tag:blogger.com,1999:blog-49876091144152055932017-03-25T14:01:51.773+00:00M-PhiA blog dedicated to mathematical philosophy.Jeffrey Ketlandhttp://www.blogger.com/profile/01753975411670884721noreply@blogger.comBlogger536125tag:blogger.com,1999:blog-4987609114415205593.post-14284388941774572272017-03-19T18:17:00.004+00:002017-03-19T18:17:51.184+00:00Aggregating incoherent credences: the case of geometric poolingIn the last few posts (<a href="http://m-phi.blogspot.co.uk/2017/03/a-dilemma-for-judgment-aggregation.html" target="_blank">here</a> and <a href="http://m-phi.blogspot.co.uk/2017/03/a-little-more-on-aggregating-incoherent.html" target="_blank">here</a>), I've been exploring how we should extend the probabilistic aggregation method of linear pooling so that it applies to groups that contain incoherent individuals (which is, let's be honest, just about all groups). And our answer has been this: there are three methods -- linear-pool-then-fix, fix-then-linear-pool, and fix-and-linear-pool-together -- and they agree with one another just in case you fix incoherent credences by taking the nearest coherent credences as measured by squared Euclidean distance. In this post, I ask how we should extend the probabilistic aggregation method of geometric pooling.<br /><br />As before, I'll just consider the simplest case, where we have two individuals, Adila and Benoit, and they have credence functions -- $c_A$ and $c_B$, respectively -- that are defined for a proposition $X$ and its negation $\overline{X}$. Suppose $c_A$ and $c_B$ are coherent. Then geometric pooling says:<br /><br /><b>Geometric pooling </b>The aggregation of $c_A$ and $c_B$ is $c$, where<br /><ul><li>$c(X) = \frac{c_A(X)^\alpha c_B(X)^{1-\alpha}}{c_A(X)^\alpha c_B(X)^{1-\alpha} + c_A(\overline{X})^\alpha c_B(\overline{X})^{1-\alpha}}$</li><li>$c(\overline{X}) = \frac{c_A(\overline{X})^\alpha c_B(\overline{X})^{1-\alpha}}{c_A(X)^\alpha c_B(X)^{1-\alpha} + c_A(\overline{X})^\alpha c_B(\overline{X})^{1-\alpha}}$</li></ul>for some $0 \leq \alpha \leq 1$.<br /><br />Now, in the case of linear pooling, if $c_A$ or $c_B$ is incoherent, then it is most likely that any linear pool of them is also incoherent. However, in the case of geometric pooling, this is not the case. Linear pooling requires us to take a weighted arithmetic average of the credences we are aggregating. If those credences are coherent, so is their weighted arithmetic average. Thus, if you are considering only coherent credences, there is no need to normalize the weighted arithmetic average after taking it to ensure coherence. However, even if the credences we are aggregating are coherent, their weighted geometric averages are not. Thus, geometric pooling requires that we first take the weighted geometric average of the credences we are pooling and then normalize the result, to ensure that the result is coherent. But this trick works whether or not the original credences are coherent. Thus, we need do nothing more to geometric pooling in order to apply it to incoherent agents.<br /><br />Nonetheless, questions still arise. What we have shown is that, if we first geometrically pool our two incoherent agents, then the result is in fact coherent and so we don't need to undertake the further step of fixing up the credences to make them coherent. But what if we first choose to fix up our two incoherent agents so that they are coherent, and then geometrically pool them? Does this give the same answer as if we just pooled the incoherent agents? And, similarly, what if we decide to fix and pool together?<br /><br />Interestingly, the results are exactly the reverse of the results in the case of linear pooling. In that case, if we fix up incoherent credences by taking the coherent credences that minimize squared Euclidean distance, then all three methods agree, whereas if we fix them up by taking the coherent credences that minimize generalized Kullback-Leibler divergence, then sometimes all three methods disagree. In the case of geometric pooling, it is the opposite. Fixing up using generalized KL divergence makes all three methods agree -- that is, pool, fix-then-pool, and fix-and-pool-together all give the same result when we use GKL to measure distance. But fixing up using squared Euclidean distance leads to three separate methods that sometimes all disagree. That is, GKL is the natural distance measure to accompany geometric pooling, while SED is the natural measure to accompany linear pooling.Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-35259569628408363342017-03-17T12:05:00.000+00:002017-03-17T12:05:56.634+00:00A little more on aggregating incoherent credencesLast week, I <a href="https://m-phi.blogspot.co.uk/2017/03/a-dilemma-for-judgment-aggregation.html" target="_blank">wrote</a> about a problem that arises if you wish to aggregate the credal judgments of a group of agents when one or more of those agents has incoherent credences. I focussed on the case of two agents, Adila and Benoit, who have credence functions $c_A$ and $c_B$, respectively. $c_A$ and $c_B$ are defined over just two propositions, $X$ and its negation $\overline{X}$.<br /><br />I noted that there are two natural ways to aggregate $c_A$ and $c_B$ for someone who adheres to Probabilism, the principle that says that credences should be coherent. You might first fix up Adila's and Benoit's credences so that they are coherent, and then aggregate them using linear pooling -- let's call that <i>fix-</i><i>then-pool</i>. Or you might aggregate Adila's and Benoit's credences using linear pooling, and then fix up the pooled credences so that they are coherent -- let's call that <i>pool-</i><i>then-fix</i>. And I noted that, for some natural ways of fixing up incoherent credences, fix-then-pool gives a different result from pool-then-fix. This, I claimed, creates a dilemma for the person doing the aggregating, since there seems to be no principled reason to favour either method.<br /><br />How do we fix up incoherent credences? Well, a natural idea is to find the coherent credences that are closest to them and adopt those in their place. This obviously requires a measure of distance between two credence functions. In last week's post, I considered two:<br /><br /><b>Squared Euclidean Distance (SED)</b> For two credence functions $c$, $c'$ defined on a set of propositions $X_1$, $\ldots$, $X_n$,$$SED(c, c') = \sum^n_{i=1} (c(X_i) - c'(X_i))^2$$<br /><br /><b>Generalized Kullback-Leibler Divergence (GKL)</b> For two credence functions $c$, $c'$ defined on a set of propositions $X_1$, $\ldots$, $X_n$,$$GKL(c, c') = \sum^n_{i=1} c(X_i) \mathrm{log}\frac{c(X_i)}{c'(X_i)} - \sum^n_{i=1} c(X_i) + \sum^n_{i=1} c'(X_i)$$<br /><br />If we use $SED$ when we are fixing incoherent credences -- that is, if we fix an incoherent credence function $c$ by adopting the coherent credence function $c^*$ for which $SED(c^*, c)$ is minimal -- then fix-then-pool gives <i>the same results</i> as pool-then-fix.<br /><br />If we use GKL when we are fixing incoherent credences -- that is, if we fix an incoherent credence function $c$ by adopting the coherent credence function $c^*$ for which $GKL(c^*, c)$ is minimal -- then fix-then-pool gives <i>different results</i> from pool-then-fix.<br /><br />Since last week's post, I've been reading <a href="https://www.princeton.edu/~osherson/papers/preddAgg.pdf" target="_blank">this</a> paper by <a href="http://www.rand.org/about/people/p/predd_joel_b.html" target="_blank">Joel Predd</a>, <a href="http://www.princeton.edu/~osherson/" target="_blank">Daniel Osherson</a>, <a href="https://www.princeton.edu/~kulkarni/" target="_blank">Sanjeev Kulkarni</a>, and <a href="http://ee.princeton.edu/people/faculty/h-vincent-poor" target="_blank">Vincent Poor</a>. They suggest that we pool and fix incoherent credences in one go using a method called the Coherent Aggregation Principle (CAP), formulated in <a href="http://www.sciencedirect.com/science/article/pii/S0899825606000613" target="_blank">this</a> paper by <a href="http://www.princeton.edu/~osherson/" target="_blank">Daniel Osherson</a> and <a href="http://www.cs.rice.edu/~vardi/" target="_blank">Moshe Vardi</a>. In its original version, CAP says that we should aggregate Adila's and Benoit's credences by taking the coherent credence function $c$ such that the sum of the distance of $c$ from $c_A$ and the distance of $c$ from $c_B$ is minimized. That is,<br /><br /><b>CAP</b> Given a measure of distance $D$ between credence functions, we should pick that coherent credence function $c$ such that minimizes $D(c, c_A) + D(c, c_B)$.<br /><br />As they note, if we take $SED$ to be our measure of distance, then this method generalizes the aggregation procedure on coherent credences that just takes straight averages of credences. That is, CAP entails unweighted linear pooling:<br /><br /><b>Unweighted Linear Pooling</b> If $c_A$ and $c_B$ are coherent, then the aggregation of $c_A$ and $c_B$ is $$\frac{1}{2} c_A + \frac{1}{2}c_B$$ <br /><br />We can generalize this result a little by taking a weighted sum of the distances, rather than the straight sum.<br /><br /><b>Weighted CAP </b>Given a measure of distance $D$ between credence functions, and given $0 \leq \alpha leq 1$, we should pick the coherent credence function $c$ that minimizes $\alpha D(c, c_A) + (1-\alpha)D(c, c_B)$.<br /><br />If we take $SED$ to measure the distance between credence functions, then this method generalizes linear pooling. That is, Weighted CAP entails linear pooling:<br /><br /><b>Linear Pooling </b>If $c_A$ and $c_B$ are coherent, then the aggregation of $c_A$ and $c_B$ is $$\alpha c_A + (1-\alpha)c_B$$ for some $0 \leq \alpha \leq 1$.<br /><br />What's more, when distance is measured by $SED$, Weighted CAP agrees with fix-then-pool and with pool-then-fix (providing the fixing is done using $SED$ as well). Thus, when we use $SED$, all of the methods for aggregating incoherent credences that we've considered agree. In particular, they all recommend the following credence in $X$: $$\frac{1}{2} + \frac{\alpha(c_A(X)-c_A(\overline{X})) + (1-\alpha)(c_B(X) - c_B(\overline{X}))}{2}$$ <br /><br />However, the story is not nearly so neat and tidy if we measure the distance between two credence functions using $GKL$. Here's the credence in $X$ recommended by fix-then-pool:$$\alpha \frac{c_A(X)}{c_A(X) + c_A(\overline{X})} + (1-\alpha)\frac{c_B(X)}{c_B(X) + c_B(\overline{X})}$$ Here's the credence in $X$ recommended by pool-then-fix: $$\frac{\alpha c_A(X) + (1-\alpha)c_B(X)}{\alpha (c_A(X) + c_A(\overline{X})) + (1-\alpha)(c_B(X) + c_B(\overline{X}))}$$ And here's the credence in $X$ recommended by Weighted CAP: $$\frac{c_A(X)^\alpha c_B(X)^{1-\alpha}}{c_A(X)^\alpha c_B(X)^{1-\alpha} + c_A(\overline{X})^\alpha c_B(\overline{X})^{1-\alpha}}$$ For many values of $\alpha$, $c_A(X)$, $c_A(\overline{X})$, $c_B(X)$, $c_B(\overline{X})$ these will give three distinct results. <br /><br /><br />Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-3826985268867587772017-03-10T17:08:00.000+00:002017-03-13T12:29:10.557+00:00A dilemma for judgment aggregationLet's suppose that Adila and Benoit are both experts, and suppose that we are interested in gleaning from their opinions about a certain proposition $X$ and its negation $\overline{X}$ a judgment of our own about $X$ and $\overline{X}$. Adila has credence function $c_A$, while Benoit has credence function $c_B$. One standard way to derive our own credence function on the basis of this information is to take a <i>linear pool</i> or <i>weighted average</i> of Adila's and Benoit's credence functions. That is, we assign a weight to Adila ($\alpha$) and a weight to Benoit ($1-\alpha$) and we take the linear combination of their credence functions with these weights to be our credence function. So my credence in $X$ will be $\alpha c_A(X) + (1-\alpha) c_B(X)$, while my credence in $\overline{X}$ will be $\alpha c_A(\overline{X}) + (1-\alpha)c_B(\overline{X})$.<br /><br />But now suppose that either Adila or Benoit or both are probabilistically incoherent -- that is, either $c_A(X) + c_A(\overline{X}) \neq 1$ or $c_B(X) + c_B(\overline{X}) \neq 1$ or both. Then, it may well be that the linear pool of their credence functions is also probabilistically incoherent. That is,<br /><br />$(\alpha c_A(X) + (1-\alpha) c_B(X)) + (\alpha c_A(\overline{X}) + (1-\alpha)c_B(\overline{X})) = $<br /><br />$\alpha (c_A(X) + c_A(\overline{X})) + (1-\alpha)(c_B(X) + c_B(\overline{X})) \neq 1$<br /><br />But, as an adherent of Probabilism, I want my credences to be probabilistically coherent. So, what should I do?<br /><br />A natural suggestion is this: take the aggregated credences in $X$ and $\overline{X}$, and then take the closest pair of credences that are probabilistically coherent. Let's call that process the <i>coherentization</i> of the incoherent credences. Of course, to carry out this process, we need a measure of distance between any two credence functions. Luckily, that's easy to come by. Suppose you are an adherent of Probabilism because you are persuaded by the so-called <a href="http://m-phi.blogspot.co.uk/2013/05/joyces-argument-for-probabilism_24.html" target="_blank">accuracy dominance arguments</a> for that norm. According to these arguments, we measure the accuracy of a credence function by measuring its proximity to the ideal credence function, which we take to be the credence function that assigns credence 1 to all truths and credence 0 to all falsehoods. That is, we generate a measure of the accuracy of a credence function from a measure of the distance between two credence functions. Let's call that distance measure $D$. In the accuracy-first literature, there are reasons for taking $D$ to be a so-called <a href="http://m-phi.blogspot.co.uk/2014/04/how-should-we-measure-accuracy-in.html" target="_blank"><i>Bregman divergence</i></a>. Given such a measure $D$, we might be tempted to say that, if Adila and/or Benoit are incoherent and our linear pool of their credences is incoherent, we should <i>not</i> adopt that linear pool as our credence function, since it violates Probabilism, but rather we should find the nearest coherent credence function to the incoherent linear pool, relative to $D$, and adopt that. That is, we should adopt credence function $c$ such that $D(c, \alpha c_A + (1-\alpha)c_B)$ is minimal. So, we should first take the linear pool of Adila's and Benoit's credences; and then we should make them coherent.<br /><br />But this raises the question: why not first make Adila's and Benoit's credences coherent, and then take the linear pool of the resulting credence functions? Do these two procedures give the same result? That is, in the jargon of algebra, does linear pooling commute with our procedure for making incoherent credences coherent? Does linear pooling commute with coherentization? If so, there is no problem. But if not, our judgment aggregation method faces a dilemma: in which order should the procedures be performed: aggregate, then make coherent; or make coherent, then aggregate.