Equivalence of class, as you say, relates to all subsets of S having the same equivalence relation (~): Taking X to be the set of all polygons, ~ to be the equivalent relation of "same number of sides" then one set would be triangles and another pentagons (among others), both of which can be identified with all polygons with the same number of sides:Coëmgenu wrote: ↑Thu Apr 22, 2021 2:06 pm 4 = 2+2
The above is the sort of equivalency you are talking about, where one side of the literal "equation" is completely identical to, exhaustively corresponding to, the other side of the equation and each side is interchangeable without loss of meaning. For instance here 2+2=4 is the same as 4=2+2 which surprises no one. However, there are also classes of equivalency. For instance, if B is the set of all balls, and ~ is the equivalence relation "is coloured identically", in that case, one particular equivalence class consists of all of the red balls, and B/~ could be identified with the set of all ball colours. The only equivalency relation that you want admitted is "is (utterly) equal to." That's fine. It's a normal usage of "equates to," but I don't think that is what the poster meant.
X/~
Whilst this is a mathematical equivalence of groups, it does not help us much here. Consider again our case above. All triangles in the subset of S would be equivalent in terms of having the same number of sides and so we can make the following statement:
"Triangles have the same number of sides"
The predicate here is describing the subject "triangle", thus standing in relation to it. We can't however say that the predicate and subject are equivalent, as in synonyms, since "triangle" and "same number of sides" do not share the same meaning. If "triangle" meant the same as "same number of sides" then we should be able to do the following:
"Squares have the same number of sides"
"Squares are triangles"
We can't do this because "triangle" and "same number of sides" are not interchangeable. If we change them the meaning is also altered. Consider further still differences within a class:
"This triangle has three 60° angles"
Here the predicate of "60°angles" is modifying the noun "triangle". If they were equivalent then we could plug in "three 60°angles" instead of "equilateral triangle" without any loss of meaning for any triangle. However, the two are not equivalent despite being within the same equivalence class since not all triangles are equilateral. Now, "equilateral" and "three 60° angles" are equivalent. We can interchange them without any loss of meaning:
"This triangle has three 60° angles"
"This triangle is equilateral"
No meaning has been lost because the words are fully interchangeable. It's important to note that this only relates to the predicate and not the subject here. Returning to our equivalent classes we have the subject "polygons" with the predicate "same number of sides" with a subset of "3 sided shape" which is a synonym for "triangle". Taking triangle as the subject, all sub-classes of 3 sided shapes would also be triangles thus being tautological and equivalent, but the adjectival predicate of each triangle within that sub-set would be different. We would also have "3 sided shape" being non-interchangeable with the set "polygons". Therefore, in any given statement which is synthetic the subject and predicate can never be equivalent, i.e. synonyms, since the predicate is modifying the noun rather than being interchangeable with it. Returning to our subject at hand, whilst kāyo rūpī can belong to an equivalent class we still cannot say that kāyo and rūpī are equivalent. Just like how "3 sided polygon" is not equivalent to "polygon", so to "kāyo rūpī" is not equivalent to "kāyo"