Why does there have to be something objective? Are you trying to imply that whatever exists empirically is objective? I thought we were talking about objective "truth"?
Whatever exists empirically is true in the a posteriori sense of truth. If you're on "Who wants to be a millionaire" and they ask you when Martin Luther King was assasinated and you answer "April 4th, 1968", that is the truth. Basically, everything that you need an outside source to confirm its validity is truth in the a posteriori sense. Like the assasination of MLK, if you don't know when it was you need some kind of source to tell you when it was. No matter how long and hard you think about it, it's impossible to get there if you don't know it.
A priori on the other hand can be worked out without an outside source. If you don't know what the answer for 2+2 is, you can work it out with your intellect. This might be easy for some and difficult for others, but it is possible. You don't need any outside source.
But no statement about objective truth can be derived from empirical truth. Even if you've spotted 1000000000000000000000 white swans, you can't say for sure that the next swan you're going to see isn't going to be blue. It's just highly unlikely. The same is true for causation. Without previous observation, it would be impossible to say what the effect of any given cause may be. You may want to read about the problem of induction: en.wikipedia.org/wiki/Problem_of_induction Or even better, Hume himself: gutenberg.org/ebooks/9662
A priori is merely they layout of the biological circuitry, that which we cannot transcend. Yes 2+2 can never equal 5, if it did it means a wire has crossed, which would not be conducive to our survival. The way we were wired is merely a tool for our survival, and we may not transcend it. Fire has always burnt me in the past, therefore I know longer touch it. Espousing knowledge in text books has brought me good grades in the past therefore I believe what they say as far as Einstein goes, and because it fits within our circuitry, the one that makes sure 2+2=4 and not 5, because if we're not right with our a priori we aren't going to get very far. It's also a priori that we are from birth conditioned to obey our experiences.
I'm honestly not sure what you're trying to say. Obviously, if you ask a question like " Does something have to be objective to be true?" you have to believe in the fact that the world - and by that I mean, everything - is based on logic. If you think that everything is irrational there is no point in asking or answering questions because you could never find out any answer. I'm sure this belief is something that most INTPs will agree with. We're so Ti-heavy and deduct all the time that it would be nearly impossible to convince us that this deductive process is not leading us to true conclusions.
What you're saying about fire and Einstein is a posteriori though. I'm sure you're aware of that, just saying.