<br /><br />It turns out that whether or not the two commute depends on the distance measure in question. First, suppose we use the so-called <i>squared Euclidean distance </i>measure. That is, for two credence functions $c$, $c'$ defined on a set of propositions $X_1$, $\ldots$, $X_n$,$$SED(c, c') = \sum^n_{i=1} (c(X_i) - c'(X_i))^2$$ In particular, if $c$, $c'$ are defined on $X$, $\overline{X}$, then the distance from $c$ to $c'$ is $$(c(X) -c'(X))^2 + (c(\overline{X})-c'(\overline{X})^2$$ And note that this generates the <i>quadratic scoring rule</i>, which is strictly proper:<br /><ul><li>$\mathfrak{q}(1, x) = (1-x)^2$</li><li>$\mathfrak{q}(0, x) = x^2$ </li></ul>Then, in this case, linear pooling commutes with our procedure for making incoherent credences coherent. Given a credence function $c$, let $c^*$ be the closest coherent credence function to $c$ relative to $SED$. Then:<br /><br /><b>Theorem 1 </b>For all $\alpha$, $c_A$, $c_B$, $$\alpha c^*_A + (1-\alpha)c^*_B = (\alpha c_A + (1-\alpha)c_B)^*$$<br /><br />Second, suppose we use the <i>generalized Kullback-Leibler divergence</i> to measure the distance between credence functions. That is, for two credence functions $c$, $c'$ defined on a set of propositions $X_1$, $\ldots$, $X_n$,$$GKL(c, c') = \sum^n_{i=1} c(X_i) \mathrm{log}\frac{c(X_i)}{c'(X_i)} - \sum^n_{i=1} c(X_i) + \sum^n_{i=1} c'(X_i)$$ Thus, for $c$, $c'$ defined on $X$, $\overline{X}$, the distance from $c$ to $'$ is $$c(X)\mathrm{log}\frac{c(X)}{c'(X)} + c(\overline{X})\mathrm{log}\frac{c(\overline{X})}{c'(\overline{X})} - c(X) - c(\overline{X}) + c'(X) + c'(\overline{X})$$ And note that this generates the following scoring rule, which is strictly proper:<br /><ul><li>$\mathfrak{b}(1, x) = \mathrm{log}(\frac{1}{x}) - 1 + x$</li><li>$\mathfrak{b}(0, x) = x$ </li></ul>Then, in this case, linear pooling <i>does not </i>commute with our procedure for making incoherent credences coherent. Given a credence function $c$, let $c^+$ be the closest coherent credence function to $c$ relative to $GKL$. Then:<br /><br /><b>Theorem 2</b> For many $\alpha$, $c_A$, $c_B$, $$\alpha c^+_A + (1-\alpha)c^+_B \neq (\alpha c_A + (1-\alpha)c_B)^+$$<br /><br /><i>Proofs of Theorems 1 and 2</i>. With the following two key facts in hand, the results are straightforward. If $c$ is defined on $X$, $\overline{X}$:<br /><ul><li>$c^*(X) = \frac{1}{2} + \frac{c(X)-c(\overline{X})}{2}$, $c^*(\overline{X}) = \frac{1}{2} - \frac{c(X) - c(\overline{X})}{2}$.</li><li>$c^+(X) = \frac{c(X)}{c(X) + c(\overline{X})}$, $c^+(\overline{X}) = \frac{c(\overline{X})}{c(X) + c(\overline{X})}$.</li></ul><br />Thus, Theorem 1 tells us that, if you measure distance using SED, then no dilemma arises: you can aggregate and then make coherent, or you can make coherent and then aggregate -- they will have the same outcome. However, Theorem 2 tells us that, if you measure distance using GKL, then a dilemma does arise: aggregating and then making coherent gives a different outcome from making coherent and then aggregating.<br /><br />Perhaps this is an argument against GKL and in favour of SED? You might think, of course, that the problem arises here only because SED is somehow naturally paired with linear pooling, while GKL might be naturally paired with some other method of aggregation such that that method of aggregation commutes with coherentization relative to GKL. That may be so. But bear in mind that there is a <a href="https://dl.dropboxusercontent.com/u/9797023/Papers/linear-pooling.pdf" target="_blank">very general argument</a> in favour of linear pooling that applies whichever distance measure you use: it says that if you do not aggregate a set of probabilistic credence functions using linear pooling then there is some linear pool that each of those credence functions expects to be more accurate than your aggregation. So I think this response won't work.Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com2tag:blogger.com,1999:blog-4987609114415205593.post-85981736128637262172017-03-01T11:54:00.000+00:002017-03-02T10:04:46.144+00:00More on the Swamping Problem for ReliabilismIn a <a href="http://m-phi.blogspot.co.uk/2017/02/the-swamping-problem-for-reliabilism.html" target="_blank">previous post</a>, I floated the possibility that we might use recent work in decision theory by Orri Stefánsson and Richard Bradley to solve the so-called Swamping Problem for veritism. In this post, I'll show that, in fact, this putative solution can't work.<br /><br />According to the Swamping Problem, I value beliefs that are both justified and true more than I value beliefs that are true but unjustified; and, we might suppose, I value beliefs that are justified but false more than I value beliefs that are both unjustified and false. In other words, I care about the truth or falsity or my beliefs; but I also care about their justification. Now, suppose we take the view, which I defend in this earlier post, that a belief in a proposition is more justified the higher the objective probability of that proposition given the grounds for that belief. Thus, for instance, if I base my belief that there was a firecrest in front of me until a few seconds ago on the fact that I saw a flash of orange as the bird flew off, then my belief is more justified the higher the objective probability that it was a firecrest given that I saw a flash of orange. And, whether there really was a firecrest in front of me, the value of my belief increases as the objective probability that there was given I saw a flash of orange increases.<br /><br />Let's translate this into Stefánsson and Bradley's version of Richard Jeffrey's decision theory. Here are the components:<br /><ul><li>a Boolean algebra $F$</li><li>a desirability function $V$, defined on $F$</li><li>a credence function $c$, defined on $F$</li></ul>The fundamental assumption of Jeffrey's framework is this:<br /><br /><b>Desirability</b> For any partition $X_1$, ..., $X_n$, $$V(X) = \sum^n_{i=1} c(X_i | X)V(X\ \&\ X_i)$$ And, further, we assume Lewis' Principal Principle, where $C^x_X$ is the proposition that says that $X$ has objective probability $x$:<br /><br /><b>Principal Principle</b> $$c(X_j | \bigwedge^n_{i=1} C^{x_i}_{X_i}) = x_i$$ Now, suppose I believe proposition $X$. Then, from what we said above, we can extract the following:<br /><ol><li>$V(X\ \&\ C^x_X)$ is a monotone increasing and non-constant function of $x$, for $0 \leq x \leq 1$</li><li>$V(X\ \&\ C^x_X)$ is a monotone increasing and non-constant function of $x$, for $0 \leq x \leq 1$</li><li>$V(X\ \&\ C^x_X) > V(\overline{X}\ \&\ C^x_X)$, for $0 \leq x \leq 1$.</li></ol>Given this, the Swamping Problem usually proceeds by identifying a problem with (1) and (2) as follows. It begins by claiming that the principle that Stefánsson and Bradley, in another context, call Chance Neutrality is indeed a requirement of rationality:<br /><br /><b>Chance Neutrality</b> $$V(X_j\ \&\ \bigwedge^n_{i=1} C^{x_i}_{X_i}) = V(X)$$ Or, equivalently:<br /><br /><b>Chance Neutrality$^*$</b> $$V(X_j\ \&\ \bigwedge^n_{i=1} C^{x_i}_{X_i}) = V(X_j\ \&\ \bigwedge^n_{i=1} C^{x'_i}_{X_i})$$ This says that the truth of $X$ swamps the chance of $X$ in determining the value of an outcome. With the truth of $X$ fixed, its chance of being true becomes irrelevant.<br /><br />The Swamping Problem then continues by noting that, if (1) or (2) is true, then my desirability function violates Chance Neutrality. Therefore, it concludes, I am irrational.<br /><br />However, as Stefánsson and Bradley show, Chance Neutrality is not a requirement of rationality. To do this, they consider a further putative principle, which they call Linearity:<br /><br /><b>Linearity</b> $$V(\bigwedge^n_{i=1} C^{x_i}_{X_i}) = \sum^n_{i=1} x_iV(X_i)$$ Now, Stefánsson and Bradley show<br /><br /><b>Theorem</b> <i>Suppose Desirability and the Principal Principle. Then Chance Neutrality entails Linearity.</i><br /><br />They then argue that, since Linearity is not a rational requirement, neither can Chance Neutrality be -- since the Principal Principle is a rational requirement, if Chance Neutrality were too, then Linearity would be; and Linearity is not because it is violated in cases of rational preference, such as in the Allais paradox.<br /><br />Thus, the Swamping Problem in its original form fails. It relies on Chance Neutrality, but Chance Neutrality is not a requirement of rationality. Of course, if we could prove a sort of converse of Stefánsson and Bradley's result, and show that, in the presence of the Principal Principle, Linearity entails Chance Neutrality, then we could show that a value function satisfying (1) is irrational. But we can't prove that converse.<br /><br />Nonetheless, there is still a problem. For we can show that, in the presence of Desirability and the Principal Principle, Linearity entails that there is no desirability function $V$ that satisfies (1). Of course, given that Linearity is not a requirement of rationality, this does not tell us very much at the moment. But it does when we realise that, while Linearity is not required by rationality, veritists who accept the reliabilist account of justification given above typically do have a desirability function that satisfies Linearity. After all, they value a justified belief because it is reliable -- that is, it has high objective expected epistemic value. That is, they value a belief at its expected epistemic value, which is precisely what Linearity says.<br /><br /><b>Theorem</b> <i>Suppose $X$ is a proposition in $F$. And suppose $V$ satisfies Desirability, Principal Principle, and Linearity. Then it is not possible that the following are all satisfied:</i><i> </i><br /><ul><li><i>(Monotonicity) $V(X\ \&\ C^x_X)$ and $V(\overline{X}\ \&\ C^x_X)$ are both monotone increasing and non-constant functions of $x$ on $(0, 1)$;</i></li><li><i>(Betweenness) There is $0 < x < 1$ such that $V(X) < V(X\ \&\ C^x_X)$</i>.</li></ul><br /><i>Proof</i>. We suppose Desirability, Principal Principle, and Linearity throughout. We proceed by reductio. We make the following abbreviations:<br /><ul><li>$f(x) = V(X\ \&\ C^x_X)$</li><li>$g(x) = V(\overline{X}\ \&\ C^x_X)$</li><li>$F = V(X)$</li><li>$G = V(\overline{X})$</li></ul>By assumption, we have:<br /><ul><li>(1f) $f$ is a monotone increasing and non-constant function on $(0, 1)$ (by Monotonicity);</li><li>(1g) $g$ is a monotone increasing and non-constant function on $(0, 1)$ (by Monotonicity);</li><li>(2) There is $0 < x < 1$ such that $F < f(x)$ (by Betweenness).</li></ul>By Desirability, we have $$V(C^x_X) = c(X | C^x_X)V(X\ \&\ C^x_X) + c(\overline{X} | C^x_X) V(\overline{X}\ \&\ C^x_X)$$ By this and the Principal Principle, we have $$V(C^x_X)= x V(X\ \&\ C^x_X) + (1 - x)V(\overline{X}\ \&\ C^x_X)$$ So $V(C^x_X) = xf(x) + (1-x)g(x)$. By Linearity, we have $$V(C^x_X) = x V(X) + (1-x)V(\overline{X})$$ So $V(C^x_X) = xF + (1-x)G$. Thus, for all $0 \leq x \leq 1$, $$x V(X) + (1-x)V(\overline{X}) = x V(X\ \&\ C^x_X) + (1 - x)V(\overline{X}\ \&\ C^x_X)$$ That is,<br /><ul><li>(3) $xF + (1-x)G = xf(x) + (1-x)g(x)$</li></ul>Now, by (3), we have $$g(x) = \frac{x}{1-x}(F - f(x)) + G$$ for $0 \leq x < 1$. Now, by (1f) and (2), there are $x < y < 1$ such that $F < f(x) \leq f(y)$. Thus, $F - f(y) \leq F - f(x) < 0$. And so $$\frac{y}{1-y}(F-f(y)) + G < \frac{x}{1-x}(F-f(x)) + G < 0$$ And thus $g(y) < g(x)$. But this contradicts (1g). Thus, there can be no such pair of functions $f$, $g$. Thus, there can be no such $V$, as required. $\Box$<br /><br /><br /><br /><br />Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com1tag:blogger.com,1999:blog-4987609114415205593.post-37519951788502376842017-02-12T19:14:00.003+00:002017-02-13T09:27:54.652+00:00Chance Neutrality and the Swamping Problem for ReliabilismReliabilism about justified belief comes in two varieties: process reliabilism and indicator reliabilism. According to process reliabilism, a belief is justified if it is formed by a process that is likely to produce truths; according to indicator reliabilism, a belief is justified if it likely to be true given the ground on which the belief is based. Both are natural accounts of justification for a veritist, who holds that the sole fundamental source of epistemic value for a belief is its truth.<br /><br />Against veritists who are reliabilists, opponents raise the Swamping Problem. This begins with the observation that we prefer a justified true belief to an unjustified true belief; we ascribe greater value to the former than to the latter; we would prefer to have the former over the latter. But, if reliablism is true, this means that we prefer a belief that is true and had a high chance of being true over a belief that is true and had a low chance of being true. For a veritist, this means that we prefer a belief that has maximal epistemic value and had a high chance of having maximal epistemic value over a belief that has maximal epistemic value and had a low chance of having maximal epistemic value. And this is irrational, or so the objection goes. It is only rational to value a high chance of maximal utility when the actual utility is not known; once the actual utility is known, this 'swamps' any consideration of the chance of that utility. For instance, suppose I find a lottery ticket on the street; I know that it comes either from a 10-ticket lottery or from a 100-ticket lottery; both lotteries pay out the same amount to the holder of the winning ticket; and I know the outcome of neither lottery. Then it is rational for me to hope that the ticket I hold belongs to the smaller lottery, since that would maximise my chance of winning and thus maximise the expected utility of the ticket. But once I know that the lottery ticket I found is the winning ticket, it is irrational to prefer that it came from the smaller lottery --- my knowledge that it's the winner 'swamps' the information about how likely it was to be the winner. This is known variously as the Swamping Problem or the Value Problem for reliabilism about justification (Zagzebski 2003, Kvanvig 2003).<br /><br />The central assumption of the swamping problem is a principle that, in a different context, H. Orri Stefánsson and Richard Bradley call Chance Neutrality (Stefánsson & Bradley 2015). They state it precisely within the framework of Richard Jeffrey's decision theory (Jeffrey 1983). In that framework, we have a desirability function $V$ and a credence function $c$, both of which are defined on an algebra of propositions $\mathcal{F}$. $V(A)$ measures how strongly our agent desires $A$, or how greatly she values it. $c(A)$ measures how strongly she believes $A$, or her credence in $A$. The central principle of the decision theory is this:<br /><br /><b>Desirability</b> If the propositions $A_1$, $\ldots$, $A_n$ form a partition of the proposition $X$, then $$V(X) = \sum^n_{i=1} c(A_i | X) V(A_i)$$<br /><br />Now, suppose the algebra on which $V$ and $c$ are defined includes some propositions that concern the objective probabilities of other propositions in the algebra. Then:<br /><br /><b>Chance Neutrality </b> Suppose $X$ is in the partition $X_1$, \ldots, $X_n$. And suppose $0 \leq \alpha_1, \ldots, \alpha_n \leq 1$ and $\sum^n_{i=1} \alpha = 1$. Then $$V(X\ \&\ \bigwedge^n_{i=1} \mbox{Objective probability of $X_i$ is $\alpha_i$}) = V(X)$$<br /><br />That is, information about the outcome of the chance process that picks between $X_1$, $\ldots$, $X_n$ `swamps' information about the chance process in our evaluation, which is recorded in $V$. A simple consequence of this: if $0 \leq \alpha_1, \alpha'_1 \ldots, \alpha_n, \alpha'_n \leq 1$ and $\sum^n_{i=1} \alpha_i = 1$ and $\sum^n_{i=1} \alpha'_i = 1$, then<br /><br />$V(X\ \&\ \bigwedge^n_{i=1} \mbox{Objective probability of $X_i$ is $\alpha_i$}) = $<br />$V(X\ \&\ \bigwedge^n_{i=1} \mbox{Objective probability of $X_i$ is $\alpha'_i$})$<br /><br />Now consider the particular case of this that is used in the Swamping Problem. I believe $X$ on the basis of ground $g$. I assign greater value to $X$ being true and justified than I do to $X$ being true and unjustified. That is, given the reliabilist's account of justification, if $\alpha$ is a probability that lies above the threshold for justification and $\alpha'$ is a probability that lies below that threshold --- for the veritist, $\alpha' < \frac{W}{R+W} < \alpha$ --- then<br /><br />$V(X\ \&\ \mbox{Objective probability of $X$ given I have $g$ is $\alpha'$}) <$<br />$V(X\ \&\ \mbox{Objective probability of $X$ given I have $g$ is $\alpha$})$<br /><br />And of course this violates Chance Neutrality. <br /><br />Thus, the Swamping Problem stands or falls with the status of Chance Neutrality. Is it a requirement of rationality? Stefánsson and Bradley argue that it is not (Section 3, Stefánsson & Bradley 2015). They show that, in the presence of the Principal Principle, Chance Neutrality entails a principle called Linearity; and they claim that Linearity is not a requirement of rationality. If it is permissible to violate Linearity, then it cannot be a requirement to satisfy a principle that entails it. So Chance Neutrality is not a requirement of rationality.