Learn English – “Listen the dear two-year-old son, this is (a) cat, and that is (a) dog.” — Leaving out the article in definitions like this

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I'm talking to my (nonexistent) two-year-old son, and he asks me: "Daddy! What's cat? What's dog?", and I point at a cat and then at a dog, saying:

This is (a) cat; that is (a) dog.

There shouldn't be a serious problem with including the indefinite article. But is it not better to leave it out in this context? after all, I'm not concerned with that individual cat; I just want to show him what the word cat means.

I guess this is the same as the situation I was in with my (imaginary) ten-year-old daughter, when she asked me: "Daddy! What does 'a schizophrenic' mean?", and I confidently answered: "Honey, if you want to know about the word, you don't need to put 'a' before it. Here's a dictionary; look 'schizophrenic' up."

So, a cat, or cat?

Addendum:
I understand that This is cat is normally an ungrammatical sentence. But here I'm presenting a context that kind of reverses the positions of Subject and Subject Complement. That is, if I say

This is 'cat'.

with a stress on This and a lowering tone for the rest of the sentence, what I actually mean is

'Cat' is this. = 'Cat' means this.

And no, I'm not asking about the adjective schizophrenic. It's the noun.

Addendum #2:
Turns out my mistake (or one of my mistakes) was in thinking that is can be used like means. I'm accepting (the first half of) @cathygomez's answer for pointing that out.

Best Answer

When you give a definition for a word, don't you usually say something like this

A cat is a four-legged animal with ...

or this?

Schizophrenic means ...

I don't think you should drop the article when you're going for the first type of response. That way, you're teaching the two-year-old about grammar rules while you're at it.

I don't think it matters whether you're referring to a specific cat or not, because "Cat is a four-legged animal ..." is not grammatically correct. You either say "A cat" or "The cat", else you say "Cats are ..."

In any case, when you showed your imaginary son the picture of a cat and a dog, it didn't matter that you weren't talking about that specific cat or dog, because at that point, you were referring to a specific cat and dog (the ones in the pictures).

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