So, you will be able to explain how it is possible for a person to think he exists even if he does not exist.
If the person is mistaken about his own nature.
This is one of the problems with Descartes' theory.
By definition, for thinking to occur the verb (think) must have a subject. Descartes simply reiterates conventions of grammar
. But we have to wonder: when does relying on conventions of grammar cease to produce accurate models of reality and begin to reinforce conventions of convenience as reality? Are appeals to conventions of grammar acceptable arguments at all?
Obviously, this depends. On a day to day level, out of the need for clear and understandable communication between people of vastly different educational, social, and philosophical backgrounds, we can safely rely on conventions of grammar. In this common use of language as convenient communication we can rightly argue that when we use a verb, the action must belong to some subject. If we use an adjective, the description must belong to some object. To speaking of thinking existing without something to think is pretty useless in conversational language.
But philosophy is not conversational language, and some of the most celebrated innovations in philosophy have been rejections of conventional language. The subject-object problem was turned on its head when Kant proposed that subjects affect the way an object is observed. What if the basic assumptions underlying sentence structure, while being effective means for most communication, are metaphysically misleading?
Look at the way you ask the question: how is it possible for a person to think he exists even if he does not exist. The very asking assumes that a person does exist. What if the person does not exist? Instead of presupposing a self in the first place, shouldn't we reformulate the question into something like: "How is it possible for thinking to occur if a self does not exist?" We have to rephrase the question in order to avoid begging the question.
For Descartes, the self was a thinking substance, thinking being the essential characteristic of the self. A given substance exists independently of the rest of reality. Thus, the self exists independently of the rest of reality. But if we, through empirical evidence, show that thinking does not exist independently of the rest of reality then we can disprove Descartes immediately. Even Descartes recognized the trouble he faced from such an objection, which he tried to deflect by proposing his medical speculations about the pituitary gland. Today, of course, Cartesian dualism has fallen out of fashion, among other reasons, because it cannot account for modern medical findings.
Descartes made no attempt to demonstrate by logic that only a self could think, or do whatever appears to be the act of thinking. Descartes argued that identity and substance could be assumed because he thought they were inherent to thinking: he believed that when a person thinks identity and substance are immediately understood. What he does not explain is why something must be true simply because people seem to assume it to be true. Descartes was familiar with philosophers who denied a static self, but he did not seem to mind ignoring their claims.