[Math] valid or invalid argument – contradicting arguments

logicpredicate-logicproof-verification

My professor asked us to assess the validity of the following argument:

Some rational numbers are powers of 5. All integers are rational. Therefore, some integers are powers of 5.

My professor went back and forth on the validity of the argument after I questioned his logic. Finally, he asserted that the argument is invalid and he gave this argument:

Now let us abstract this argument, letting R represent the set of rational numbers, P the set of powers of five, and G the set of integers. Then the premises and conclusion become:
1) Some element of R belongs to P.
2) All elements of G belong to R.
Conclusion: Some element of G belongs to P.

As a model of this argument, let R = { a, b, c }; P = { c }; and G = { a, b }.
Are the premises true? 1) Some element of R belongs to P.
2) All elements of G belong to R.
Does the conclusion now follow: Some element of G belongs to P?

I've spoken to another professor who says the argument is valid and I've seen different answers to this problem including this one:

from some textbook - name unknown

Link here:
see exercise 6

So I ask, what is the truth?

EDIT:

In addition, for any open sentence P(x), is
$$\exists x \in \mathbb{Q} P(x)$$
not equivalent to,
$$\exists x (x \in \mathbb{Q} \Rightarrow P(x))$$
and if these are not equivalent, then why does the author, in the example below, rephrase the following quantified statement as an implication?
enter image description here

EDIT 2:
This post here is similar to my last question.

thank you.

Best Answer

The argument is invalid. Here is a refutation by logical analogy:

Some coins are dimes

All nickels are coins

Therefore, some nickels are dimes

The argument based on formal logic notation fails, since it uses the wrong symbolizations. For example, some rational numbers are powers of five needs to be symbolized as:

$$\exists x (Q(x) \land R(x))$$

and not as:

$$\exists x (Q(x) \rightarrow R(x))$$

So ... either the text was asking you to find the error in the 'Solution' ... or the text provided a horribly mistaken Solution! Given how everything else labeled 'Solution' seems to be treated as the actual answerk to the exercises, I fear it's the latter .. what text is this?!

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