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 August 5th, 2013, 07:21 AM #1 Newbie   Joined: Aug 2013 Posts: 2 Thanks: 0 writing the sentence, as a statement Hello I will describe my problem, and some things from my book, then I will ask a question. Assume that the universe of discourse for the variable x is the collection of all persons. 1. Then my book says that we can write the sentence. "Some students are clever", as: $\exists$ (x is a student $\wedge$ x is clever) 2. And they say that we can write the sentence "All students are clever" as $\forall$ (x is a student $\rightarrow$ x is clever) By looking at these two ways of wrinting things, I am wondering if the first one, could be written as the second one, that is, can we write: "Some students are clever", as: $\exists$ (x is a student $\rightarrow$ x is clever)
August 5th, 2013, 04:55 PM   #2
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Re: writing the sentence, as a statement

Quote:
 Originally Posted by lanzo By looking at these two ways of wrinting things, I am wondering if the first one, could be written as the second one, that is, can we write: "Some students are clever", as: $\exists$ (x is a student $\rightarrow$ x is clever)
This says: there is some x; if x is a student, then x is clever.

For example, the statement is true if there is something (say, a cactus) which is not a student. But that's not what you want; you want that there exists x such that x is a student and x is also clever.

 August 5th, 2013, 05:28 PM #3 Math Team   Joined: Apr 2012 Posts: 1,579 Thanks: 22 Re: writing the sentence, as a statement CRG is completely correct. There IS a style of predicate logic, predicate logic with restricted quantification, where the sentence receive syntactically parallel treatment: (some x: student x)(clever x) (all x: student x)(clever x) and the implicitly conditional sense of the latter and conjunctive sense of the former is handled wholly in the semantics. There's no doubt something online about it.
August 6th, 2013, 03:52 AM   #4
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Re: writing the sentence, as a statement

Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by lanzo By looking at these two ways of wrinting things, I am wondering if the first one, could be written as the second one, that is, can we write: "Some students are clever", as: $\exists$ (x is a student $\rightarrow$ x is clever)
This says: there is some x; if x is a student, then x is clever.

For example, the statement is true if there is something (say, a cactus) which is not a student. But that's not what you want; you want that there exists x such that x is a student and x is also clever.
But if your argument creates a problem here. Does it not also create a problem in the second sentence "All students are clever". There may not even exist any students, and it is also this sentence will also be vacuously for a cactus?

August 6th, 2013, 05:47 AM   #5
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Re: writing the sentence, as a statement

Quote:
Originally Posted by lanzo
Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by lanzo By looking at these two ways of wrinting things, I am wondering if the first one, could be written as the second one, that is, can we write: "Some students are clever", as: $\exists$ (x is a student $\rightarrow$ x is clever)
This says: there is some x; if x is a student, then x is clever.

For example, the statement is true if there is something (say, a cactus) which is not a student. But that's not what you want; you want that there exists x such that x is a student and x is also clever.
But if your argument creates a problem here. Does it not also create a problem in the second sentence "All students are clever". There may not even exist any students, and it is also this sentence will also be vacuously for a cactus?
Universal statements can indeed be "vacuously true" when nothing satisfies the predicate in the antecedent clause (and this holds for restricted quantification as well, even though there is no overt conditional statement involved). This is why universal statements, though intuitively "stronger" than the corresponding existential statement, don't actually entail the existential statement. (All x)(Fx->Gx) will be true and (some x)(Fx&Gx) false if there is no x that is an F.

While some people consider this a real problem, a vacuously true universal conditional is nowhere near as perverse as a vacuously true existential conditional. For 'all students are clever' to be vacuously true, there has to be no students at all. For (some x)(student x -> clever x) to be vacuously true, there merely has to be one thing in the world that isn't a student. So while there are some pretty high standards to meet for a universal condition to be true but vacuous, it is almost impossible for an existential conditional to avoid being vacuous, hence true but in a thoroughly unilluminating way

 August 6th, 2013, 02:35 PM #6 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: writing the sentence, as a statement I agree with johnr.

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