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Let the universe for all quantified variables be the set of all novels. Assume the following predicates and constant symbols:

  • $W(x,y) :\; x$ wrote novel $y$
  • $L(x,y) : \;x$ is longer than $y$
  • $h :$ Hardy
  • $a :$ Austen

Given these specifications, which of the predicate logic formulas below represent the sentence, 'Hardy wrote a novel which is longer than any of Austen's' in predicate logic?

  1. $\forall x \exists y (L(x,y) \rightarrow W(h,y) \wedge W(a,x))$
  2. $\forall x \forall y (W(h,x) \wedge W(a,y) \rightarrow L(x,y)))$
  3. $\exists x (W(h,x) \wedge \forall y ( W(a,y) \rightarrow L(x,y)))$
  4. $\exists x \forall y (W(h,x) \rightarrow W(a,y) \wedge L(x,y))$
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2 votes
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Option A Explanation:
The formula $\forall x \exists y (L(x,y) \rightarrow W(h,y) \wedge W(a,x))$

  • literally says 'For all novels $x,$ there is some novel $y$ such that: if $x$ is longer than $y,$ then Hardy wrote novel $y$ and Austen wrote novel $x'.$
  • This sentence is rather "mathematical" and does not really have a more elegant alternative of stating it.

Not only does this sentence not say what we want, but it is false even if hardy wrote some novel longer than  all of austen’s novels, if, say, jerry wrote a novel longer than all other novels.

Option B Explanation:
The formula $\forall x  \forall y (W(h,x) \wedge W(a,y) \rightarrow  L(x,y)))$

  • literally says 'For all novel $x$ and all novel $y,$ if Hardy wrote $x$ and Austen wrote $y,$ then $x$ is longer than $y'.$
  • This sentence can be re-expressed more elegantly as 'All of Hardy's novels are longer than any of Austen's'.

Although this sentence comes close to expressing what we want, it claims that the desired property holds for all novels that Hardy wrote, not just for at least one of them!

Option C Explanation:
The formula $\exists x (W(h,x) \wedge \forall y ( W(a,y) \rightarrow  L(x,y)))$

  • literally says 'There is some novel $x$ and which Hardy wrote such that for all novels $y$ and written by Austen, $x$ is longer than $y'.$
  • This sentence can be re-expressed more elegantly as 'Hardy wrote a novel which is longer than any of Austen's'.

Option D Explanation:
The formula $\exists x \forall y (W(h,x) \rightarrow W(a,y) \wedge  L(x,y))$

  • literally says 'There is some novel $x$ such that for all novels $y$ we have the following: If Hardy wrote $x,$ then Austen wrote $y$ and $x$ is longer than $y.$
  • It is hard to re-express this sentence more elegantly, as it describes a rather "mathematical" situation.

This sentence does not come close to expressing what we want. For one thing, if the "witness" for $x$ is indeed a novel that Hardy wrote, then this sentence suggests that all novels $y$ are written by Austen and shorter than $x.$

Another wrong expression would be:
The formula $\forall x (W(h,x) \rightarrow L(x,a)))$

  • literally says 'For all novels $x,$ if Hardy wrote $x,$ then $x$ is longer than Austen'.
  • We can express this sentence more elegantly as 'Any novels that Hardy wrote are longer than Austen'.

Not only does this sentence not say what we want, but it treats Austen as if she/he were a novel, not an author, lacking sense and sensibility. :-)))

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Option D has rather similar translation.
One way to think about it is : If domain is not specified consider it having relevant, irrelevant elements.
Irrelevant element will make this option True. Which will not be in sync with our English statement.

I think this is dependent on question maker if we think about context of this grammar then it may contain only set of novels written by Hardy and Austin.
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