishes contrariety and reciprocation, adding further consideration on simultaneity.

All relatives are 'reciprocated', but no all of them have 'contraries'; simultaneity is optional.

7a22 All relatives, then, if properly defined, have a correlative.

this is the conclusion of a short discussion but earlier it is noted:

6b16 virtue has a contrary, vice, these both being relatives; knowledge, too, has a contrary, ignorance. But this is not the mark of all relatives; 'double' and 'triple' have no contrary, nor indeed has any such term.

So it seems that logical inversion generates symmetrical relations while the rest do not have this property. The contrary of ignorance is knowledge and its contrary is ignorace. 'Master of' and 'slave of' are reciprocal.

Ref: Pamela M. Hood, Aristotle on the Category of Relation 2004

William Rowan Hamilton's discovery of quaternions in the 19th century may be the first studied non-commutative relations. Here is Wikipedia:

In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative.

Others may have anticipated this discovery:

Important precursors to this work included Euler's four-square identity (1748) and Olinde Rodrigues' parameterization of general rotations by four parameters (1840), but neither of these writers treated the four-parameter rotations as an algebra. Carl Friedrich Gauss had also discovered quaternions in 1819, but this work was not published until 1900.

One may also look at the symmetry of subject and predicate in the syllogism. According to Irving Copi if one can interchange subject and predicate and maintain validity this would be an immediate inference: (page 190)

It is called conversion and is perfectly valid in the case of E and I propositions.

Identifying where conversion does not work could be viewed as a study of assymetry between subject and predicate.

Copi, I. M. Introduction to Logic. 1982. Macmillan.

Wikipedia contributors. (2019, June 16). Quaternion. In Wikipedia, The Free Encyclopedia. Retrieved 20:09, July 5, 2019, from https://en.wikipedia.org/w/index.php?title=Quaternion&oldid=902089865

• Relation is one of Aristotle's categories (I'm not restricting relation to mathematical logic here). I'm sure asymmetric relations were studied before the 19th century. – Geremia Jul 5 '19 at 20:23
• @Geremia There may be people who earlier studied such relationships. I am only listed the one I am aware of. – Frank Hubeny Jul 5 '19 at 20:25
• @Geremia I've added a reference to conversion in syllagisms. When conversion does not work this could be a form of assymetry. – Frank Hubeny Jul 5 '19 at 20:43
• Oh, yes, I forgot about those terms: •simple conversion •conversion per accidens •metathesis (interchanging the premises) •obversion •convertio syllogism (indirect conversion) – Geremia Jul 5 '19 at 20:48