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Related: Cached Thoughts
Last summer I was talking to my sister about something. I don't remember the details, but I invoked the concept of "truth", or "reality" or some such. She immediately spit out a cached reply along the lines of "But how can you really say what's true?".
Of course I'd learned some great replies to that sort of question right here on LW, so I did my best to sort her out, but everything I said invoked more confused slogans and cached thoughts. I realized the battle was lost. Worse, I realized she'd stopped thinking. Later, I realized I'd stopped thinking too.
I went away and formulated the concept of a "Philosophical Landmine".
I used to occasionally remark that if you care about what happens, you should think about what will happen as a result of possible actions. This is basically a slam dunk in everyday practical rationality, except that I would sometimes describe it as "consequentialism".
The predictable consequence of this sort of statement is that someone starts going off about hospitals and terrorists and organs and moral philosophy and consent and rights and so on. This may be controversial, but I would say that causing this tangent constitutes a failure to communicate the point. Instead of prompting someone to think, I invoked some irrelevant philosophical cruft. The discussion is now about Consequentialism, the Capitalized Moral Theory, instead of the simple idea of thinking through consequences as an everyday heuristic.
It's not even that my statement relied on a misused term or something; it's that an unimportant choice of terminology dragged the whole conversation in an irrelevant and useless direction.
That is, "consequentialism" was a Philosophical Landmine.
In the course of normal conversation, you passed through an ordinary spot that happened to conceal the dangerous leftovers of past memetic wars. As a result, an intelligent and reasonable human was reduced to a mindless zombie chanting prerecorded slogans. If you're lucky, that's all. If not, you start chanting counter-slogans and the whole thing goes supercritical.
It's usually not so bad, and no one is literally "chanting slogans". There may even be some original phrasings involved. But the conversation has been derailed.
So how do these "philosophical landmine" things work?
It looks like when a lot has been said on a confusing topic, usually something in philosophy, there is a large complex of slogans and counter-slogans installed as cached thoughts around it. Certain words or concepts will trigger these cached thoughts, and any attempt to mitigate the damage will trigger more of them. Of course they will also trigger cached thoughts in other people, which in turn... The result being that the conversation rapidly diverges from the original point to some useless yet heavily discussed attractor.
Notice that whether a particular concept will cause trouble depends on the person as well as the concept. Notice further that this implies that the probability of hitting a landmine scales with the number of people involved and the topic-breadth of the conversation.
Anyone who hangs out on 4chan can confirm that this is the approximate shape of most thread derailments.
Most concepts in philosophy and metaphysics are landmines for many people. The phenomenon also occurs in politics and other tribal/ideological disputes. The ones I'm particularly interested in are the ones in philosophy, but it might be useful to divorce the concept of "conceptual landmines" from philosophy in particular.
Here's some common ones in philosophy:
Landmines in a topic make it really hard to discuss ideas or do work in these fields, because chances are, someone is going to step on one, and then there will be a big noisy mess that interferes with the rather delicate business of thinking carefully about confusing ideas.
My purpose in bringing this up is mostly to precipitate some terminology and a concept around this phenomenon, so that we can talk about it and refer to it. It is important for concepts to have verbal handles, you see.
That said, I'll finish with a few words about what we can do about it. There are two major forks of the anti-landmine strategy: avoidance, and damage control.
Avoiding landmines is your job. If it is a predictable consequence that something you could say will put people in mindless slogan-playback-mode, don't say it. If something you say makes people go off on a spiral of bad philosophy, don't get annoyed with them, just fix what you say. This is just being a communications consequentialist. Figure out which concepts are landmines for which people, and step around them, or use alternate terminology with fewer problematic connotations.
If it happens, which it does, as far as I can tell, my only effective damage control strategy is to abort the conversation. I'll probably think that I can take those stupid ideas here and now, but that's just the landmine trying to go supercritical. Just say no. Of course letting on that you think you've stepped on a landmine is probably incredibly rude; keep it to yourself. Subtly change the subject or rephrase your original point without the problematic concepts or something.
