There is also the irrational/sloppy reductionism where you mistakenly believe that something irreducible is reducible rather than illusionary. E.g. you can believe in some absolute truth, because you feel that there is absolute truth, and you are 'reductionist' as in ideology, and so have to believe that absolute truth is reducible (belief in a verbal statement). Whereas a reductionist may think whenever absolute truth can be reduced to something more basic, and see no way of doing that, and thus declare truth relative and illusionary (so that the only the feeling of absolute truth is reducible to interactions between neurons). edit: and an anti-reductionist would take the feeling that there is absolute truth, proclaim that there really is absolute truth, see no way to reduce absolute truth to anything more basic, and then proclaim that reductionism is false.

Ultimately two latter things are the only internally sound approaches, while first is not coherent.

edit: actually, it's best reworded with soul example. Reductionist feels that he has soul, doesn't see a way to reduce soul to e.g. elementary math (and sees impossibility of such reduction), and claims that feeling of soul is some sort of illusion (and seeks a way to reduce the feeling of soul). That is internally consistent.

The anti-reductionist feels that he has soul, proclaims that soul really exists, doesn't see a way to reduce soul to e.g. elementary math, doesn't buy into the feeling being an illusion, proclaims that reductionism is false and the soul is out of this world or something. This is also internally consistent.

The incoherent reductionist feels that he has soul, proclaims that soul really exists, proclaims that soul absolutely must be reducible to elementary math (while not really having reduced it to elementary math or even seen any way to), and doesn't notice that it would suffice to reduce the feeling of soul. That is just sloppy.