 Okay, it's a pleasure to have Taukatzi among us from the University of Iria in Georgia. He's visiting for three weeks as an invited professor. And we'll do a little talk, a research talk about formal theories on the different contexts of sensitivity. Thank you. Well, yeah, I'm going to be talking about formal theories of belief-credence interaction and the problems that with some subset of the service you rise to. So to introduce this topic, I'll just briefly explain what the theories are theories of and what problems the theories are designed to solve and what are the problems they do rise. So when we think about doxastic attitudes of human thinkers, it seems to come in two general types. We can think about belief as one of the fundamental epistemic doxastic attitudes as a categorical attitude towards a proposition. So belief is a binary attitude towards a proposition. You can either believe proposition or not. For instance, you use a belief that it will rain tomorrow or you don't have this belief. So it's a categorical binary attitude towards a proposition. But human thinkers also have various levels of confidences towards propositions. And I'm going to assume that confidences have numerical structure or at least can be represented as numerical attitudes. So when we're going to be talking about confidences, I would have in mind degrees of belief or Bayesian credences towards a proposition. And we can have both belief senses, numerical attitude towards the same propositions. For example, I can both believe that it will rain tomorrow and have, say, 80% confidence in that proposition. Now, plausibly, there should be some kind of connection between these two, between our confidences in a proposition and our categorical attitudes. For instance, it seems highly unreasonable to have a high confidence in a proposition. But I would try to disbelieve it. For instance, if I have a high confidence that it will rain tomorrow, it seems inappropriate for me to invest very low degree of confidence in that proposition. So this kind of plausible source motivates this possible research program to find general precise coherency conditions which beliefs and credences should satisfy. So this should be a mutual constraint on belief and credences or confidences. And we want to find some kind of bridges, maybe, between beliefs and credences, such that rational agents should have beliefs and credences that satisfy these normative constraints. So this thought can be motivated by rather ordinary thinking about our psychology. We both have confidences and beliefs. These two must be hanged together in certain kinds of cases, as I've already given. So we want to find more general precise principles to relate the two. So this will be purely formal principles. I came to how we think about logical norms of one belief, for instance, like principles like consistency, that rational agent should try to arrange consistency. So we want to find similar formal principles on how these two should hang together. So the most obvious way to do this is via the so-called Locant thesis that you can see in your handout in point four. So Locant thesis says that belief should correspond to a sufficiently high degree of confidence in a proposition. So this Locant thesis is not meant to reduce belief to confidences or reduce confidences to beliefs. It just states that these two should hang together in a way that you believe something if and only if you have a high confidence in that proposition. So what exactly do we mean when we say high confidence? Well, we may have some specific number in mind, which is strictly greater than point five, but less than one, and this degree of sufficiently high degree must be somewhere in between. So it should be less than one, but greater than point one. And we can say that you should believe it if and only if your probability in X is greater or equal to this threshold value S. So we possibly have some threshold value between point five and one and say that belief corresponds to having this confidence equal to this threshold one. So idea is quite plausible. I think it's most natural comes to mind when we think about how these ingredients should interact. So that's the principle that is most discussed within a literature in one way or another. Okay, now I will introduce why is this obvious way to connecting beliefs and prejudices. I can rise to a problem. I will explain two problems. One is most well-known, I would say, and it has to do with the lottery paradox. So lottery paradox tells us that the basic assumptions we make about, at least in the literature, within the epistemology, basic assumptions we can make about belief and credence and how these two should come together are not consistent with each other. So lottery paradox can be explained in this way. So imagine there is a fair lottery, let's say consisting of 100 tickets, let's say. What amount of tickets is irrelevant here, but for simplicity and specificity let's take 100 tickets. And we know the lottery is fair in a way that exactly one ticket will win. So we have beliefs that is a ticket one will win, ticket two will win, and so on, up to the 100th ticket. And because it's fair, it seems rational and required to be equally confident that each ticket will win. So because we have 100 tickets, we will think about the uniform probability distribution versus tickets winning, so we will have 1 over 100 probability in each ticket winning. So these two assumptions are quite uncontroversial. So then we assume Lockean thesis with specific threshold, threshold 0.99, which seems to be quite high anyway. So according to this version of Lockean thesis with threshold 99.99, we will proposition if its freedom is 99% or 0.99 probability. So given this scenario, Lockean thesis demands us to believe about each individual ticket that it will lose. So it's rational to believe that ticket one will lose because its probability is 0.99 and so on, up to the 100th ticket. So already these three principles, this is one assumption that you believe that some ticket will win and the assumption of fairness with the Lockean thesis gives us a strange result that you believe propositions which cannot simultaneously be true. Because from your overall perspective, because you believe that this ticket first will lose and ticket second will lose, that you also believe that some ticket will win. So there is already an inconsistency, but we can show more. If we postulate another possible principle, which is conjunctive closure, if we think that belief should be closed under conjunction, meaning that if you believe proposition X and if you believe proposition Y, then you should also believe X and Y's are conjunction. And if we further assume that you should not believe explicit contradictions, that is you should not believe proposition X and its negation, then we get inconsistency between these principles. And it is fairly easy to see. So given conjunctive closure as a rule of no contradiction, now given Lockean thesis and conjunctive closure, you believe that this big conjunction in all links that ticket one will lose, ticket two will lose and so on. But you also believe that some ticket will win, so you will have a belief that no ticket will win and some ticket will win. So which of course violates no contradiction rule. So these principles are not mutually consistent, and that's what the plot recovery shows. I have a question. You say explicitly that you don't, that no contradiction principle says don't believe an explicit contradiction, not just a contradiction but an explicit contradiction. And I don't see the explicit contradiction as far as I see. I mean is there no sentence A and not A that directly follows from how you set it up? Yeah, I see. Yeah, so now if we don't get too many viewer propositions here and we're just syncing tactically as sentences, then we should also add that if you believe a proposition and another one is logically equivalent to the proposition, then you should also believe that. Then we should... So I like the Morgan's Law. We get explicit contradiction. But if you set this scenario within possible worlds, a counter proposition, then you automatically get... I agree, but then my question I guess is why the word explicit? If you don't think about it in a syntactic way, what does explicit mean? Yeah, okay. So we imagine if I reject conjunctive closure, so I can have... I can believe set of propositions which are inconsistent. That is there is no possible world in which all of them are true. But I still don't believe explicit contradiction because I don't close my beliefs under conjunction. So I can have inconsistent beliefs but don't believe explicit contradiction. So in lottery scenario it will