A third prong could be playing "philosophical bomb squad", which means permanently defusing landmines by supplying satisfactory nonconfusing explanations of things without causing too many explosions in the process. Needless to say, this is quite hard. I think we do a pretty good job of it here at LW, but for topics and people not yet defused, avoid and abort.
ADDENDUM: Since I didn't make it very obvious, it's worth noting that this happens with rationalists, too, even on this very forum. It is your responsibility not to contain landmines as well as not to step on them. But you're already trying to do that, so I don't emphasize it as much as not stepping on them.
One of the few things that I really appreciate having encountered during my study of philosophy is the Gettier problem. Paper after paper has been published on this subject, starting with Gettier's original "Is Justified True Belief Knowledge?" In brief, Gettier argues that knowledge cannot be defined as "justified true belief" because there are cases when people have a justified true belief, but their belief is justified for the wrong reasons.
For instance, Gettier cites the example of two men, Smith and Jones, who are applying for a job. Smith believes that Jones will get the job, because the president of the company told him that Jones would be hired. He also believes that Jones has ten coins in his pocket, because he counted the coins in Jones's pocket ten minutes ago (Gettier does not explain this behavior). Thus, he forms the belief "the person who will get the job has ten coins in his pocket."
Unbeknownst to Smith, though, he himself will get the job, and further he himself has ten coins in his pocket that he was not aware of-- perhaps he put someone else's jacket on by mistake. As a result, Smith's belief that "the person who will get the job has ten coins in his pocket" was correct, but only by luck.
While I don't find the primary purpose of Gettier's argument particularly interesting or meaningful (much less the debate it spawned), I do think Gettier's paper does a very good job of illustrating the situation that I refer to as "being right for the wrong reasons." This situation has important implications for prediction-making and hence for the art of rationality as a whole.
Simply put, a prediction that is right for the wrong reasons isn't actually right from an epistemic perspective.
If I predict, for instance, that I will win a 15-touch fencing bout, implicitly believing this will occur when I strike my opponent 15 times before he strikes me 15 times, and I in fact lose fourteen touches in a row, only to win by forfeit when my opponent intentionally strikes me many times in the final touch and is disqualified for brutality, my prediction cannot be said to have been accurate.
Where this gets more complicated is with predictions that are right for the wrong reasons, but the right reasons still apply. Imagine the previous example of a fencing bout, except this time I score 14 touches in a row and then win by forfeit when my opponent flings his mask across the hall in frustration and is disqualified for an offense against sportsmanship. Technically, my prediction is again right for the wrong reasons-- my victory was not thanks to scoring 15 touches, but thanks to my opponent's poor sportsmanship and subsequent disqualification. However, I likely would have scored 15 touches given the opportunity.
In cases like this, it may seem appealing to credit my prediction as successful, as it would be successful under normal conditions. However, I we have to resist this impulse and instead simply work on making more precise predictions. If we start crediting predictions that are right for the wrong reasons, even if it seems like the "spirit" of the prediction is right, this seems to open the door for relying on intuition and falling into the traps that contaminate much of modern philosophy.
What we really need to do in such cases seems to be to break down our claims into more specific predictions, splitting them into multiple sub-predictions if necessary. My prediction about the outcome of the fencing bout could better be expressed as multiple predictions, for instance "I will score more points than my opponent" and "I will win the bout." Some may notice that this is similar to the implicit justification being made in the original prediction. This is fitting-- drawing out such implicit details is key to making accurate predictions. In fact, this example itself was improved by tabooing "better" in the vague initial sentence "I will fence better than my opponent."
In order to make better predictions, we must cast out those predictions that are right for the wrong reasons. While it may be tempting to award such efforts partial credit, this flies against the spirit of the truth. The true skill of cartography requires forming both accurate and reproducible maps; lucking into accuracy may be nice, but it speaks ill of the reproducibility of your methods.
