How to convert ... to logical symbols?

Question

If Bluenose is guilty then no witness is lying unless he is fearful. There is a witness who is fearful. Therefore, Bluenose is not guilty.

$$B\to[(\forall x)(\,(Wx\land\lnot Fx)\to(\lnot Lx)\,)] \tag{1}$$

$$(\exists x)(Wx\land Fx) \tag{2}$$

$$\lnot B \tag{3}$$

Is this interpretation correct ?

I am asked to deduce the conclusion which is $\lnot B$ in this case.

** T.E stands for tautological equivalence

My try is:

$$3.\quad \lnot B \lor [(\forall x)(\,(Wx\land \lnot Fx)\to (\lnot Lx)\,)] \quad1 \; T.E\\4. \quad\lnot B \lor [\lnot (\exists x)\lnot(\,(Wx\land \lnot Fx)\to (\lnot Lx)\,)] \quad \; 3Q1$$

I am not able to deduce the conclusion, I think I misinterpret the sentence.

The following is my last try:

$$\tag{1} B\to (\nexists x)(Wx\land Lx)\oplus (\exists x)(Wx\land Fx) \qquad P$$ $$\tag{2} (\exists x)(Wx\land Fx) \qquad P$$ $$\tag{3} B\to (\exists x)(Wx\land Lx) \qquad 1,2$$

I am still struggling.

Answer 1

There is some English word play going on. I think the author means it as follows.

No witness is lying unless the witness is fearful.

To work out the "unless" operator: $$A\mbox{ unless }B \equiv \neg A\rightarrow B \qquad (\equiv A\lor B)$$ is how it should work. In this topic the unless operator is also tackled.

Accordingly, sentence 1. translates to $$B\rightarrow \neg\exists x\left (Wx\ \land\ (\neg Lx\rightarrow Fx)\right ) \equiv B\rightarrow \neg\exists x(Wx\ \land\ (Lx\lor Fx))$$

To deduce $\neg B$ it suffices to show that the implication $$[B\rightarrow \neg\exists x(Wx\ \land\ (Lx\lor Fx))]\land [\exists x (Wx\land Fx)]\rightarrow \neg B\tag{i} $$ is a tautology.

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For the sake of completeness. Assume implication (i) is false. Then we have two conflicting conditions $$\neg\exists x(Wx\ \land\ (Lx\lor Fx))\quad\mbox{and}\quad \exists x(Wx\ \land\ Fx). $$ The left hand condition is equivalent to $\forall x(\neg Wx\ \lor\ (\neg Lx\ \land\ \neg Fx))$. The right hand condition provides for some $a$ that $Wa\ \land\ Fa$ is true ( i.e there exists a fearful witness). The left hand condition, however, states that all witnesses are truthful and fearless, in particular witness $a$. This is impossible.

Answer 2

.. no witness is lying unless he is fearful.

The common English meaning of this sentence is satisfied if every witness tells the truth. It also is satisfied if some witnesses lie, but every lying witness is fearful.

In short, this clause says that every witness either tells the truth or is fearful:

$$ (\forall x)(Wx \to (\lnot Lx \lor Fx)).$$

We put this together with "If Bluenose is guilty" and we have

$$ B \to (\forall x)(Wx \to (\lnot Lx \lor Fx)). \tag1$$

Now, $(Wx \to (\lnot Lx \lor Fx)) \equiv (\lnot Wx \lor (\lnot Lx \lor Fx)).$ Therefore $(1)$ is equivalent to the following:

$$ B \to (\forall x)(\lnot Wx \lor \lnot Lx \lor Fx). \tag{$1'$}$$

Also, $((Wx\land\lnot Fx)\to(\lnot Lx)) \equiv (\lnot(Wx\land\lnot Fx)\lor(\lnot Lx)) \equiv (\lnot Wx \lor Fx \lor \lnot Lx)$, so your formulation is also equivalent to $(1')$. That is, your formulation is correct (in the sense that it has the same logical consequences as the sentence in English), though it was difficult for me to see this by direct translation.

Another equivalent way to translate the first sentence is $$ B \to \lnot(\exists x)(Wx \land Lx \land \lnot Fx). \tag{$1''$}$$

The rest of your formulation is obviously correct: $$(\exists x)(Wx \land Fx) \tag2$$ $$\lnot B \tag3$$

As has been observed in comments, this system is satisfied by a model in which Bluenose is guilty, there is exactly one witness, and the witness is fearful.

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I think you were intended to translate the first sentence as follows: $$ B \to \lnot(\exists x)(Wx \land (\lnot Lx \lor Fx)). \tag{1a}$$

This might be the meaning of the first sentence if it had the following logical structure:

If (Bluenose is guilty) then no witness is (lying unless he is fearful).

But that is not the way the English language works. A more logic-friendly phrasing of the English sentence is this:

If (Bluenose is guilty) then (no witness is lying) unless (every witness who lied is fearful).

Take away the parentheses and the sentence still means the same thing.