Correct transcription to predicate logic

Question

Consider the following predicates: $$T(x) = x \text{ is a train}$$ $$P(x,y) = x \text{ is a train of type } y$$ Now if I want to convert the sentence: there exists a train of type $A$ into predicate logic, which of the following would be correct? $$\exists x(T(x) \land P(x, A)) $$ or $$\exists x P(x, A)$$

I think it should be the first one, but since the definition of $P$ says that $x$ is a train, maybe the second one is also valid.

Answer 1

Indeed, if $P(x,A)$ says "$x$ is a train of type $A$", then also declaring $T(x)$, "$x$ is a train", would be redundant. $$\forall x~.(P(x,A)\to T(x))$$

So both are correct, though the second is all you need.

There exists a train that is of type $A$: $\quad\exists x~.P(x,A)$

Answer 2

I would reformulate P as follows:

$P(x,y)=x $ is of type $y$

And then just use the first formula. The reason for this reformulation is that in transcribing in First Order Logic we want to have as many relations as possible, not overcrowding a relation with more than one adjective. This helps in checking for valid arguments. If we don't follow this rule some arguments may appear to be not valid if we check them formally.