 I greatly suggest that you make tabooing a five-second skill, and better still recognizing when you need to apply it to your own processes. It pays great dividends in terms of precise thought.
Part of the sequence: Rationality and Philosophy
Hitherto the people attracted to philosophy have been mostly those who loved the big generalizations, which were all wrong, so that few people with exact minds have taken up the subject.
I've complained before that philosophy is a diseased discipline which spends far too much of its time debating definitions, ignoring relevant scientific results, and endlessly re-interpreting old dead guys who didn't know the slightest bit of 20th century science. Is that still the case?
You bet. There's some good philosophy out there, but much of it is bad enough to make CMU philosopher Clark Glymour suggest that on tight university budgets, philosophy departments could be defunded unless their work is useful to (cited by) scientists and engineers — just as his own work on causal Bayes nets is now widely used in artificial intelligence and other fields.
How did philosophy get this way? Russell's hypothesis is not too shabby. Check the syllabi of the undergraduate "intro to philosophy" classes at the world's top 5 U.S. philosophy departments — NYU, Rutgers, Princeton, Michigan Ann Arbor, and Harvard — and you'll find that they spend a lot of time with (1) old dead guys who were wrong about almost everything because they knew nothing of modern logic, probability theory, or science, and with (2) 20th century philosophers who were way too enamored with cogsci-ignorant armchair philosophy. (I say more about the reasons for philosophy's degenerate state here.)
As the CEO of a philosophy/math/compsci research institute, I think many philosophical problems are important. But the field of philosophy doesn't seem to be very good at answering them. What can we do?
Why, come up with better philosophical methods, of course!
Part of the sequence: Rationality and Philosophy
Philosophy is notable for the extent to which disagreements with respect to even those most basic questions persist among its most able practitioners, despite the fact that the arguments thought relevant to the disputed questions are typically well-known to all parties to the dispute.
The goal of philosophy is to uncover certain truths... [But] philosophy continually leads experts with the highest degree of epistemic virtue, doing the very best they can, to accept a wide array of incompatible doctrines. Therefore, philosophy is an unreliable instrument for finding truth. A person who enters the field is highly unlikely to arrive at true answers to philosophical questions.
After millennia of debate, philosophers remain heavily divided on many core issues. According to the largest-ever survey of philosophers, they're split 25-24-18 on deontology / consequentialism / virtue ethics, 35-27 on empiricism vs. rationalism, and 57-27 on physicalism vs. non-physicalism.
Sometimes, they are even divided on psychological questions that psychologists have already answered: Philosophers are split evenly on the question of whether it's possible to make a moral judgment without being motivated to abide by that judgment, even though we already know that this is possible for some people with damage to their brain's reward system, for example many Parkinson's patients, and patients with damage to the ventromedial frontal cortex (Schroeder et al. 2012).1
Why are physicists, biologists, and psychologists more prone to reach consensus than philosophers?2 One standard story is that "the method of science is to amass such an enormous mountain of evidence that... scientists cannot ignore it." Hence, religionists might still argue that Earth is flat or that evolutionary theory and the Big Bang theory are "lies from the pit of hell," and philosophers might still be divided about whether somebody can make a moral judgment they aren't themselves motivated by, but scientists have reached consensus about such things.
Part of the sequence: Rationality and Philosophy
Consider these two versions of the famous trolley problem:
Stranger: A train, its brakes failed, is rushing toward five people. The only way to save the five people is to throw the switch sitting next to you, which will turn the train onto a side track, thereby preventing it from killing the five people. However, there is a stranger standing on the side track with his back turned, and if you proceed to thrown the switch, the five people will be saved, but the person on the side track will be killed.
Child: A train, its brakes failed, is rushing toward five people. The only way to save the five people is to throw the switch sitting next to you, which will turn the train onto a side track, thereby preventing it from killing the five people. However, there is a 12-year-old boy standing on the side track with his back turned, and if you proceed to throw the switch, the five people will be saved, but the boy on the side track will be killed.
