in Mathematical Logic recategorized by
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3 votes
3 votes

Assume the following predicate and constant symbols.

  • $W(x,y) :\; x$ wrote $y$
  • $L(x,y) :\; x$ is longer than $y$
  • $h :$ Hardy
  • $a :$ Austen
  • $j :$ Jude the Obscure
  • $p :$ Pride and Predjudice

Which of the following represents the sentence,

''Jude the Obscure' is not longer than 'Pride and Predjudice'', in predicate logic?

  1. $W(h,j) \wedge W(a,p)$
  2. $\neg \forall x \forall y \neg L(x,y)$
  3. $L(p,j)$
  4. $\neg L(j,p)$
in Mathematical Logic recategorized by
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1 Answer

10 votes
10 votes

Option D Explanation:

The formula $\neg L(j,p)$

  • literally says 'It is not the case that 'Jude the Obscure' is longer than 'Pride and Predjudice''.
  • Expressing this more elegantly, we uncover the sought sentence ''Jude the Obscure' is not longer than 'Pride and Predjudice''.

Option A Explanation:

The formula $W(h,j) \wedge  W(a,p)$

  • literally says 'Hardy wrote 'Jude and Obscure', and Austen wrote 'Pride and Predjudice''.

Obviously, this sentence has a different meaning than ''Jude the Obscure' is not longer than 'Pride and Predjudice''.

Option B Explanation:

The formula $\neg \forall x \forall y \neg L(x,y)$

  • literally says 'It is not the case that for all x and y the x is not longer than y''.
  • A less convoluted way of putting this, making use of the quantifier equivalences, is ''There is some $x$ and some $y$ such that $x$ is longer than $y$''.

Obviously, this sentence has a different meaning than ''Jude the Obscure' is not longer than 'Pride and Predjudice''.

Option C Explanation:

The formula $L(p,j)$

  • literally says ''Pride and Predjudice' is longer than 'Jude the Obscure''.

While this comes very close to expressing ''Jude the Obscure' is not longer than 'Pride and Predjudice'', there is a difference in the amount of information that these sentences reveal. If 'Pride and Predjudice' is longer than 'Jude the Obscure', then 'Jude the Obscure' is clearly not longer than 'Pride and Predjudice'. But if we only know the latter, we cannot infer the former, for they could have equal length!

For Practice:

The formula $L(h,a)$

  • literally says ''Hardy is longer than Austen''.

Obviously, this is a long shot from expressing that ''Jude the Obscure' is not longer than 'Pride and Predjudice'', as it confounds, among other things, the authors with their most famous novels.

edited by

2 Comments

@GO Classes line 2: The formula ~L(j,p). You have missed the negation sign.

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@Yuvraj Raghuvanshi Corrected. Thank you for pointing out.

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