<br /><br />In this context, the Principal Principle runs as follows:<br /><br /><b>Principal Principle</b> $$c(X_i | \bigwedge^n_{i=1} \mbox{Objective probability of $X_i$ is $\alpha_i$}) = \alpha_i$$<br /><br />That is, an agent's credence in $X_i$, conditional on information that gives the objective probability of $X_i$ and other members of a partition to which it belongs, should be equal to the objective probability of $X_i$. And Linearity is the following principle:<br /><br /><b>Linearity</b> $$V(\bigwedge^n_{i=1} \mbox{Objective probability of $X_i$ is $\alpha_i$}) = \sum^n_{i=1} \alpha_iV(X_i)$$<br /><br />That is, an agent should value a lottery at the expected value of its outcome. Now, as is well known, real agents often violate Linearity (Buchak 2014). The most famous violations are known as the Allais preferences (Allais 1953). Suppose there are 100 tickets numbered 1 to 100. One ticket will be drawn and you will be given a prize depending on which option you have chosen from $L_1$, $\ldots$, $L_4$:<br /><ul><li>$L_1$: if ticket 1-89, £1m; if ticket 90-99, £1m; if ticket 100, £1m.</li><li>$L_2$: if ticket 1-89, £1m; if ticket 90-99, £5m; if ticket 100, £0m</li><li>$L_3$: if ticket 1-89, £0m; if ticket 90-99, £1m; if ticket 100, £1m</li><li>$L_4$: if ticket 1-89, £0m; if ticket 90-99, £5m; if ticket 100, £0m </li></ul>I know that each ticket has an equal chance of winning --- thus, by the Principal Principle, $c(\mbox{Ticket $n$ wins}) = \frac{1}{100}$. Now, it turns out that many people have preferences recorded in the following desirability function $V$: $$V(L_1) > V(L_2) \mbox{ and } V(L_3) < V(L_4)$$<br /><br />When there is an option that guarantees them a high payout (\pounds 1m), they prefer that over something with 1% chance of nothing (\pounds 0) even if it also provides 10% chance of much greater payout (£5m). On the other hand, when there is no guarantee of a high payout, they prefer the chance of the much greater payout (\pounds 5m), even if there is also a slightly greater chance of nothing (£0). The problem is that there is no way to assign values to $V(£0)$, $V(£1m)$, and $V(£5m)$ so that $V$ satisfies Linearity and also these inequalities. Suppose, for a reductio, that there is. By Linearity,<br />$$V(L_1) = 0.89V(£1\mathrm{m}) + 0.1 V(£1\mathrm{m}) + 0.01 V(£1\mathrm{m})$$<br />$$V(L_2) = 0.89V(£1\mathrm{m}) + 0.1 V(£5\mathrm{m}) + 0.01 V(£0\mathrm{m}) $$<br />Then, since $V(L_1) > V(L_2)$, we have: $$0.1 V(£1\mathrm{m}) + 0.01 V(£1\mathrm{m}) > 0.1 V(£5\mathrm{m}) + 0.01 V(£0\mathrm{m})$$ But also by Linearity, $$V(L_3) = 0.89V(£0\mathrm{m}) + 0.1 V(£1\mathrm{m}) + 0.01 V(£1\mathrm{m})$$<br />$$V(L_4) = 0.89V(£0\mathrm{m}) + 0.1 V(£5\mathrm{m}) + 0.01 V(£0\mathrm{m})$$<br />Then, since $V(L_3) < V(L_4)$, we have: $$0.1 V(£1\mathrm{m}) + 0.01 V(£1\mathrm{m}) < 0.1 V(£5\mathrm{m}) + 0.01 V(£0\mathrm{m})$$<br />And this gives a contradiction. In general, an agent violates Linearity when she has any risk averse or risk seeking preferences.<br /><br />Stefánsson and Bradley show that, in the presence of the Principal Principle, Chance Neutrality entails Linearity; and they argue that there are rational violations of Linearity (such as the Allais preferences); so they conclude that there are rational violations of Chance Neutrality. So far, so good for the reliabilist: the Swamping Problem assumes that Chance Neutrality is a requirement of rationality; and we have seen that it is not. However, reliabilism is not out of the woods yet. After all, the veritist's version of reliabilism that in fact assumes Linearity! They say that a belief is justified if it is likely to true. And they say this because a belief that is likely to be true has high expected epistemic value on the veritist's account of epistemic value. And so they connect justification to epistemic value by taking the value of a belief to be its expected epistemic value --- that is, they assume Linearity. Thus, if the only rational violations of Chance Neutrality are also rational violations of Linearity, then the Swamping Problem is revived. In particular, if Linearity entails Chance Neutrality, then reliabilism cannot solve the Swamping Problem.<br /><br />Fortunately, even in the presence of the Principal Principle, Linearity does not entail Chance Neutrality. Together, the Principal Principle and Desirability entail:<br /><br />$V(\mbox{Objective probability of $X$ given I have $g$ is $\alpha$}) =$<br /><br />$\alpha V(X\ \&\ \mbox{Objective probability of $X$ given I have $g$ is $\alpha$}) + $<br /><br />$(1-\alpha) V(\overline{X}\ \&\ \mbox{Objective probability of $X$ given I have $g$ is $\alpha$})$<br /><br />And Linearity entails:<br /><br /> $V(\mbox{Objective probability of $X$ given I have $g$ is $\alpha$}) = \alpha V(X) + (1-\alpha) V(\overline{X})$<br /><br />So<br />$\alpha V(X) + (1-\alpha) V(\overline{X}) =$<br /><br />$\alpha V(X\ \&\ \mbox{Objective probability of $X$ given I have $g$ is $\alpha$}) + $<br /><br />$(1-\alpha) V(\overline{X}\ \&\ \mbox{Objective probability of $X$ given I have $g$ is $\alpha$})$<br /><br />And, whatever the values of $V(X)$ and $V(\overline{X})$, there are values of $$V(X\ \&\ \mbox{Objective probability of $X$ given I have $g$ is $\alpha$})$$ and $$V(\overline{X}\ \&\ \mbox{Objective probability of $X$ given I have $g$ is $\alpha$})$$<br />such that the above equation holds. Thus, it is at least possible to adhere to Linearity, yet violate Chance Neutrality. Of course, this does not show that the agent who adheres to Linearity but violates Chance Neutrality is rational. But, now that the intuitive appeal of Chance Neutrality is undermined, the burden is on those who raise the Swamping Problem to explain why such cases are irrational.<br /><br /><h2>References</h2><br /><ul><li>Allais, M. (1953). Le comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l'école Amáricaine. Econometrica, 21(4), 503–546.</li><li>Buchak, L. (2013). Risk and Rationality. Oxford University Press.</li><li>Kvanvig, J. (2003). The Value of Knowledge and the Pursuit of Understanding. Cambridge: Cambridge University Press.</li><li>Stefánsson, H. O., & Bradley, R. (2015). How Valuable Are Chances? Philosophy of Science, 82, 602–625.</li><li>Zagzebski, L. (2003). The search for the source of the epistemic good. Metaphilosophy, 34(12-28).</li></ul><br />Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-6713211078660404562017-02-06T23:50:00.005+00:002017-02-08T08:55:29.400+00:00What is justified credence?Aafira and Halim are both 90% confident that it will be sunny tomorrow. Aafira bases her credence on her observation of the weather today and her past experience of the weather on days that follow days like today -- around nine out of ten of them have been sunny. Halim bases his credence on wishful thinking -- he's arranged a garden party for tomorrow and he desperately wants the weather to be pleasant. Aafira, it seems, is justified in her credence, while Halim is not. Just as one of your full or categorical beliefs might be justified if it is based on visual perception under good conditions, or on memories of recent important events, or on testimony from experts, so might one of your credences be; and just as one of your full beliefs might be unjustified if it is based on wishful thinking, or biased stereotypical associations, or testimony from ideologically driven news outlets, so might your credences be. In this post, I'm looking for an account of justified credence -- in particular, I seek necessary and sufficient conditions for a credence to be justified. Our account will be reliabilist. <br /><br />Reliabilism about justified beliefs comes in two varieties: process reliabilism and indicator reliabilism. Roughly, process reliabilism says that a belief is justified if it is formed by a reliable process, while indicator reliabilism says that a belief is justified if it is based on a ground that renders it likely. Reliabilism about justified credence also comes in two varieties; indeed, it comes in the same two varieties. And, indeed, of the two existing proposals, <a href="http://academic.depauw.edu/jeffreydunn_web/" target="_blank">Jeff Dunn</a>'s is a version of process reliabilism (<a href="http://link.springer.com/article/10.1007%2Fs11098-014-0380-2" target="_blank">paper</a>) while <a href="http://profile.nus.edu.sg/fass/phitwh/" target="_blank">Weng Hong Tang</a> offers a version of indicator reliabilism (<a href="https://academic.oup.com/mind/article-abstract/125/497/63/2563643/Reliability-Theories-of-Justified-Credence" target="_blank">paper</a>). As we will see, both face the same objection. If they are right about what justification is, it is mysterious why we care about justification, for neither of the accounts connects justification to a source of epistemic value. We will call this the <i>Connection Problem</i>.<br /><br />I begin by describing Dunn's process reliabilism and Tang's indicator reliabilism. I argue that, understood correctly, they are, in fact, extensionally equivalent. That is, Dunn and Tang reach the top of the same mountain, albeit by different routes. However, I argue that both face the Connection Problem. In response, I offer my own version of reliabilism, which is both process and indicator, and I argue that it solves that problem. Furthermore, I show that it is also extensionally equivalent to Dunn's reliabilism and Tang's.<br /><br /><h2>Reliabilism and Dunn on reliable credence</h2><br />Let us begin with Dunn's process reliabilism for justified credences. Now, to be clear, Dunn takes himself only to be providing an account of reliability for credence-forming processes. He doesn't necessarily endorse the other two conjuncts of reliabilism, which say that a credence is justified if it is reliable, and that a credence is reliable if formed by a reliable process. Instead, Dunn speculates that perhaps being reliably formed is but one of the epistemic virtues, and he wonders whether all of the epistemic virtues are required for justification. Nonetheless, I will consider a version of reliabilism for justified credences that is based on Dunn's account of reliable credence. For reasons that will become clear, I will call this the calibrationist version of process reliabilism for justified credence. Dunn rejects it based on what I will call below the <i>Graining Problem</i>. As we will see, I think we can answer that objection.<br /><br />For Dunn, a credence-forming process is perfectly reliable if it is well calibrated. Here's what it means for a process $\rho$ to be well calibrated:<br /><ul><li>First, we construct a set of all and only the outputs of the process $\rho$ in the actual world and in nearby counterfactual scenarios. An output of $\rho$ consists of a credence $x$ in a proposition $X$ at a particular time $t$ in a particular possible world $w$ -- so we represent it by the tuple $(x, X, w, t)$. If $w$ is a nearby world and $t$ a nearby time, we call $(x, X, w, t)$ a <i>nearby output</i>. Let $O_\rho$ be the set of nearby outputs -- that is, the set of tuples $(x, X, w, t)$, where $w$ is a nearby world, $t$ is a nearby time, and $\rho$ assigns credence $x$ to proposition $X$ in world $w$ at time $t$.</li><li>Second, we say that the truth-ratio of $\rho$ for credence $x$ is the proportion of nearby outputs $(x, X, w, t)$ in $O_\rho$ such that $X$ is true at $w$ and $t$.</li><li>Finally, we say that $\rho$ is well calibrated (or nearly so) if, for each credence $x$ that $\rho$ assigns, $x$ is equal to (or approximately equal to) the truth-ratio of $\rho$ for $x$.</li></ul>For instance, suppose a process only ever assigns credence 0.6 or 0.7. And suppose that, 60% of the time that it assigns 0.6 in the actual world or a nearby world it assigns it to a proposition that is true; and 70% of the time it assigns 0.7 it assigns it to a true proposition. If, on the other hand, 59% of the time that it assigns 0.6 in the actual world or a nearby world it assigns it to a proposition that is true, while 71% of the time it assigns 0.7 it assigns it to a true proposition, then that process is not well calibrated, but it is nearly well calibrated. But if 23% of the time that it assigns 0.6 in the actual world or a nearby world it assigns it to a proposition that is true, while 95% of the time it assigns 0.7 it assigns it to a true proposition, then that process is not even nearly well calibrated.<br /><br />This, then, is Dunn's calibrationist account of the reliability of a credence-forming process. Any version of reliabilism about justified credences that is based on it requires two further ingredients. First, we must use the account to say when an individual credence is reliable; second, we must add the claim that a credence is justified iff it is reliable. Both of these moves creates problems. We will address them below. But first it will be useful to present Tang's version of indicator reliabilism for justified credence. It will provide an important clue that helps us solve one of the problems that Dunn's account faces. And, having it in hand, it will be easier to see how these two accounts end up coinciding.<br /><br /><h2>Tang's indicator reliabilism for justified credence</h2><br />According to indicator reliabilism for justified belief, a belief is justified if the ground on which it is based is a good indicator of the truth of that belief. Thus, beliefs formed on the basis of visual experiences tend to be justified because the fact that the agent had the visual experience in question makes it likely that the belief they based on it is true. Wishful thinking, on the other hand, usually does not give rise to justified belief because the fact that an agent hopes that a particular proposition will be true -- which in this case is the ground of their belief -- does not make it likely that the proposition is true.<br /><br />Tang seeks to extend this account of justified belief to the case of credence. Here is his first attempt at an account:<br /><br /><b>Tang's Indicator Reliabilism for Justified Credence (first pass)</b> A credence of $x$ in $X$ by an agent $S$ is justified iff<br />(TIC1-$\alpha$) $S$ has ground $g$;<br />(TIC2-$\alpha$) the credence $x$ in $X$ by $S$ is based on ground $E$;<br />(TIC3-$\alpha$) the objective probability of $X$ given that the agent has ground $g$ approximates or equals $x$ -- we write this $P(X | \mbox{$S$ has $g$}) \approx x$.<br /><br />Thus, just as an agent's full belief in a proposition is justified if its ground makes the objective probability of that proposition close to 1, a credence $x$ in a proposition is justified if its ground makes the objective probability of that proposition close to $x$. There is a substantial problem here in identifying exactly to which notion of objective probability Tang wishes to appeal. But we will leave that aside for the moment, other than to say that he conceives of it along the lines of hypothetical frequentism -- that is, the objective probability of $X$ given $Y$ is the hypothetical frequency with which propositions like $X$ are true when propositions like $Y$ are true. <br /><br />However, as Tang notes, as stated, his version of indicator reliabilism faces a problem. Suppose I am presented with an empty urn. I watch as it is filled with 100 balls, numbered 1 to 100, half of which are white, and half of which are black. I shake the urn vigorously and extract a ball. It's number 73 and it's white. I look at its colour and the numeral printed on it. I have a visual experience of a white ball with '73' on it. On the basis of my visual experience of the numeral alone, I assign credence 0.5 to the proposition that ball 73 is white. According to Wang's first version of indicator reliabilism for justified credence, my credence is justified. My ground is the visual experience of the number on the ball; I have that ground; I base my credence on that ground; and the objective probability that ball 73 is white given that I have a visual experience of the numeral '73' printed on it is 50% -- after all, half the balls are white. Of course, the problem is that I have not used my total evidence -- or, in the language of grounds, I have not based my belief on my most inclusive ground. I had the visual experience of the numeral on the ball as a ground; but I also had the visual experience of the numeral on the ball <i>and the colour of the ball</i> as a ground. The resulting credence is unjustified because the objective probability that ball 73 is white given I have the more inclusive ground is not 0.5 -- it is close to 1, since my visual system is so reliable. This leads Tang to amend his account of justified credence as follows:<br /><br /><b>Tang's Indicator Reliabilism for Justified Credence</b> A credence of $x$ in $X$ by an agent $S$ is justified iff<br />(TIC1) $S$ has ground $g$;<br />(TIC2) the credence $x$ in $X$ by $S$ is based on ground $g$;<br />(TIC3) the objective probability of $X$ given that the agent has ground $g$ approximates or equals $x$ -- that is, $P(X | \mbox{$S$ has $g$}) \approx x$;<br />(TIC4) there is no more inclusive ground $g'$ such that (i) $S$ has $g'$ and (ii) the objective probability of $X$ given that the agent has ground $g'$ does not equal or approximate $x$ -- that is, $P(X | \mbox{$S$ has $g'$}) \not \approx x$.