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'John met the student' would translate as:

(∃x student(x) ⋀ (∀y (student(y) → y = x)) ⋀ met(j,x))

where, j stands for John.

We have the existential quantifier and the universal quantifier; hence, translation of 'a (or, some) student' and 'all students' would be easy but how could you express 'some students,' 'many students,' etc. in logical formulas?

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  • What is "etc."? There are infinitely many possible quantifiers; if you want to know about more than the ones you listed, you need to tell us concretely which ones.
    – lemontree
    Jun 26 '20 at 14:17
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  • To express "some students" in the sense of "more than one", you could say that there exist at least two distinct individuals both of which are students and met by John (and yes, the duplication is necessary):

    ∃x ∃y (x ≠ y ⋀ student(x) ⋀ student(y) ⋀ met(j,x) ⋀ met(j,y))

  • "at least one but not all" would be "There is at least one student who John met, but not every student was met by John":

    ∃x (student(x) ⋀ met(j,x)) ⋀ ¬∀y (student(y) → met(j,y))

  • "Many", "most", and "more than" can not be expressed in standard (first-order) predicate logic. This requires comparing sizes of sets, and it can be proven that this is not possible in first-order logic.

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  • McCawley has a section in his logic book on "The Logicians' Favorite Quantifiers", pointing out that ∃ and ∀ are useful but not anywhere close to the complete story.
    – jlawler
    Jun 26 '20 at 17:51

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