Thanks for that. Is their a simple term to describe the taxonomy that has Aves as a Class to distinguish it from the cladistic one?
Just "pre-cladistic classification" would get the idea across. I almost answered "Linnaean", because that's how the guy who invented the system categorized them, but someone would probably have argued that even with modifications such as extra levels being inserted and some species & groups being moved around, the system is still fundamentally Linnaean as long as it uses words like "class" and "order" and "phylum" and puts them in the right order.
I think I see the point you are making except that under the old-style taxonomy we were still able to call dinosaurs Reptilia, so it can't be completely correct that no account was taken of extinct species.
Extinct species can be fit into it after they're discovered, but that doesn't mean that it was set up with extinct species in mind. If it had been, then birds could never have been a class on their own. They only ended up as a distinct class in the first place because of the lack of anything else really similar to them to put in a group with them, and that lack of close relatives was due to extinctions.
It's the equivalent of all mammals but the cloven-hoofed ones going extinct. Someone (not a human of course) classifying living things in such a world (in the same kind of system as Linnaeus's) would have put the cloven-hoofed mammals as a class because they'd be different enough from all other living vertebrates (birds, reptiles, amphibians, & fish) to warrant being separated from them, and there'd be nobody else around to put in a group together with them. It would take the discovery of fossils of other things we now call mammals to change the cloven-hoofed animals' context and move their taxonomic level.
Really, the problem is that old Reptilia was paraphyletic because the historical taxonomists saw birds as being very different from reptiles, but internally consistent. Actually the latter is true, birds are monophyletic, but the former is not, if you exclude them from reptiles then old-style reptiles turn out not to be monophyletic.
Right so far?
Yes. But there's one more catch. To be sure that something is monophyletic, you have to not only check for fracturing within the group, as we've been dealing with so far in this case, but also check the larger group that it all fits within, see if there's any fracturing above as well as below. The "reptile" case happens to be a handy example of that.
Birds, non-avian dinosaurs, crocodilians, lizards, snakes, tuataras, pterosaurs, and ichthyosaurs all fit into a group called "diapsids". Mammals and some extinct critters (of which
Dimetrodon is the most famous) make a group called "synapsids". Those two groups can be identified by the number of pits on each side of the head. (Yours are behind and a bit above the eyes, where the muscles you use in chewing are.) But there's one more group of amniotes (vertebrates that aren't fish or amphibians; they have amniotic eggs that don't need to be laid in water) I didn't mention: turtles. Their skulls lack pits, making them anapsids. And they're reptiles. So that would mean reptiles, even if you include all of the diapsids, still aren't necessarily monophyletic. Since it includes both diapsids and anapsids, a clade including them all must include the common ancestor of both, and all of that ancestor's descendants, which could very well mean including the synapsids (which means you've managed to narrow it down to all amniotes). Nobody knew at first which one of the three surviving amniote groups had split off from the other two first, before the remaining two split off from each other, so there were three possible arrangements, and under two of them, one of the two non-mammal groups could have ended up more closely related to mammals than to the other non-mammal group. For a while there, we seemed to have a two-thirds chance of getting one clade including some reptiles, and another one including mammals and the remaining reptiles.
As it turns out, that's all a could-have-been scenario. Biochemists found that turtle proteins & DNA look a lot like diapsid proteins & DNA, and could even match them to a particular group of diapsids (birds & crocodilians) more closely than to the others. This caused anatomists, whose discipline had said turtles were a distinct third 'apsid group in the first place, to say "Wait, did we miss something?" and recheck the anatomical work and come back with the answer "Yes, we did". It turned out that the pitless skull had been given too much "weight" in turtles' classification, and the weight of other clues in other parts of the body was consistent with a diapsid origin for turtles. Their skulls just smoothed out sometime after they split off from other diapsids. This mistake might have been influenced by the existence of fossils of true anapsids, a true third amniote group, which, since it doesn't include turtles after all, is entirely extinct.
So, even including all diapsids together, you still wouldn't necessarily have had a clade because of the anapsid issue, but, with turtles as archosaurs, it is. Also, even if turtles were anapsids, the question of whether they and diapsids together were a clade separate from synapsids would have been simply unanswered, until some more fossils or biochemical comparisons showed us the order of the splitting among those three clades. Then it would become known whether or not the group we're talking about had been a clade all along.
If Reptilia is the class containing birds, do we have a word that describes the paraphyletic group that would represent was we call reptiles in common language? Is there any use for such a word?
No. When there's already a common word that covers the meaning, and the established pattern is for scientific terms to apply to clades, there's no need for new scientific terms for groups that aren't clades.
"Systematics"
Is that what we call the non-cladistics version of taxonomy. And does it mean looking at the morphology and trying to pick out features to use as markers?
Yes, but in some cases human convenience & tradition set which morphological features are considered and which are ignored. For example, morphologically, we know that vultures come in two groups, each of which has some traits in common with some other group of non-vultures but not with the other group of vultures. But sometimes when we're talking about vultures, we really mean both groups of vultures and no non-vultures, so we're only concerned with traits that vultures have in common with each other and not with non-vultures. In that case, which morphological traits we decide matter and which ones we decide don't matter depends on the nature of the group we want to end up with.