<br /><br />This, then, is Tang's version of indicator reliabilism for justified credences.<br /><br /><h2>Same mountain, different routes</h2><br />Thus, we have now seen Dunn's process reliabilism and Tang's indicator reliabilism for justified credences. Is either correct? If so, which? In one sense, both are correct; in another, neither is. Less mysteriously: as we will see in this section, Dunn's process reliablism and Tang's indicator reliabilism are extensionally equivalent -- that is, the same credences are justified on both. What's more, as we will see in the final section, both are extensionally equivalent to the correct account of justified credence, which is thus a version of both process and indicator reliabilism. However, while they get the extension right, they do so for the wrong reasons. A justified credence is not justified because it is formed by a well calibrated process; and it is not justified because it matches the objective chance given its grounds. Thus, Dunn and Tang delimit the correct extension, but they use the wrong intension. In the final section of this post, I will offer what I take to be the correct intension. But first, let's see why it is that the routes that Dunn and Tang take lead them both to the top of the same mountain.<br /><br />We begin with Dunn's calibrationist account of the reliability of a credence-forming process. As we noted above, any version of reliabilism about justified credences that is based on this account requires two further ingredients. First, we must use the calibrationist account of reliable credence-forming processes to say when an individual credence is reliable. The natural answer: when it is formed by a reliable credence-forming process. But then we must be able to identify, for a given credence, the process of which it is an output. The problem is that, for any credence, there are a great many processes of which it might be the output. I have a visual experience of a piece of red cloth on my desk, and I form a high credence that there is a piece of red cloth on my desk. Is this credence the output of a process that assigns a high credence that that there is a piece of red cloth on my desk whenever I have that visual experience? Or is it the output of a process that assigns a high credence that there is a piece of red cloth on my desk whenever I have that visual experience <i>and the lighting conditions in my office are good</i>, while it assigns a middling credence that there is a piece of red cloth on my desk whenever I have that visual experience <i>and the lighting conditions in my office are bad</i>? It is easy to see that this is important. The first process is poorly calibrated, and thus unreliable on Dunn's account; the second process is better calibrated and thus more reliable on Dunn's account. This is the so-called <i>Generality Problem</i>, and it is a challenge that faces any version of reliabilism. I will offer a version of Juan Comesaña's solution to this problem below -- as we will see, that solution also clears the way for a natural solution to the Graining Problem, which we consider next.<br /><br />Dunn provides an account of when a credence-forming process is reliable. And, once we have a solution to the Generality Problem, we can use that to say when a credence is reliable -- it is reliable when formed by a reliable credence-forming process. Finally, to complete the version of process reliablism about justified credence that we are basing on Dunn's account, we just need the claim that a credence is justified iff it is reliable. But this too faces a problem, which we call the <i>Graining Problem</i>. As we did above, suppose I am presented with an empty urn. I watch as it is filled with 100 balls, numbered 1 to 100, half of which are white, and half of which are black. I shake the urn vigorously and extract a ball. I look at its colour and the numeral printed on it. I have two processes at my disposal. Process 1 takes my visual experience of the numeral only, say '$n$', and assigns the credence 0.5 to the proposition that ball $n$ is white. Process 2 takes my visual experience of the numeral, '$n$', <i>and my visual experience of the colour of the ball</i>, and assigns credence 1 to the proposition that ball $n$ is white if my visual experience is of a white ball, and assigns credence 1 to the proposition that ball $n$ is black if my visual experience is of a black ball. Note that both processes are well calibrated (or nearly so, if we allow that my visual system is very slightly fallible). But we would usually judge the credence formed by the second to be better justified than the credence formed by the first. Indeed, we would typically say that a Process 1 credence is unjustified, while a Process 2 credence is justified. Thus, being formed by a well calibrated or nearly well calibrated process is not sufficient for justification. And, if reliability is calibration, then reliability is not justification and reliabilism fails. It is this problem that leads Dunn to reject reliabilism about justified credence. However, as we will see below, I think he is a little hasty.<br /><br />Let us consider the Generality Problem first. To this problem, <a href="http://comesana.arizona.edu/" target="_blank">Juan Comesaña</a> offers the following solution (<a href="http://link.springer.com/article/10.1007%2Fs11098-005-3020-z" target="_blank">paper</a>). Every account of doxastic justification -- that is, every account of when a given doxastic attitude of a particular agent is justified for that agent -- must recognize that two agents may have the same doxastic attitude and the same evidence while the doxastic attitude of one is justified and the doxastic attitude of the other is not, because their doxastic attitudes are not based on the same evidence. The first might base her belief on the total evidence, for instance, whilst the second ignores that evidence and bases his belief purely on wishful thinking. Thus, Comesaña claims, every theory of justification needs a notion of the grounds or the basis of a doxastic attitude. But, once we have that, a solution to the Generality Problem is very close. Comesaña spells out the solution for process reliabilism about full beliefs:<br /><br /><b>Well-Founded Process Reliablism for Justified Full Beliefs</b> A belief that $X$ by an agent $S$ is justified iff<br />(WPB1) $S$ has ground $g$;<br />(WPB2) the belief that $X$ by $S$ is based on ground $g$;<br />(WPB3) the process <i>producing a belief state $X$ based on ground $g$</i> is a reliable process.<br /><br />This is easily adapted to the credal case:<br /><br /><b>Well-Founded Process Reliablism for Justified Credences</b> A credence of $x$ in $X$ by an agent $S$ is justified iff<br />(WPC1) $S$ has ground $g$;<br />(WPC2) the credence $x$ in $X$ by $S$ is based on ground $g$;<br />(WPC3) the process <i>producing a credence of $x$ in $X$ based on ground $g$</i> is a reliable process.<br /><br />Let us now try to apply Comesaña's solution to the Generality Problem to help Dunn's calibrationist reliabilism about justified credences. Recall: according to Dunn, a process $\rho$ is reliable if it is well calibrated (or nearly so). Consider the process <i>producing a credence of $x$ in $X$ based on ground $g$</i> -- for convenience, we'll write it $\rho^g_{X,x}$. There is only one credence that it assigns, namely $x$. So it is well calibrated if that truth-ratio of $\rho^g_{X,x}$ for $x$ is equal to $x$. Now, $O_{\rho^g_{X,x}}$ is the set of tuples $(X, x, w, t)$ where $w$ is a nearby world and $t$ a nearby time where $\rho^g_{X,x}$ assigns credence $x$ to proposition $X$. But, by the definition of $\rho^g_{X,x}$, those are the nearby worlds and nearby times at which the agent has the ground $g$. Thus, the truth-ratio of $\rho^g_{X,x}$ for $x$ is the proportion of those nearby worlds and times at which the agent has the ground $g$ at which $X$ is true. And that, it seems to me, is the something like the objective probability of $X$ conditional on the agent having ground $g$, at least given the hypothetical frequentist account of objective probability of the sort that Tang favours. As above, we denote the objective probability of $X$ conditional on the agent $S$ having grounds $g$ as follows: $P(X | \mbox{$S$ has $g$})$. Thus, $P(X | \mbox{$S$ has $g$})$ is the truth-ratio of $\rho^g_{p,x}$ for $x$. And thus, a credence $x$ in $X$ based on ground $g$ is reliable iff $x$ is close to $P(X | \mbox{$S$ has $g$})$. That is,<br /><br /><b>Well-Founded Calibrationist Process Reliabilism for Justified Credences (first attempt)</b> A credence of $x$ in $X$ by an agent $S$ is justified iff<br />(WCPC1) $S$ has ground $g$;<br />(WCPC2) the credence $x$ in $X$ by $S$ is based on ground $g$;<br />(WCPC3) the process <i>producing a credence of $x$ in $X$ based on ground $g$</i> is a (nearly) well calibrated process -- that is, $P(X | \mbox{$S$ has $g$}) \approx x$. <br /><br />But now compare Well-Founded Calibrationist Process Reliabilism, based on Dunn's account of reliable processes and Comesaña's solution to the Generality Problem, with Tang's first attempt at Indicator Reliabilism. Consider the necessary and sufficient conditions that each imposes for justification: TIC1 = WCPC1; TIC2 = WCPC2; TIC3 = WCPC3. Thus, these are the same account. However, as we saw above, Tang's first attempt to formulate indicator reliabilism for justified credence fails because it counts as justified a credence that is not based on an agent's total evidence; and we also saw that, once the Generality Problem is solved for Dunn's calibrationist process reliabilism, it faces a similar problem, namely, the Graining Problem from above. Tang amends his version of indicator reliabilism by adding the fourth condition TIC4 from above. Might we amend Dunn's calibrationist process reliabilism is a similar way?<br /><br /><b>Well-Founded Calibrationist Process Reliabilism for Justified Credences</b> A credence of $x$ in $X$ by an agent $S$ is justified iff<br />(WCPC1) $S$ has ground $g$;<br />(WCPC2) the credence $x$ in $X$ by $S$ is based on ground $g$;<br />(WCPC3) the process <i>producing a credence of $x$ in $X$ based on ground $g$</i> is a (nearly) well calibrated process -- that is, $P(X | \mbox{$S$ has $g$}) \approx x$;<br />(WCPC4) there is no more inclusive ground $g'$ and credence $x' \not \approx x$, such that the process <i>producing a credence of $x'$ in $X$ based on ground $g'$</i> is a (nearly) well calibrated process -- that is, $P(X | \mbox{$S$ has $g'$}) \approx x'$.<br /><br />Since TIC4 is equivalent to WCPC4, this final version of process reliabilism for justified credences is equivalent to Tang's final version of his indicator reliabilism for justified credences. Thus, Dunn and Tang have reached the top of the same mountain, albeit by different routes<br /> <br /><h2>The third route up the mountain</h2><br />Once we have addressed certain problems with the calibrationist version of process reliabilism for justified credence, we see that it agrees with the current best version of indicator reliabilism. This gives us a little hope that both have hit upon the correct account of justification. In the end, I will conclude that both have indeed hit upon the correct <i>extension</i> of the concept of justified credence. But that have done so for the wrong reasons, for they have not hit upon the correct <i>intension</i>.<br /><br />There are two sorts of route you might take when pursuing an account of justification for a given sort of doxastic attitude, such as a credence or a full belief. You might look to intuitions concerning particular cases and try to discern a set of necessary and sufficient conditions that sort these cases in the same way that your intuitions do; or, you might begin with an account of epistemic value, assume that justification must be linked in some natural way to the promotion of epistemic value, and then provide an account of justification that vindicates that assumption. Dunn and Tang have each taken a route of the first sort; I will follow a route of the second sort.<br /><br />I will adopt the veritist's account of epistemic value. That is, I take accuracy to be the sole fundamental source of epistemic value for a credence, where a credence in a true proposition is more accurate the higher it is; a credence in a false proposition is more accurate the lower it is. Given this account of epistemic value, what is the natural account of justification? Well, at first sight, there are two: one is process reliabilist; the other is indicator reliabilist. But, in a twist that should come as little surprise given the conclusions of the previous section, it will turn out that these two accounts coincide, and indeed coincide with the final versions of Dunn's and Tang's accounts that we reached above. Thus, I too will reach the top of the same mountain, but by yet another route.<br /><br /><h3>Epistemic value version of indicator reliabilism</h3><br />In the case of full beliefs, indicator reliabilism says this: a belief in $X$ by $S$ on the basis of grounds $g$ is justified iff the objective probability of $X$ given that $S$ has grounds $g$ is high --- that is, close to 1. Tang generalises this to the case of credence, but I think he generalises in the wrong direction; that is, he takes the wrong feature to be salient and uses that to formulate his indicator reliabilism for justified credence. He takes the general form of indicator reliabilism to be something like this: a doxastic attitude $s$ towards $X$ by $S$ on the basis of grounds $g$ is justified iff the attitude $s$ 'matches' the objective probability of $X$ given that $S$ has grounds $g$. And he takes the categorical attitude of belief in $X$ to 'match' high objective probability of $X$, and credence $x$ in $X$ to 'match' objective probability of $x$ that $X$. The problem with this account is that it leaves mysterious why justification is valuable. Unless we say that matching objective probabilities is somehow epistemic valuable in itself, it isn't clear why we should want to have justified doxastic attitudes in this sense.<br /><br />I contend instead that the general form of indicator reliabilism is this:<br /><br /><b>Indicator reliabilism for justified doxastic attitude (epistemic value version)</b> Doxastic attitude $s$ towards proposition $X$ by agent $S$ is justified iff<br />(EIA1) $S$ has $g$;<br />(EIA2) $s$ in $X$ by $S$ is based on $g$;<br />(EIA3) if $g' \subseteq g$ is a ground that $S$ has, then for every doxastic attitude $s'$ of the same sort as $s$, the expected epistemic value of attitude $s'$ towards $X$ given that $S$ has $g'$ is at most (or not much above) the expected epistemic value of attitude $s$ towards $X$ given that $S$ has $g'$.<br /><br />Thus, attitude $s$ towards $X$ by $S$ is justified if $s$ is based on a ground $g$ that $S$ has, and $s$ is the attitude towards $X$ that has highest expected accuracy relative to the most inclusive grounds that $S$ has.