Here it is: a standard-form philosophical thought experiment. In standard analytic philosophy, the next step is to engage in conceptual analysis — a process in which we use our intuitions as evidence for one theory over another. For example, if your intuitions say that it is "morally right" to throw the switch in both cases above, then these intuitions may be counted as evidence for consequentialism, for moral realism, for agent neutrality, and so on.
Alexander (2012) explains:
Philosophical intuitions play an important role in contemporary philosophy. Philosophical intuitions provide data to be explained by our philosophical theories [and] evidence that may be adduced in arguments for their truth... In this way, the role... of intuitional evidence in philosophy is similar to the role... of perceptual evidence in science...
Is knowledge simply justified true belief? Is a belief justified just in case it is caused by a reliable cognitive mechanism? Does a name refer to whatever object uniquely or best satisfies the description associated with it? Is a person morally responsible for an action only if she could have acted otherwise? Is an action morally right just in case it provides the greatest benefit for the greatest number of people all else being equal? When confronted with these kinds of questions, philosophers often appeal to philosophical intuitions about real or imagined cases...
...there is widespread agreement about the role that [intuitions] play in contemporary philosophical practice... We advance philosophical theories on the basis of their ability to explain our philosophical intuitions, and appeal to them as evidence that those theories are true...
In particular, notice that philosophers do not appeal to their intuitions as merely an exercise in autobiography. Philosophers are not merely trying to map the contours of their own idiosyncratic concepts. That could be interesting, but it wouldn't be worth decades of publicly-funded philosophical research. Instead, philosophers appeal to their intuitions as evidence for what is true in general about a concept, or true about the world.
Part of the sequence: Rationality and Philosophy
In my last post, I showed that the brain does not encode concepts in terms of necessary and sufficient conditions. So, any philosophical practice which assumes this — as much of 20th century conceptual analysis seems to do — is misguided.
Next, I want to show that human abstract thought is pervaded by metaphor, and that this has implications for how we think about the nature of philosophical questions and philosophical answers. As Lakoff & Johnson (1999) write:
If we are going to ask philosophical questions, we have to remember that we are human... The fact that abstract thought is mostly metaphorical means that answers to philosophical questions have always been, and always will be, mostly metaphorical. In itself, that is neither good nor bad. It is simply a fact about the capacities of the human mind. But it has major consequences for every aspect of philosophy. Metaphorical thought is the principal tool that makes philosophical insight possible, and that constrains the forms that philosophy can take.
To understand how fundamental metaphor is to our thinking, we must remember that human cognition is embodied:
We have inherited from the Western philosophical tradition a theory of faculty psychology, in which we have a "faculty" of reason that is separate from and independent of what we do with our bodies. In particular, reason is seen as independent of perception and bodily movement...
The evidence from cognitive science shows that classical faculty psychology is wrong. There is no such fully autonomous faculty of reason separate from and independent of bodily capacities such as perception and movement. The evidence supports, instead, an evolutionary view, in which reason uses and grows out of such bodily capacities.
Consider, for example, the fact that as neural beings we must categorize things:
We are neural beings. Our brains each have 100 billion neurons and 100 trillion synaptic connections. It is common in the brain for information to be passed from one dense ensemble of neurons to another via a relatively sparse set of connections. Whenever this happens, the pattern of activation distributed over the first set of neurons is too great to be represented in a one-to-one manner in the sparse set of connections. Therefore, the sparse set of connections necessarily groups together certain input patterns in mapping them across to the output ensemble. Whenever a neural ensemble provides the same output with different inputs, there is neural categorization.
To take a concrete example, each human eye has 100 million light-sensing cells, but only about 1 million fibers leading to the brain. Each incoming image must therefore be reduced in complexity by a factor of 100. That is, information in each fiber constitutes a "categorization" of the information from about 100 cells.
Moreover, almost all our categorizations are determined by the unconscious associative mind — outside our control and even our awareness — as we interact with the world. As Lakoff & Johnson note, "Even when we think we are deliberately forming new categories, our unconscious categories enter into our choice of possible conscious categories."
