Should we also use the inverse of that? I.e., those who don't contribute in taxes should not get a vote? If not, why?
Regarding your other point, that "living somewhere should be reason enough" that is really just a restatement of the conclusion. i.e. "Non citizen residents should be allowed to vote because non citizen residents should be allowed to vote." It's tautological.
To me, it doesn't make sense to say that long term residents who have invested in the community, potentially over generations, should have an equal vote to newcomers who are not even citizens.
Where someone lives is the biggest investment they can make. It cannot be collected by rich people, and even poor people invest where they spend their time.
People putting money in to something is often out of self-interest. Those people already have the control afforded to them via money - an inequality that should be reduced, not exacerbated, by voting.
All of these arguments apply to citizenship too, even more strongly. I've lived the majority of a year on three different continents, but have never changed my citizenship. It's also not accurate to say that the rich are less able to take advantage of this: the fact that I was able to try these things out is precisely because I'm rich.
I don't think the newcomer/old-timer distinction mentioned upthread is especially useful, for the reasons you already outlined. But formal citizenship maps almost precisely to being invested in a place: it's a commitment that you identify with this society and its future, (mostly) to the exclusion of any other society.
People who immigrate and become citizens, yes - they are invested. I don't think people who are natural citizens are as invested (except by default). And I definitely don't think people who are non-citizens are not invested -- they aren't allowed to "invest" in the same way.
For local residency, it's a little different since it's relatively easy to switch residency (at least here in the US).
Agreed, this is a significant mitigating factor, which might make the policy make sense in any given situation.
Enfranchising citizens is also a message to citizens that they should care about the long term of their country, state, or city. If the state will care about you and your posterity then you should care about it. Conversely, if the state won't especially care for you and yours then you don't need to care for the state.
Original Claim: Because I pay taxes I should have a say in how that money is used. Restatement: If I pay taxes, then I should have a say in how that money is used.
If P then M.
You suggested that the OP is making a mistake because he is rejecting the inverse of his statement and that somehow his original claim is weakened as a result.
You say that OP is wrong to reject "If Not P, then Not M", however the inverse of a conditional statement is NOT logically equivalent to the original statement and as such it's rejection has nothing to do with the validity of the original claim.
The inverse of a conditional does not follow from the original conditional. What follows from the original conditional statement "If I pay taxes, then I should have a say in how that money is used" is the contrapositive: "If I do not have a say in how the [tax] money is used, then I do not pay taxes". That is all, nothing more. You cannot infer anything beyond that from a simple conditional.
I suspect you have committed the inverse error. https://en.wikipedia.org/wiki/Denying_the_antecedent
Another way of looking at it is, suppose the original comment said "I am tall, therefore I deserve to vote." If someone replied "Do you think short people deserve to vote?" What this line of questioning is trying to pin down is, what is it, exactly, that entitles people to vote. If short or tall people can vote then it seems that height is not really related to the right to vote. Analogously, if tax payers and non-payers alike can vote, then it seems you think paying taxes is unrelated to the right to vote.
To return to symbolic logic - "If P then M" and "If not P then M" means that P isn't really related to M. P says nothing about whether or not M. That's the point I was trying to make by asking about the inverse.