<br /><br />Let's consider this in the full belief case. We have:<br /><br /><b>Indicator reliabilism for justified belief (epistemic value version)</b> A belief in proposition $X$ by agent $S$ is justified iff<br />(EIB1) $S$ has $g$;<br />(EIB2) $s$ in $X$ by $S$ is based on $g$;<br />(EIB3) if $g' \subseteq g$ is a ground that $S$ has, then<br /><ol><li>the expected epistemic value of <i>disbelief</i> in $X$, given that $S$ has $g'$, is at most (or not much above) the expected epistemic value of <i>belief</i> in $X$, given that $S$ has $g'$;</li><li>the expected epistemic value of <i>suspension</i> in $X$, given that $S$ has $g'$, is at most (or not much above) the expected epistemic value of <i>belief</i> in $X$, given that $S$ has $g'$.</li></ol><br />To complete this, we need only an account of epistemic value. Here, the veritist's account of epistemic value runs as follows. There are three categorical doxastic attitudes towards a given proposition: belief, disbelief, and suspension of judgment. If the proposition is true, belief has greatest epistemic value, then suspension of judgment, then disbelief. If it is false, the order is reversed. It is natural to say that a belief in a truth and disbelief in a falsehood have the same high epistemic value -- following <a href="http://www.kennyeaswaran.org/" target="_blank">Kenny Easwaran</a> (<a href="http://onlinelibrary.wiley.com/doi/10.1111/nous.12099/full" target="_blank">paper</a>), we denote this $R$ (for `getting it Right'), and assume $R >0$. And it is natural to say that a disbelief in a truth and belief in a falsehood have the same low epistemic value -- again following Easwaran, we denote this $-W$ (for `getting it Wrong'), and assume $W > 0$. And finally it is natural to say that suspension of belief in a truth has the same epistemic value as suspension of belief in a falsehood, and both have epistemic value 0. We assume that $W > R$, just as Easwaran does. Now, suppose proposition $X$ has objective probability $p$. Then the expected epistemic utility of different categorical doxastic attitudes towards $X$ is given below:<br /><ul><li>Expected epistemic value of belief in $X$ = $p\cdot R + (1-p)\cdot(-W)$.</li><li>Expected epistemic value of suspension in $X$ = $p\cdot 0 + (1-p)\cdot 0$.</li><li>Expected epistemic value of disbelief in $X$ = $p\cdot (-W) + (1-p)\cdot R$. </li></ul>Thus, belief in $X$ has greatest epistemic value amongst the possible categorical doxastic attitudes to $X$ if $p > \frac{W}{R+W}$; disbelief in $X$ has greatest epistemic value if $p < \frac{R}{R+W}$; and suspension in $X$ has greatest value if $\frac{R}{R+W} < p < \frac{W}{R+W}$ (at $p = \frac{W}{R+W}$, belief ties with suspension; at $p = \frac{R}{R+W}$, disbelief ties with suspension). With this in hand, we have the following version of indicator reliabilism for justified beliefs:<br /><br /><b>Indicator reliabilism for justified belief (veritist version)</b> A belief in $X$ by agent $S$ is justified iff<br />(EIB1$^*$) $S$ has $g$;<br />(EIB2$^*$) the belief in $X$ by $S$ is based on $g$;<br />(EIB3$^*$) the objective probability of $X$ given that $S$ has $g$ is (nearly) greater than $\frac{W}{R+W}$;<br />(EIB4$^*$) there is no more inclusive ground $g'$ such that (a) $S$ has $g'$ and (b) the objective probability of $X$ given that $S$ has $g'$ is <i>not</i> (nearly) greater than $\frac{W}{R+W}$.<br /><br />And of course this is simply a more explicit version of the standard version of indicator reliabilism. It is more explicit because it gives a particular threshold above which the objective probability of $X$ given that $S$ has $g$ counts as 'high', and above which (or not much below which) the belief in $X$ by $S$ counts as justified --- that threshold is $\frac{W}{R+W}$.<br /><br />Note that this epistemic value version of indicator reliabilism for justified doxastic states also gives a straightforward account of when a suspension of judgment is justified. Simply replace (EIB3$^*$) and (EIB4$^*$) with:<br /><br />(EIS3$^*$) the objective probability of $X$ given that $S$ has $g$ is (nearly) between $\frac{W}{R+W}$ and $\frac{R}{R+W}$;<br />(EIS4$^*$) there is no more inclusive ground $g'$ such that (a) $S$ has $g'$ and (b) the objective probability of $X$ given that $S$ has $g'$ is <i>not</i> (nearly) between $\frac{W}{R+W}$ and $\frac{R}{R+W}$.<br /><br />And when a disbelief is justified. This time, replace (EIB3$^*$) and (EIB4$^*$) with:<br /><br />(EID3$^*$) the objective probability of $X$ given that $S$ has $g$ is (nearly) less than $\frac{R}{R+W}$;<br />(EID4$^*$) there is no more inclusive ground $g'$ such that (a) $S$ has $g'$ and (b) the objective probability of $X$ given that $S$ has $g'$ is <i>not</i> (nearly) less than $\frac{R}{R+W}$.<br /><br />Next, let's turn to indicator reliabilism for justified credence. Here's the epistemic value version:<br /><br /><b>Indicator reliabilism for justified credence (epistemic value version)</b> A credence of $x$ in proposition $X$ by agent $S$ is justified iff<br />(EIC1) $S$ has $g$;<br />(EIC2) credence $x$ in $X$ by $S$ is based on $g$;<br />(EIC3) if $g' \subseteq g$ is a ground that $S$ has, then for every credence $x'$, the expected epistemic value of credence $x'$ in $X$ given that $S$ has $g'$ is at most (or not much above) the expected epistemic value of credence $x$ in $X$ given that $S$ has $g'$.<br /><br />Again, to complete this, we need an account of epistemic value for credences. As noted above, the veritist holds that the sole fundamental source of epistemic value for credences is their accuracy. There is a lot to be said about different potential measures of the accuracy of a credence -- see, for instance, <a href="http://www-personal.umich.edu/~jjoyce/" target="_blank">Jim Joyce</a>'s 2009 paper <a href="http://link.springer.com/chapter/10.1007%2F978-1-4020-9198-8_11" target="_blank">'Accuracy and Coherence'</a>, chapters 3 & 4 of <a href="https://richardpettigrew.wordpress.com/" target="_blank">my</a> 2016 book <a href="https://global.oup.com/academic/product/accuracy-and-the-laws-of-credence-9780198732716?cc=gb&lang=en&" target="_blank"><i>Accuracy and the Laws of Credence</i></a>, or <a href="http://www.levinstein.org/" target="_blank">Ben Levinstein</a>'s forthcoming paper <a href="https://www.dropbox.com/s/7dga9rlxertbz5g/Schervish%20draft%202.0.pdf?dl=0" target="_blank">'A Pragmatist's Guide to Epistemic Utility'</a>. But here I will say only this: we assume that those measures are <i>continuous</i> and <i>strictly proper</i>. That is, we assume: (i) we assume that the accuracy of a credence is a continuous function of that credence; and (ii) any probability $x$ in a proposition $X$ expects credence $x$ to be more accurate than it expects any other credence $x' \neq x$ in $X$ to be. These two assumptions are widespread in the literature on accuracy-first epistemology, and they are required for many of the central arguments in that area. Given veritism and the continuity and strict propriety of the accuracy measures, (EIC3) is provably equivalent to the conjunction of:<br /><br />(EIC3$^*$) the objective probability of $X$ given that the agent has ground $g$ approximates or equals $x$ -- that is, $P(X | \mbox{$S$ has $g$}) \approx x$;<br />(EIC4$^*$) there is no more inclusive ground $g'$ such that (i) $S$ has $g'$ and (ii) the objective probability of $X$ given that the agent has ground $g'$ does not equal or approximate $x$ -- that is, $P(X | \mbox{$S$ has $g'$}) \not \approx x$.<br /><br />But of course EIC3 = TIC3 and EIC4 = TIC4 from above. Thus, the veritist version of indicator reliabilism for justified credences is equivalent to Tang's indicator reliabilism, and thus to the calibrationist version of process reliabilism. <br /><br /><h2>Epistemic value version of process reliabilism</h2><br />Next, let's turn to process reliabilism. How might we give an epistemic value version of that? The mistake made by the calibrationist version of process reliabilism is of the same sort as the mistake made by Tang in his formulation of indicator reliabilism -- both generalise from the case of full beliefs in the wrong way by mistaking an accidental feature for the salient feature. For the calibrationist, a full belief is justified if it is formed by a reliable process, and a process is reliable if a high proportion of the beliefs it produces are true. Now, notice that there is a sense in which such a process is calibrated: a belief is associated with a high degree of confidence, and that matches, at least approximately, the high truth-ratio of the process. In fact, we want to say that this process is belief-reliable. For it is possible for a process to be reliable in its formation of beliefs, but not in its formation of disbeliefs. So a process is disbelief-reliable if a high proportion of the disbeliefs it produces are false. And we might say that a process is suspension-reliable if a middling proportion the suspensions it forms are true and a middling proportion are false. In each case, we think that, corresponding to each sort of categorical doxastic attitude $s$, there is a fitting proportion $x$ such that a process is $s$-reliable if $x$ is (approximately) the proportion of truths amongst the propositions to which it assigns $s$. Applying this in the credal case gives us the calibrationist version of process reliabilism that we have already met -- a credence $x$ in $S$ is justified if it is formed by a process whose truth-ratio for a given credence is equal to that credence. However, being the product of a belief-reliable process is not the feature of a belief in virtue of which it is justified. Rather, a belief is justified if it is the product of a process that has high expected epistemic value.<br /><br /><b>Process reliabilism for justified doxastic attitude (epistemic value version)</b> Doxastic attitude $s$ towards proposition $X$ by agent $S$ is justified iff<br />(EPA1-$\beta$) $s$ is produced by a process $\rho$;<br />(EPA2-$\beta$) If $\rho'$ is a process that is available to $S$, then the expected epistemic value of $\rho'$ is at most (or not much more than) the expected epistemic value of $\rho$.<br /><br />That is, a doxastic attitude is justified for an agent if it is the output of a process that maximizes or nearly maximizes expected epistemic value amongst all processes that are available to her. To complete this account, we must say which processes count as available to an agent. To answer this, recall Comesaña's solution to the Generality Problem. On this solution, the only processes that interest us have the form, <i>process producing doxastic attitude $s$ towards $X$ on basis of ground $g$</i>. Clearly, a process of this form is available to an agent exactly when the agent has ground $g$. This gives<br /><br /><b>Process Reliabilism about Justified Doxastic Attitudes (Epistemic value version)</b> Attitude $s$ towards proposition $X$ by $S$ is justified iff<br />(EPA1-$\alpha$) $s$ is produced by process $\rho^g_{s, X}$;<br />(EPA2-$\alpha$) If $g' \subseteq g$ is a ground that $S$ has, then for every doxastic attitude $s'$, the expected epistemic value of process $\rho^{g'}_{s', X}$ is at most (or not much more than) the expected epistemic value of process $\rho^{g}_{s, X}$.<br /><br />Thus, in the case of full beliefs, we have:<br /><br /><b>Process reliabilism for justified belief (epistemic value version)</b> A belief in proposition $X$ by agent $S$ is justified iff<br />(EPB1) Belief in $X$ is produced by process $\rho^g_{\mathrm{bel}, X}$;<br />(EPB2) if $g' \subseteq g$ is a ground that $S$ has, then<br /><ol><li>the expected epistemic value of process $\rho^g_{\mathrm{dis}, X}$ is at most (or not much more than) the expected epistemic value of process $\rho^g_{\mathrm{bel}, X}$;</li><li>the expected epistemic value of process $\rho^g_{\mathrm{sus}, X}$ is at most (or not much more than) the expected epistemic value of process $\rho^g_{\mathrm{bel}, X}$;</li></ol><br />And it is easy to see that (EPB1) = (EIB1) + (EIB2), since belief in $X$ is produced by process $\rho^g_{\mathrm{bel}, X}$ iff $S$ has ground $g$ and a belief in $X$ by $S$ is based on $g$. Also, (EPB2) is equivalent to (EIB3). Thus, as for the epistemic version of indicator reliabilism, we get:<br /><br /><b>Indicator reliabilism for justified belief (veritist version)</b> A belief in $X$ by agent $S$ is justified iff<br />(EPB1) $S$ has $g$;<br />(EPB2) the belief in $X$ by $S$ is based on $g$;<br />(EPB3) the objective probability of $X$ given that $S$ has $g$ is (nearly) greater than $\frac{W}{R+W}$;<br />(EPB4) there is no more inclusive ground $g'$ such that (a) $S$ has $g'$ and (b) the objective probability of $X$ given that $S$ has $g'$ is <i>not</i> (nearly) greater than $\frac{W}{R+W}$.<br /><br />Next, consider how the epistemic value version of process reliabilism applies to credences.<br /><br /><b>Process reliabilism for justified credence (epistemic value version)</b> A credence of $x$ in proposition $X$ by agent $S$ is justified iff<br />(EPC1) the credence in $x$ is produced by process $\rho^g_{x, X}$;<br />(EPC2) if $g' \subseteq g$ is a ground that $S$ and $x'$ is a credence, then the expected epistemic value of process $\rho^{g'}_{x', X}$ is at most (or not much more than) the expected epistemic value of process $\rho^g_{x, X}$.<br /><br />As before, we see that (EPC1) is equivalent to (EIC1) + (EIC2). And, providing the measure of accuracy is strictly proper and continuous, we get that (EPC2) is equivalent to (EIC3). So, once again, we arrive at the same summit. The routes taken by Tang, Dunn, and the epistemic value versions of process and indicator reliabilism lead to the same spot, namely, the following account of justified credence:<br /><br /><b>Reliabilism for justified credence (epistemic value version)</b> A credence of $x$ in proposition $X$ by agent $S$ is justified iff<br />(ERC1) $S$ has $g$;<br />(ERC2) credence $x$ in $X$ by $S$ is based on $g$;<br />(ERC3) the objective probability of $X$ given that the agent has ground $g$ approximates or equals $x$ -- that is, $P(X | \mbox{$S$ has $g$}) \approx x$;<br />(ERC4) there is no more inclusive ground $g'$ such that (i) $S$ has $g'$ and (ii) the objective probability of $X$ given that the agent has ground $g'$ does not equal or approximate $x$ -- that is, $P(X | \mbox{$S$ has $g'$}) \not \approx x$.<br /><br /><br />Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-76013994225603478572017-01-31T15:09:00.002+00:002017-01-31T15:09:27.280+00:00Fifth Reasoning Club Conference @ Turin EXTENDED DEADLINE<div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;">The <span style="border: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-weight: 700; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Fifth Reasoning Club Conference</span> will take place at the Center for Logic, Language, and Cognition in Turin on May 18-19, 2017.</div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><br /></div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;">Keynote speakers:</div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><br /></div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><a href="http://fitelson.org/" style="border: 0px; color: #2c67d1; font-family: inherit; font-size: inherit; font-style: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;" target="_blank">Branden FITELSON</a> (Northeastern University, Boston)</div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><a href="http://www.rug.nl/staff/jeanne.peijnenburg/research" style="border: 0px; color: #2c67d1; font-family: inherit; font-size: inherit; font-style: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;" target="_blank">Jeanne PEIJNENBURG</a> (University of Groningen)</div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><a href="https://www5.unitn.it/People/en/Web/Persona/PER0003393#INFO" style="border: 0px; color: #2c67d1; font-family: inherit; font-size: inherit; font-style: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;" target="_blank">Katya TENTORI</a> (University of Trento)</div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><a href="http://paulegre.free.fr/" style="border: 0px; color: #2c67d1; font-family: inherit; font-size: inherit; font-style: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;" target="_blank">Paul EGRÉ</a> (Institut Jean Nicod, Paris)</div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><br /></div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;">Organizing committee: Gustavo Cevolani (Turin), Vincenzo Crupi (Turin), Jason Konek (Kent), and Paolo Maffezioli (Turin).