The syllogism is logically valid.....

Yes, it is. So what?

1) All cats are blue.

2) I own a cat.

3) Therefore my cat is blue.

is ALSO an entirely valid argument. Problem is, it is not a SOUND argument, since at least one of its premises are false.

Perhaps only validity counts to you? If so then fine - your argument is valid, well done, have 100 Quatloos. But you'll have to give them straigh back to me because my argument on blue cats is precisely as valid.

For there to be ANY point to your argument, you need to prove that your premises are true, which would then - and ONLY then - make the argument sound. At the moment, its as if I am requiring YOU to prove that all cats are NOT blue (AND that I do NOT own a cat), rather than me proving my premises to you, which is the usual method where there is disagreement.