One of the core aims of the philosophy of probability is to explain the relationship between frequency and probability. The frequentist proposes identity as the relationship. This use of identity is highly dubious. We know how to check for identity between numbers, or even how to check for the weaker copula relation between particular objects; but how would we test the identity of frequency and probability? It is not immediately obvious that there is some simple value out there which is modeled by probability, like position and mass are values that are modeled by Newton's Principia. You can actually check if density * volume = mass, by taking separate measurements of mass, density and volume, but what would you measure to check a frequency against a probability?
There are certain appeals to frequentest philosophy: we would like to say that if a bag has 100 balls in it, only 1 of which is white, then the probability of drawing the white ball is 1/100, and that if we take a non-white ball out, the probability of drawing the white ball is now 1/99. Frequentism would make the philosophical justification of that inference trivial. But of course, anything a frequentist can do, a Bayesian can do (better). I mean that literally: it's the stronger magic.
A Subjective Bayesian, more or less, says that the reason frequencies are related to probabilities is because when you learn a frequency you thereby learn a fact about the world, and one must update one's degrees of belief on every available fact. The subjective Bayesian actually uses the copula in another strange way:
Probability is subjective degree of belief.
and subjective Bayesians also claim:
Probabilities are not in the world, they are in your mind.
These two statements are brilliantly championed in Probability is Subjectively Objective. But ultimately, the formalism which I would like to suggest denies both of these statements. Formalists do not ontologically commit themselves to probabilities, just as they do not say that numbers exist; hence we don't allocate probabilities in the mind or anywhere else; we only commit ourselves to number theory, and probability theory. Mathematical theories are simply repeatable processes which construct certain sequences of squiggles called "theorems", by changing the squiggles of other theorems, according to certain rules called "inferences". Inferences always take as input certain sequences of squiggles called premises, and output a sequence of squiggles called the conclusion. The only thing an inference ever does is add squiggles to a theorem, take away squiggles from a theorem, or both. It turns out that these squiggle sequences mixed with inferences can talk about almost anything, certainly any computable thing. The formalist does not need to ontologically commit to numbers to assert that "There is a prime greater than 10000.", even though "There is x such that" is a flat assertion of existence; because for the formalist "There is a prime greater than 10000." simply means that number theory contains a theorem which is interpreted as "there is a prime greater than 10000." When you say a mathematical fact in English, you are interpreting a theorem from a formal theory. If under your suggested interpretation, all of the theorems of the theory are true, then whatever system/mechanism your interpretation of the theory talks about, is said to be modeled by the theory.
So, what is the relation between frequency and probability proposed by formalism? Theorems of probability, may be interpreted as true statements about frequencies, when you assign certain squiggles certain words and claim the resulting natural language sentence. Or for short we can say: "Probability theory models frequency." It is trivial to show that Komolgorov models frequency, since it also models fractions; it is an algebra after all. More interestingly, probability theory models rational distributions of subjective degree of believe, and the optimal updating of degree of believe given new information. This is somewhat harder to show; dutch-book arguments do nicely to at least provide some intuitive understanding of the relation between degree of belief, betting, and probability, but there is still work to be done here. If Bayesian probability theory really does model rational belief, which many believe it does, then that is likely the most interesting thing we are ever going to be able to model with probability. But probability theory also models spatial measurement? Why not add the position that probability is volume to the debating lines of the philosophy of probability?
Why are frequentism's and subjective Bayesianism's misuses of the copula not as obvious as volumeism's? This is because what the Bayesian and frequentest are really arguing about is statistical methodology, they've just disguised the argument as an argument about what probability is. Your interpretation of probability theory will determine how you model uncertainty, and hence determine your statistical methodology. Volumeism cannot handle uncertainty in any obvious way; however, the Bayesian and frequentest interpretations of probability theory, imply two radically different ways of handling uncertainty.
