edited by
4,253 views
33 votes
33 votes

If $Mr.M$ is guilty, then no witness is lying unless he is afraid. There is a witness who is afraid. Which of the following statements is true?

(Hint: Formulate the problem using the following predicates

  • $G - Mr.M$ is guilty
  • $W(x) - x$ is a witness
  • $L(x) - x$ is lying
  • $A(x) - x$ is afraid )
  1. $Mr.M$ is guilty.
  2. $Mr.M$ is not guilty.
  3. From these facts one cannot conclude that $Mr.M$ is guilty.
  4. There is a witness who is lying.
  5. No witness is lying.
edited by

5 Answers

Best answer
36 votes
36 votes

$\newcommand{m}{\text{Mr. } M}$If $\m$ is guilty, then if we pick a witness, we know that the witness won't lie unless he is afraid. If the witness is afraid, it may lie or it may lot lie (nothing is guaranteed).

However, unless we know what the victim said in the court (whether he said that $\m$ was guilty or not guilty), we can't say anything about $\m$.

All we know is that we've a witness who is afraid, so he may or may not lie in the court. We haven't been told anything about what actually happened in the court proceeding.

So, we can't logically conclude anything about $\m$ being guilty or not guilty.

Thus, options (a) and (b) are False.

Furthermore, that witness who was afraid, he may or may not lie. Since he is afraid, we know that he "can" lie, but we're not guaranteed that he will lie.

Thus, options (d) and (e) are False too.

This leaves option c, and as we have seen earlier, we cannot conclude anything about $\m$ being guilty or not guilty.

Hence, option (c) is the correct answer. 


Although not necessary, the logic equivalent of the given statement will be:

$\Big[G\Rightarrow\neg\exists x:(W(x)\wedge L(x)\wedge\neg A(x))\Big]\equiv$

$\Big[G\Rightarrow\forall x:(W(x)\Rightarrow(\neg A(x)\Rightarrow\neg L(x)))\Big]$

edited by
15 votes
15 votes

It is one those questions which are easier to answer if you use propositional and predicate logic.

Given the argument:

1. G ⟹ ∀x (W(x) ∧ ¬A(x) ⟹ ¬L(x))

2. ∃x (W(x) ∧ A(x))

Since statement 2 is true, W(x) ∧ ¬A(x) in statement 1 is false. So, W(x) ∧ ¬A(x) ⟹ ¬L(x) is always true for any x no matter L(x) is true or false for any value of x. Also, since ∀x (W(x) ∧ ¬A(x) ⟹ ¬L(x)) is true, therefore, statement 1 is true no matter G is true or false.

Hence, option C is correct.

2 votes
2 votes
"((If Mr.M is guilty, then no witness is lying) unless he is afraid)" :
=G->¬∃x(W(x)∧L(x)) unless ∃x(A(x))
=¬∃x(A(x))->G->¬∃x(W(x)∧L(x))
=(¬∃x(A(x))∧G)->¬∃x(W(x)∧L(x))
=¬∃x((A(x))∧G)->(W(x)∧L(x)))
=¬∃x((A(x))∧G)->W(x)) ∧ ¬∃x((A(x))∧G)->L(x))

"There is a witness who is afraid" :
=∃x(W(x)∧A(x))
=∃xW(x) ∧ ∃xA(x)

¬∃x(((A(x))∧G)->W(x)) ...(1)
¬∃x(((A(x))∧G)->L(x))   ...(2)
∃xW(x)                                      ...(3)
∃xA(x)                                        ...(4)

using (1) and (3) we get
¬∃x((A(x))∧G))        ...(5)

No further simplication happen
Hence Ans is C
1 votes
1 votes

if p then q unless r =   $(p ∧¬r)⟹q$

If Mr. M is guilty then no witness is lying unless he is afraid.=  

 ∃x(G ∧¬A(x))⟹¬∃x:(W(x)∧L(x))

There is a witness who is afraid.  =

∃x:(W(x)∧A(x))

Right or not??

Answer:

Related questions

26 votes
26 votes
5 answers
1
makhdoom ghaya asked Oct 30, 2015
2,076 views
For a person $p$, let $w(p)$, $A(p, y)$, $L(p)$ and $J(p)$ denote that $p$ is a woman, $p$ admires $y$, $p$ is a lawyer and $p$ is a judge respectively. Which of the foll...
23 votes
23 votes
5 answers
3