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Identify What's Wrong with the Argument (Logic and Proofs)

  1. Sep 30, 2015 #1
    1. The problem statement, all variables and given/known data
    Taken from Discrete Mathematics and its Applications, Seventh Edition:
    "What is wrong with this argument? Let [itex]S(x, y)[/itex] be '[itex]x[/itex] is shorter than [itex]y[/itex].' Given the premise [itex]\exists s S(s, Max)[/itex], it follows that [itex]S(Max, Max)[/itex]. Then by existential generalization it follows that [itex]\exists x S(x, x)[/itex], so that someone is shorter than himself."

    2. Relevant equations
    A similarly worded question (i.e., it begins with, "What is wrong with this argument?...") has its answer in the back of the book. The answer is as follows:

    "We know that some [itex]x[/itex] exists that makes [itex]H(x)[/itex] true, but we cannot conclude that [itex]Lola[/itex] is one such [itex]x[/itex]."

    3. The attempt at a solution
    Prior to looking at the answer to the similar question in the back of the book, I thought that perhaps I was being asked to identify what logical error was being made (e.g., fallacy of affirming the conclusion, fallacy of denying the hypothesis, etc.). However, the answer to the similarly worded question is very plainspoken (for lack of a better term).

    So I figured they'd be looking for an answer that looks like one of the following:
    (1)-[itex]s[/itex] isn't apart of the given domain ([itex]x[/itex] or [itex]y[/itex]).
    (2)-It doesn't follow that [itex]S(Max,Max)[/itex] given [itex]∃sS(s,Max)[/itex].
    (3)-A person can't be shorter than his or herself (i.e., [itex]x \not< x[/itex]). That's impossible.
    (4)-[itex]x[/itex] is distict from [itex]y[/itex] (i.e., [itex]x \not= y[/itex]).
    (5, which is what I'd say is the answer)-Some combination of answers 3 and 4.

    (I realize that some of these are kind of saying the same thing.)

    A student peer tutor has said that none of those are the correct answers.

    What's wrong with these answers? If the answer, "We don't know that [itex]Lola \in x[/itex]," will satisfy a similar question, what's wrong with these? As I said, my first choice for my answer would be (5).
     
  2. jcsd
  3. Sep 30, 2015 #2

    Mark44

    Staff: Mentor

    I like it that their chosen answers are plainspoken (as you put it) rather than logical jargon such as "fallacy of affirming the conclusion," etc.

    Think about what ##\exists s : S(s, Max)## means, in ordinary words, and why S(Max, Max) does not necessarily follow.
     
  4. Sep 30, 2015 #3
    Should I say something like, "[itex]∃s:S(s,Max)[/itex] means that there is some person [itex]s[/itex] that is shorter than [itex]Max[/itex], but [itex]Max[/itex] cannot be said to be one such [itex]s[/itex]."?
     
  5. Sep 30, 2015 #4

    Mark44

    Staff: Mentor

    Yeah, something like that is what I had in mind. The fact that there is someone who is shorter than Max does not mean that that someone is Max.
     
  6. Sep 30, 2015 #5
    It's probably a question for my TA, but do you think the other answers are even partially correct? i.e., Are they the kinds of answers you might see as receiving partial credit on a test, or are they completely bogus answers?
     
  7. Sep 30, 2015 #6

    Mark44

    Staff: Mentor

    (1) doesn't make sense, because the domain is not just x and y.
    (2) does make sense, and is about what I said before.
    (3) is true, but I don't think that's the answer they're probably looking for.
    (4) there is no information given that x and y are distinct. Of course, for the relation "is shorter than", two things being compared in the relation can't be the same.
     
  8. Sep 30, 2015 #7
    Alrighty then. Cool beans. Thanks.
     
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