I finally figured out the real key difference between Talmudic thinking and secular legal thinking. Well, not the only difference, but the important one for... what I'm thinking about. You'll see what I mean.
It isn't the kind of logic used. It isn't the fineness of the hairs being split. It's the goal, the aim of the canon of law being produced. Secular law aims to produce a canon of law that can be followed by people more or less like the ones who promulgated it (with all their faults). So when it comes to, say, the validity of a contract, if a litigant contests the validity of the signatures on a contract, the law will say (or even leave unsaid, since it is “obvious”) something like “the party contesting the contract shall present what evidence it has supporting forgery, the other party shall attempt to demonstrate that the signatures are valid, they should bring testimony from handwriting and document-tampering experts, blah blah blah...” (though the “blah blah blah” might be in Latin.) The details of how the case should be judged is left up to the presumed competence of the judge adjudicating the case, and also whatever precedence and case-law that has built up over the years.
In contrast, the Talmud goes into exhaustive detail about what kinds of tampering is suspect, how the signatories can approve erasures and changes to the text (N.B. Talmudic contracts are signed by witnesses, attesting to the transaction, and not by the parties to the transaction). So it discusses the exact effects of the placement of the witnesses' signatures, and how much blank space can be around them, and when we may suspect that the contract was tampered with by the addition of an extra line of text... What happens if there is one blank line, what if there are two, what if there's space for just one and a half lines... This is just from skimming quickly from Tractate Baba Batra, Chapter 10 (in English at sacred-texts.com). There is a very clear attempt to try to cover all the possible situations that may arise, so that a future judge need only look up the answer and not have to apply too much actual judgment. They talk about exactly which arguments or claims by litigants in various cases should be or should not be believed, and why, as though future judges in these cases need all the help they can get. I suppose in some sense it's like trying to establish case-law, but it isn't: they talk in hypotheticals, suggest possible arguments and claims and propose rules to decide when they should or should not be believed! See, for example, some of the discussion in this explanation of part of Tractate Ketubot. The terminology is highly “Yeshivish,” mixing a lot of Aramaic with the English, but maybe you can get the gist. The actual details of the cases aren't even the point, just the way that they are handled.
Combine this with the ancestor-veneration in Judaism which I have complained about before, and you have a system that is not only resistant to change, it's positively resistant to remaining itself. Especially now, with notions floating around that the Talmud rabbis were somehow inerrant, and when they talk about things that are contrary to science as we know it, it's not because they didn't know better, but because we don't know better, and they're still right (even though they themselves admit that they don't know everything. See the discussion in Pesachim 94b that I discussed at the end of this post almost two years ago).
What do I mean by “resistant to remaining itself”? I suppose in many ways this post is a repeat of a lot of what I said in the one linked at the end of the last paragraph (I guess the main new insight is the way the Talmud tries to lessen the role of future judges). Because Judaism does not confer onto itself the authority to decide and change and rule itself, and because the Talmud is so structured as to make such decisions unnecessary, Judaism has difficulty adapting to the changing world (which is nothing new, though it must in fairness be remarked that for something that has trouble adapting to the changing world, it is doing a pretty good job of it anyway), but this doesn't mean that by remaining true to these old principles, Judaism is maintaining its identity. On the contrary, if the Talmud-rabbis were alive today, I am certain they would have all kinds of rulings on modern conundrums that are very different than the stringencies that we have adopted in their name. Read how they dealt with the world in their time, and imagine how they would have dealt with ours.
(This turned out to be more of a downer than I thought it would).