be where... I believe about each individual ticket's attitude to lose, but I don't believe conjunction of such propositions. I see. I'm not sure that it's going to work with more possible worlds, but I see. So you'll have to do something... Propositions... You wouldn't understand propositions to be such or possible. Otherwise there will be the same... Exactly. No, actually it will work because if you don't close your beliefs under conjunction, so you will believe separate propositions but not their conjunction, right? Okay, okay. So we have two accessibility relations. Let's say we consider two atomic propositions, and you believe each, but you don't believe their conjunction. So it cannot be modeled. You cannot give simple possible worlds amount to it. So we would... Yeah, but it's good that we qualified this. So people who don't accept conjunctive rules usually can also, like, don't accept conjunction, but accept no explicit contradictions. So it would only... Because we get explicit contradiction when we better close your beliefs under conjunction. Another... Any other question about lottery paradox? So another is a quite interesting argument against Locant's thesis. So this problem comes when we think about belief in the diachronic way. So when we consider how beliefs should change when we acquire new evidence. And as we'll see on Locant's thesis, you may start believing a proposition and then learn something you already believe. And learning what you already believe can undermine your belief, some other belief. So this seems to be very counterintuitive, like learning something you already believe. Why should it undermine your other beliefs, right? So the case that I explained this problem with is on page two. So imagine you are in a situation when you think in terms of street proposition, you are concerned with street proposition. So you're wondering whether or not the loan is wet and you consider two potential causes of this proposition. One is that it is arraigned or the user left the sprinkler on and says that in this situation your Locant's threshold is 0.55. So in this scenario you believe something if its probability is above or equal to 0.55. And as you can see in this scenario you will believe that loan is wet because its probability is roughly 0.7. You also believe that it won't rain and you also believe that sprinkler is on. All of these beliefs are rational, all of this Locant's thesis was 0.55 threshold. Now suppose that you learn what you believe that it hasn't rained, right? You would learn it to be true that it hasn't rained. But even without making explicit probability calculations it seems clear that once you boost this big support for your beliefs that it has rained you should no longer believe that loan is wet. This case should be intuitive but it can be also modelled with probability assignments. So this is a case in which you originally believe certain things, there's something you believe but lost your other beliefs as long as wet because its probability dropped to 0.48. Is this example clear? So yeah, that's another problem with the Locant's thesis. You can discuss other problems as well but it's an interesting dichronic dimension which shows problems with the Locant's thesis as well. Okay, now I'll explain what will come next in my talk. So I'll explain another approach to belief-critic interaction which rejects the simple Locant's thesis but still retains the core aspect of it. Then I will show that it raised this problem of context sensitivity or partition sensitivity and I'll suggest a way of avoiding this problem. Yeah, I'll say more about this when we turn to that section in my talk. Okay, so what are contemporary approaches? I will, as a representative example, I'll take Hannes Leiteb's stability theory but a similar problem arises for other similar theories as well like Lin and Kelly's theory of belief-criticist interaction so we'll only focus on Leiteb's theory. So the philosophical idea of this theory is quite simple. So Leiteb's proposal is motivated by the views that rational belief should be stable with respect to a certain defeaters that you may learn. Defeaters are potential evidence or knowledge that you may gain in the future which will cancel your original belief. So Leiteb's proposal is this that your belief should be stable with respect to a certain set of defeaters. So that's the idea. So why do we want belief to be stable in this way? So we can give a couple of motivation to this view. So one thing is that beliefs like what function of role belief should have when we think about this it seems that stability should be a property of rational belief. For instance, if I believe something but I expect this belief to be easily lost during acquisition of new information then this belief won't be so useful in navigating and in doing extended actions. For example, if I believe that it will be a good weather tomorrow and I don't expect to have this belief in the next hour or so and it doesn't seem that this belief would be of any use to me. So belief should have some kind of stability. And another motivation can come for thinking about the role of suppositional reasoning. So I can add certain propositions to my stock of knowledge and convey reasoning from their own. For example, I can say something like supposing the age of the earth is three million years or whatever and reason from their own. So if that's a quite drastic suppositional reason but let's take an easier one like supposing the weather will be good tomorrow. I make this supposition. I want some of my core beliefs to stay intact after I make certain suppositions. I don't want suppositions to cancel all my other beliefs otherwise suppositional reasoning won't be of much use. So these two things will show that some kind of stability should be an aspect for questioning at least that's how we can be motivated. Light week has other motivations as well but I think these two will be enough for our purposes. So then he proposes a specific way to explicate this idea this stability idea of rational belief and the explication he endorses it's called the Humian thesis with a specific interpretation but I'll just call it the Humian thesis here. So Humian thesis says the following that the set of potential defeaters should be a set of all propositions you don't disbelieve. So you believe should be stable with respect to a set of propositions you don't disbelieve. So it need not be stable with respect to propositions you disbelieve only with propositions you don't disbelieve. So it includes both beliefs and the propositions you suspend judgments. And precisely so he calls this the post-variant of Humian thesis so post-variant means that he considers propositions which are doxastically possible from your point of view. So either you believe them or don't disbelieve them. So this version of Humian thesis is a specific statement of thesis in 2.1 so it says that any rational agents belief and probability function should be such that you believe proposition X if and only if for all propositions Y if Y is doxastically possible for you the probability of X given Y should be greater than R where R is a number which is equal or greater than 2.5. So the idea is so basic idea is quite simple like when you condition your beliefs to propositions where you don't disbelieve the probability of this belief proposition should not drop too low it should be higher than 0.5. So that's the explanation. The R is supposed to be like a variable or is it something that you fix for extra logical reasons or something? You fix it for extra logical reasons but the weakest version of the rest of the money will just be more than 0.5. It's a parameter of this thesis so this statement here still has undefined parameters the same sense in which we have undefined parameter in Locan's thesis so here we have this R undefined but we can assume that this is just 0.5 more than 0.5. So Leibniz also considers other way to explicate this which takes other sets of the features instead of proposition you disbelieve for example you may demand that your belief should be stable with respect to proposition you assign certain probability to that's one way another way will be to demand stability only with respect to beliefs you have but this is more demanding than that it demands stability to be with respect to propositions you don't disbelieve So from this human thesis with this interpretation that I've given he showed quite remarkable results for instance he shows that human thesis can tell standard principles of doxastic logic like principles like conjunctive closure principles like that you should believe tautologists principles like you should not believe contradiction so all standard principles