Results 1 to 7 of 7

Math Help - English to logic translation

  1. #1
    Newbie
    Joined
    Aug 2011
    Posts
    18

    English to logic translation

    I am going back and reviewing some elementary material in logic/set theory. Among the problems in the quantifier section is the following english sentence to logic sentence translation:

    Everyone likes Mary, except Mary herself.

    Now, my attempt was: (∀x)[(x≠m → L(x,m)) ∧ (x=m → L(x,m))], where m stands for "mary," and L(a,b) stands for "a likes b."

    The solution to the exercise in the text was given as: (∀x)(x≠m → L(x,m)). I was a bit skeptical of this and searched the problem online and found another, different solution:
    (∀x)(x≠m ↔ L(x,m)).

    My question is, which of these three is correct? (or, if they are all/some equivalent, how?)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,657
    Thanks
    1613
    Awards
    1

    Re: English to logic translation

    Quote Originally Posted by Syrus View Post
    I am going back and reviewing some elementary material in logic/set theory. Among the problems in the quantifier section is the following english sentence to logic sentence translation:
    Everyone likes Mary, except Mary herself.
    Note that "Everyone likes Mary, except Mary herself" does not mean that it is necessary that Mary does not like herself.
    All that says is that "if x is not Mary then x likes Mary".
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2011
    Posts
    18

    Re: English to logic translation

    Ah, that is very subtle. Isn't it true, however, that either Mary likes herself, or Mary does not like herself (since (P OR P) is a tautology)? And since everyone likes mary, EXCEPT Mary herself, it would have to be the case that NOT L(x,m). At least this is what comes to mind initially.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,536
    Thanks
    778

    Re: English to logic translation

    Quote Originally Posted by Syrus View Post
    Ah, that is very subtle. Isn't it true, however, that either Mary likes herself, or Mary does not like herself (since (P OR P) is a tautology)?
    Yes.

    Quote Originally Posted by Syrus View Post
    And since everyone likes mary, EXCEPT Mary herself, it would have to be the case that NOT L(x,m).
    That would follow if the statement gave an exhaustive set of people who likes Mary. In that case, it would follow that L(m,m) is false and, by the law of excluded middle (LEM), ~L(m,m). However, it is claimed that the statement says nothing about Mary, neither L(m,m) nor ~L(m,m), so the LEM does not help.

    However, saying that some property holds for a smaller set of elements than it actually does is unfair to the reader even if it may be pedantically correct. For example, saying that the set {1, 2, 3} has two elements is misleading. For a real math example, Picard theorem from complex analysis is sometimes formulated as follows: "If a complex function is entire and non-constant, then it assumes all complex points except possibly one." It may be technically correct to say that it assumes all points except one, about which no claim is made, i.e., the function may assume it or it may not. However, the word "possibly" is added to describe the complete image of the function in all cases.

    Therefore, I would count (∀x)[(x≠m → L(x,m)) ∧ (x=m → L(x,m))] and (∀x)(x≠m ↔ L(x,m)), which are equivalent, as correct.

    Finally, a joke. A passerby noticed that the sheep kept by a certain man were well fed and healthy, so he asked him:
    - What do you give these sheep for food?
    - Which sheep: the black or the white?
    - The white, said the passerby.
    - Oh, those eat the grass there.
    - What about the black then?
    - They eat the same grass.
    Puzzled, the passerby asks:
    - What about drink then?
    - Excuse me, are you asking about the black sheep or the white ones?
    - The white, I guess.
    - Well, normal water from that river.
    - OK, what about the black then?
    - Same water.
    Frustrated, the passerby yells at the man:
    - If everything is the same why do you keep asking me about white and black?
    - The white sheep are mine.
    - Oh, I'm sorry... And the black sheep?
    - They are also mine.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Aug 2011
    Posts
    18

    Re: English to logic translation

    Thank you both for your responses, Plato and emakarov. I finished up the remainder of the exercises in the section and only came upon three others which I was uncertain about:

    1) "Anyone who has bought a Rolls Royce with cash must have a rich uncle."
    My question with this problem was less with the translation and more with the use of predicates. My translation was: (∀x)[B(x,r) → R(x)], where r stands for "Rolls Royce," B(a,b) stands for "a has bought b with cash," and R(c) stands for "c has a rich uncle." In particular, I was wondering if using either of the predicates B(x) or B(x,r,c) for "x has bought a Rolls Royce with cash" and "x has bought a Rolls Royce (denoted r) with cash (denoted c)," respectively, are acceptable (that is, is what is 'included' in the predicate up to one's discretion)?

