If I get to ignore real-world social constraints, I would approach this problem by more or less ignoring the courts, and concentrating on the schools.
Raise a generation of students who have been taught about decision theory and cognitive bias, who have been taught to analyze an argument for evidence to roughly the same degree that students today are taught to analyze a sentence for its subject and predicate, who have been taught to identify common fallacies of reasoning to roughly the same level that students today are taught to identify the years of famous historical events. Wait two generations and see what happens to the average quality of courtroom reasoning.
I'd probably also try to raise the floor... that is, establish some standard for arguments so common-sense fallacious that for a lawyer to make those arguments in a court of law is grounds for appeal. But I suspect I would fail, as regulatory capture would no doubt set in.
I have a low prior for the success of this idea because it does not address people's motives. If people still want to deceive (others and themselves), then making them cleverer at noticing and articulating evidence should have no net effect on accuracy, especially in court. If everyone became adept at noticing now-common fallacies, lawyers would have a strong incentive to find subtler fallacies or more effective obfuscatory techniques.
I liken this suggestion to the idea of giving advanced weapons to both sides in a war. A priori there's no way to say what effect it would have, and certainly there's no reason to suppose the "right" side will win.
I'm interested in how courts and juries might use rational techniques to arrive at correct decisions on guilt.
In a complex case, it would seem to sensible to assess each component of the prosecution and defence case, and estimate the relative likelihood. If the prosecution case is (say) 100 times more likely than the defence case, then you can say the defendant is guilty beyond reasonable doubt.
I never heard of this being done though. I recently made an analysis of the Massei report into the Amanda Knox case. It looked like this ( see http://massei-report-analysis.wikispaces.com/ for the entire analysis and some insight into the numbers below ).
This is perhaps a bit vague. It's not a great example, because in the end I didn't find any credible prosecution evidence. It's not entirely clear what the "probability" numbers here actually are, and whether two columns are needed. But hopefully it shows that the Massei's account of the murder is quite improbable, and there is considerable doubt.
I'm interested in possibly devising a more complete framework for how such an assessment should be done, the pitfalls that need to be guarded against (how uncertain are the probability estimates?), and even views as to how "reasonable doubt" should be quantified.
Perhaps readers would like to make an assessment of other interesting cases, to explore the issues.
Or how would you approach this problem?