Could the "Evil Genius" deceive you regarding the "Cogito"?

One of the first questions that must surely arise from Descartes' infamous statement "Cogito, ergo sum" must be why it is that Descartes believes that that evil genius cannot deceive you about the validity of this argument in exactly the same way as she is able to deceive you about all the other seemingly self-evident "facts" that you believe you know? Could it not be the case that, while you believe it to be valid to deduce your existence from the belief (and hence thought) that you think, it really is not; and that it is the evil genius that is leading you to believe otherwise?

Consider the situation that arises if this is indeed the case; that is, assume that the evil genius can deceive you into believing that the "cogito" is correct when in fact it is invalid, and consider what happens when she does try to do so.

Two main observations arise from the consideration; firstly, you are being deceived, and secondly, the "cogito" is an invalid argument. The first of these observations requires implicitly that you exist, and that you think one thing is true when it is in fact false. Thus you have from the first observation as true statements about yourself "I exist" and "I think."

The second of these two observations is that the "cogito" is invalid. In order for an argument to be invalid it must be possible for the premises to be true and the conclusion false. However, from the first observation we have that you exist at any time the evil genius is trying to deceive you into believing the "cogito." From this can be concluded that the only time the "cogito" can be shown to be invalid is when the evil genius (or anybody else) is not deceiving you. Now, if you are not being deceived, the "cogito" is valid, because you believe it to be valid and there is no deception. Thus, the "cogito" is valid when the evil genius is trying to deceive you.

Hence you have the following situation: Given the premise that the evil genius is trying to deceive you into believing that the "cogito" is valid when it is in fact invalid, we can logically deduce that it is valid. Since the conclusion contradicts the premise, the premise must be false. therefore, the evil genius cannot deceive you into believing that the "cogito" is valid when it is invalid.

What of the possibility that the evil genius is trying to deceive you into believing the "cogito" is invalid when it is valid? This cannot be the case. Assume that the evil genius is deceiving you into believing that the "cogito" is invalid. By the same reasoning as above it can be demonstrated that the "cogito" is valid, and thus you cannot remain a reasoning person and still believe the "cogito" to be invalid.

Thus it has been shown that whenever the evil genius (or anybody else) tries to deceive you regarding the "cogito," the "cogito" is valid and you must believe it is valid by virtue of the above reasoning. Thus, the evil genius cannot deceive you regarding the "cogito."