of doxastic logic can be deduced from this human thesis so it's quite interesting you don't need to assume this from the get go so you can deduce this from the stability account of belief which was quite remarkable and surprisingly he also shows that this human thesis entails the core idea of the doxastic thesis as well so earlier we took the doxastic paradox as showing that you cannot have both doxastic logic and doxastic thesis simultaneously but what now I'll explain is that you can still retain the core idea of doxastic thesis and the trick is to make doxastic threshold context sensitive so we don't fix one doxastic threshold which is good for all contexts but we value doxastic threshold with respect to context that we are interested in for instance in the lottery case where we have 99 sorry 100 tickets 100 fare tickets in this context doxastic threshold should be higher than 0.99 but less than 1 if we have this kind of threshold then no contradiction will be believed but in some other contexts like the more coarse-grained context which I give an example with this rain and sprinkler and low-wing wet in this context the doxastic threshold can be considerably lower so we just fix doxastic threshold relative to which probability assignment we are focusing on so that's the trick how he manages to get the doxastic context dependent version of doxastic thesis and doxastic logic so that's how the trick is done okay so any questions until now? okay so we have this context dependency of doxastic threshold but we also have other context sensitive variable in lightning theory which as I argue is much more problematic and this has been noticed by many other people and lightning also discusses this as a problem so it's quite well recognized so on his theory belief is also partition sensitive by partition sensitivity I meant the following that whether or not agent is rational in believing in proposition X depends not only this proposition but also other propositions that you happen to consider right? so first I'll say why this aspect of his theory may be a good-making feature of his theory like advantage of his theory then as well why it also has extremely implausible consequences so this feature of his theory allows lightning to give a solution to the lottery paradox which is quite neat so here is the explanation so in lottery paradox we have two conflicting intuitions one intuition is that it is rational to believe that lottery ticket will lose because probability is so high another intuition is that we shouldn't believe it because otherwise we will at least believe in consistent set of propositions so we have this conflicting intuition about this I know that some people don't have intuitions that you should believe lottery tickets but yeah let's close that there is such intuition so his theory can explain why we have this conflicting intuition about this scenario he says that when we just focus on whether or not this ticket will lose or not let's say ticket 1 here we are focusing on very coarse-grained set of possibilities possibilities in which this ticket will win or lose okay so the parameter which explicitly states how many possibilities we are considering is capital W so capital W is set of all possibilities we take into account so in a coarse-grained context when we only only consider two possibilities this is let's say that ticket 1 will win this is that ticket 1 will lose here this capital W is a set of two possibilities but we can also think about which ticket will win or lose like having into mind like all hundred possibilities if we are thinking about hundred lottery consisting of hundred tickets so then we have different sets of possibilities so it will be W1, W2, W2, W1 okay so on his account it is rationally permissible to believe that ticket 1 will lose in this context but not in this context okay because on this context you cannot you cannot satisfy this humans thesis so this so I also want to comment on how we can figure out like whether or not humans thesis is satisfied or not fortunately finding out whether or not set of beliefs and probability function satisfies their humanities it's very computationally simple to determine this we only need to look whether the following condition which likely because outclassing condition is satisfied so condition is quite simple you can have a set of possibilities and if there is a proposition with respect to this possibilities such that each world in which this proposition is true is still more probable than the negation of this proposition than this proposition is rational to believe and basically you can logically you can get deductive closure of this proposition and this will give you belief that here's what I mean like take this fine-grained context so to find out whether or not some proposition is rational to believe here we can think in this way like is there any proposition X such that if X is true in say WI then probability of this proposition should be greater than probability of not X so this is what we should look at so in lottery cases because we have a uniform probability distribution every possibility is of equal probability or associated equal probability we don't get any such proposition because each world in which any proposition is true there is another world outside it which is equally probable so this outclassing condition always sets right so proposition which sets right this outclassing condition is called stable proposition to unlikely so we won't get any stable proposition here so computation is quite simple to determine with respect to any finite at least for computers it's very easy to find out whether or not this outclassing condition sets right so here is this context sensitivity of Lightgear's approach is working on his favor so he can say that look I have a solution to lottery paradox in a way that keeps this intuition we had that it's both reasonable to believe that there is no such ticket to lose and also reasonable to think that we should not believe anything so it works in his favor in this case but all is not so good with this context sensitivity which I'll explain with the following simple example so I'm now in the fourth range of finding out look at section 2.6 and we will first consider the example examples that I give you so the example is as follows suppose you are thinking about whether your colleague will say hold them or not and call this proposition H so you are 0.66 confidence that both will hold them and if you are only focusing on this proposition H then it's rational according to Lightgear's theory to believe that it's H because it has a stable proposition in this coarse grain situation it has sufficiently high probability now suppose for some reason we don't need to get into you start considering completely irrelevant proposition first of all whether or not a coin then it has or not and suppose you are 50% confident that it will lend heads or tails now let's find grain in our reasoning context include this proposition as well and as it happens in this context it's no longer rational like in theory to believe that H because this proposition won't be stable there won't be any stable proposition here except the trivial propositions that sum up this possibility so our problem is this considering irrelevant propositions it shouldn't influence our beliefs at all but here I start considering irrelevant proposition about I'm losing my beliefs so a coin nicely summarizes this impossibility of this so what Dominic says and many others also are in this is that if stability theory is supposed to be a theory of rational belief we must have some kind of story of which context are relevant to considering which not so you just say that anything goes as far as context as course says that we get this awfully bad scenarios where just considering irrelevant propositions defeats our beliefs I was going to say that that was actually it's funny because that's that question about where you focus your attention that was my initial reaction to your long case because I was thinking that seems weird how can that be right but then I thought to myself wait what I feel like I would probably do in that situation is I would keep my belief that the long was wet but then if you were like but remember your belief that this sprinkler was on was not very strong then I would go back and say alright that wasn't pretty weak belief I probably should now believe that the long is dry I would go back so revising in light of relevant evidence makes perfect sense but revising in light of your relevant evidence seems like a big problem that's just really neat you're going to get the same problem regardless of whether or not the change in partition actually has to do with the