Solving this logical puzzle by resolution doesn't work for me

Question

In this document there is a logical puzzle:

If the unicorn is mythical, then it is immortal. If the unicorn is not mythical, then it is a mortal mammal. If the unicorn is either immortal or a mammal, then it is horned. The unicorn is magical if it is horned.

They go on by defining:

$M: \text{unicorn is mythical}$

$I: \text{unicorn is immortal}$

$L: \text{unicorn is mammal}$

$H : \text{unicorn is horned}$

$G : \text{unicorn is magical}$

Putting the definitions and the text together they get to this formula

$$(M \rightarrow I) \wedge (\neg M \rightarrow (\neg I \wedge L)) \wedge ((I \vee L) \rightarrow H) \wedge (H \rightarrow G) $$

which is the first thing that doesn't make sense to me, since the text says "either immortal or a mammal". Shouldn't it be an xor instead of $\vee$?

Anyway, they continue by resolving the formula in this way:

\begin{align*} &(M \rightarrow I) \wedge (\neg M \rightarrow (\neg I \wedge L)) \wedge ((I \vee L) \rightarrow H) \wedge (H \rightarrow G) \\ &(\neg M \vee I) \wedge (M \vee (\neg I \wedge L)) \wedge ((I \vee L) \rightarrow H) \wedge (H \rightarrow G) \\ &(\neg M \vee I) \wedge (M \vee \neg I) \wedge ( M \vee L) \wedge ((I \vee L) \rightarrow H) \wedge (H \rightarrow G) \\ &(I \vee L) \wedge ((I \vee L) \rightarrow H) \wedge (H \rightarrow G) \\ &H \wedge G \end{align*}

This confuses me even more:

First, how did they resolve $(\neg M \vee I) \wedge (M \vee \neg I) \wedge ( M \vee L)$ to $(I \vee L)$? If we put together $(\neg M \vee I)$ and $( M \wedge L)$ we get $(I \vee L)$. Putting that together with $( M \vee \neg I)$ we get $(M \vee L)$.

Second, when we rewrite the forelast line and resolve we get

\begin{align*} &(I \vee L) \wedge ((I \vee L) \rightarrow H) \wedge (H \rightarrow G) \\ &(I \vee L) \wedge ((\neg I \wedge \neg L) \vee H) \wedge (\neg H \vee G) \\ &(I \vee L) \wedge (\neg I \vee H) \wedge (\neg L \vee H) \wedge (\neg H \vee G) \\ &(H \vee L) \wedge (\neg L \vee G) \\ &H \vee G \end{align*}

so not $H \wedge G$.. I'm confused. Am I right or wrong?

Answer 1

It’s true that either ... or often means exclusive or, but that is by no means always the case. Consider the following sentence:

If John is either a veteran or over the age of 65, he is entitled to a discount at XYZ Market.

The normal reading of this sentence is that John is entitled as long as he meets at least one of the qualifications: he’s a veteran, or he’s over 65, or he’s a veteran who is over 65. This interpretation, with so-called inclusive or, is the normal reading of any sentence of this type. In particular, it’s the normal reading of this sentence:

If the unicorn is either immortal or a mammal, then it is horned.

Thus, the symbolic translation is indeed $(I\lor L)\to H$.

You’re quite right in thinking that $(\neg M\lor I)\land(M\lor\neg I)\land(M\lor L)$ is not equivalent to $I\lor L$: the first expression is false when $M$ is false and $I$ is true, while the second is true. However, that’s not what the author is claiming: he’s claiming only that the first implies the second. And that is true: as you noted, $(\neg M\lor I)\land(M\lor L)$ gives you $I\lor L$, and it turns out that we simply don’t care that $M\lor\neg I$ is also true.

That takes us down to

$$(I\lor L)\land\big((I\lor L)\to H\big)\land(H\to G)\;.$$

The first two conjuncts then give us $H$, so we can deduce $H\land(H\to G)$, and from that we get $H\land G$.

Answer 2

The problem is easily solved from the text version.

1. If the unicorn is mythical, then it is immortal.
2. If the unicorn is not mythical, then it is a mortal mammal.
3. If the unicorn is either immortal or a mammal, then it is horned.
4. The unicorn is magical if it is horned.

From (1+2) it follows that the unicorn is either immortal or a mortal mammal. Therefore the if-condition in sentence (3) is true, which implies that the unicorn is horned. Sentence (4) then asserts that the unicorn is horned.

In conclusion: H is true and G is true.

Answer 3

As Brian M Scott have brilliantly answered above, the trick of the solution is to note that

$(¬M∨I)∧(M∨L) \Rightarrow I \vee L$

Recall that logical reasoning is all about entailment, and if certain sentence, say, $\alpha$, entails other, say, $\beta$, it does not follow that both sentences are logically equivalent, although it can happens to be the case if $\beta$ entails $\alpha$ as well.

It might be helpful to get a insight of the above assertion by noting that:

$$\begin{align} (¬M∨I)∧(M∨L) & \Rightarrow (¬M∧M)∨(I∧L) \tag{1} \\ &\Rightarrow \bot \vee (I∧L) \tag{2} \\ &\Rightarrow I∧L \tag{3} \\ &\Rightarrow I∨L \tag{4}\\ \\ \end{align}$$

That is, we can distribute $¬M∨I$ over $M∨L$, eventually making some simplifications to get our desired result $I \wedge L$. Together with $((I∨L)→H)$ and $H \wedge G$ get $H$ and $G$ by modus ponens, respectively. Hence the document's answer:

$G \wedge H: \text{the unicorn is horned and magical}$