</div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><br /></div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"> </div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-weight: 700; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">CALL FOR ABSTRACTS</span></div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><br /></div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;">The submission deadline for the Fifth Reasoning Club Conference has been <span style="border: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-weight: 700; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">EXTENDED to 15 February 2017</span>. The final decision on submissions will be made by 15 March 2017.</div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><br /></div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;">All PhD candidates and early career researchers with interests in reasoning and inference, broadly construed, are encouraged to submit an abstract of up to 500 words (prepared for blind review) via Easy Chair at <a href="https://easychair.org/conferences/?conf=rcc17" style="border: 0px; color: #2c67d1; font-family: inherit; font-size: inherit; font-style: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;" target="_blank">https://easychair.org/conferences/?conf=rcc17</a>. We especially welcome members of groups that are underrepresented in philosophy to submit. We are committed to promoting diversity in our final programme.</div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><br /></div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;">Grants will be available to help cover travel costs for contributed speakers. To apply for a travel grant, please send a CV and a short travel budget estimate in a single pdf file to <a class="mailto" href="mailto:reasoningclubconference2017@gmail.com" style="border: 0px; color: #2c67d1; font-family: inherit; font-size: inherit; font-style: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;" target="_blank">reasoningclubconference2017@gmail.com</a>.</div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><br /></div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;">More information is available at <a href="http://www.llc.unito.it/notizie/reasoning-club-2017-llc-call-papers-now-open" target="_blank">http://www.llc.unito.it/notizie/reasoning-club-2017-llc-call-papers-now-open</a>. For any queries please contact Vincenzo Crupi (<a class="mailto" href="mailto:vincenzo.crupi@unito.it" style="border: 0px; color: #2c67d1; font-family: inherit; font-size: inherit; font-style: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;" target="_blank">vincenzo.crupi@unito.it</a>) or Jason Konek (<a href="mailto:J.Konek@kent.ac.uk">J.Konek@kent.ac.uk</a>).</div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><br /></div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;">The <a href="https://www.kent.ac.uk/secl/researchcentres/reasoning/club/index.html" style="border: 0px; color: #2c67d1; font-family: inherit; font-size: inherit; font-style: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;" target="_blank">Reasoning Club</a> is a network of institutes, centres, departments, and groups addressing research topics connected to reasoning, inference, and methodology broadly construed. It issues the monthly gazette <a href="http://blogs.kent.ac.uk/thereasoner/about/" style="border: 0px; color: #2c67d1; font-family: inherit; font-size: inherit; font-style: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;" target="_blank">The Reasoner</a>.</div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><br /></div><div style="background-color: white; border: 0px; color: #1c2024; font-family: Roboto, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 14px; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;">Earlier editions of the meeting were held in <a href="http://www.vub.ac.be/CLWF/RC2012/" style="border: 0px; color: #2c67d1; font-family: inherit; font-size: inherit; font-style: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;" target="_blank">Brussels</a>, <a href="http://reasoningclubpisa.weebly.com/" style="border: 0px; color: #2c67d1; font-family: inherit; font-size: inherit; font-style: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;" target="_blank">Pisa</a>, <a href="https://reasoningclubkent.wordpress.com/" style="border: 0px; color: #2c67d1; font-family: inherit; font-size: inherit; font-style: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;" target="_blank">Kent</a>, and <a href="http://www.maths.manchester.ac.uk/news-and-events/events/fourth-reasoning-club-conf/" style="border: 0px; color: #2c67d1; font-family: inherit; font-size: inherit; font-style: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;" target="_blank">Manchester</a>. </div>Jason Konekhttp://www.blogger.com/profile/01750769966011528630noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-10905532510983655542017-01-21T08:09:00.000+00:002017-01-25T17:15:45.082+00:00More on the Principal Principle and the Principle of IndifferenceLast week, I <a href="http://m-phi.blogspot.co.uk/2017/01/the-principal-principle-does-not-imply.html" target="_blank">posted</a> about a recent paper by <a href="http://james-hawthorne.oucreate.com/" target="_blank">James Hawthorne</a>, <a href="https://jlandes.wordpress.com/" target="_blank">Jürgen Landes</a>, <a href="https://kent.academia.edu/ChristianWallmann" target="_blank">Christian Wallmann</a>, and <a href="http://blogs.kent.ac.uk/jonw/" target="_blank">Jon Williamson</a> called <a href="http://bjps.oxfordjournals.org/content/early/2015/07/13/bjps.axv030.abstract" target="_blank">'The Principal Principle implies the Principle of Indifference'</a>, which was published in the <i>British Journal for the Philosophy of Science </i>in 2015. In that post, I read the HLWW paper a particular way. I took their argument to run roughly as follows:<br /><br /><i>The Principal Principle, as Lewis stated it, includes an admissibility condition. Any adequate account of admissibility should entail Conditions 1 and 2 (see below). Together with Conditions 1 and 2, the Principal Principle entails the Principle of Indifference. Thus, the Principal Principle entails the Principle of Indifference.</i><br /><br />Read like this, my response to the argument ran thus:<br /><br /><i>There is an account of admissibility -- namely, Levi-admissibility -- that is adequate and on which Condition 2 is not generally true. Levi-admissibility is adequate since has all of the features that Lewis required of admissibility, and it is very natural when we consider a close relative of Lewis' Principal Principle, namely, Levi's Principal Principle, which follows from Lewis' Principal Principle given some natural assumptions about admissibility that Lewis accepts.</i><br /><br />However, there is another reading of the HLWW argument, and indeed it seems that some of H, L, W, and W favour it. On this alternative reading, it is not assumed that Conditions 1 and 2 follow from any adequate account of admissibility. Rather Conditions 1 and 2 are not taken to be consequences of the Principal Principle at all. Rather, they are intended to be plausible further constraints on credences that are independent of the Principal Principle. Thus, on this reading, the conclusion of the HLWW is not that the Principal Principle implies the Principle of Indifference. Rather, it is that the Principal Principle, together with two further norms (namely, Conditions 1 and 2), implies the Principle of Indifference.<br /><br />In this post, I will raise an objection to this alternative argument.<br /><br />The HLWW argument turns on a mathematical theorem. It takes certain constraints -- (I), (II), (III) below -- and shows that, if an agent's credence function satisfies those constraints, then it must satisfy a particular instance of the Principle of Indifference.<br /><br /><b>Theorem 1</b> If there is $0 < x < 1$ such that<br />(I) $P(F | X) = P(F)$<br />(II) $P(A | FX) = x$<br />(III) $P(A | X (A \leftrightarrow F)) = x$<br />then <br />(IV) $P(F) = 0.5$.<br /><br />Now, the instance of the Principle of Indifference that HLWW wish to infer using this theorem is this:<br /><br /><b>Principle of Indifference (atomic case)</b> Suppose $F$ is an atomic proposition and $P_0$ is our agent's initial credence function. Then $P_0(F) = 0.5$.<br /><br />Thus, to obtain this from Theorem 1, we need the following: for each atomic $F$, there is $A$, $X$, and $0 < x < 1$ that satisfy (I), (II), and (III). Conditions 1 and 2 are intended to obtain this, but I think the argument is clearest if we argue for them directly, using the considerations found in HLWW.<br /><br />Thus, suppose $F$ is atomic. Then the idea is this. Pick a proposition $X$ with two features: (a) if you were to learn $X$ and nothing more as your first piece of evidence, it would place a very strict constraint on your credence in $A$ --- it would require you to have credence $x$ in $A$; (b) $X$ provides no information about $F$ nor about the relationship between $A$ and $F$. Now, providing that $A$ is not logically related to $F$, we might take $X$ to be the proposition $C^A_x$ that says that the objective chance of $A$ is $x$. By the Principal Principle, $C^A_x$ has the first feature (a): $P_0(A | X) = x$. What's more, since $A$ is logically independent of $F$, $C^A_x$ also has the second feature (b): in the absence of further evidence, and in particular evidence about the relationship between $A$ and $F$, $C^A_x$ provides no information about $F$ nor about the relationship between $A$ and $F$.<br /><br />Now, with $A$, $X$, $x$ in hand, we appeal to two principles concerning the way that we should respond to evidence:<br /><br />(Ev1): If your credence function is $P$ and your evidence does not provide any information about the connection between $B$ and $C$, then $P(B | C) = P(B)$.<br /><br />In slogan form, this says: <i>Ignorance entails irrelevance</i>. <br /><br />(Ev2): If you have strong evidence concerning $B$ and no evidence concerning $C$, then $P(B | B \leftrightarrow C) = P(B)$.<br /><br />In slogan form, as we will see: <i>Credences supported by stronger evidence are more resilient</i>. <br /><br />Now, from (Ev1), we immediately obtain (I) for our agent's initial credence function $P_0$ with $F$ atomic and $X = C^A_x$. After all, if you have no evidence, your evidence certainly does not provide any information about the connection between $C^A_x$ and $F$.<br /><br />From (Ev1) and the Principal Principle, we obtain (II) for $P_0$ with $F$ atomic and $X = C^A_x$. Suppose you first learn $C^A_x$ as evidence. So your credence function is $P_1(-) = P_0(-|C^A_x)$. Now, by hypothesis, $C^A_x$ provides no information about the connection between $F$ and $A$. Then, by (Ev1), $P_1(A | F) = P_1(A)$. So $P_0(A | F\ \&\ C^A_x) = P_0(A | C^A_x)$. And, by the Principal Principle, $P_0(A | C^A_x) = x$. So $P_0(A | F\ \&\ C^A_x) = x$.<br /><br />Finally, from (Ev2) and the Principal Principle, we (III) for $P_0$ with $F$ atomic and $X = C^A_x$. Again, suppose you learn $C^A_x$. So $P_1(-) = P_0(-|C^A_x)$. You thus have strong evidence concerning $A$ and no evidence concerning $F$. Thus, by (Ev2), $P_1(A | A \leftrightarrow F) = P_1(A)$. That is, $P_0(A | C^A_x\ \&\ (A \leftrightarrow F)) = P_0(A | C^A_x)$. And by the Principal Principle, $P_0(A | C^A_x) = x$. So $P_0(A | C^A_x\ \&\ (A \leftrightarrow F)) = x$.<br /><br />Thus, the plausibility of the HLWW argument turns on the plausibility of (Ev1) and (Ev2). Unfortunately, both beg the question concerning the Principle of Indifference. As a result, they cannot be assumed in a justification of that norm. Let's consider each in turn.<br /><br />First, (Ev1). If your evidence does not provide any information about the connection between $B$ and $C$, then this evidence leaves open the possibility that $B$ is positively relevant to $C$; it leaves open the possibility that $B$ is negatively relevant to $C$; and it leaves open the possibility that $B$ is irrelevant to $C$. But (Ev1) demands that we deny the first two possibilities and take $B$ to be irrelevant to $C$. But why? Without further argument, it seems that we would be equally justified in taking $B$ to be positively relevant to $C$ and equally justified in taking $C$ to be negatively relevant to $C$.<br /><br />Second, (Ev2). The idea is this: When I learn that two propositions, $B$ and $C$, are equivalent, there are many ways I might respond. I might retain my prior credence in $B$ and bring my credence in $C$ into line with that. Or I might retain my prior credence in $C$ and bring my credence in $B$ into line with that. Or I might do many other things. (Ev2) says that, if I have strong evidence concerning $B$ and no evidence concerning $C$, then I should opt for the first response and retain my prior credence in $B$ -- which was formed in response to the strong evidence concerning $B$ -- and bring my credence in $C$ into line with that -- since my prior credence in $C$ was, in any case, formed in response to no relevant evidence at all.<br /><br />Now, on the face of it, this seems like a reasonable constraint on our response to evidence. It says, essentially, that credence formed in response to stronger evidence should be more resilient than credence formed in response to weaker evidence. And, as a limiting case, credence formed in response to strong evidence, such as evidence about the chances, should be maximally resilient when compared to credence formed in response to no evidence. (Note that a similar way of thinking might give an alternative motivation for (II), since this is also a principle of resilient credence.)<br /><br />However, unfortunately, (Ev2) threatens to be inconsistent. After all, it is easy to suppose that there are propositions $B$, $C$, and $D$ such that you have strong evidence for $B$, but no evidence concerning $C$ or $D$ or $C\ \&\ D$ or $C\ \&\ \neg D$. But, in that situation, (Ev2) entails:<br /><br /><ul><li>$P(B | B \leftrightarrow C) = P(B)$</li><li>$P(B | B \leftrightarrow (C\ \&\ D)) = P(B)$</li><li>$P(B | B \leftrightarrow (C\ \&\ \neg D)) = P(B)$ </li></ul><br />And unfortunately these are inconsistent constraints on a probability function. To avoid this inconsistency, the defender of (Ev2) must say that, in fact, our lack of evidence concerning $C$, $D$, $C\ \&\ D$ and $C\ \&\ \neg D$ indeed counts as no evidence concerning $C$ and $D$, but does count as evidence concerning $C\ \&\ D$ and $C\ \&\ \neg D$. How might they do that? Well, they might note that, while $C$ and $D$ are each true in half the possible worlds, since they are atomic, $C\ \&\ D$ and $C\ \&\ \neg D$ are true only in a quarter of the possible worlds. And thus a lack of evidence is in fact evidence against them. But of course this line of argument appeals to the Principle of Indifference. Only if you think that every world should receive equal credence will you think that a lack of evidence counts as no evidence for a proposition that is true at half of the possible worlds, but counts as genuine evidence against a proposition that is true at only a quarter of the worlds.<br /><br />Thus, I conclude that the HLWW argument fails. While (Ev1) and (Ev2) may be true, we cannot appeal to them in order to justify the Principle of Indifference, since they can only be defended by appealing to the Principle of Indifference itself.Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com8tag:blogger.com,1999:blog-4987609114415205593.post-70283494038819106682017-01-17T10:52:00.000+00:002017-01-25T17:14:43.250+00:00The Principal Principle does not imply the Principle of IndifferenceRecently, <a href="http://james-hawthorne.oucreate.com/" target="_blank">James Hawthorne</a>, <a href="https://jlandes.wordpress.com/" target="_blank">Jürgen Landes</a>, <a href="https://kent.academia.edu/ChristianWallmann" target="_blank">Christian Wallmann</a>, and <a href="http://blogs.kent.ac.uk/jonw/" target="_blank">Jon Williamson</a> published a <a href="https://bjps.oxfordjournals.org/content/early/2015/07/13/bjps.axv030.abstract" target="_blank">paper</a> in the <i>British Journal of Philosophy of Science</i> in which they claim that the Principal Principle entails the Principle of Indifference -- indeed, the paper is called 'The Principal Principle implies the Principle of Indifference'. In this post, I argue that it does not.<br /><br />All Bayesian epistemologists agree on two claims. The first, which we might call <i>Precise Credences</i>, says that an agent's doxastic state at a given time $t$ in her epistemic life can be represented by a single credence function $P_t$, which assigns to each proposition $A$ about which she has an opinion a precise numerical value $P_t(A)$ that is at least 0 and at most 1. $P_t(A)$ is the agent's credence in $A$ at $t$. It measures how strongly she believes $A$ at $t$, or how confident she is at $t$ that $A$ is true. The second point of agreement, which is typically known as <i>Probabilism</i>, says that an agent's credence function at a given time should be a probability function: that is, for all times $t$, $P_t(\top) = 1$ for any tautology $\top$, $P_t(\bot) = 0$ for any contradiction $\bot$, and $P_t(A \vee B) = P_t(A) + P_t(B) - P_t(AB)$ for any propositions $A$ and $B$.<br /><br />So Precise Credences and Probabilism form the core of Bayesian epistemology. But, beyond these two norms, there is little agreement between its adherents. Bayesian epistemologists disagree along (at least) two dimensions. First, they disagree about the correct norms concerning updating on evidence learned with certainty --- some say they are diachronic norms concerning how an agent should in fact update; others say that there are only synchronic norms concerning how an agent should plan to update; and others think there are no norms concerning updating at all. Second, they disagree about the stringency of the synchronic norms that don't concern updating. Our concern here is with the latter. Some candidates norms of this sort: the Principal Principle, which says how an agent's credences in propositions concerning the objective chances should relate to her credences in other propositions (Lewis 1980); the Reflection Principle, which says how an agent's current credences in propositions concerning her future credences should relate to her current credences in other propositions (van Fraassen 1984, Briggs 2009); and the Principle of Indifference, which says, roughly, that an agent with no evidence should divide her credences equally over all possibilities (Keynes 1921, Carnap 1950, Jaynes 2003, Williamson 2010, Pettigrew 2014). Those we might call <i>Radical Subjective Bayesians</i> adhere to Precise Credences and Probabilism, but reject the Principal Principle, the Reflection Principle, and the Principle of Indifference. Those we might call <i>Moderate Subjective Bayesians</i> adhere to Precise Credences, Probabilism, and the Principal Principle (and also, quite often, the Reflection Principle), but they reject the Principle of Indifference. And the <i>Objective Bayesians</i> accept all of the principles.<br /><br />In a recent paper, Hawthorne et al. (2015) (henceforth, HLWW) argue that Moderate Subjective Bayesianism is an inconsistent position, because the Principal Principle (and, indeed the Reflection Principle) entails the Principle of Indifference. Thus, it is inconsistent to accept the former and reject the latter. We must either reject the Principal Principle, as the Radical Subjective Bayesian does, or accept it together with the Principle of Indifference, as the Objective Bayesian does.<br /><br />Notoriously, as Lewis originally stated it, the Principal Principle includes an <i>admissibility condition</i> (266-7, Lewis 1980). Equally notoriously, Lewis did not provide a precise account of this condition, thereby leaving his formulation of the principle similarly imprecise. HLWW do not give a precise account either. But they do appeal to two principles that they take to follow intuitively from the Principal Principle. And from these two principles, together with the Principal Principle itself, they derive what they take to be an instance of the Principle of Indifference. The first principle to which they appeal --- their Condition 1 --- is in fact provable, as they note. The second --- their Condition 2 --- is not. Indeed, as we will see, on the correct understanding of admissibility, it is false. Thus, the HLWW argument fails. What's more, its conclusion is not true. It is possible to satisfy the Principal Principle without satisfying the Principle of Indifference, as we will see below. Moderate Subjective Bayesianism is a coherent position.<br /><br /><br /><h2>Introducing the Principal Principle</h2><br />We begin by introducing the Principal Principle. To aid our statement, let me introduce a piece of notation. Given a proposition $A$ and a real number $0 \leq x \leq 1$, let $C^A_x$ be the following proposition: <i>The current objective chance of $A$ is $x$</i>. And we will let $P_0$ be the credence function of our agent at the very beginning of her epistemic life --- when she is, as Lewis would say, a <i>superbaby</i>; that is, she is not yet in receipt of any evidence. Then, as Lewis originally formulates the Principal Principle, it says this:<br /><br /><b>Lewis' Principal Principle</b> Suppose $A$, $E$ are propositions and $0 \leq x \leq 1$. Then it should be the case that $$P_0(A | C^A_xE) = x $$providing (i) $P_0(C^A_xE) > 0$, and (ii) $E$ is admissible for $A$.<br /><br />In this version, the principle applies to an agent only at the beginning of her epistemic life; it governs her initial credence function. In this situation, the principle says, her credence in a proposition $A$ conditional on the conjunction of some proposition $E$ and a chance proposition that says that the chance of $A$ is $x$ should be $x$, providing the conditional probability is well-defined and $E$ is admissible for $A$.<br /><br />The motivation for the admissibility condition is this. Suppose $E$ entails $A$. Then we surely don't want to demand that $P_0(A | C^A_xE) = x$. After all, if $x < 1$, then such a demand would conflict with Probabilism, since it is a consequence of Probabilism that, if $E$ entails $A$, then $P_0(A | C^A_xE) = 1$. Thus, we must at least restrict the Principal Principle so that it does not apply when $E$ entails $A$. But there are other cases in which the Principal Principle should not be imposed, even if such an application would not be outright inconsistent with other norms such as Probabilism. For instance, suppose that $E$ entails that the chance of $A$ at some time in the future is $x' \neq x$. Then, again, we don't want to require that $P_0(A | C^A_xE) = x$. The moral is this: if $E$ contains information about $A$ that <i>overrides</i> the information that the current chance of $A$ gives about $A$, then it is inadmissible. Clearly any proposition that logically entails $A$ provides information that overrides the current chance information about $A$; and so does a proposition that entails something about the future chance of $A$. So much for propositions that are inadmissible. Are there any we can be sure are admissible? According to Lewis, there are, namely, propositions solely concerning the past or the present. Thus, Lewis does not give a precise account of admissibility: he gives a heuristic --- $E$ is admissible for $A$ if $E$ does not provide information about $A$ that overrides the information contained in propositions about the current chance of $A$ --- and he gives examples of propositions that do and do not provide such information --- I've recalled some of Lewis' examples here.<br /><br />Now, as Lewis himself noted, the Principal Principle has implausible consequences when the chances are self-undermining --- that is, when the chances assign a positive probability to outcomes in which the chances are different. This happens, for instance, for Lewis' own favoured account of chance, the Humean account or Best System Analysis. This lead to reformulations of the Principal Principle, such as Thau's and Hall's New Principle (Lewis 1994, Thau 1994, Hall 1994) and Ismael's General Recipe (Ismael 2008). HLWW say nothing explicitly about whether or not chances are self-undermining. But, since they are interested in investigating the Principal Principle and not the New Principle or the General Recipe, I take them to assume that chances are not self-undermining. I will do likewise.<br /><br /><h2><b>The HLWW argument</b></h2><br />However imprecise Lewis' account of admissibility is, HLWW take it to be precise enough to allow us to be confident of the following principles:<br /><br /><b>Condition 1 </b> If<br />(1a) $E$ is admissible for $A$, and<br />(1b) $C^A_xE$ contains no information that renders $F$ relevant to $A$,<br />then<br />(1c) $EF$ is admissible for $A$.<br /><br />Now, HLWW propose to make (1b) precise as follows: $$P_0(A | FC^A_xE) = P_0(A | C^A_xE)$$ That is, $C^A_xE$ contains no information that renders $F$ relevant to $A$ just in case $C^A_xE$ renders $A$ probabilistically independent of $F$. With that explication in hand, Condition 1 now actually follows logically from Lewis' Principal Principle, as HLWW note. After all, by (1a) and Lewis' Principal Principle, $P_0(A | C^A_xE) = x$. And, by the explication of (1b), $P_0(A | C^A_xE) = P_0(A | FC^A_xE)$. Daisychaining these identities together, we have $P_0(A | FC^A_xE) = x$, which is (1c).<br /><br /><b>Condition 2</b> If<br />(2a) $E$ is admissible for $A$, and<br />(2b) $C^A_xE$ contains no information that renders $F$ relevant to $A$,<br />then<br />(2c) $E(A \leftrightarrow F)$ is admissible for $A$.<br /><br />This is not provable. Indeed, as we will see below, it is false. Nonetheless, together with Lewis' Principal Principle, Conditions 1 and 2 entail a constraint on an agent's credence function that HLWW take to be the constraint imposed by the Principle of Indifference.<br /><br /><b>Proposition 1 </b>Suppose Lewis' Principal Principle together with Conditions 1 and 2 hold. And suppose that there are propositions $A$, $E$, and $F$ and $0 < x < 1$ such that $E$ is admissible for $A$. Suppose further that $F$ is atomic and contingent. Then<br /><br />(i) If $C^A_xE$ contains no information that renders $F$ relevant to $A$, then the following is required of the agent's initial credence function: $P_0(F | C^A_xE) = 0.5.$<br /><br />(ii) If $C^A_xE$ contains no information whatsoever about $F$ (so that $P_0(F | C^A_xE) = P_0(F)$), then the following is required of the agent's initial credence function: $P_0(F) = 0.5$<br /><br />HLWW take Proposition 1 to show that the Principle of Indifference follows from the Principal Principle. After all, Condition 1 is simply a theorem. And they take Condition 2 to be a consequence of the Principal Principle, given the correct understanding of admissibility. So if you assume the Principal Principle, you get all of the hypotheses of the theorem. However, as we will see in the next two sections, Condition 2 is in fact false.<br /><br /><h2>Levi's Principal Principle and Levi-Admissibility</h2><br />Above, we stated the Principal Principle as follows:<br /><br /><b>Lewis' Principal Principle</b> $P_0(A | C^A_xE) = x$, providing (i) $P_0(C^A_xE) > 0$, and (ii) $E$ is admissible for $A$.<br /><br />Now suppose we make the following assumption about admissibility:<br /><br /><b>Current Chance Admissibility</b> Propositions about the current objective chances are admissible.<br /><br />Thus, for instance, $P_0(A | C^A_xC^B_y) = x$, providing $P_0(C^A_xC^B_y) > 0$, which also ensures that $C^A_x$ and $C^B_y$ are compatible.<br /><br />Now suppose that, if $ch$ is a probability function defined over all the propositions about which the agent has an opinion, $C_{ch}$ is the proposition that says that the objective chances are given by $ch$. Then it follows from the Principal Principle and Current Chance Admissibility that $P_0(A | C_{ch}) = ch(A)$. But it also follows from this that:<br /><br /><b>Levi's Principal Principle</b> (Bodgan 1984, Pettigrew 2012) $P_0(A | C_{ch}E) = ch(A | E)$, providing $P_0(C_{ch}E), ch(E) > 0$.<br /><br />This is a version of the Principal Principle that makes no mention of admissibility. From it, something close to Lewis' Principal Principle follows: If $P_0(C^{A|E}_x E) > 0$, then $$P_0(A | C^{A|E}_x E) = x$$ where $C^{A|E}_x$ is the proposition: <i>The current objective chance of $A$ conditional on $E$ is $x$</i>. What's more, while Levi's version does not mention admissibility, since it applies equally when the proposition $E$ is not admissible, it does suggest a precise account of admissibility. And it is possible to show that, if we take the version of Lewis' Principal Principle that results from understanding admissibility in this way, it is a consequence of Levi's Principal Principle.<br /><br /><b>Levi-Admissibility</b> <i>$A$ is Levi-admissible for $E$</i> if, for all possible chance functions $ch$, $ch(A | E) = ch(A)$. <br /><br />That is, on this account $A$ is admissible for $E$ if every chance function renders $A$ and $E$ stochastically independent. Three points are worthy of note:<br /><ol><li>All propositions providing future information about the chance of $A$ or information about the truth value of $A$ are Levi-inadmissible, since $A$ will be stochastically dependent on such propositions according to all possible current chance functions. So this account of admissibility agrees with the examples of clearly inadmissible propositions that we gave above.</li><li>All propositions solely about the past are Levi-admissible, since all such propositions will now be true or false and will be assigned chance 1 or 0 accordingly by all possible current chance functions. So this account of admissibility agrees with the examples of clearly admissible propositions that we gave above.</li><li>If $A$ is Levi-admissible for $E$, then $P_0(A | C^A_xE) = P_0(A | C^{A|E}_xE ) = x$. That is, Lewis' Principal Principle follows from Levi's version if we understand Lewis' notion of admissibility as Levi-admissibility.</li></ol>Taken together, (1), (2), and (3) entail that Levi-admissibility has all of the features that Lewis wished admissibility to have.<br /><br />Now, although Levi's account of admissibility recovers Lewis' examples, it might seem to be too demanding. Suppose, for instance, that $A$ is a proposition concerning the toss of a coin in Quito --- it says that it will lands heads --- while $E$ is a proposition concerning tomorrow's weather in Addis Ababa --- it says that it will rain. Then, intuitively, $E$ is admissible for $A$. But $E$ is not Levi-admissible for $A$. After all, we are considering an agent at the beginning of her epistemic life. And so there are certainly possible chance functions --- probability functions that, for all she knows, give the objective chances --- that do not render $E$ and $A$ stochastically independent.