problem that's really cool but in long cases I should also point out that when you find out that it hasn't rained your probability that long is wet will come down to less than 0.5 so here I was implicitly assuming the view that you shouldn't be the proposition which you are not more confident than the negation but in long cases these propositions that we are considering are all causally relevant so this has some kind of unity to it but light evidence has a story about which contexts are relevant or not like it doesn't consider it at all he thinks it is solely pragmatic so then people pointed out that it has a very implausible consequences did he react to that? did he react to Duven's criticism? about context sensitivity yes but I cannot seem to him commenting on this kind of counter-examples but everybody is impressed by the results he got formal results but all reviews find this to be highly implausible aspect of his theory so just any context goes and your relief just crumbles like let me put it this way so stability let's say stability is a good-making feature if we accept this but why not worry about stability across context, right? so he is only worried about stability within a context but whenever I change my partition of possibilities reviews can crumble and that doesn't seem to be correct as I said he thinks that there are plausible applications of this in lottery at the end he recognizes that this is the cost of the theory but things that overall it's a cost we should be willing to pay to get these nice results but he doesn't respond to the relevant proposition example as far as I know so far okay so all I think this problem should be quite clear that considering the relevant proposition cannot alter my views it's just not how rational reviews should work okay so so we have this we have this problem there are so yeah so some people others and no one also ask for like we should have some kind of theory at least if we make completely context sensitive well what should we pay attention to or what not so just saying that belief is context sensitive won't gonna work and given that we have such immediate problems it's not a plausible option just to accept that belief is context sensitive now I want to point out that other theories of belief-credence interaction like Lehman-Haley's theory have similar problems with context sensitive not with this example but their stories are also context sensitive in the lottery case for example in many other cases they actually proved a general result that all theories will be context sensitive if they want to have some kind of logical norms will lead to it so this will be a general feature of formal theories if they want to satisfy other views as well okay another problem another problem that I want to point out is this second problem on page 4 so this argument was made by Donovan Roth so they have the same problem so life history and other theories as well are built on toy examples like we only consider a couple of possibilities in the lottery case we consider many but still it's a toy example because we are only focusing on lottery and they wanted to know how the situation will be if you take a more realistic approach a lot of more possibilities and they are curious to find out whether or not life history and other theories as well will support any beliefs which are not certain in this more realistic context and they got a formal result which says that it's unlikely that a random distribution of probability will get you any no certain beliefs if we expand the sorts of possibilities so the problem is this that the stability theory becomes very skeptical once you include a lot of propositions in your context of reasoning ok it can work with 3-4 possibilities or with the sprinkle example we only have 8 examples and we don't even need that much there but when we take into account our beliefs we have in our background which we don't focus on explicitly then the lighting theory is very similar to what it's called certainty proposal certainty proposal is believe something which you are certain of and nothing else so it becomes more or less the same so that's another problem I will also discuss do you mean by more or less the same I mean that you need very for instance let's take a probability assignment over probabilities and think like randomly assigning some probability values to them we consider hundreds, thousands of probability assignments and the probabilities that any such probability assignment will give you any non-skeptical belief will be very low so most probability assignment won't support any non-skeptical belief so they have a Monte Carlo simulation of these scenarios and consider like how this will work and the interesting part is that light-gibber is more skeptical in such situations but Lina Kelly for example is not that skeptical so they think it's a good evidence against light-gibber so yeah so that's another problem in this account once we find great sets of possibilities like when we focus on more beliefs will crumble like light-gibber to crumble so I'm going to be addressing both of these problems next and I will begin with I will begin with a second actually okay so now I'm on the part 4 of my handout okay so let's take this issue about like whether or not we should be defined over many sets of possibilities so what gives this estimate is that a person's declarative knowledge base eventually approximates 1 million pieces of knowledge so it takes this to be like beliefs and not knowledge but that's not crucial so we can think like this that there are million independent propositions which represent our complete belief set if these beliefs are independent as we are assuming so there will be 2 to the power of million possibilities associated with this okay now I'm not sure that this supports their persistence in the ways that they think because think about computationally like if we have such a vast beliefs in our head and think about whether or not there is a probability assignment over all these possibilities right so if we accept that then we have to assign probabilities or let's like computer assign or our brain assign probabilities to this absolute vast number of possibilities that doesn't seem to be computationally realistic actually reflecting on cases like this today so that something different is more probable to be going on here so he suggests that there is no like even when you think about rational agents here now I'm speaking now this is my opinion when we are speaking of a rational agent it doesn't seem to be computational to define one probability assignment over this vast space of possibilities so first suggestion is this like we have much better grasp on probabilities that are conditional rather than this long conjunctive probabilities which has to be assigned to worlds so it seems that there are several probability distributions that represent our knowledge not just one and these probability distributions are about like closely connected sets of propositions so I don't think it's realistic to think that there is this one big picture of all we have like separate interests, separate purposes, separate topics and maybe it's not that bad that beliefs are defined in a rather small possibilities so I'm not sure let's move the evidence for the claim so indeed this also this point is also striking when we think about what is the point of belief is the first place like I'm speaking about categorical belief here like so many people think that categorical beliefs simplified our reasoning so compared to probabilistic reasoning right probabilistic reasoning is notoriously hard for humans and even absence of psychological fact propositional reasoning seems to be much simpler like we just think categorically like assume this is true or like I believe this will fall from this so on so if belief will simplify reasoning we would expect and we want belief reasoning to be constrained by small number of propositions otherwise if you consider a lot of propositions reasoning with them won't be reasonable at all because for example let's say I have an argument for some philosophical proposition let's say this argument demands me to ascend to 20 propositions now probably the rejection of this 20 propositions won't be very high so arguments like if argument is cashed out in propositional terms the more premises I need less likely that my reasoning is probabilistically reliable so beliefs if they are useful so it will be useful in a small constrained number of reasoning contexts and not in a large way so demanding to be able to cohere with all our sets of possibilities would be abandoning this simplified rule so otherwise beliefs will like this error probabilities would accumulate at all and