@Jessica phil,
But couldn't something like "pure thought" exist? (For example, in a "thought generator" created/evolved far in our future?) And if this unembodied thought nonetheless possessed an internally-conceptualized self-concept, of a singular identity or "I", which did not in fact point to something actually existing, then the "pure thought" would be wrong in that case.
To make this clear, we must understand that the human brain harbors internally-represented concepts, some of which point to things that actually exist -- like the plants and animals and world around us -- and some of which point to things that do not actually exist -- like Santa Claus, Fairies, Unicorns, etc.
One's self-concept in some cases is accurate -- if I consider I am a mortal human being, etc. -- and in some cases not accurate -- if I consider I am a god, or an android, or an alien in human skin, etc.
So, if "pure thought" existed, without an actual self or unique "I" -- then the "pure thought" would merely be thought itself, and therefore "I exist" would be incorrect, while "thought exists" would be correct. That is, "I think" is begging the question by attributing existing thought to an "I" or self. So, if thought exists, how does that thought know that an "I" exists, and that the thought belongs solely to this "I"?
And, it is begging the question to claim that "I doubt my own existence" since, again, the "I" is assumed to exist, then it is concluded from that that "I must exist to doubt". To avoid that logical fallacy, this must be worded, "Doubt exists about that 'I' which is assumed to originate these thoughts", and so on.