Is it normal to seemingly not care?

ive noticed for a while now that i have REALLY strange emotions. the only way i can put it is, i care, but at the same time i dont. heres an example. for the past couple of months, my cat has been horribly sick. we found out on monday that he had a failing liver and was getting jaundice from it. the vet, someone my dad knows very well, said that he cant lie, and that there was probably no saving the cat at this rate. i felt really shitty. i had that weird like gut wrench feeling, like "this is gonna happen and i cant stop it. this is it." today will be the first full day without him. i havent cried, or had crazy emotional breakdowns, but i know i care. i know im not a complete asshole. this cat was my life, he loved me so much and i felt the same for him, yet i dont feel absolutely heart broken.
im alot like my dad, i feel. hes really strong with that kind of stuff
is this normal? i know its natural to let out emotions, but i havent really held back anything. it just hasnt come to me. id hate to seem like a dick for not seeming to care about anything.