The easiest way to understand the philosophical dispute between the frequentist and the subjective Bayesian is to look at the classic biased coin:
A subjective Bayesian and a frequentist are at a bar, and the bartender (being rather bored) tells the two that he has a biased coin, and asks them "what is the probability that the coin will come up heads on the first flip?" The frequentist says that for the coin to be biased means for it not have a 50% chance of coming up heads, so all we know is that it has a probability that is not equal 50%. The Bayesain says that that any evidence I have for it coming up heads, is also evidence for it coming up tails, since I know nothing about one outcome, that doesn't hold for its negation, and the only value which represents that symmetry is 50%.
I ask you. What is the difference between these two, and the poor souls engaged in endless debate over realism about sound in the beginning of Making Beliefs Pay Rent?
If a tree falls in a forest and no one hears it, does it make a sound? One says, "Yes it does, for it makes vibrations in the air." Another says, "No it does not, for there is no auditory processing in any brain."
One is being asked: "Are there pressure waves in the air if we aren't around?" the other is being asked: "Are there auditory experiences if we are not around?" The problem is that "sound" is being used to stand for both "auditory experience" and "pressure waves through air". They are both giving the right answers to these respective questions. But they are failing to Replace the Symbol with the Substance and they're using one word with two different meanings in different places. In the exact same way, "probability" is being used to stand for both "frequency of occurrence" and "rational degree of belief" in the dispute between the Bayesian and the frequentist. The correct answer to the question: "If the coin is flipped an infinite amount of times, how frequently would we expect to see a coin that landed on heads?" is "All we know, is that it wouldn't be 50%." because that is what it means for the coin to be biased. The correct answer to the question: "What is the optimal degree of belief that we should assign to the first trial being heads?" is "Precisely 50%.", because of the symmetrical evidential support the results get from our background information. How we should actually model the situation as statisticians depends on our goal. But remember that Bayesianism is the stronger magic, and the only contender for perfection in the competition.
For us formalists, probabilities are not anywhere. We do not even believe in probability technically, we only believe in probability theory. The only coherent uses of "probability" in natural language are purely syncategorematic. We should be very careful when we colloquially use "probability" as a noun or verb, and be very careful and clear about what we mean by this word play. Probability theory models many things, including degree of belief, and frequency. Whatever we may learn about rationality, frequency, measure, or any of the other mechanisms that probability models, through the interpretation of probability theorems, we learn because probability theory is isomorphic to those mechanisms. When you use the copula like the frequentist or the subjective Bayesian, it makes it hard to notice that probability theory modeling both frequency and degree of belief, is not a contradiction. If we use "is" instead of "model", it is clear that frequency is not degree of belief, so if probability is belief, then it is not frequency. Though frequency is not degree of belief, frequency does model degree of belief, so if probability models frequency, it must also model degree of belief.
Part of the sequence: Rationality and Philosophy
Philosophy in the Flesh, by George Lakoff and Mark Johnson, opens with a bang:
The mind is inherently embodied. Thought is mostly unconscious. Abstract concepts are largely metaphorical.
These are three major findings of cognitive science. More than two millennia of a priori philosophical speculation about these aspects of reason are over. Because of these discoveries, philosophy can never be the same again.
When taken together and considered in detail, these three findings... are inconsistent with central parts of... analytic philosophy...
This book asks: What would happen if we started with these empirical discoveries about the nature of mind and constructed philosophy anew?
...A serious appreciation of cognitive science requires us to rethink philosophy from the beginning, in a way that would put it more in touch with the reality of how we think.
So what would happen if we dropped all philosophical methods that were developed when we had a Cartesian view of the mind and of reason, and instead invented philosophy anew given what we now know about the physical processes that produce human reasoning?
What emerges is a philosophy close to the bone. A philosophical perspective based on our empirical understanding of the embodiment of mind is a philosophy in the flesh, a philosophy that takes account of what we most basically are and can be.
