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!

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.

Axioms are not true or false. They either model what we intended them to model, or they don't. In puzzle 1, assuming you have carefully checked both proofs, confidence that (F, P1, P2, P3) implies T and (F, P1, P2, P3) implies ~T are both justified, rendering (F, P1, P2, P3) an uninteresting model that probably does not reflect the system that you were trying to model with those axioms. If you are trying to figure out whether or not T is true within the system you were trying to model, then of course you cannot be confident one way or the other, since you aren't even confident of how to properly model the system. The fact that your proof of T relied on fewer axioms would seem to be some evidence that T is true, but is not particularly strong.

puzzle 2: (ME) points both ways. While it certainly seems to be strong evidence against the reliability of (RM), since she just reasoned from clearly inconsistent axioms, it can't prove that F is the axiom you should throw away. Consider the possibility that you could construct a proof of ~T given only F, P1, and P2. Now, (ME) could not possibly say anything different about F and P3.

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 think a lot of the replies here suggesting that Bayesian epistemology easily dissolves the puzzles are mistaken. In particular, the Bayesian-equivalent of (1) is the problem of logical omniscience. Traditional Bayesian epistemology assumes that reasoners are logically omniscient at least with respect to propositional logic. But (1), suitably understood, provides a plausible scenario where logical omniscience fails.

I do agree that the correct understanding of the puzzles is going to come from formal epistemology, but at present there are no agreed-upon solutions that handle all instances of the puzzles.

Puzzle 1

• RM is irrelevant.

The concept of "defeat", in any case, is not necessarily silly or inapplicable to a particular (game-based) understanding of reasoning, which has always been known to be discursive, so I do not think it is inadequate as an autobiographical account, but it is not how one characterizes what is ultimately a false conclusion that was previously held true. One need not commit oneself to a particular choice either in the case of "victory" or "defeat", which are not themselves choices to be made.

Puzzle 2

• Statements ME and AME are both false generalizations. One cannot know evidence for (or against) a given theorem (or apodosis from known protases) in advance based on the supposition that the apodosis is true, for that would constitute a circular argument. I.e.:

T is true; therefore, evidence that it is false is false. This constitutes invalid reasoning, because it rules out new knowledge that may in fact render it truly false. It is also false to suppose that a human being is always capable of reasoning correctly under all states of knowledge, or even that they possess sufficient knowledge of a particular body of information perfectly so as to reason validly.

• MF is also false as a generalization.

In general, one should not be concerned with how "misleading" a given amount of evidence is. To reason on those grounds, one could suppose a given bit of evidence would always be "misleading" because one "knows" that the contrary of what that bit of evidence suggests is always true. (The fact that there are people out there who do in fact "reason" this way, based on evidence, as in the superabundant source of historical examples in which they continue to believe in a false conclusion, because they "know" the evidence that it is false is false or "misleading", does not at all validate this mode of reasoning, but rather shores up certain psychological proclivities that suggest how fallacious their reasoning may be; however, this would not itself show that the course of necessary reasoning is incorrect, only that those who attempt to exercise it do so very poorly.) In the case that the one is dealing with a theorem, it must be true, provided that the reasoning is in fact valid, for theorematic reasoning is based on any axioms of one's choice (even though it is not corollarial). !! However, if the apodosis concerns a statement of evidence, there is room for falsehood, even if the reasoning is valid, because the premisses themselves are not guaranteed to be always true.

The proper attitude is to understand that the reasoning prior to exposure of evidence/reasoning from another subject (or one's own inquiry) may in fact be wrong, however necessary the reasoning itself may seemingly appear. No amount of evidence is sufficient evidence for its absolute truth, no matter how valid the reasoning is. Note that evidence here is indeed characteristic of observational criteria, but the reasoning based thereon is not properly deductive, even if the reasoning is essentially necessary in character. Note that deductive logic is concerned with the reasoning to true conclusions under the assumption that the relevant premisses are true; if one is taking into account the possibility of premisses which may not always be true, then such reasoning is probabilistic (and necessary) reasoning.

!! This, in effect, resolves puzzle 1. Namely, if the theorem is derived based on valid necessary reasoning, then it is true. If it isn't valid reasoning, then it is false. If "defeat" consists in being shown that one's initial stance was incorrect, then yes, it is essential that one takes the stance of having been defeated. Note that puzzle 2 is solved in fundamentally the same manner, despite the distracting statements ME, AME, and MF, on account of the nature of theorems. Probabilities nowhere come into account, and the employment of Bayesian reasoning is an unnecessary complication. If one does not take the stance of having been defeated, then there is no hope for that person to be convinced of anything of a logical (necessary) character.