- May 2020
R is transitive iff ∀∀\forall x ∀∀\forall y ∀∀\forall z [xRy ∧ yRz → xRz].
For any x, y, and z in a set, if x is related to y and y is related to z, then z must also be related to z for the relation to be transitive.
We have the set [1 2 3].
For a relation to be transitive, if we have (1 2) and (2 3), we must have (1 3). Likewise, any reflexive pair is transitive.
Additionally, if we do not have xRy and yRz (even if x, y, or z are the same element), the relation is vacuously transitive.
- Feb 2020
Image Credit: Detail from "The School of Athens" by Raffaello Sanzio da Urbino (c. 1509–1511).
Euclid's common notions appear to be grounds for many of Marx's arguments in Ch. 1, but also throughout the book.
Near the beginning of Ch. 1 of the Elements Euclid lists them [PDF]:
- Things that are equal to the same thing are also equal to one another (the Transitive property of a Euclidean relation).
- If equals are added to equals, then the wholes are equal (Addition property of equality).
- If equals are subtracted from equals, then the differences are equal (Subtraction property of equality).
- Things that coincide with one another are equal to one another (Reflexive property).
- The whole is greater than the part.
Regarding the fifth, also see Aristotle, Metaphysics 8.6 [=1045a]; Topics 6.13 (=150a15-16);
On the concept of the "whole-before-the-parts" (along with the "whole of the parts" and the "whole in the part"), also see Proclus, El. Theol., prop. 67.
- Aug 2016
You have another function g that takes a B and returns a C. You can compose them by passing the result of f to g. You have just defined a new function that takes an A and returns a C.
That clears that. Arrow A->B and arrow B->C are not necessarily the same arrow (function).
Okay, so what are we doing here? Category is a set (?) of objects with arbitrary functions between them. The only property that this needs to satisfy is one of transitivity.