# Compound proposition from Alice in Wonderland

I am analyzing the following section from Alice in wonderland: If it makes me grow larger, I can reach the key; and if it makes me grow smaller, I can creep under the door; so either way I'll get into the garden, and I don't care which happens!.

I define the following propositional variables: $\\$

L= makes me grow larger

K= reach the key

S= makes me grow smaller

D= creep under the door

G= get into the garden

My task is to detemrine the compound propustition. I have done it this way:

$((L \rightarrow K) \mathrm{V} (S \rightarrow D)) \rightarrow G$

Does that look right ?

• Are you sure it need to be one statement? I see it as three 1)$L\to K$ 2)$S\to D$ and C) $L \lor S \to G$. and there is a fourth unstated hypothesis 2) $K \lor D \to G$ – fleablood Mar 4 '18 at 18:14
• That is not correct because it is't $L\to K$ nor $S\to D$ that implies $G$. It is $K$ and $D$ that imply $G$. And also its not a choose of $L\to D$ or $S\to D$. She knows both of those are true. It's a case of $L$ or $S$. – fleablood Mar 4 '18 at 18:17
• If I had to put it in 1 sentence: I'd say "I know L or S and I know L-> K and S-> D therefore G" with the unstated "I assume it goes without saying that K or D -> G". – fleablood Mar 4 '18 at 18:20
• Of course Alice's reasoning is flawed in that $L\to \lnot G$. Whether $K$ or not. It's only $(K\land \lnot L)\to G$..... But that's not part of this exercise. – fleablood Mar 4 '18 at 18:25

No: in order for Alice to get into the garden it needs to be both true that being larger she can get the key, and that being smaller she can creep under the door. So:

$((L \rightarrow K) \color{red}\land (S \rightarrow D)) \rightarrow G$

I think you're trying to use the $\lor$ because you're thinking of:

$(K \lor D) \rightarrow G$

which, given $L \rightarrow K$ and $S \rightarrow D$, would mean that:

$(L \lor S) \rightarrow G$

In fact, to make that inference you could also use that:

$(L \lor S) \rightarrow (K \lor D)$

And all of those constructions are compatible with Alice's reasoning.

To further see and understand the confusion you have, please note that:

$(K \lor D) \rightarrow G \Leftrightarrow (K \rightarrow G) \land (D \rightarrow G)$

So yes, it's easy to confuse the use of the $\lor$ with the $\land$ in these kinds of constructions!

Finally, though, I would make all this into an argument, rather than a single statement. That way, you can make some of the implicit assumptions in her reasoning explicit, such that she will get either smaller or larger, and that she can get into the garden with the key or by creeping under the door.

$L \lor S$

$L \rightarrow K$

$S \rightarrow D$

$K \rightarrow G$

$D \rightarrow G$

$\therefore G$

Or maybe:

$L \rightarrow K$

$S \rightarrow D$

$K \rightarrow G$

$D \rightarrow G$

$\therefore (L \lor S) \rightarrow G$

• If we had to do one statemet would you agree with $[(L\lor S)\land [(L\to K)\land (S\to D)]\land [(K\lor D\to G]]\to G$? – fleablood Mar 4 '18 at 18:22
• @fleablood Sounds good! Allow me to add that to my post? I'll credit you! – Bram28 Mar 4 '18 at 18:23
• Thanks for the great clarification @Bram28 – Elias S. Mar 4 '18 at 18:43
• @EliasS. You're welcome! :) – Bram28 Mar 4 '18 at 18:45