The argument on the linked site is somewhat out there, but the reasoning is based on a sound premise, it is impossible to truthfully assert "I am dead", if you are truly dead you will not be in a position to assert it. The move from that to immortality is a leap however, unless one adopts a stance that is both solipsistic and legalistic: X is true if I can never truthfully assert its negation. So if I can never truthfully assert that I am dead I must always be not dead.
In this context however the meaning of "always" has been illicitly redefined to beg the question, and so has the meaning of "immortality". Normally when someone says that a person is dead they surely assume that the person wouldn't be there to assert otherwise, so they surely do not have in mind this kind of 'immortality'.
Heidegger in Being and Time gives a sophisticated version of such reasoning, but distinguishes the two meanings as "now" and "then":
"Of course, only as long as Dasein is, that is, the ontical possibility of the understanding of being is, 'is there' being. If Dasein does not exist... such a thing is then neither understandable nor not understandable... Then it can be said neither that entities are, nor that they are not. Nevertheless, it can now be said - as long as the understanding of being, and thereby the understanding of occurrentness are - that then entities will continue to be".
Then again, perhaps the now-immortality is all that should matter.