Formulate the problem using the given predicates.

Question

Statement : If Mr.M is guilty, then no witness is lying unless he is afraid

Also given in question : There is a witness who is afraid.

Given predicates :

• G−Mr.M is guilty
• W(x)−x is a witness
• L(x)−x is lying
• A(x)−x is afraid

Answer according to me is : G⟹∀x(¬L(x)⟹¬A(x))

However according to this answer site , the answer is : \begin{array}{c} G \implies \lnot \exists x: \Bigl (W(x) \land L(x) \land \lnot A(x) \Bigr )\\[1em] \equiv\\[1em] G \implies \forall x: \Biggl (W(x) \implies \Bigl ( \lnot A(x) \implies \lnot L(x) \Bigr ) \Biggr ) \end{array}

If I am wrong, kindly explain where my logic went wrong?

Answer 1

You made two mistakes:

1. You forgot to say that you are talking about witnesses. So, you need to add a $W(x)$ predicate

2. You need to say that for all witnesses $x$: '$x$ is not lying unless $x$ is afraid', which is equivalent to '$x$ is not lying if $x$ is not afraid' (whenever you see 'unless', just substitute 'of not'!) , which is symbolized as $\neg A(x) \to \neg L(x)$ You have $\neg L(x) \to \neg A(x)$, which is just the other way around