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In the paper, they dismiss the example of "if you score 90 and above" as incorrect use of "and" (or too vague to turn into any formal logic).

However, looking at your summary, it suddenly sticks out that this issue could actually be connected with the tendency to use set manipulation. "If you score 90 and above", and I suspect many other seemingly abusive uses of "and", can turn out to be perfectly valid if you consider them as (infinite) set manipulations. However, I'm not sure which of the two explanations is closer to the actual cognitive processes behind such a phrase. Seems to me that humans are naturally comfortable with many set manipulations, while current computers require fairly elaborate abstractions in order to deal with them as sets, especially infinite. This might be one of the gnarly parts of human -> machine translation.




To elaborate on this point, "if you score 90 and above" could be parsed in two different ways:

  1. "if [you score 90] and [you score above 90]"
  2. "if you score in [{90} 'and' {x: x > 90}]"
[1] is unsatisfiable. [2] is still ambiguous, as it's unclear in natural language whether 'and' is a set union or intersection.

In mathematical terminology, 'and' in this countext would mean set intersection, but I don't think it's necessarily "incorrect" to have this mean set union in natural language.

To elaborate, take: C = A union B. Here are two propositions about C:

  I. forall c in C. (c in A) OR (c in B)
  II. (forall a in A. a in C) AND (forall b in B. b in C)
These propositions are not equivalent. [I] actually implies C is a subset of (A union B), and [II] implies that it's a superset. Note that set builder notation for C, {c: (c in A) OR (c in B)} is structurally very similar to [I].

I think [II] is the interpretation of 'and' that is intended through the natural language use. It's essentially a form of set construction: I am constructing a set; it contains 90, and it contains the numbers above 90. As a set construction it also adds an implicit constraint that the new set can't contain anything not in the operands, so that resolves the superset ambiguity (it would be patently absurd in natural language to claim that 55 could be in the set "90 and above").


I don't read it that way. To me the use of the word "AND" is a red herring, as it wasn't meant to imply a logical operation, but rather in the usual sense (and meaning "plus that") to denote the range as half open.

  [90, infinity)
as opposed to:

  (90, infinity)
https://en.wikipedia.org/wiki/Interval_%28mathematics%29#Inc...


The ACTUAL meaning:

3. "if [you score 90] do [x] and if you score [above 90] also do [x]"


I think about it as:

"90 and above" is the smallest set X satisfying both:

* 90 is in X

AND

* Above(90) is a subset of X

Another description of this set is:

An element x is in the set "90 and above" if x is 90 OR if x is in Above(90).

AND/OR are dual to each other, and it's just a matter of perspective on whether you're building up the set(OR) or constraining the set(AND).




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