Before I read Probability is in the Mind and Probability is Subjectively Objective I was a realist about probabilities; I was a frequentest. After I read them, I was just confused. I couldn't understand how a mind could accurately say the probability of getting a heart in a standard deck of playing cards was not 25%. It wasn't until I tried to explain the contrast between my view and the subjective view in a comment on Probability is Subjectively Objective that I realized I was a subjective Bayesian all along. So, if you've read Probability is in the Mind and read Probability is Subjectively Objective but still feel a little confused, hopefully, this will help.
I should mention that I'm not sure that EY would agree with my view of probability, but the view to be presented agrees with EY's view on at least these propositions:
- Probability is always in a mind, not in the world.
- The probability that an agent should ascribe to a proposition is directly related to that agent's knowledge of the world.
- There is only one correct probability to assign to a proposition given your partial knowledge of the world.
- If there is no uncertainty, there is no probability.
And any position that holds these propositions is a non-realist-subjective view of probability.
Imagine a pre-shuffled deck of playing cards and two agents (they don't have to be humans), named "Johnny" and "Sally", which are betting 1 dollar each on the suit of the top card. As everyone knows, 1/4 of the cards in a playing card deck are hearts. We will name this belief F1; F1 stands for "1/4 of the cards in the deck are hearts.". Johnny and Sally both believe F1. F1 is all that Johnny knows about the deck of cards, but sally knows a little bit more about this deck. Sally also knows that 8 of the top 10 cards are hearts. Let F2 stand for "8 out of the 10 top cards are hearts.". Sally believes F2. John doesn't know whether or not F2. F1 and F2 are beliefs about the deck of cards and they are either true or false.
So, sally bets that the top card is a heart and Johnny bets against her, i.e., she puts her money on "Top card is a heart." being true; he puts his money on "~The top card is a heart." being true. After they make their bets, one could imagine Johnny making fun of Sally; he might say something like: "Are you nuts? You know, I have a 75% chance of winning. 1/4 of the cards are hearts; you can't argue with that!" Sally might reply: "Don't forget that the probability you assign to '~The top card is a heart.' depends on what you know about the deck. I think you would agree with me that there is an 80% chance that 'The top card is a heart' if you knew just a bit more about the state of the deck."
To be undecided about a proposition is to not know which possible world you are in; am I in the possible world where that proposition is true, or in the one where it is false? Both Johnny and Sally are undecided about "The top card is a heart."; their model of the world splits at that point of representation. Their knowledge is consistent with being in a possible world where the top card is a heart, or in a possible world where the top card is not a heart. The more statements they decide on, the smaller the configuration space of possible worlds they think they might find themselves in; deciding on a proposition takes a chunk off of that configuration space, and the content of that proposition determines the shape of the eliminated chunk; Sally's and Johnny's beliefs constrain their respective expected experiences, but not all the way to a point. The trick when constraining one's space of viable worlds, is to make sure that the real world is among the possible worlds that satisfy your beliefs. Sally still has the upper hand, because her space of viably possible worlds is smaller than Johnny's. There are many more ways you could arrange a standard deck of playing cards that satisfies F1 than there are ways to arrange a deck of cards that satisfies F1 and F2. To be clear, we don't need to believe that possible worlds actually exist to accept this view of belief; we just need to believe that any agent capable of being undecided about a proposition is also capable of imagining alternative ways the world could consistently turn out to be, i.e., capable of imagining possible worlds.
For convenience, we will say that a possible world W, is viable for an agent A, if and only if, W satisfies A's background knowledge of decided propositions, i.e., A thinks that W might be the world it finds itself in.
