Using logical expressions to represent statements

Question

I having trouble with a question on my assignment and I've exhausted other avenues of help. So here's the question (my prof speaks English as a second language and it shows in the wording of this question):

One day Lady Smith returned home and was surprised by a big birthday cake prepared by her three children, named Ada, Bob, and Chris. She was reliably informed that only one of them made this special gift for her. Regarding which child made the cake children requested Lady Smith to figure it out. For this purpose Ada and Bob made their following statements:

Ada: I did not make the cake. Bob: I actually made.

Lady Smith then asked Chris: "Whether this cake really made by Bob?" Chris answered: Yes, he did!

Lady Smith knows that at least two of the children did not tell the truth. Please help Lady Smith to prove that Ada made the cake by using Resolution and Unification.

So the part I'm stuck on is part a of the question which asks us to "use logical expressions to represent essential statements above".

I was thinking something like this, but I'm not sure that's a valid way to actually do this.

Says (x, y) = x says y Bake(x) = x baked the cake

$Bake(Ada) \lor Bake(Bob) \lor (Bake(Chris) \equiv T$ $Says(Ada, \lnot (Bake(Ada)) \to Bake(Bob) \lor Bake(Chris)$ $Says(Bob), Bake(Bob)) \to Bake(Bob) \land \lnot Bake(Ada) \land \lnot Bake(Chris)$ $Says(Chris, Bake(Bob) \to Bake(Bob) \land \lnot Bake(Ada) \land \lnot Bake(Chris)$

Answer 1

HINT

As Henning already said, you really don't want to use a Says predicate, as it will complicate things, and unnecessarily so. in fact, I would just use propositional logic here: A: Ada baked the cake, B: Bob baked the cake, C: Chris baked the cake

Also, make sure that exactly one of them baked the cake, so instead of just using $A \lor B \lor C$ (which allow for multiple bakers) you should rule out two (or more) bakers, so you get the following statements:

$A \lor B \lor C$

$\neg (A \land B)$

$\neg (A \land C)$

$\neg (B \land C)$

To represent that at least two of the children were not telling the truth, you can represent the claims that it is not possible to have two children telling the truth.

Now, Ada says $\neg A$, Bob says $B$, and Chris says $B$. So, we must have:

$\neg (\neg A \land B)$ (Ada and Bob are not both telling the truth)

$\neg (\neg A \land B)$ (Ada and Chris are not both telling the truth)

$\neg (B \land B)$ (Bob and Chris are not both telling the truth)

So, in sum, you get:

$A \lor B \lor C$

$\neg (A \land B)$

$\neg (A \land C)$

$\neg (B \land C)$

$\neg (\neg A \land B)$ (duplicate removed)

$\neg (B \land B)$

These becomes the following clauses:

1. $A, B, C$

2. $\neg A, \neg B$

3. $\neg A, \neg C$

4. $\neg B, \neg C$

5. $A, \neg B$

6. $\neg B$

2,4, and 5 are subsumed by 6, so can be removed:

1. $A, B, C$

2. $\neg A, \neg C$

3. $\neg B$

1 and 6 resolve to:

7. $A,C$

... and that's as far as you can get (resolving 3 and 7 only gives you tautologies).

So: it seems there are 2 solutions: Ada baked the cake, or Chris baked the cake!

Answer 2

A simple notation: $X$=$\{X$ made the Cake$\}$, $Y(X)$=$\{$Y sais X made the cake$\}$. If $Y(X)$ is true then $X$ made the cake. If $Y(X)$ is false then $X$ did not make the cake. $\ell (Y(X))$ will mean that $Y(X)$ is a lie; $Y(X)$ is false that is $\lnot X$ is true

We have that $$A(\lnot A)\land B(B) \land C(B)$$ There is no contradiction. However, we know that at least two of the guys lied.

So, we have the following possibilities

$$\begin{bmatrix} \ell(A(\lnot A))&\land &\ell(B(B))&\land&\ell(C(B))\\ A(\lnot A)&\land&\ell(B(B))&\land&\ell(C(B))\\ \ell(A(\lnot A))&\land&B(B)&\land&\ell(C(B))\\ \ell(A(\lnot A))&\land&\ell(B(B))&\land&C(B)\\\end{bmatrix}= \begin{bmatrix} A&\land&\lnot B&\land&\lnot B\\ \lnot A&\land&\lnot B&\land&\lnot B\\ A&\land&B&\land&\lnot B\\ A&\land&\lnot B&\land&B\end{bmatrix}.$$

It is only the first line that doesn't contain a contradiction. They all are liars!