| 17744 | De Morgan started the study of relations and their properties [De Morgan, by Walicki] |
| Full Idea: De Morgan started the sustained interest in the study of relations and their properties. | |
| From: report of Augustus De Morgan (On the Syllogism IV [1859]) by Michal Walicki - Introduction to Mathematical Logic History D.1.1 |
| 13501 | De Morgan found inferences involving relations, which eluded Aristotle's syllogistic [De Morgan, by Hart,WD] |
| Full Idea: There was a prejudice against relations (in favour of properties) but De Morgan and others that impeccable inferences turn on relations and elude Aristotle's syllogistic. Thus: All horses are animals. Hence, all heads of horses are heads of animals. | |
| From: report of Augustus De Morgan (On the Syllogism IV [1859]) by William D. Hart - The Evolution of Logic 4 | |
| A reaction: This is actually an early example of modern analytic philosophy in action. You start with the inferences, and then work back to the ontology and the definition of concepts. But in pinning down such concepts, do we miss their full meaning? |
| 19238 | The logic of relatives relies on objects built of any relations (rather than on classes) [Peirce] |
| Full Idea: In the place of the class ...the logic of relatives considers the system, which is composed of objects brought together by any kind of relations whatsoever. | |
| From: Charles Sanders Peirce (Reasoning and the Logic of Things [1898], III) | |
| A reaction: Peirce's logic of relations might support the purely structural view of reality defended by Ladyman and Ross. Modern logic standardly expresses its semantics in terms of set theory. Peirce pioneered relations in logic. |
| 8492 | Relations are functions with two arguments [Frege] |
| Full Idea: Functions of one argument are concepts; functions of two arguments are relations. | |
| From: Gottlob Frege (Function and Concept [1891], p.39) | |
| A reaction: Nowadays we would say 'two or more'. Another interesting move in the aim of analytic philosophy to reduce the puzzling features of the world to mathematical logic. There is, of course, rather more to some relations than being two-argument functions. |
| 10036 | In 'Principia' a new abstract theory of relations appeared, and was applied [Russell/Whitehead, by Gödel] |
| Full Idea: In 'Principia' a young science was enriched with a new abstract theory of relations, ..and not only Cantor's set theory but also ordinary arithmetic and the theory of measurement are treated from this abstract relational standpoint. | |
| From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Kurt Gödel - Russell's Mathematical Logic p.448 | |
| A reaction: I presume this is accounting for relations in terms of ordered sets. |
| 21698 | All relations, apart from ancestrals, can be reduced to simpler logic [Quine] |
| Full Idea: Much of the theory of relations can be developed as a virtual theory, in which we seem to talk of relations, but can explain our notation in terms {finally] of just the logic of truth-functions, quantification and identity. The exception is ancestrals. | |
| From: Willard Quine (Lecture on Nominalism [1946], §8) | |
| A reaction: The irreducibility of ancestrals is offered as a reason for treating sets as universals. |
| 10816 | We can use mereology to simulate quantification over relations [Lewis] |
| Full Idea: We can simulate quantification over relations using megethology. Roughly, a quantifier over relations is a plural quantifier over things that encode ordered pairs by mereological means. | |
| From: David Lewis (Mathematics is Megethology [1993], p.18) | |
| A reaction: [He credits this idea to Burgess and Haven] The point is to avoid second-order logic, which quantifies over relations as ordered n-tuple sets. |
| 8525 | Relations need terms, so they must be second-order entities based on first-order tropes [Campbell,K] |
| Full Idea: Because there cannot be relations without terms, in a meta-physic that makes first-order tropes the terms of all relations, relational tropes must belong to a second, derivative order. | |
| From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §8) | |
| A reaction: The admission that there could be a 'derivative order' may lead to trouble for trope theory. Ostrich Nominalists could say that properties themselves are derivative second-order abstractions from indivisible particulars. Russell makes them first-order. |
| 9926 | A relation is either a set of sets of sets, or a set of sets [Burgess/Rosen] |
| Full Idea: While in general a relation is taken to be a set of ordered pairs <u, v> = {{u}, {u, v}}, and hence a set of sets of sets, in special cases a relation can be represented by a set of sets. | |
| From: JP Burgess / G Rosen (A Subject with No Object [1997], II.C.1.a) | |
| A reaction: [See book for their examples, which are <, symmetric, and arbitrary] The fact that a relation (or anything else) can be represented in a certain way should never ever be taken to mean that you now know what the thing IS. |
| 9561 | The mathematics of relations is entirely covered by ordered pairs [Chihara] |
| Full Idea: Everything one needs to do with relations in mathematics can be done by taking a relation to be a set of ordered pairs. (Ordered triples etc. can be defined as order pairs, so that <x,y,z> is <x,<y,z>>). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.2) | |
| A reaction: How do we distinguish 'I own my cat' from 'I love my cat'? Or 'I quite like my cat' from 'I adore my cat'? Nevertheless, this is an interesting starting point for a discussion of relations. |
| 23627 | 'Before' and 'after' are not two relations, but one relation with two orders [Hossack] |
| Full Idea: The reason the two predicates 'before' and 'after' are needed is not to express different relations, but to indicate its order. Since there can be difference of order without difference of relation, the nature of relations is not the source of order. | |
| From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 10.3) | |
| A reaction: This point is to refute Russell's 1903 claim that order arises from the nature of relations. Hossack claims that it is ordered series which are basic. I'm inclined to agree with him. |