Results 1 to 9 of 9

Math Help - Confused about logical equivalence of some statements

  1. #1
    Member
    Joined
    Oct 2010
    From
    Mumbai, India
    Posts
    203

    Confused about logical equivalence of some statements

    Hi

    I am little confused about the following statements.

    1) \;\left[\exists x P(x)\right]\Rightarrow M(x)

    and

    2) \forall x \left[P(x)\Rightarrow M(x)\right]

    Its obvious that they are not logically equivalent. But lets take some examples.

    let P(x) = x is majoring in maths
    M(x)= x is mad.

    so the statement 2 means that all math majors are mad and
    statement 1 means that if there is a math major then he is mad

    here it looks like they are equivalent in meaning. so whats happening ?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member abhishekkgp's Avatar
    Joined
    Jan 2011
    From
    India
    Posts
    495
    Thanks
    1

    Re: Confused about logical equivalence of some statements

    Quote Originally Posted by issacnewton View Post
    Hi

    I am little confused about the following statements.

    1) \;\left[\exists x P(x)\right]\Rightarrow M(x)

    and

    2) \forall x \left[P(x)\Rightarrow M(x)\right]

    Its obvious that they are not logically equivalent. But lets take some examples.

    let P(x) = x is majoring in maths
    M(x)= x is mad.

    so the statement 2 means that all math majors are mad and
    statement 1 means that if there is a math major then he is mad
    no, statement 1 means there is at least one math major who is mad
    here it looks like they are equivalent in meaning. so whats happening ?
    ...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,561
    Thanks
    785

    Re: Confused about logical equivalence of some statements

    Quote Originally Posted by issacnewton View Post
    1) \;\left[\exists x P(x)\right]\Rightarrow M(x)
    This formula is not a proposition (either true or false) because x in M occurs freely, and so the formula's truth value depends on the value of x. In contrast, \forall x \left[P(x)\Rightarrow M(x)\right] is a proposition. So, the concept of equivalence does not apply to them.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Oct 2010
    From
    Mumbai, India
    Posts
    203

    Re: Confused about logical equivalence of some statements

    Hi abhishek,

    ok. so the key word is at least one. But I am still having difficulty
    understanding the difference between the two statements. Statement 1 guarantees that if we have even one math major then he is mad , am I right ? But does statement 1 mean that we can have 1 mad math major and rest of the math majors are not necessarily mad ?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Oct 2010
    From
    Mumbai, India
    Posts
    203

    Re: Confused about logical equivalence of some statements

    Quote Originally Posted by emakarov View Post
    This formula is not a proposition (either true or false) because x in M occurs freely, and so the formula's truth value depends on the value of x. In contrast, \forall x \left[P(x)\Rightarrow M(x)\right] is a proposition. So, the concept of equivalence does not apply to them.
    Oh... but what if x is a mad person who is not a math major ? then there does not exist any math major (antecedent is false) but there is a mad person
    (consequent is true). by rules of implication, the statement is true. is it not ?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,561
    Thanks
    785

    Re: Confused about logical equivalence of some statements

    Oh... but what if x is a mad person who is not a math major ? then there does not exist any math major (antecedent is false)
    I see no reason why ∃x P(x) is false.

    but there is a mad person
    (consequent is true). by rules of implication, the statement is true. is it not ?
    Yes. I don't see the connection between [∃x P(x)] => M(x) being true for some particular x and the equivalence of [∃x P(x)] => M(x) and ∀x [P(x) => M(x)] ("equivalence" is not applicable to these two formulas).

    Quote Originally Posted by issacnewton
    Statement 1 guarantees that if we have even one math major then he is mad , am I right ?
    No, it says that if we have even one math major, then x is mad, where x is undefined so far.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Oct 2010
    From
    Mumbai, India
    Posts
    203

    Re: Confused about logical equivalence of some statements

    Quote Originally Posted by emakarov View Post
    No, it says that if we have even one math major, then x is mad, where x is undefined so far.
    I think I am beginning to see some sense after all. The problem is , x in antecedent is bound variable and x in the consequent is free variable, right ?

    I have to read again about these variables. They are so confusing. Any online source would you recommend ?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,561
    Thanks
    785

    Re: Confused about logical equivalence of some statements

    The PlanetMath article is a little formal but not too hard. This presentation [PDF] seems nice. See also this page written by some very qualified people.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member
    Joined
    Oct 2010
    From
    Mumbai, India
    Posts
    203

    Re: Confused about logical equivalence of some statements

    haha, I was reading that page at cnx.org at the time you replied.......
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 7
    Last Post: November 27th 2011, 12:39 PM
  2. Replies: 1
    Last Post: July 8th 2011, 06:21 AM
  3. Logical Equivalence
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: October 22nd 2010, 02:43 AM
  4. Logical statements
    Posted in the Discrete Math Forum
    Replies: 20
    Last Post: August 17th 2010, 02:55 AM
  5. Logical Equivalence Help
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: February 2nd 2010, 02:06 PM

/mathhelpforum @mathhelpforum