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I'm trying to work out how (∀y (student(y) → y = x) represents uniqueness. How come we need that formula there? Doesn't just ∃x student(x) ⋀ met(j,x)) suffice? Or, would it be the expression for I John met a student? EDIT: I >>> John

According to the (neo) Russellian theory of definite descriptions [Russell, 1 905; Neale, 1990], a definite description combines an assertion of existence with an assertion of uniqueness. Thus, the sentence John met the student asserts that there is a student, that this student is unique, and John met him/her. These truth conditions can be summarized as in 2.

  1. John met the student.
  2. (∃x student(x) ⋀ (∀y (student(y) → y = x)) ⋀ met(j,x))

[Poesio, M. (1994, November). Weak definites. In Semantics and Linguistic Theory (Vol. 4, pp. 282-299)]

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The existential quantifier doesn't mean "one", it means "at least one". So ∃x(student(x) ⋀ met(j,x)) translates as "John met at least one student". This formalization is consistent with structures in which there exist several students, but this should be ruled out, given that the English sentence uses the definite article "the".
"Structure" here means "context relevant to the discourse" -- of course, as Greg Lee points out, we don't claim that there exists only one student in the entire world, but the use of "the" suggests that there is only one student that comes into question as a relevant referent in the current discussion, and this situation is the one the formalization is supposed to adequately describe.
To assert that there is one unique student, we need to specify that there is no other individual which is also a student but different from that first introduced student: ¬∃y(student(y) ⋀ y ≠ x). You can easily verify that this can logically equivalently be expressed as ∀y(student(y) → y = x).

Whether the individual constant j means John or you depends on how you want to read it. A constant in a logical formula can't unambiguously refer to a concrete individual; its interpretation will depend on what we define. By convention, when translating to a formal representation we will usually choose meaningful names for non-logical symbols (= individual constants and predicates) and have an intended interpretation in mind; but on the logical side, there is nothing that prevents us from interpreting "j" as 2Lady Gaga" and "Student(x)" as "x is an elephant". Logic doesn't make predictions about the meaning of non-logical symbols; their interpretation is subject purely to convention. So to answer your second question, the formula could also mean "I met a student". But the use of the letter "j" makes it likely that John is indeed the intended interpretation.

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  • Much obliged, lemontree. (1) I'm having difficulty deciding how I should see which 'x' is a variable and which 'x' is an individual, specific 'x.' Same thing is true with 'y.' (2) When you arrive at ∀y(student(y) → y = x), do you derive it by first visiting ∃y(student(y) ⋀ y ≠ x), then negating it and applying some established rules to it? (3) Ooops, I should have said 'John met a student' in my second question. (4) I'm curious how you would represent in a logical formula 'John met some students' and 'John met many students'? ∃ and ∀ only cover part of all possibilities, don't they?
    – Sssamy
    Jun 26 '20 at 0:35
  • (2) Or, do you reach the formula straight?
    – Sssamy
    Jun 26 '20 at 0:51
  • (4) Also, thinking ∃ means 'at least one,' I can't see how it relates to 'uniquely one.' y = x. All y's won't narrow down to one x but one or more x's if ∃ means 'at least one.' ???
    – Sssamy
    Jun 26 '20 at 5:09
  • (1) If you struggle with the basics of predicate logic such as the use of variables, I suggest you grab a good textbook, and ask those kinds of questions over at Math SE. This can not be comprehensively explained in a comment. (2) However you want. Maybe Russel came up with it straight; I thought the ¬∃y formulation the more intuitive one. It does not matter which one one starts with; both are equivalent. And yes, one can arrive at the ∀y formulation by rewriting equivalence rules. (3) Yes, "John met a student" (where "a" = "at least one") is what ∃x(student(x) ⋀ met(j,x)) means.
    – lemontree
    Jun 26 '20 at 9:17
  • (4) This is best suited as a new question. (5) The y = x with the universal quantifier is exactly what does the narrowing down. ∀y(student(y) → y = x) says that everyone who is a student in the discussion is that same person x. Or equivalently, there is noone who is also a student but different from that person x.
    – lemontree
    Jun 26 '20 at 9:23
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Suppose you've phoned me about some issue or other, and I tell you what it says in SPE, which I say I am looking at right now, open on the bed in front of me. What am I referring to as "the bed"? Have I implied that there is only one bed in my neighborhood, or in my house, or in this room I'm in right now? None of those. It's the unique bed in the mental picture I'm drawing for you -- me crouched on a bed peering at an open volume of SPE. That bed.

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