beliefs won't be reasonably reliable so this has been pointed out by Richard Foley things that he was considering reduction types of argument he says a different argument has a lot of premises which are also not very well theoretically connected I won't be so worried about this reduction because maybe there are some errors there like not in terms of reasoning but in terms of some terms not being true but wherever we have like a reduction which has small number of premises all are theoretically intertwined and of course this reduction is much more significant so the same goes with other types of arguments like if I have an argument which requires all premises to be true whenever a number of premises increases reliability of reasoning decreases so beliefs if they are useful will be only useful for a situation which like we've considered in all team where finding great situation if you say a lot of premises do you mean it in a really like numerical way that seems like maybe this meant more something for the discussion this is very contraintuitive to me that it would be a matter of counting or something like that I completely grew it into openness I mean as soon as premises are very non like non related areas I work forward to have very unrelated beliefs and reasons it's going to be problematic to just use them as premises together or you're not going to be assuming that it's going to work anyway but maybe I'm really at this from a mathematical point of view but it seems like you can always reformulate premises so that they become they get a higher number or something like that or there are like schemas in mathematics or in other system that are actually in infinity of sentences infinity of premises that are very efficiently formulated as a schema which of course an infinity is a very large number of premises but it's one schema so it's about one topic and you should accept it as a whole and so it's very like natural to be touched by your arguments based on one schema you see my point the number intuitively doesn't matter maybe you don't agree with that but then I would like to No I totally agree No I wanted to make sure that many premises whether you meant it in a numerical way that you could count or it's more a matter of many different premises very good, yeah of course this sheer number is not relevant that's my question but because like in mathematical reasoning there are deductive relationships even with the premise that there can be many premise arguments and in ordinary reasoning premises are not usually strongly interrolling together so because there are smaller probabilities we see each premises their probability of the conclusion will increase so number will be relevant because this very tight interconnectedness is not present in ordinary non deductive reasoning but only number of course, yeah only numbers are not relevant only numbers are not relevant we have yeah because in usual reasoning it's very safe to assume that adding a premise is adding another error possibility of error so that's why number may be relevant but in mathematical context yes that won't be relevant at all yeah good so we simplify reasoning and this simplification comes with a price and the price is that when we reason with large set of propositions that are not very strongly intertwined such reasoning is often very unreliable because the joint probability of all premises can go very low so this is what happens in a way in a lottery case scenarios actually you get a negative correlation between premises so you get zero probability from the argument which is deductively valid so I think this shows that in ordinary reasoning for example reasoning which involves postulating several different potential causes of some phenomena like in this loan example which is considered as a window screen call or whatever inside cases premises are not usually very strongly connected so a small number of propositions to be considered is more colorful now I will move to the substantial part which is a positive suggestion about how we can think about which context are relevant so this proposal follows the criticism so his challenge is as follows that we need some kind of story which context are relevant and which are not otherwise we don't have a third rational belief so first my proposal is very basic philosophical ideas that rational belief is based on some evidence for some reasons as a rational the rational agent doesn't just believe anything there is a reason for supporting propositions which he has ok of course we don't need to consider the full structure of justification but for usual beliefs we are interested in whether or not the treatment will be effective whether or not it will rain tomorrow or whatever there is a reason for holding beliefs that we have so in the example in the figure 3 example for example when Tracy believes that her loan is wet so her reasons or doesn't believe it her reason has to do with whether or not she believes that it will rain or it has rained or whether or not sprinkled or something ok so proposal is in a general philosophical context which I will explain next will be as follows the context which are very relevant whether or not you believe a propositional doubt are context which represents the structure of your evidence so that's a basic idea this is a very big idea because it doesn't really help us to put down like exact context which you have to consider so a way to explicate this proposal is the following which uses some terms from the network theory so this is a 0.4 0.4 so I find this term called basic partition so when we consider proposition L for example Tracy's loan is wet so basic partition with respect to this proposition will be a set of all propositions that are immediately explanatory prior to L sorry it should be T so when I want to know whether which context should I consider with respect to this proposition Tracy's loan is wet I should consider propositions which are immediately explanatory prior so what does this term mean so we will think about this term in terms of a network which you can see in figure screen so so in this network we have two potential causes of loan being wet one is raining and another is a screen that is wet and there is also another proposition in this network which is that Jack's loan is wet now Jack's loan is wet there is not a cause of Tracy's loan being wet like they can share the same cause but they are not causes of each other right so here we have a network which is directly graph or dark for short and this is a graph which doesn't have any loops in it so intuitively it should represent cause of structure or the situation we will consider and a dark so this graph without loops which which satisfies the following condition which is called Markov condition it's called the Bayesian network so Markov condition is quite simple it's as a following that the dark obeys the Markov condition if and only if parents are like the ones who cause error to the child right so so these are these are parents and this is a child here so it's as the following that we have this Markov condition here if we assume parents of this variable then adding which is non-descendant of this I mean it's not a parent or descendant of this this won't be probabilistically relevant so whenever we have the satisfaction of this principle we have a dark which satisfies Markov condition so condition our parents and all this probabilistically independent of its non-descendance so Markov condition is very it's highly duty when we think about causal terms because causes in this kind of scenario are local for example suppose there is a cause of this here some other cause so if I already know this this one is not relevant right so it's local I need to only know like prior causes and it is also one directional like if there is a causal variable here there is no back so one directionality and locality is what we are having here and it's how it's called when we think about this dark representing causal structure of the situation so going back to my suggestion set of propositions that are immediately explanatory prior by this here I mean these two variables these are ones who are immediately explanatory prior and this term and this kind approach is influenced by paper by by Dabin Gaman Haga I'm not sure if I pronounce the name correctly but he has this approach for another purpose for defining what he calls basic probability which is not relevant here at all so this is the suggestion so when we think about basic context with respect to this proposition this gives us basic context and notes this this is a bit arbitrary at this point but I'll try to motivate this a bit so this proposition is only relevant here for this because it influences