I am looking for the propositional logic translation of "even if", not the epistemic logic meaning.
Specifically, how do I translate
On
Let's actually make a truth table for "even if". We get: \begin{array}{cc|ccc} P&Q&P&\text{ even if }&Q\\\hline T&T&T&\mathbf{T}&T\\ T&F&T&\mathbf{T}&F\\ F&T&F&\mathbf{F}&T\\ F&F&F&\mathbf{F}&F \end{array} where the first two are trivial, and $P$ being false and $Q$ being true would be something like "Grass is blue even if it's green" and the last would be something like "Grass is blue even if it's red", which are both obviously false, because grass is green.
So after all, $P$ even if $Q$ is really just saying $P$. So if you're saying: "I'm going to die even if I'm careful" really just comes down to "I'm going to die".
On
I think the answer with the truth table nails the bare semantics of it: P even if Q is strictly-semantically the same as Q.
It's okay if that seems to leave something out. Logic doesn't capture what linguists call implicature. Implicature is the inferred meaning taken from the situation and the fact that the speaker bothered to say it. The rub of it is that can you pass the salt doesn't just convey 'do you have the ability to pass the salt' (or perhaps 'is it possible for you to pass the salt'), even though a strictly semantic interpretation produces that paraphrase. The additional meaning is given by the fact that the speaker said something and the assumption that people say things for reasons. There is some interesting work on implicature and game theory: there is a rigorous deductive way to show how implicature works.
So to your question again: the fact that the speaker bothers saying even if I am careful can convey something that I will die does not convey, and it can do so even if its antecedent and the whole even-if conditional have the same truth conditions. (If I had to guess, P even if Q conveys 'P, oh and contrary to what you might believe, Q doesn't change the fact that P', which you could express with doxastic logic.)
On
D= your dying
C= being careful
I have uncovered an aporia within Logic by that the proper way to translate "I will die even if I am careful." is D & ~[(C v ~C)→ D], as you have effectively stated the contradiction, D & ~D.
When you say "I will die even if I am careful", you are effectively stating that you will die and that it doesn't matter as to whether or not you are careful. You're both making the claim that you will die and that you're being careful has no bearing upon your dying. The negation of the conditional, within most systems of Logic, however, seems to result in the exact opposite of what you are attempting to express. This, I believe, should result in the creation of an entirely other field of Logic, or a radical reconception of it as it stands, but, if you are asked this question on a test, just write "D", as that is what every Logic textbook will tell you to do.
P (even if) Q <=> P and (Q or not Q) <=> P