Yes, if he would've said the two form some kind of biconditional, then I would just argue with the truth of the premises. But the way the argument is laid out, and they way he responded to me pointing it out makes it obvious he takes it to mean the fallacy. He's trying to tie together purpose and rationality (or no purpose and no rationality (irrationality).

If there is purpose there is mind.
"No mind means irrational."
There is no purpose. (What we 'believe')
Therefore there is no mind (everything is irrational).
[The implicit premise is 'if something is based on chance it has no purpose' otherwise he can't logically make the jump from purpose to chance]
"If chance is ultimate, since chance is irrational, the universe has no ultimate purpose or meaning."

This quote means:
If something is based on chance, it is irrational.
The universe is based on chance. (What he's attributing to us)
[Implicit premise comes to work here:
If something is based on chance then it has no purpose.
The universe is based on chance.
Therefore the universe has no purpose.]
Therefore the universe is irrational (no purpose--> no mind [all is irrational])

He actually begs the question (assumes what he's trying to prove) here as well. He assumes (or he's saying we 'assume') no purpose as one of his premises. Then concludes "therefore no ultimate purpose or meaning."
Even if he didn't mean to commit negating the antecedent and wanted it as just a biconditional premise, he concludes with 'therefore no purpose.' Question begging logic.
His argument for why our belief is illogical is itself illogical.

Any philosophers/logicians here think I'm off the mark?
If I'm wrong I'll tuck my tail between my legs and scurry away...