What is Reductionism?

Today BV is discussing reductionism versus eliminativism in the context of philosophy of mind.  Here is his paragraph (5):
5.  Let's note two obvious logical points.  The first is that identity is a symmetrical relation.  The second is that reduction is asymmetrical.  Thus,
I.  Necessarily, for any x, y, if x = y, then y = x.
R. Necessarily, for any x, y, if x reduces to y, then it is not the case that y reduces to x.
It is clear, then, that identity and reduction are not the same relation.  And yet if particular a reduces to particular b, then a is nothing other than b, and is therefore identical to b.  If you think about it, reduction is a strange and perhaps incoherent notion.  For if a reduces to b, a is identical to b, but, since reduction is asymmetrical,  b is not identical to a!  Reduction is asymmetrical identity.  And that smacks of radical incoherence.   This is what inclines me to say that reduction collapses into elimination.  For if a reduces to b, and is therefore identical to b, while b is not identical to a, then it follows that there simply is no a.  And so if my present mental state reduces to a pattern of electrical activity in a network of neurons, then my mental state does not exist; all that exists is the electrical activity.
Now, there is surely something rather wrong about this, especially if it leads to the conclusion that reduction is radically incoherent.  Is reduction to be seen as a relation between particulars?  I have always thought of it as a relation between concepts.  For example, earlier in the post Bill comments that 'the gene got reduced' (presumably to snippets of DNA).  What this means to me is that talk of genes, their properties and behaviours was found to be translatable into talk about DNA molecules and their properties and behaviours.  This was all very well because genes were previously hypothetical entities put forward to account for the observed facts of inheritance.  For all we knew they might be some exotic entity outside the domain of physics and chemistry.  But no, they turn out to be molecules of a large and complicated kind but with some interesting symmetries.  But there is no sense in which genes are no longer thought to exist.  Indeed a gene is now identified with a DNA molecule consisting of a certain sequence of nucleotides.  But this is an identity, or better, an equivalence, between concepts, not between particulars, concepts F and G being equivalent when everything that falls under F also falls under G, and vice versa.  Bill is making a category mistake here, I think.

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