Taxa Without Organisms?

In my 2007 paper, I built up an array of phylogenetic operations (precedence, parentage, minimal, maximal, all descendants, common ancestors, synapomorphic ancestors, clade, node-based ancestors, crown, etc.) relying on these definitions/assumptions:

1. Organisms are discrete units.
2. Ancestry can be encapsulated as discrete relations.
3. Taxa are sets of organisms.

Although I argued for this discrete, organism-level approach (pp. 608–609), there are certain problems with it.

Organisms are discrete units.

This works well enough for the organisms I tend to focus on (amniotes). But there are many cases where it's not clear. Is a slime mold growth a multicellular organism, or is it a colony of unicellular organisms? Are plastids and mitochondria separate organisms, or organelles that are descended from other organisms? Is a lichen a mutualistic association of organisms, or one composite organism? Is a Portuguese Man o'War a colony of individuals, or a superorganism made up of zooids? What about an ant colony? And are viruses organisms?

Apart from these problems of interpretation, there's also a problem to do with taxonomic conventions. Basically, there's no common method in taxonomy to denote an individual organism. The closest thing we have is specimen identifiers, but, as I pointed out in the paper, this is really just another way of indicating a taxon, since a specimen may represent any number of organisms: "An individual organism may have multiple specimens which represent it (e.g., an organism may have several parts catalogued as different specimens), or a specimen may represent no organism (e.g., a mineralogical sample) or multiple organisms (e.g., a microbe slide)." (p. 612)

Ancestry can be encapsulated as discrete relations.

Again, this works well enough for many taxa, but it's problematic elsewhere. Is lateral transfer among bacteria a form of ancestry? What about transfer of mitochondrial genes to the nucleus? Gene-splicing? Introgression of viral DNA?

Taxa are sets of organisms.

How can this be, if there is no such objective unit as "the organism"? And how can phylogenetic taxa be based on ancestry if there is no single objective definition for "ancestry"?

A solution?

I hinted at an escape route for some of these problems in a parenthetical note: "It should be noted, however, that the methods presented here will work for any type of entity which has parents and/or children. Thus, they may be applicable to some types of ‘population-level’ organism groupings." (p. 609; emphasis added)

Basically, it doesn't matter what the units are. Taxa are sets. It doesn't matter if the units are genes, genomes, organisms, colonies, populations, or whatever. Taxa can still overlap, be disjoint, include each other, or be identical to each other.

This still leaves the difficulty of what, exactly, "ancestry" is. You can say that all members of taxon A are ancestral to all members of taxon B, but what exactly does that mean? Clearly it has something to do with the replication of the discrete molecular patterns in genetic material, but can it be stated explicitly and objectively? I'm not sure. For now, the only solution I can offer is to let that be a taxonomic decision—the only taxonomic decision you have to make when applying phylogenetic nomenclature.