Revise sort.md and docstrings in sort.jl (#48363)

Co-authored-by: Jeremie Knuesel <knuesel@gmail.com>
(cherry picked from commit a134076)

base/sort.jl

@@ -2013,9 +2013,9 @@ struct MergeSortAlg     <: Algorithm end
 """
     PartialQuickSort{T <: Union{Integer,OrdinalRange}}
 
-Indicate that a sorting function should use the partial quick sort
-algorithm. Partial quick sort returns the smallest `k` elements sorted from smallest
-to largest, finding them and sorting them using [`QuickSort`](@ref).
+Indicate that a sorting function should use the partial quick sort algorithm.
+`PartialQuickSort(k)` is like `QuickSort`, but is only required to find and
+sort the elements that would end up in `v[k]` were `v` fully sorted.
 
 Characteristics:
   * *not stable*: does not preserve the ordering of elements that
@@ -2024,7 +2024,7 @@ Characteristics:
   * *in-place* in memory.
   * *divide-and-conquer*: sort strategy similar to [`MergeSort`](@ref).
 
-  Note that `PartialQuickSort(k)` does not necessarily sort the whole array. For example,
+Note that `PartialQuickSort(k)` does not necessarily sort the whole array. For example,
 
 ```jldoctest
 julia> x = rand(100);

doc/src/base/sort.md

@@ -1,7 +1,7 @@
 # Sorting and Related Functions
 
-Julia has an extensive, flexible API for sorting and interacting with already-sorted arrays of
-values. By default, Julia picks reasonable algorithms and sorts in standard ascending order:
+Julia has an extensive, flexible API for sorting and interacting with already-sorted arrays
+of values. By default, Julia picks reasonable algorithms and sorts in ascending order:
 
 ```jldoctest
 julia> sort([2,3,1])
@@ -11,7 +11,7 @@ julia> sort([2,3,1])
  3
 ```
 
-You can easily sort in reverse order as well:
+You can sort in reverse order as well:
 
 ```jldoctest
 julia> sort([2,3,1], rev=true)
@@ -36,8 +36,8 @@ julia> a
  3
 ```
 
-Instead of directly sorting an array, you can compute a permutation of the array's indices that
-puts the array into sorted order:
+Instead of directly sorting an array, you can compute a permutation of the array's
+indices that puts the array into sorted order:
 
 ```julia-repl
 julia> v = randn(5)
@@ -65,7 +65,7 @@ julia> v[p]
   0.382396
 ```
 
-Arrays can easily be sorted according to an arbitrary transformation of their values:
+Arrays can be sorted according to an arbitrary transformation of their values:
 
 ```julia-repl
 julia> sort(v, by=abs)
@@ -101,9 +101,12 @@ julia> sort(v, alg=InsertionSort)
   0.382396
 ```
 
-All the sorting and order related functions rely on a "less than" relation defining a total order
-on the values to be manipulated. The `isless` function is invoked by default, but the relation
-can be specified via the `lt` keyword.
+All the sorting and order related functions rely on a "less than" relation defining a
+[strict weak order](https://en.wikipedia.org/wiki/Weak_ordering#Strict_weak_orderings)
+on the values to be manipulated. The `isless` function is invoked by default, but the
+relation can be specified via the `lt` keyword, a function that takes two array elements
+and returns `true` if and only if the first argument is "less than" the second. See
+[`sort!`](@ref) and [Alternate Orderings](@ref) for more information.
 
 ## Sorting Functions
 
@@ -165,22 +168,17 @@ Base.Sort.defalg(::AbstractArray{<:Union{SmallInlineStrings, Missing}}) = Inline
 
 ## Alternate Orderings
 
-By default, `sort` and related functions use [`isless`](@ref) to compare two
-elements in order to determine which should come first. The
-[`Base.Order.Ordering`](@ref) abstract type provides a mechanism for defining
-alternate orderings on the same set of elements: when calling a sorting function like
-`sort`, an instance of `Ordering` can be provided with the keyword argument `order`.
-
-Instances of `Ordering` define a [total order](https://en.wikipedia.org/wiki/Total_order)
-on a set of elements, so that for any elements `a`, `b`, `c` the following hold:
-
-* Exactly one of the following is true: `a` is less than `b`, `b` is less than
-  `a`, or `a` and `b` are equal (according to [`isequal`](@ref)).
-* The relation is transitive - if `a` is less than `b` and `b` is less than `c`
-  then `a` is less than `c`.
-
-The [`Base.Order.lt`](@ref) function works as a generalization of `isless` to
-test whether `a` is less than `b` according to a given order.
+By default, `sort`, `searchsorted`, and related functions use [`isless`](@ref) to compare
+two elements in order to determine which should come first. The
+[`Base.Order.Ordering`](@ref) abstract type provides a mechanism for defining alternate
+orderings on the same set of elements: when calling a sorting function like
+`sort!`, an instance of `Ordering` can be provided with the keyword argument `order`.
+
+Instances of `Ordering` define an order through the [`Base.Order.lt`](@ref)
+function, which works as a generalization of `isless`.
+This function's behavior on custom `Ordering`s must satisfy all the conditions of a
+[strict weak order](https://en.wikipedia.org/wiki/Weak_ordering#Strict_weak_orderings).
+See [`sort!`](@ref) for details and examples of valid and invalid `lt` functions.
 
 ```@docs
 Base.Order.Ordering