our probability of this for example if I learn that this is true this will change probability of this and it gets influenced from here so only things which are directly influencing my probabilities is immediate parents so if we think about so motivation is this like why it's not arbitrary to focus on immediate parents because causality is local and it's not cyclic we only need to look at parents so if we take this approach like look at page 6 it's the last page the problem of irrelevant proposition would be avoided why because irrelevant propositions are something like this they are not connected to this network at all so we avoid considering like we have an explanation why we need to need like why we need to be boring by this other proposition and also it explains why stability is important with respect to this network but not with respect to propositions which are not part of the network because they want influence probability of this at all this proposition and another point is that because because this causal structure of the situation usually represented by only a handful of propositions not always but usually it is far well-attuned with this idea that benefit is only useful in such kind of cause-wraining situation and not in very fine rate situation ok this is the basic proposal which will become clearer once I consider some objection which is next yeah usually use the causal paragraph in the causal relation between belief while the relation between propositions are logic relation and not causal so then it may be not suit the causal analysis may be not can not be used for the relation between your beliefs yeah so basic networks which satisfy Markov condition usually are used for causal representation of causal reasoning of course representation of causal reasoning is done by propositions we are still considering propositions and the probabilities we define even by causal relationships right it also can be that in the direction A to B it tells B not cause then if I have A then definitely I have B while in causal relation as counter probability of B is dependent on A not perfectly yeah when we have a logical relationship yeah so it is not mandatory for basic networks to represent causal structures so it can be representation of possible relationship for example like for example acceleration mass and gravitation can be also represented by basic network so part of relationship can be represented as a stop as well causal interpretation is most used but it is not mandatory so the relationship here which I am interested in in general are epistemic relations not only causal relations yeah and that is why basic networks are useful other than causal scenarios but here I put my claim in terms of explanation what is direct explanation of what so if there are other explanations other than causal explanation which probably there are right so we don't always need to consider we don't always need to consider causal interpretation of the network but if for example yeah so light is called humus thesis if humus is right and causal connection are most important for our beliefs it will be a useful useful approach even if it is not fully general ok so first objection is we can call into questions this demand to focus only on immediate parents ok here on it going back to what you said also so here I mean things which we think are most directly related to this one ok so every other influences which comes from T from other side which comes from this other than finding out this directly of course so direct that is my priorities when I'm interested ok so one is like why only focus on this what is wrong with this as well like it's not arbitrary to focus on parents and even our non-descendants which are also part of basic network so yeah this is this is a fair point so at this point I think that like really with respect to basic propositions can define weaker notion of belief in belief is table with respect to whole network that's good stronger sense of belief but we can still get some special partitioning of propositions which are crucial and they are special because all influences on this should come from from their parents they still have special role but I would concede that if we have a network that is still relevant in some sense ok I have not looked out detail of this but I would still think that this is a more basic considering of parents ok and second objection is as follows what is causal structure or evidential structure of situation is not given whether or not which network represents correctly which network represents correctly my evidence so what should we do what should we do then so that will be that will be an issue in this case the formal details of this is is not it's not very important here so we should take into account some kind of higher work and uncertainty about which variables are parents or which are not but even if we get something where there is this causal knowledge or evidential knowledge it still will be something so that will be my answer and and final question will be whether whether this approach can be applied to what are your preference parables now what are your parables what are your parables is it's unclear to me which partition which partition should we focus on there because there is now tickets so imagine we only have three ticket lottery so of course probability of tickets three winning there is something which these are related in the sense that if we assume these two to be false of course it changes probability of tickets three using it but it's I'm not sure whether or not this character represents causal structure of our evidence I would say more like we have this evidence that ticket has been drawn and there are three possible ways to look at it if we think about this way then this approach recommends that we live in proposition one losing is rational but I would admit that it doesn't really translate well to this kind of situation but better translation will be in preference parable situation I'm curious what you think about this so so context sensitive approaches I think difficulty with preface paradox so preface paradox just to briefly say what it is when the author of a book like large book let's say historical book here I have an example of Darwin's original pieces when the author publish a book this book creates a specific context which unifies all the assertions he made so as a paradox it will be like if you write each and every sentence there and if you are a reasonable person you wouldn't write something in an affirmative way if you don't believe what you write but as a modest person you would accept that there will be some mistakes in the book so if a book creates one context then Leipzig's proposal has a problem there he has a solution so the problem is that as Leipzig writes by writing and publishing the book the author seems to include some sort of commitment to all of the statements is the book taken as a whole so this book creates a context of each other so here one way this approach can be useful with us all so the author doesn't when she considers each and every sentence she made in the book so there is an obvious sense in which she shouldn't use it all of the work too so this brings me back to the number issue even all statements are somehow related because the relation is not as strong as in mathematical proof so the error probability will accumulate so that's how numbers will be important so my approach will be as follows so while a book creates special context not all claims in the book are of equal importance for the general claim the argument of the book so I think context sensitivities can still be applied there so for example if we take a simple example like Darwin's book he makes a lot of claims in the book so some of the claims has to do with biology geography and so on not all of them are very crucial to his main argument so what I would say here is that an author of the book must have a strong belief or at least belief in a main claim of the book in Darwin's case it's quite easy because it's a mechanism of natural selection right so this should also this can be also represented as a network here for a simple network we take different variables of course the full network will be much more complicated so with respect to this network we can demand that the author's belief should be stable but not with respect to the whole book which will create very fine-graining sets of possibilities so that would be one way to justify a context sensitive approach to practice paradoxes as well to differentiate which statements all times create a basic network so yeah that's what we are trying to teach thank you very much do we take five minutes break or do we need to go to this same time I would