Suppose you claim that it is impossible for a debt to create a burden on future generations unless: it causes a reduction in investment that reduces future output; the higher future taxes cause disincentive effects that reduce future output; it is owed to foreigners so we have to pay part of our future output to foreigners and have less future output for ourselves.
Suppose I then come up with one counter-example, with no investment, no disincentives, and no foreigners. Where the debt can't reduce future output, because I'm holding future output constant. But where the debt does make some future generation worse off.
Doesn't that prove that your claim is wrong?
You don't start complaining that my counter-example is "unrealistic", because I'm assuming output is constant, or making some other assumption you don't think is realistic. It's a counter-example.
Here's my (old) counterexample:
Assume: closed economy; no investment or real capital of any kind; lump-sum non-distorting taxes with zero collection costs; positive real interest rate and zero real growth; exogenous full-employment level of output; apples are the only output good; apples cannot be stored; identical agents; overlapping generations; no funny stuff.
Suppose the government makes a transfer of 100 apples to the current cohort, financed by borrowing. Does that create a burden on future generations?
The government borrows 100 apples from each of cohort A, then gives each person in cohort A a transfer payment of 100 apples. It is exactly as if the government had simply given each person in cohort A an IOU for 100 apples. That IOU is a bond.
So far there is no change in cohort A's consumption of apples.
Cohort A then sells the bonds to the younger members of cohort B. So each person in cohort A gets an extra 110 apples (assume 10% interest per generation), which he eats. Cohort A then dies.
Cohort A is better off. Each member of cohort A eats an extra 110 apples. In present value terms, those extra 110 apples are worth 100 apples at the time the transfer payment is made.
Cohort B eats 110 fewer apples when young, but 121 extra apples when old, and they sell their bonds to cohort C. Although cohort B eats 11 more apples in their lifetimes, the present value of their total consumption of apples is the same. The rate of interest must be high enough to persuade them to eat fewer apples when young and more apples when old, otherwise they wouldn't have bought the bonds from cohort A. So cohort B is not worse off.
But (given my assumption) the debt is rising faster than GDP. The government knows this is unsustainable. It cannot rollover the debt forever, because eventually the next cohort will be unable to buy the bonds from the older cohort. So the government decides to pay off the debt by imposing a tax of 121 apples on each young person in cohort C, which it uses to buy back the bonds from cohort C.
Each member of cohort C eats 121 fewer apples.
Cohort A eats more apples, and cohort C eats fewer apples. It is exactly as if apples travelled back in time, out of the mouths of cohort C into the mouths of cohort A. (With interest subtracted as they travel back in time through the time machine.)