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August 5th, 2013, 06:21 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: (x is a student x is clever) 2. And they say that we can write the sentence "All students are clever" as (x is a student 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: (x is a student x is clever) 
August 5th, 2013, 03:55 PM  #2  
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 Quote:
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, 04:28 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, 02:52 AM  #4  
Newbie Joined: Aug 2013 Posts: 2 Thanks: 0  Re: writing the sentence, as a statement Quote:
 
August 6th, 2013, 04:47 AM  #5  
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: writing the sentence, as a statement Quote:
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, 01:35 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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