say we just keep rolling since we're half an hour from the end since we interrupted a lot this was also part of the question exactly we did some of it in advance I'm really interested by this the idea of what this does for the lottery paradox I think this is really interesting because this actually it's a little bit at one's question about what is it that's relevant about the structure of the lottery what's weird about the lottery is it almost seems like there's a second level where it's not just that we have the arrows from the draw to the tickets it's also that we have constraints between what values the arrows can take depending on what else you can't just you can't just free a sign even if you the arrows aren't independent from each other in a very weird and complicated way and I just don't know this seems to be the kind of thing that the Carnegie Mellon Bayes Net people would have thought about how to model I just have no idea how they did it if you tried to see how what are the proposals on offer about how to model the connections in that kind of thing this can be represented by Bayes without problem that won't be an issue the issue is that it won't represent causal structure there is some kind of structure represented like a structure of causal dependency because of the information we have but not like so tickets free meaning I don't like it's not that these two things are causal causes of but it can be very easily represented by Bayes network it just doesn't represent causal structure in the same way as the city but with hungry tickets it's like you'll get a lot of errors but with five tickets it's quite it's easily easily done so Bayes network can be applied in cases in which we don't have really this kind of causal influences right it can model evidential connections as well but for the causal structure I would say there is some draw that's a cause of which ticket will win or not others will be just informations about what's the draw will be and influence from the the thing is that we don't get any interesting Bayes network from place it's not useful to represent lottery cases won't simplify anything won't make anything more obvious sure yeah I see that makes sense you have the sort of draw sub network that shows the causal influence and then you have all these other non-causal constraint networks that spell the constraints out but yeah that doesn't really teach us anything sure I see what you mean yeah okay that's helpful I have a couple of questions first of all I really like the proposal how to ask my questions a bit so something with Bayesian that that I always like in case where you apply to epistemology and it seems kind of relevant to you even more than elsewhere I've been talking about lottery and so on and it's related to the trust question I guess and maybe it's just a confusion in my head but there seems to be a huge difference in sort of epistemic causes like the causes why you have certain beliefs or why you would have certain beliefs where you confronted with some other information that you don't in fact have but there's some like the way your brain works independent of anything existing outside of there it's like the causal stuff going on in your head whether that's the kind of thing you really want to model for such kind of problems because we're interested in beliefs and not so much in in defense or whether we really want to have beliefs about the objective causes in the world or objective other relations that are explanatory I'm not sure what my question is very clear but it seems that the two kind of approaches you might have to to the lottery paradox where you see like one draw is the cause of the thing happening then it's really about the causes in a ordinary kind of sense in an almost objective kind of sense while to get the right kind of context you have to sort of write sort of a symbolical things dropping out I'd expect more a network that is about how agents deal with information like oh there's that draw that's going to change my grievances that way and that's also sort of causation but it's not causation of things in the world it's causation of thoughts or something, beliefs normative one right? normative? normative per se it's about how the thing that happens in my head if I'm getting confronted with some new information it's going to affect something else happening in my head and there's certain rational agents or if I'm if my thinking is kind of structured there's going to be some causal principles that take me from one mental state to another mental state in view of this new information independent of whether anything of what I think is true or is it anyway related to causes or to grounding relations or whatever there's something the laws of the mental stuff that is happening seems to be kind of the thing that we want to understand better in such proposals and that's a causation of another completely different level I agree and I always wonder like are we really perfectly clear whether we are now talking about the events, the draw or are we talking about the information we have about the draw and what that would cause in our head rather than what the draw causes in the real world and what we think about that that seems to do you have any help there where I should see these propositions are they like mental entities or are they it's a very broad question yeah no I see so we have a causal order and mental order and they may not match often they often they don't cause on me that might not be any cause in the world exactly so basically that's what I applied to artificial intelligence systems to model a certain reasoning yeah so they assume this mental stuff like not mental but this proposition somehow represents a real causal order right but I don't want to say it's a similar thing here in my case so that's why I want to focus within the agent's perspective what are immediately epistemically epistemically like what are directly epistemically into this proposition what the agent considers to be directly epistemically so when I try to convey this with examples I give examples which I would think map real causal structures in certain way because even I'm no expert in this but like the causes of event doesn't seem to be correct like there are many causes depending on which what we are really interested the cause is not the thing so I would say that like within the agent's mental state what is directly explanatory to what it may not map the causal structure yeah so when we apply basic systems to make an inference for example that's also what we look like we look at the structure of our knowledge right like when we have a basic network consisting of like 8 variables which has to give pressuring to someone who broke his hand in a certain way they might have an expert knowledge system like put certain variables like how old is this person what is structure like and there are other variables as well and get some kind of answer that represents our state of knowledge about what's relevant for this treatment and what's not so this I would look at this from an internalistic point of view like structure of our knowledge so when we talk about beliefs I wanted to be structure of our beliefs not the structure outside our heads and because we are focusing on rational person normaturity can help us to simplify certain things so because this structure of belief should be rational yeah we can assume certain kind of like not how kind of connection would work in that way so yeah I would say this should be analysis of structure of our knowledge ok we don't we bring it to a second question I so if it's that kind of structure it reminds me some of the things you said remind me a lot of beliefs or intuitions that I might all believe maybe just also because that's a nice visual structure but it could also be a way to define a nice context and to see that some sort of things you don't want to be relevant at all to this specific part of the partition because it is very different somewhere in the web of belief if it's very like central in the belief you don't want to revise it even if you get the reasons to revise it you won't rather put into doubt things that are more on the app so it's like a very open question how how would you relate what do you think about the web of beliefs would it be any way related to sort of structures you have in mind you might want maybe that the things that your arrows come from are typically like more central I guess use much more I guess reliable stable information to come to a certain conclusion in a lot of ways of course so do you see any relations with web of beliefs so connection should be there because network is a you can specify finding an idea as a huge Asian network did people do that