need feedback on some simple discrete math proofs

Question

I have a couple of proofs that I am not sure if I have solved correctly and was wondering if someone could give me some feedback. You guys are the best! Here are the proofs:

Using rules of inference, determine whether the argument being made is correct or incorrect. Provide a sequence of premises and conclusions justifying your answer. Be sure to state all of your premises and domains.

1.) All calicos like milk. My cat, Mr Boots, is not a calico. Therefore, Mr Boots does not like milk.

C(x) = x is a calico

M(x) = x likes milk

1. ∀x(C(x) ∧ M(x))

2. ∃x(¬C(x))

3. C(x) ∧ M(x) universal instantiation

4. ¬C(x) ∧ ¬M(x) inverse law 3

5. ¬M(x) ∧ ¬C(x) commutative law

6. ¬M(x) simplification 5

7. ∃x(¬M(x)) Existential Generalization

2.) All tennis players are fast. John is a tennis player. Therefore, John is fast.

P(x) = ”x is a tennis player”

F(x) = ”x is fast”

1. ∀x(P(x) ∧ F(x))

2. ∃x(P(x))

3. P(x) ∧ F(x) Universal Instantiation 1

4. F(x) ∧ P(x) commutative law

5. F(x) simplification 4

6. ∃x(F(x)) Existential Generalization

Answer 1

For 1. Premise 1 should be $\forall x (C(x) \to M(x))$, because if you use $\forall x (C(x) \land M(x))$, you end up saying that everything is a calico and likes milk, which is far too strong. You want to just say that all calicos like milk, which is to say that if something is a calico, then it likes milk.

Then, I would use $b$ as an individual constant denoting Mr. Boots, so you have $\neg C(b)$ as premise 2, and the conclusion is $\neg M(b)$

Now, think about this: does that sound like a valid argument to you? I mean, I am not a calico, so does that mean I don't like milk? No, this argument is invalid.

Indeed, you made a mistake in your proof on line 4: I don't know how your 'Inverse Law' is defined, but I am certain it does not allow you to infer $\neg C(x) \land \neg M(x)$ from $C(x) \land M(x)$, since that does not follow ... not even close!

For 2: Premise 1 should be $\forall x (P(x) \to F(x))$, just as with the first argument. Also, I would use $j$ for John, and so premise 2 is $P(j)$, and the conclusion is $F(j)$

This one is valid, but note how in your proof you never use premise 2, which should strike you as really weird, because premise 2 is clearly necessary for the validity of the argument. So, what happened? It is because you used a $\land$ in premise 1, which again should be a $\to$. Change that, and you'll find that you will need premise 2 to prove the conclusion.

Answer 2

1. Add B(x) for x is Booth.
   1.1 Should be for all x, (C(x) implies M(x)).

2. Same mistakes. Add J(x) for x is John.
   2.1 Should be for all x, (P(x) implies F(x)).

With the failure of the first step, the rest needs to be revamped.