Over the river and through the woods . . .
In common parlance, the word "Talmudic" has come to connote reasoning or argumentation that is excessively arcane, esoteric, convoluted, hair-splitting, etc.
I just started a summer program in which I am studying Talmud every day, and my overwhelming sense is quite the opposite.
Just as a differential equations textbook would be arcane and esoteric to someone who had not studied some math, Talmud is certainly daunting and seemingly-irrelevant to those not engrossed in Jewish texts. Obviously, many sugyot (discussions) in the Talmud are also highly concerned with the specific details of observance or law, because that's the Gemara's basic job-- to explore what it means to live day-to-day while trying to follow the principles and rules established in the Mishnah.
In the past few days, I've really felt how folksy the Talmud is, how mundane-- how much the materials collected in the Talmud are a product of the everyday discussions and disagreements of a group of (albeit very intelligent and learned) guys.
For example, when a rabbi hasn't explicitly stated an opinion about a particular topic (or even sometimes when he has), the discussion sometimes tries to figure out what he thought by bringing anecdotes about him. One of my favorites from Friday was from a sugya about whether or not a "taste" from a plate that has been used for one type of food will transfer to different hot food that is later placed on that plate. The story tells that a rabbi goes to his grandfather's house (the grandfather also being a great rabbi). Grandpa rabbi's eyes hurt, so grandson rabbi prepares him an ointment in a bowl. Later, grandson rabbi brings grandpa rabbi stew in the same bowl (presumably after it's been washed) and Grandpa complains, "Even after all this time I can still taste the ointment!"
Doesn't this sound like your grandpa?
The Talmud is filled with little stories, petty arguments and ego plays, domestic dramas, and the details of day-to-day life... not "Talmudic" at all, in that sense...