it would be cool I just seen an article preparing for this talk two days ago they had a new paper which constructed a network of scientific theories explaining things like how they interact with so yeah some people would do this in philosophy of science for sure so my thinking here will be that at least when we think about belief I don't think we should think about the web of belief for the reasons I give like if we have this vast knowledge we should be fragmented so in a way I would think about different webs of belief which don't really interact much in our everyday life at all because the word much seems to be quite important they don't interact typically because they're very far apart but given that in the end we are living these lives where these things are important in some way they will interact I mean even just by us have been confronted with both kind of contexts so I don't think Quine would be he wouldn't find that a reason to have multiple sets of beliefs sets of webs of beliefs certain parts of these webs are extremely unrelated but they are still related because at least they share certain charts for something for for like so there will be things like connecting very distinct positions like very core things like like Zeyers didn't just appear one second ago we don't have this kind of explicit beliefs but there will be still connection so my argument will be very normative one but I would say that from the purpose of having rational beliefs this huge integration of different webs into a coherent all won't be won't have any purpose we want to think like what is the purpose of belief purpose of belief is to give us reliable reasoning within the relevant context there is this extended unification but that's yeah so the kind approach that I have will be more even if I have a thought about this will be more in tune with this fragmented mind this goes in the direction of the one other kind of comment question that I had so I will jump in half of what I know about epistemology I learned from a grad seminar with Bob Audi and one thing that he always pounded a table about is like stop giving people so many current beliefs people don't have that many current beliefs really and this I think is a place where this kind of locks in could help like in some of these complicated cases where you're worried about context exploding when you were talking about that your skepticism about this argument is sure maybe I have I know that I do because the mathematical case guarantees it I might have a disposition to believe a million things I sure as heck don't have a million current beliefs my head's not fat enough to hold that many current beliefs so I think that plays right into this if you make that distinction and say you know what we've got to be really careful about a lot of these worries seem to your proposal plays very nicely into this idea that like I don't need to assume that people have that many current beliefs if I can say sure I have a disposition to believe lots of stuff like I I didn't, two seconds ago I didn't have a current belief about where my dog is, like I had a disposition and I was thinking of an example I had a disposition to have one, now probably I have one because I've just been thinking about it but like I didn't bother to think about it until now and I feel like that's very normal that's kind of how we work and so I think you're moved to say these distant parts of the network we can model them maybe but they don't pose a problem for us precisely because it's in a sense they're not really there yeah and I also want to add that even if they're worse there but I fully agree that they're not there even if they're worse there it won't pose a problem because we have a weight of differentiating and favoring certain contexts over as they have us so if we are happy that beliefs cannot be just conjunctively closed overall of our career we should all accept this if we reject this kind of conjunction rule and we go with partition sensitivity then a domain is right that we need some way of treating certain partitions in a in a more important way rather than others I think this network it's also nice if it's how people are parallel I've seen about this there are certain networks we don't need to consider everything I think this is most obvious we want to be this partition sensitive so other more will be what we study and say belief is just not closed under conjunction we have different beliefs sometimes we connect them sometimes not but I still want to know why we connect certain beliefs and other notes maybe this gives a nice picture why we connect others so one is like psychological question or about the current beliefs and which are fully accept and other is like even if we had many occurred beliefs I have a current belief like some of it just consider this proposition whether or not it's heads or not but this shouldn't come to this context of for somebody wanting their home if I don't see a connection so it needs to be fleshed out more but I'm thinking that this may be one way to meet Dona's challenge but it might not work at the end but I'll try is it very a kind of problem that I could see popping up for such ways to mobile it but I'm not sure but it could be very representation sensitive like if you consider slightly different like different propositions in your deck the kind of context would be described differently and you'll get a different result do you think that will be the case yeah I think that and would you find that a problem yeah I think that's what we present and this is a potential problem that's why I think we need some kind of notion of weaker notion of belief I believe it means that there is some appropriate context and this is not arbitrary because in this context reasoning will be reliable so it has some use maybe if there are other representation of situation in which belief is no longer rational that's not ideal but I wouldn't be very upset about this because we still have some idea of belief which is not as partition sensitive as I think because we have some idea of why we need this because there is a special partition but I would expect that I don't get very nice story in which there is only one very nice network and I would I don't expect that I would I would this kind of cases can be easily made even this needs justifications why don't consider non-descendants more justifications as there can be more parents considered etc but because I have a there is an answer to this in principle answer is if there is a well structured network in which belief is useful it means that you have belief maybe there are other contexts in which the same belief is not useful but still there is some type of belief that you have so I haven't thought much about this but maybe it gives an idea to differentiate belief which goes beyond the probabilities and say that I believe this is stronger because it can be embedded in more and more wider and wider webs of belief and some propositions may have high probability but it's not embeddable I think lottery is it's not embeddable in a context for the reasons that it's not related to any other proposition except other lottery and the issue I want to have also which I haven't thought about is the role of believing in practical decision making because in practical decision making I can do very different propositions together so this proposal looks at reasoning like truth oriented reasoning but in practical reasoning I can then we see a good thing which I also consider like whether or not coin landed heads or tails and horned off I don't know for why but yeah so to go back to you also I don't expect there to be very nice formal making sure that it will be messy but I will be you have to say that even if there is like some set of partition in which this belief is stable and useful then you believe it you and if it's not embedded in all relevant context it means that it's only weakly believed but I need to figure this out so but yeah so stronger sense of belief will be this which will be embedded in all different relevant context but I'm not sure whether or not we have such a belief which will be so stable so if it needs to consider a lot of propositions so I would say belief is no longer useful we need probabilities so for example Jeffers point will be relevant so that belief is not useful in very serious exact thinking because of every probability so we need to base it on preferences or comparative probabilities so I think belief is a humble humble belief we shouldn't expect it to be so stable and useful but this is very rough so I need to sit down and think more about this so this is my initial initial idea about this thank you of course we're pretty much in time thanks again thank you