Consider: The frogs are green. Which of these rewrites to first-order logic are equivalent to "The frogs are green."?
Rewrite A.
1. For all X, if X is a frog, then X is green.
(No claim of existence.)
Rewrite B.
1. For all X, if X is a frog, then X is green.
2. There exists an X such that X is a frog. (At least 1 frog exists.)
Rewrite C.
1. For all X, if X is a frog, then X is green.
2. There exist X and Y such that X is a frog, Y is a frog, and X ≠ Y. (At least 2 frogs exist.)
I have a strong preference (which I will keep secret temporarily, to not bias anybody) but I doubt there is a universally accepted rewrite. Forestidia disagrees with my choice here and has a strong background in philosophical logic. The classical examples on the internet don't help either, Russell's essay is purely about the singular "the".
-- Simon
I don't have any background in formal logic, but I will make an attempt at this, as I don't really have anything better to do right now and I find this to be an interesting thought exercise.
It seems to me that there probably can't be a universal rewrite for this sentence, because there's insufficient context to determine the exact bounds of the subset "the frogs."
"The frogs are green" can be true or false depending on what "the frogs" refers to. We need context to determine that. Suppose there is a box containing frogs. I point to it and say, "The frogs are green." This extra context removes the ambiguity as to which frogs we are referring to - now it's the frogs that are in the hypothetical box, rather than an unspecified group subset of frogs. But now we must check to see if the statement is actually correct. It is possible for all the frogs in the box to be green
[1][2]. However, it is also possible that at least one of the frogs in the box is NOT green
[3][4]. For this reason, I can create a set of frogs for which the statement "The frogs are green" is false.
This is where I can no longer really work with the given information without more background knowledge. I am unsure of the exact meaning of "For all X, if X is a frog, then X is green." Is this statement defining the object selection, to state that all the frogs under consideration are green (i.e. if an object is a frog, and it is not green, it is not X)? Or is it a statement that, for any given object X, if X is a frog, it is green (i.e. stating that all frogs are green)?
Based on these two interpretations, I'd say the following:
If "For all X, if X is a frog, then X is green" is equivalent to the statement "If an object is a frog, and it is not green, it is not X" then I'd say that Rewrite C is best, simply because it clarifies that there is more than one frog, which is implied in the plain language "The frogs are green." Rewrite B also allows for the possibility that there is one frog, which is NOT a possibility implied by the plain language, and Rewrite A does not specify that any frogs exist.
If "For all X, if X is a frog, then X is green" is equivalent to the statement "All frogs are green," then, due to the pictures shown in
[3] and
[4], then Rewrites A, B, and C are all demonstrably false.
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Regardless of the proper formalization of "The frogs are green," if we make the assumption that the statement is true, then the following items must be accounted for to avoid anything being lost in translation:
1) "The frogs" refers to at least 2 frogs, all of which are green.
2) The statement becomes false if "the frogs" is changed to "all frogs."
3) It is not possible to fully ascertain what frogs are being referred to by "the frogs" without additional context.