I can conceive the puzzle as one where all the relevant beliefs - (R1), (T), (AME), etc, - have degree 1.

Well, in that case, learning RM & TM leaves these degrees of belief unchanged, as an agent who updates via conditionalization cannot change a degree of belief that is 0 or 1. That's just an agent with an unfortunate prior that doesn't allow him to learn.

More generally, I think you might be missing the point of the replies you're getting. Most of them are not-very-detailed hints that you get no such puzzles once you discard traditional epistemological notions such as knowledge, belief, justification, defeaters, etc. (or change the subject from them) and adopt Bayesianism (here, probabilism & conditionalization & algorithmic priors). I am confident this is largely true, at least for your sorts of puzzles. If you want to stick to traditional epistemology, a reasonable-seeming reply to puzzle 2 (more within the traditional epistemology framework) is here: http://www.philosophyetc.net/2011/10/kripke-harman-dogmatism-paradox.html

If you downvoted, maybe offer constructive criticism? I feel like you're shooting the messenger, when we should really be shooting (metaphorically) mainstream philosophy for not recognizing the places where these questions have already been solved, rather than publishing more arguments about Gettier problems.

Both of these puzzles fall apart if you understand the concepts in Argument Screens Off Authority, A Priori, and Bayes Theorem. Essentially, the notion of "defeat" is extremely silly. In Puzzle 1, for example, what you should really be doing is updating your level of belief in T based on the mathematician's argument. The order in which you heard the arguments doesn't matter--the two Bayesian updates will still give you the same posterior regardless of which one you update on first.

Puzzle 2 is similarly confused about "defeat"; the notion of "misleading evidence" in Puzzle 2 is also wrong. If you look at things in terms of probabilities instead of the "known/not known" dichotomy presented in the puzzle, there is no confusion. Just update on the mathematician's argument and be done with it.

I second Manfred's suggestion about the use of beliefs expressed as probabilities.

In puzzle (1) you essentially have a proof for T and a proof for ~T. We don't wish the order in which we're exposed to the evidence to influence us, so the correct conclusion is that you should simply be confused*. Thinking in terms of "Belief A defeats belief B" is a bit silly, because you then get situations where you're certain T is true, and the next day you're certain ~T is true, and the day after that you're certain again that T is true after all. So should beliefs defeat each other in this manner? No. Is it rational? No. Does the order in which you're exposed to evidence matter? No.

In puzzle (2) the subject is certain a proposition is true (even though he's still free to change his mind!). However, accepting contradicting evidence leads to confusion (as in puzzle 1), and to mitigate this the construct of "Misleading Evidence" is introduced that defines everything that contradicts the currently held belief as Misleading. This obviously leads to Status Quo Bias of the worst form. The "proof" that comes first automatically defeats all evidence from the future, therefore making sure that no confusion can occur. It even serves as a Universal Counterargument ("If that were true I'd believe it and I don't believe it therefore it can't be true"). This is a pure act of rationalization, not of rationality.

*) meaning that you're not completely confident of T and ~T.

Yo, deductive logic is a special case of probabilistic logic in the limit that your probabilities for things go to 0 and 1, i.e. you're really sure of things. If I'm really sure that Socrates is a man, and I'm really sure that all men are mortal, then I'm really sure that Socrates is mortal. However, if I am 20% sure that Socrates is a space alien, my information is no longer well-modeled by deductive logic, and I have to use probabilistic logic.

The point is that the conditions for deductive logic have always broken down if you can deduce both T and ~T. This breakdown doesn't (always) mean you can no longer reason. It does mean you should stop trying to use deductive logic, and use probabilistic logic instead. Probabilistic logic is, for various reasons, the right way to reason from incomplete information - deductive logic is just an approximation for when you're really sure of things. Try phrasing your problems with degrees of belief expressed as probabilities, follow the rules, and you will find that the apparent problem has vanished into thin air.

Welcome to LessWrong!

Good afternoon, morning or night! I'm a graduate student in Epistemology. My research is about epistemic rationality, logic and AI. I'm actually investigating about the general pattern of epistemic norms and about their nature - if these norms must be actually accessed by the cognitive agent to do their job or not; if these norms in fact optimize the epistemic goal of having true beliefs and avoiding false ones, or rather if these norms just appear to do so; and still other questions. I was navigating through the web and looking for web-based softwares to calculate probabilites, so that I found LW, and guess what! I started to read it and couldn't stop - each link sounds exciting and interesting (bias, probability, belief, bayesianism...). So, I happily made an account, and I'm eager to discuss with you guys! Hope I can contribute to LW some way. We (me and my research partners) have a blog (https://fsopho.wordpress.com) on epistemology and reasoning. We're all together in the search for knowledge, fighting bias and requiring evidence! see ya =]