Sorry, I think I still don't understand your reasoning.

First, I have the beliefs P1, P2 and P3, then I (in an apparently deductively valid way) reason that [C1] "T is a theorem of P1, P2, and P3", therefore I believe T.

Either my reasoning that finds out [C1] is valid or invalid. I do think it's valid, but I am fallible.

Then the Authority asserts F, I add F to the belief pool, and we (in an apparently deductively valid way) reason [C2] "~T is a theorem of F, P1, P2, and P3", therefore we believe ~T.

Either our reasoning that finds out [C2] is valid or invalid. We do think it's valid, but we are fallible.

• Is it possible to conclude C2 without accepting I made a mistake when reasoning C1 (therefore we were wrong to think that line of reasoning was valid)? Otherwise we would have both T and ~T as theorems of F, P1, P2, and P3, and we should conclude that the promises lead to contradiction and should be revised; we wouldn't jump from believing T to believing ~T.
• But the story doesn't say the Authority showed a mistake in C1. It says only that she made a (apparently valid) reasoning using F in addition to P1, P2, and P3.
• If the argument of the Authority doesn't show the mistake in C1, how should I decide whether to believe C1 has a mistake, C2 has a mistake, or the promises F, P1, P2, and P3 actually lead to contradiction, with both C1 and C2 being valid?

I think Bayesian reasoning would inevitably enter the game in that last step.

Here's one that comes to mind:

I really don't know anything about baseball, so if I'm going to bet on either the Red Socks or the Yankees, I'd have to go fifty-fifty on it. Therefore, the chance that either will win is fifty percent.

(Right at the "therefore" is the fallacy put forward as a veritable property of either of the teams winning, when in fact it is merely indicative of the ignorance of the gambler. The actual probability is most likely not 50-50.)

EDIT: Others might enjoy reading this PDF ("Probability Theory as Logic") for additional background and ideas. There you'll also see a bon mot by Montaigne: "Man is surely mad. He cannot make a worm; yet he makes Gods by the dozen."

I'm interested in what you have to say, and I'm sympathetic (I think), but I was hoping you could restate this in somewhat clearer terms. Several of your sentences are rather difficult to parse, like "And to be committed to false statements as being not-false would be absurd, such that it would alone be proper to aver that one has been defeated in having previously been committed to the truth of T despite that that committment was fundamentally invalid."

In essence, if a counterargument comes along demonstrating that T is false, then it necessarily would involve demonstrating that invalid reasoning was somewhere committed in someone's having arrived at the (fallacious) truth of T.

I think I see your point, but if you allow for the possibility that the original deductive reasoning is wrong, i.e. deny logical omniscience, don't you need some way to quantify that possibility, and in the end that would mean treating the deductive reasoning itself as bayesian evidence for the truth of T?

Unless you assume that you can't make a mistake at the deductive reasoning, T being a theorem of the promises is a theory to be proven with the Bayesian framework, with Bayesian evidence, not anything special.

And if you do assume that you can't make a mistake at the deductive reasoning, I think theres no sense in paying attention to any contrary evidence.

we are not justified in assigning probability 1 to the belief that 'A=A' or to the belief that 'p -> p'? Why not?

"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."

Actually, I think if "I know T is true" means you assign probability 1 to T being true, and if you ever were justified in doing that, then you are justified in assigning probability 1 that the evidence is misleading and not even worth to take into account. The problem is, for all we know, one is never justified in assigning probability 1 to any belief. So I'd say the problem is a wrong question.

Edited: I meant probability 1 of misleading evidence, not 0.

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.

I wonder whether or not there might be a prime example of the game of general expertise par excellence out there, one that touches on many domains simultaneously...

Probably not. While in video game design there are general competencies you can rely on, there are both mutually exclusive challenges: fast paced FPS games like Quake 3 cannot be played like slower paced FPS games like Call of Duty, players who attempt to transfer their skills without understanding this don't succeed; and balance problems, where the addition of game elements overshadow others like in Alien Swarm where there are five effective weapons even though there are fifteen other options and some of them are dismissed unfairly because they are introduced to players who haven't seen a need for the skills they ask. Both of these factors, however, mean that challenges and tradeoffs go hand in hand in your game's design.

That all said, people do try. Spore is the readiest example of this to me: the mishmash of different games doesn't really work, the way they tried to address the challenge balancing issues means that four fifths of the game design is effectively useless, but it's an instructive game nonetheless.

I think the main issue here is that expertise must be conceptualized with respect to a particular activity or set of activities in order for it to maintain its essential meaning. The nature of expertise is also restricted to a specific range of tools the brain embodies (as in "embodied cognition"); in other words, it is not the hand the knows what to type, but rather the keyboard that knows what to type. To be clear, my cognitive capacity is effectively extended and reshaped by the interaction with the keyboard, so in effect the nature of the expertise will be limited specifically to the final cause (in the philosophical sense) of the activity itself. I like to think of it as the mind further approximating the function of the game, or activity, over time serving as a kind of analogy to the ever-accumulating expertise therein.

Taking the example of chess versus a modern-day computer-enhanced strategy game, the modes of embodiment are vastly different, and so the kinds of expertise to be expected should naturally diverge. However, I would not be so pollyannaish as to assert that playing StarCraft 2 (or Chess) would be "really useful", unless you're playing for money to help you in some specific goal outside of the game itself. That is going a bit too far, in my opinion. We already know that the nature of expertise is such that it only operates at the level of the activity one is engaged in, and will not generalize (or transfer) far from that domain of activity. For instance, the expertise in knowing the layout of a keyboard and being able to type commands without a second thought (being constantly honed by a game that demands it) will transfer to the tasks (of other games) that require the same input on a keyboard (and will differentially benefit from those quick reflexes), but the specific tactics and techniques learned in-game will generally not find much use beyond that game, and I do believe that is what we're getting at with a game like SC2 insofar as "expertise" is a concern here. Similarly with chess: one might very well have excellent reflexes, honed in certain other tasks, and know many strategies and techniques for other things, but they won't apply to the space of chess, and so vice versa for chess to other activities. (And we already know that typical memorization techniques used in chess really don't help with memorizing anything else.)

Having said all that, I wonder whether or not there might be a prime example of the game of general expertise par excellence out there, one that touches on many domains simultaneously... Perhaps the Glass Bead Game? Ah, never mind. But, in all seriousness, the way of the game is probably the only way we'll ever find out if such a thing exists and will permit the mind to approximate the function of life all the more perfectly.

By the way, I don't know how it is the researchers in the article don't think there hasn't been such a "satellite view" of expertise before, particularly on the note of chess. Hasn't anyone told them of the Chess Tactics Server? ( http://chess.emrald.net/ ) Chumps to champs aplenty there.