* One increment would be to add aliasing or passing by reference, as illustrated above. In the illustration, we relied on the combination of passing by reference and mutation to demonstrate how you could get different behavior depending on whether an argument was passed to a function by reference or instead passed in the more familiar way (called "passing by value"). However, it would be possible to have passing by reference in a language without having mutation. For it to make any difference whether an argument is passed by reference or by value, such a language would have to have some primitive predicates which are sensitive to whether their arguments are aliased or not. In Jim's paper linked above, he calls such predicates "hyper-evaluative."
- The simplest such predicate we might call "hyperequals": `y hyperequals w` should evaluate to true only when the arguments `y` and `w` are aliased.
+ The simplest such predicate we might call "hyperequals": `y hyperequals w` should evaluate to true when and only when the arguments `y` and `w` are aliased.
* Another increment would be to add implicit-style mutable variables, as we explained above. You could do this with or without also adding passing-by-reference.
Isn't this interesting? Intuitively, elsewhere in math, you might think that qualitative indicernibility always suffices for numerical identity. Well, perhaps this needs discussion. In some sense the imaginary numbers ι and -ι are qualitatively indiscernible, but numerically distinct. However, arguably they're not *fully* qualitatively indiscernible. They don't both bear all the same relations to ι for instance. But then, if we include numerical identity as a relation, then `ycell` and `xcell` don't both bear all the same relations to `ycell`, either. Yet there is still a useful sense in which they can be understood to be qualitatively equal---at least, at a given stage in a computation.
- Terminological note: in OCaml, `=` and `<>` express the qualitative (in)discernibility relations, also expressed in Scheme with `equal?`. In OCaml, `==` and `!=` express the numerical (non)identity relations, also expressed in Scheme with `eq?`. `=` also has other syntactic roles in OCaml, such as in the form `let x = value in ...`. In other languages, like C and Python, `=` is commonly used just for assignment (of either of the sorts we've now seen: `let x = value in ...` or `change x to value in ...`). The symbols `==` and `!=` are commonly used to express qualitative (in)discernibility in these languages. Python expresses numerical (non)identity with `is` and `is not`. What an unattractive mess. Don't get me started on Haskell (qualitative non-identity is `/=`) and Lua (physical (non)identity is `==` and `~=`).
+ Terminological note: in OCaml, `=` and `<>` express the qualitative (in)discernibility relations, also expressed in Scheme with `equal?`. In OCaml, `==` and `!=` express the numerical (non)identity relations, also expressed in Scheme with `eq?`. `=` also has other syntactic roles in OCaml, such as in the form `let x = value in ...`. In other languages, like C and Python, `=` is commonly used just for assignment (of either of the sorts we've now seen: `let x = value in ...` or `change x to value in ...`). The symbols `==` and `!=` are commonly used to express qualitative (in)discernibility in these languages. Python expresses numerical (non)identity with `is` and `is not`. What an unattractive mess. Don't get me started on Haskell (qualitative discernibility is `/=`) and Lua (physical (non)identity is `==` and `~=`).
Note that neither of the equality predicates here being considered are the same as the "hyperequals" predicate mentioned above. For example, in the following (fictional) language:
in let (adder', setter') = factory 1
in ...
- Of course, in most languages you wouldn't be able to evaluate a comparison like `getter = getter'`, because in general the question whether two computations always return the same values for the same argument is not decidable. So typically languages don't even try to answer that question. However, it would still be true that `getter` and `getter'` (and `adder` and `adder'`) were extensionally equivalent.
+ Of course, in most languages you wouldn't be able to evaluate a comparison like `getter = getter'`, because in general the question whether two functions always return the same values for the same arguments is not decidable. So typically languages don't even try to answer that question. However, it would still be true that `getter` and `getter'` (and `adder` and `adder'`) were extensionally equivalent.
However, they're not numerically identical, because by calling `setter 2` (but not calling `setter' 2`) we can mutate the function value `getter` (and `adder`) so that it's *no longer* qualitatively indiscernible from `getter'` (or `adder'`).