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I am trying to learn QL on my own and I wish I could get some help translating the two sentences below.

I have given two sets of answers: Ans A is the textbook one and Ans B is my own attempt. Please tell me whether Ans B is acceptable, and if it's not, what mistake I've made. Thanks!

universe of discourse: candies
$Cx:$ $x$ has chocolate in it.
$Bxy:$ $x$ is better than $y.$

  1. The very best candy is chocolate.
  2. Any candy with chocolate is better than any candy without it.

Ans A:

  1. $∃x[Cx\&¬∃yByx]$
  2. $∀x[Cx → ∀y(¬Cy→Bxy)]$

Ans B:

  1. $∀x[Cx→¬∃yByx]$
  2. $∀x[Cx → ¬∃y(¬Cy\&Byx)]$
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  • $\begingroup$ B1 is wrong. It must be: $¬∀x[Cx→∃yByx]$ $\endgroup$ Sep 20 at 10:36
  • $\begingroup$ B2 must be: $∀x[Cx → ¬∃y(¬Cy \& ¬Byx)]$ $\endgroup$ Sep 20 at 10:44
  • $\begingroup$ @MauroALLEGRANZA would you be able to say $( \forall x)( \forall y)[(Cx & Cy) \rightarrow Bxy]$ for B2? $\endgroup$ Sep 23 at 16:51
  • $\begingroup$ Welcome to the site! If the problem has been resolved, consider accepting and/or upvoting answers: it scores points, signals resolution, and prevents bumping. $\endgroup$
    – Ryan G
    Oct 7 at 7:28
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universe of discourse: candies
Cx: x has chocolate in it.
Bxy: x is better than y.

  1. The very best candy is chocolate.

    textbook: $∃x[Cx \& ¬∃yByx]$
    mine: $∀x[Cx→¬∃yByx]$

Either candies don't exist, or chocolate candy must exist and non-chocolate candy is not better (note that "non-chocolate candy is worse" may be inaccurate). In other words, $$\forall \top \lor \exists x (Cx\land\forall y\lnot Byx);$$ this corresponds to the textbook's answer.

Your answer isn't equivalent to the textbook's, since they have opposite truth values

  • in a universe containing only gummy candy, and
  • in a universe where pure chocolate candy is better than chocolate-caramel candy.
  1. Any candy with chocolate is better than any candy without it.

    textbook: ∀x[Cx → ∀y(¬Cy→Bxy)]
    mine: ∀x[Cx → ¬∃y(¬Cy&Byx)]

Again, the two answers aren't equivalent, since they have opposite truth values

  • in a universe where chocolate and gummy candy are equally good.
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