Of the possible worlds that satisfy F1, i.e., of the possible worlds where "1/4 of the cards are hearts" is true, 3/4 of them also satisfy "~The top card is a heart." Since Johnny holds that F1, and since he has no further information that might put stronger restrictions on his space of viable worlds, he ascribes a 75% probability to "~The top card is a heart." Sally, however, holds that F2 as well as F1. She knows that of the possible worlds that satisfy F1 only 1/4 of them satisfy "The top card is a heart." But she holds a proposition that constrains her space of viably possible worlds even further, namely F2. Most of the possible worlds that satisfy F1 are eliminated as viable worlds if we hold that F2 as well, because most of the possible worlds that satisfy F1 don't satisfy F2. Of the possible worlds that satisfy F2 exactly 80% of them satisfy "The top card is a heart." So, duh, Sally assigns an 80% probability to "The top card is a heart." They give that proposition different probabilities, and they are both right in assigning their respective probabilities; they don't disagree about how to assign probabilities, they just have different resources for doing so in this case. P(~The top card is a heart|F1) really is 75% and P(The top card is a heart|F2) really is 80%.
This setup makes it clear (to me at least) that the right probability to assign to a proposition depends on what you know. The more you know, i.e., the more you constrain the space of worlds you think you might be in, the more useful the probability you assign. The probability that an agent should ascribe to a proposition is directly related to that agent's knowledge of the world.
This setup also makes it easy to see how an agent can be wrong about the probability it assigns to a proposition given its background knowledge. Imagine a third agent, named "Billy", that has the same information as Sally, but say's that there's a 99% chance of "The top card is a heart." Billy doesn't have any information that further constrains the possible worlds he thinks he might find himself in; he's just wrong about the fraction of possible worlds that satisfy F2 that also satisfy "The top card is a heart.". Of all the possible worlds that satisfy F2 exactly 80% of them satisfy "The top card is a heart.", no more, no less. There is only one correct probability to assign to a proposition given your partial knowledge.
The last benefit of this way of talking I'll mention is that it makes probability's dependence on ignorance clear. We can imagine another agent that knows the truth value of every proposition, lets call him "FSM". There is only one possible world that satisfies all of FSM's background knowledge; the only viable world for FSM is the real world. Of the possible worlds that satisfy FSM's background knowledge, either all of them satisfy "The top card is a heart." or none of them do, since there is only one viable world for FSM. So the only probabilities FSM can assign to "The top card is a heart." are 1 or 0. In fact, those are the only probabilities FSM can assign to any proposition. If there is no uncertainty, there is no probability.
The world knows whether or not any given proposition is true (assuming determinism). The world itself is never uncertain, only the parts of the world that we call agents can be uncertain. Hence, Probability is always in a mind, not in the world. The probabilities that the universe assigns to a proposition are always 1 or 0, for the same reasons FSM only assigns a 1 or 0, and 1 and 0 aren't really probabilities.
In conclusion, I'll risk the hypothesis that: Where 0≤x≤1, "P(a|b)=x" is true, if and only if, of the possible worlds that satisfy "b", x of them also satisfy "a". Probabilities are propositional attitudes, and the probability value (or range of values) you assign to a proposition is representative of the fraction of possible worlds you find viable that satisfy that proposition. You may be wrong about the value of that fraction, and as a result you may be wrong about the probability you assign.
We may call the position summarized by the hypothesis above "Modal Satisfaction Frequency theory", or "MSF theory".
Anyone who does not believe mental states are ontologically fundamental - ie anyone who denies the reality of something like a soul - has two choices about where to go next. They can try reducing mental states to smaller components, or they can stop talking about them entirely.
In a utility-maximizing AI, mental states can be reduced to smaller components. The AI will have goals, and those goals, upon closer examination, will be lines in a computer program.
But in the blue-minimizing robot, its "goal" isn't even a line in its program. There's nothing that looks remotely like a goal in its programming, and goals appear only when you make rough generalizations from its behavior in limited cases.
Philosophers are still very much arguing about whether this applies to humans; the two schools call themselves reductionists and eliminativists (with a third school of wishy-washy half-and-half people calling themselves revisionists). Reductionists want to reduce things like goals and preferences to the appropriate neurons in the brain; eliminativists want to prove that humans, like the blue-minimizing robot, don't have anything of the sort until you start looking at high level abstractions.
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