I misread your post and rescind the strawman comment.
Appreciate and respect that.
And that is mathematically false how?
Because it assumes the person writing in would automatically be a vote for Hillary Clinton, since you indicated it was 1 less for her and 1 less than Trump had to overcome. On the contrary, it could be one less vote for Trump and 1 more for Hillary to overcome. There is zero guarantee that just because they're voting for Bernie Sanders, that Hillary would be their vote if they were forced to vote for one of the two party candidates.
I assume, is the knowledge that you voted for someone who most closely matches your principles.
Decent assumption, and for most going 3rd party it's probably accurate. I've explained mine elsewhere but here's my general take.
The first criteria for me is do I believe that the person has a high probability of significantly damaging the foundational principles of this country as I see them or significantly damaging the well being of the country, beyond simply a disagreement on policy from an ideological statement. As long as I can reconcile that part in my head, then I can move onto step two.
Step two is looking at those candidates that I feel meet criteria one, and determining first...which are most likely to win (ie, one of the two major parties). Then, once I determine that, it's which one is most likely to support the most things I support.
The issue is if neither major party candidate meets criteria 1 for me, then it ends. Essentially, on a scale to 0 to 10, if you don't meet criteria 1 then you're a 0. There's no lower number than 0, and there's no going higher than zero if you can't pass the first criteria. If there's an election where I feel both major party candidates can't get past criteria 1, then neither are going to get my vote.
TL;DR: The knowledge that that my vote, my support, is being given to someone that I at least find "acceptable" is what's important to me.
I don't believe that either of our goals are inherently wrong if we're honest about the outcomes we desire.
Agree 100%
But mathematically, I don't see how you can argue against the statement that a vote for a candidate that cannot win takes a vote away from a candidate who can potentially win.
If someone told me that voting 3rd party or staying home takes a vote away from BOTH candidates, or even if they said EITHER candidate, then I could at least stomach that and agree with that from a mathematical stand point. The issue is this automatic assumption that the vote would go to [x].
Is it LIKELY that most libertarian 3rd partiers, if a gun was put to their head, would vote for Trump over Hillary? Sure. Is it LIKELY that most green or Bernie write in people, if a gun was put on their head, would vote for Hillary over Trump? Sure. But it's not guaranteed. There's people who may simply despise their "normal" parties candidate so much that they would vote the other way. There's people who would possibly vote the other way out of spite due to the party passing on their guy. There may be some who go with the theory that it's better to have the opposition party in, because then there's someone to right against and rally against in 4 years instead of sticking to supporting crap for the next 8. There may be some who want to vote for Trump because the attraction to Bernie is his anti-establishment nature, and they like that in Trump more than they like Hillary's policies. There may be some who want to vote for Hillary, because she may actually be more for "free trade" than Trump is. And it can go on and on.
I have no issue mathematically stating "Voting for a 3rd party candidate takes a vote away from the two main party candidates.". That's essentially accurate at 100% of the time when you're at least giving SOME kind of vote. My issue is with claiming that it's specifically taking it away from ONE candidate or another.
If I vote 3rd party in Virginia and Trump wins, I didn't cause him to win or cause Hillary to lose, because I caused them BOTH to lose a potential voter. Similarly, if Hillary wins Virginia and Trump wins, I didn't cause her ot win or cause Trump to lose, because I caused them BOTH to lose a potential voter.
Without a statement of certainty, or near certainty, from me of "If I were to vote major party, my vote would be [x]", ultimately who I'm "costing" a vote is a complete variable, and thus can only be spoken about as a POTENTIAL loss for both, which essentially negates itself. It is only once I've made it clear that my vote would go to one or the other IF it was to go to either that you can definitively say I cost [x], and [x] alone, a vote.