    2) If nobody failed the test, then everybody who got an A will tutor someone who got a D.
    I simply felt unsure about this one for some reason: (∃w)F(w) → (∀x)[A(x) → ((∃y)D(y) → (∃z)T(x,z))], where F(x) stands for "x failed the test," A(y) stands for "y got an A on the test," y(z) stands for "z got a D on the test," and T(a,b) stands for "a tutors b."

    Lastly, "x is prime" where the universe of discourse is the natural numbers.

    My attempt was: x > 1 ∧ (∀j)(∀k)(x = jk → (j = 1 or j = x))

    The solution had a similar statement, except that they stated that there do not exist natural numbers y,z such that x =yz and y and z are both less than x. To me these both seem correct. Is this true?
    Last edited by Syrus; December 19th 2011 at 09:10 PM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,657
    Thanks
    1613
    Awards
    1

    Re: English to logic translation

    Quote Originally Posted by Syrus View Post
    Lastly, "x is prime" where the universe of discourse is the natural numbers.
    My attempt was: x > 1 ∧ (∀j)(∀k)(x = jk → (j = 1 or j = x))
    The solution had a similar statement, except that they stated that there do not exist natural numbers y,z such that x =yz and y and z are both less than x. To me these both seem correct. Is this true?
    I will let the logician comment on the first two.
    However, I do agree with you on the last.
    You were clever in forcing k to be 1 or x.

    A prime number is a integer with exactly two divisors.
    That gets around the bogus definition that one sees in print (math-ed material) "a prime number is an integer is divisible only by itself and 1".
    But of course by that definition 1 is prime.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,536
    Thanks
    778

    Re: English to logic translation

    I agree with 1).

    Quote Originally Posted by Syrus View Post
    In particular, I was wondering if using either of the predicates B(x) or B(x,r,c) for "x has bought a Rolls Royce with cash" and "x has bought a Rolls Royce (denoted r) with cash (denoted c)," respectively, are acceptable (that is, is what is 'included' in the predicate up to one's discretion)?
    This depends on the problem requirement or other external circumstances. At one extreme, you can denote the whole proposition by one letter. I think, a reasonable and slightly mode detailed formalization is (∀x)[B(x,r) → (∃y)(U(x,y) ∧ R(y))] where U(x,y) means "y is x's uncle" and R(y) means "y is rich."

    Quote Originally Posted by Syrus View Post
    2) If nobody failed the test, then everybody who got an A will tutor someone who got a D.
    I simply felt unsure about this one for some reason: (∃w)F(w) → (∀x)[A(x) → ((∃y)D(y) → (∃z)T(x,z))], where F(x) stands for "x failed the test," A(y) stands for "y got an A on the test," y(z) stands for "z got a D on the test," and T(a,b) stands for "a tutors b."
    Should be

    (∃w)F(w) → (∀x)[A(x) → (∃y)(D(y) ∧ T(x,y))]

    As it is, the person x who got an A tutors some z who did not necessarily got a D. Also, if some people got As but nobody got either a D or an F, then the original English statement is false while the formula in the quote is true since the premise (∃y)D(y) is false.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Translating English to FO Logic
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: August 3rd 2010, 06:32 AM
  2. Formalizing Predicate Logic (English to PL)
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: November 22nd 2009, 11:18 AM
  3. Predicate Logic converted to English Statements
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: September 24th 2009, 12:47 AM
  4. English to Predicate Logic
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: November 4th 2008, 04:28 AM
  5. Logic translation
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: October 30th 2007, 12:41 PM

Search Tags


/mathhelpforum @mathhelpforum