<br /><br />However, in fact, on closer inspection, the Levi-admissibility verdict is exactly right. Consider my credence in $A$ conditional on $E$ and the chance hypothesis $C^A_{0.5}$, which says that the coin in Quito is fair and so the unconditional chance of $A$ is 0.5. Amongst the chance functions that are epistemically possible for me, some make $E$ irrelevant to $A$, some make it positively relevant to $A$ and some make it negatively relevant to $A$. Indeed, we might suppose that the possible chances of $A$ conditional on $E$ run the full gamut of values from 0 to 1. In that case, surely we don't want to say that $E$ is admissible for $A$ and thereby impose, via the Principal Principle, the demand that our agent's credence in $A$ conditional on $E$ and $C^A_{0.5}$ is 0.5. After all, if I choose to place most of my prior credence on the chance hypotheses on which $E$ is positively relevant to $A$, then my credence in $A$ conditional on $E$ and $C^A_{0.5}$ should not be 0.5 --- it should be something greater than 0.5. If I choose to place most of my prior credence on the chance hypotheses on which $E$ is negatively relevant to $A$, then my credence in $A$ conditional on $E$ and $C^A_{0.5}$ should not be 0.5 --- it should be something less than 0.5. Of course, we might think that it is irrational for our agent, a superbaby with no evidence one way or the other, to favour the positive relevance hypotheses over those that posit neutral relevance and negative relevance. We might think that she should spread her credences equally over all of the possibilities, in which case their effects will cancel out, and her credence in $A$ conditional on $E$ and $C^A_{0.5}$ will indeed be 0.5. But of course to do this is to assume the Principle of Indifference and beg the question.<br /><br /><h2>The failure of Condition 2</h2><br />With this precise account of admissibility in hand, we can now test to see whether or not it vindicates Condition 2 --- recall, HLWW claim that this is a consequence of the Principal Principle. As we saw above, Condition 2 runs as follows:<br /><br /><b>Condition 2</b> If<br />(2a) $E$ is admissible for $A$, and<br />(2b) $C^A_xE$ contains no information that renders $F$ relevant to $A$,<br />then<br />(2c) $E(A \leftrightarrow F)$ is admissible for $A$.<br /><br />Now suppose that Lewis' Principal Principle is true, and assume that admissibility means Levi-admissibility. Then this is equivalent to:<br /><br /><b>Condition 2$^*$</b> If $ch$ is a possible chance function, and<br />(2a$^*$) $ch(A | E) = ch(A)$, and<br />(2b$^*$) $ch(A | FE) = ch(A | E)$,<br />then<br />(2c$^*$) $ch(A | E(A \leftrightarrow F)) = ch(A)$.<br /><br />However, this is false. Indeed, we can show the following:<br /><br /><b>Proposition</b> <b>2</b> For any value $0 \leq y \leq 1$, there is a chance function $ch$ such that (2a$^*$) and (2b$^*$) hold, but $$ch(A | E(A \leftrightarrow F)) = y$$<br /><br />Thus, (2a$^*$) and (2b$^*$) impose no constraints whatsoever on the chance of $A$ conditional on $E(A \leftrightarrow F)$.<br /><br />Thus, it is possible that $E$ is Levi-admissible for $A$ and that $C^A_xE$ carries no information whatsoever about $F$, and yet $E(A \leftrightarrow F)$ is not Levi-admissible for $A$. Thus, Condition 2 is false and the HLWW argument fails.<br /><br /><h2>Levi's Principal Principle and the Principle of Indifference</h2><br />Of course, the failure of an argument does not entail the falsity of its conclusion. It might yet be the case that the Principal Principle entails the Principle of Indifference, even if the HLWW argument does not show that. But in fact we can show that this is not true. To see this, we note a sufficient condition for satisfying Levi's Principal Principle:<br /><br /><b>Proposition 3</b> Suppose $C$ is the set of all possible chance functions. Then, if $P_0$ is in the convex hull of $C$, then $P_0(A | C_{ch} E) = ch(A | E)$.<br /><br />Now, if Levi's Principal Principle entails the Principle of Indifference, and the Principle of Indifference entails that every atomic proposition has probability 0.5, then it follows that every member of the convex hull of the set of possible chance functions must assign probability 0.5 to every atomic proposition. But it is easy to see that this is not true. Let $F$ be the atomic proposition that says that a sample of uranium will decay at some point in the next hour. In the absence of evidence, the possible chances of $F$ range over the full unit interval from 0 to 1. Thus, there are members of the convex hull of the set of possible chance functions that assign probabilities other than 0.5 to $F$. And, by Proposition 3, these members will satisfy Levi's Principal Principle.<br /><br /><h2>Applying Levi's Principal Principle</h2><br />A possible objection: Levi's Principal Principle is all well and good in theory, but it is not applicable. Suppose we are interested in a proposition $A$; and we have collected evidence $E$. How might we apply Levi's Principal Principle in order to set our credence in $A$? In the case of Lewis' version of the principle, we need only know the chance of $A$ and the fact that $E$ is admissible for $A$, and we often know both of these. But, in order to apply Levi's version, we must know the chance of $A$ <i>conditional on our evidence $E$</i>. And, at least for large and varied bodies of evidence, we never know this. Or so the objection goes.<br /><br />But the objection fails. In fact, Levi's Principal Principle may be applied in those cases. You don't have to know the chance of $A$ conditional on $E$ in order to set your credence in $A$ when you have evidence $E$. You simply have to have opinions about the different possible values that that conditional chance might take. You then apply Levi's Principal Principle, together with the Law of Total Probability, which jointly entail that your credence in $A$ given $E$ should be your expectation of the chance of $A$ given $E$. Of course, neither Levi's Principal Principle nor the Law of Total Probability will tell you how to set your credences in the different possible values that the conditional chance of $A$ given $E$ might take. But that's not a problem for the Moderate Subjective Bayesian, who doesn't expect her evidence to pin down a unique credal response. Only the Objective Bayesian would expect that. You pick your probability distribution over those possible conditional chance values and Levi's Principal Principle does the rest via the Law of Total Probability.<br /><br /><h2>Conclusion</h2><br /><br />The HLWW argument purports to show that the Principal Principle entails the Principle of Indifference. But it fails because, on the correct understanding of admissibility, Condition 2 is not a consequence of the Principal Principle; and indeed it is false. What's more, we can see that there are credence functions that satisfy the correct version of the Principal Principle --- namely, Levi's Principal Principle --- that do not satisfy the Principle of Indifference. The logical space is therefore safe once again for Moderate Subjective Bayesians, that is, those who accept Precise Credences, Probabilism, the Principal Principle (and perhaps the Reflection Principle), but who deny the Principle of Indifference.<br /><br /><br /><h2>References</h2><ul><li>Bogdan, R. (Ed.) (1984). <i>Henry E. Kyburg, Jr. and Isaac Levi.</i> Dordrecht: Reidel.</li><li>Briggs, R. (2009). Distorted Reflection. <i>Philosophical Review</i>, 118(1), 59–85.</li><li>Carnap, R. (1950). <i>Logical Foundations of Probability</i>. Chicago: University of Chicago Press.</li><li>Hall, N. (1994). Correcting the Guide to Objective Chance. <i>Mind</i>, 103, 505–518.</li><li>Hawthorne, J., Landes, J., Wallman, C., & Williamson, J. (2015). The Principal Principle Implies the Principle of Indifference. <i>The British Journal for the Philosophy of Science</i>. </li><li>Ismael, J. (2008). Raid! Dissolving the Big, Bad Bug. <i>Noûs</i>, 42(2), 292–307.</li><li>Jaynes, E. T. (2003). <i>Probability Theory: The Logic of Science</i>. Cambridge, UK: Cambridge University Press.</li><li>Keynes, J. M. (1921). <i>A Treatise on Probability</i>. London: Macmillan.</li><li>Lewis, D. (1980). A Subjectivist’s Guide to Objective Chance. In R. C. Jeffrey (Ed.) <i>Studies in Inductive Logic and Probability</i>, vol. II. Berkeley: University of California Press.</li><li>Lewis, D. (1994). Humean Supervenience Debugged. <i>Mind</i>, 103, 473–490.</li><li>Pettigrew, R. (2012). Accuracy, Chance, and the Principal Principle. <i>Philosophical<br />Review,</i> 121(2), 241–275.</li><li>Pettigrew, R. (2014). Accuracy, Risk, and the Principle of Indifference. <i>Philosophy<br />and Phenomenological Research</i>.</li><li>Thau, M. (1994). Undermining and Admissibility. <i>Mind</i>, 103, 491–504.</li><li>van Fraassen, B. C. (1984). Belief and the Will. <i>Journal of Philosophy</i>, 81, 235–56.</li><li>Williamson, J. (2010). <i>In Defence of Objective Bayesianism</i>. Oxford: Oxford University Press. </li></ul>Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-56419652168966516382016-12-20T12:58:00.000+00:002016-12-20T12:58:55.274+00:00Assistant professorship in mathematical philosophy, University of Gdansk<div dir="ltr" style="text-align: left;" trbidi="on"><b style="text-align: justify;"><br class="Apple-interchange-newline" />Assistant Professorship</b><span style="text-align: justify;"> (“adiunkt” in Polish terminology) in the Chair of Logic, Philosophy of Science and Epistemology is available at the Department of Philosophy, Sociology and Journalism, University of Gdansk, Poland. The position is to start sometime between July 1 and September 1, 2017, for a fixed period of time with the possibility of extension. Decisions about the exact beginning date of the contract and the number of years will be made during the hiring process. No knowledge of Polish is required.</span><br /><span style="text-align: justify;"><br /></span><span style="text-align: justify;">Details available <a href="http://entiaetnomina.blogspot.in/2016/12/assistant-professorship-in-mathematical.html">here</a>.</span></div>Rafal Urbaniakhttp://www.blogger.com/profile/10277466578023939272noreply@blogger.com2tag:blogger.com,1999:blog-4987609114415205593.post-41751717144951664482016-12-18T09:07:00.001+00:002016-12-18T09:07:38.944+00:00Call for submissions: PhDs in Logic IX, Bochum, 2nd - 4th May 2017<div dir="ltr" style="text-align: left;" trbidi="on"><div class="p1"><span class="s1">PhDs in Logic is an annual graduate conference organised by local graduate students. This interdisciplinary conference welcomes contributions to various topics in mathematical logic, philosophical logic, and logic in computer science. It involves tutorials by established researchers as well as short (20 minutes) presentations by PhD students, master students and first-year postdocs on their research.</span></div><div class="p1"><span class="s1">We are happy to announce that the ninth edition of PhDs in Logic will take place at the Ruhr University Bochum, Germany, during 2nd - 4th May 2017.</span></div><div class="p2"><span class="s1"></span><br /></div><div class="p1"><span class="s1">Confirmed tutorial speakers are :</span></div><div class="p1"><span class="s1">Petr Cintula (Czech Academy of Sciences)</span></div><div class="p1"><span class="s1">María Manzano (University of Salamanca)</span></div><div class="p1"><span class="s1">João Marcos (University of Natal)</span></div><div class="p1"><span class="s1">Gabriella Pigozzi (Paris Dauphine University)</span></div><div class="p1"><span class="s1">Christian Straßer (Ruhr-University Bochum)</span></div><div class="p1"><span class="s1">Heinrich Wansing (Ruhr-University Bochum)</span></div><div class="p2"><span class="s1"></span><br /></div><div class="p1"><span class="s1">Abstract submission:</span></div><div class="p1"><span class="s1">PhD students, master students and first-year postdocs in logic from disciplines, that include but are not limited to philosophy, mathematics and computer science are invited to submit an extended abstract on their research. Submitted abstracts should be between 2 and 3 pages, including the relevant references. Each abstract will be anonymously reviewed by the scientific committee. Accepted abstracts will be presented by their authors in a 20-minute presentation during the conference. The deadline for abstract submission is 2nd February 2017. Please submit your blinded abstract via: <a href="https://easychair.org/conferences/?conf=phdsinlogic9"><span class="s2">https://easychair.org/conferences/?conf=phdsinlogic9</span></a></span></div><div class="p2"><span class="s1"></span><br /></div><div class="p1"><span class="s1">For more information please see:</span></div><div class="p3"><span class="s3"><a href="http://www.ruhr-uni-bochum.de/phdsinlogicix">http://www.ruhr-uni-bochum.de/phdsinlogicix</a></span></div><div class="p2"><span class="s1"></span><br /></div><div class="p1"><span class="s1">Local organisers:</span></div><style type="text/css">p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 13.0px Helvetica; -webkit-text-stroke: #000000} p.p2 {margin: 0.0px 0.0px 0.0px 0.0px; font: 13.0px Helvetica; -webkit-text-stroke: #000000; min-height: 16.0px} p.p3 {margin: 0.0px 0.0px 0.0px 0.0px; font: 13.0px Helvetica; color: #4787ff; -webkit-text-stroke: #4787ff} span.s1 {font-kerning: none} span.s2 {text-decoration: underline ; font-kerning: none; color: #4787ff; -webkit-text-stroke: 0px #4787ff} span.s3 {text-decoration: underline ; font-kerning: none} </style> <br /><div class="p1"><span class="s1">Christopher Badura, AnneMarie Borg, Jesse Heyninck and Daniel Skurt</span></div></div>Rafal Urbaniakhttp://www.blogger.com/profile/10277466578023939272noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-90066853208597734362016-10-27T16:23:00.000+01:002016-10-27T16:23:35.757+01:00Assistant Professorship at the MCMPLudwig-Maximilians-University Munich is seeking applications for one<br /><br /><b>Assistant Professorship position in Logic and Philosophy of Language</b><br />(for three years, with the possibility of extension)<br /><br />at the Chair of Logic and Philosophy of Language (Professor Hannes Leitgeb) and the Munich Center for Mathematical Philosophy (MCMP) at the Faculty of Philosophy, Philosophy of Science, and Study of Religion. The position, which is to start on April 1st 2017, is for three years with the possibility of extension.<br /><br />The appointee will be expected (i) to do philosophical research, especially in logic and philosophy of language, (ii) to teach five hours a week in areas relevant to the chair, and (iii) to participate in the administrative work of the MCMP.<br /><br />The successful candidate will have a PhD in philosophy or logic, will have teaching experience in philosophy and logic, and will have carried out research in logic and related areas (such as philosophy of logic, philosophy of language, philosophy of mathematics, formal epistemology).<br /><br />Women are currently underrepresented in the Faculty, therefore we particularly welcome applications for this post from suitably qualified female candidates. Furthermore, given equal qualification, severely physically challenged individuals will be preferred.<br /><br />Applications (including CV, certificates, list of publications), a description of planned research projects (1000-1500 words), and letters of reference of two referees should be sent either by email (ideally all requested documents in just one PDF document) or by mail to<br /><br />Ludwig-Maximilians-Universität München<br />Faculty of Philosophy, Philosophy of Science and Study of Religion<br />Chair of Logic and Philosophy of Language / MCMP<br />Geschwister-Scholl-Platz 1<br />80539 München<br />E-Mail: <a href="mailto:office.leitgeb@lrz.uni-muenchen.de" target="_blank">office.leitgeb@lrz.uni-muenchen.de</a><br /><br />by<br /><br /><b>December 1st, 2016</b>.<br /><br />If possible at all, we very much prefer applications by email.<br /><br />Contact for informal inquiries: office.leitgeb@lrz.uni-muenchen.de<br /><br />More information about the MCMP can be found at <a href="http://www.mcmp.philosophie.uni-muenchen.de/index.html" target="_blank">http://www.mcmp.philosophie.uni-muenchen.de/index.html</a>.<br /><br />The German description of the position is to be found at <a href="http://www.uni-muenchen.de/aktuelles/stellenangebote/wissenschaft/20161017140416.html" target="_blank">http://www.uni-muenchen.de/aktuelles/stellenangebote/wissenschaft/20161017140416.html</a>. <br /><br />*****<br /><br />Vincenzo Crupihttp://www.blogger.com/profile/08069145846190162517noreply@blogger.com3tag:blogger.com,1999:blog-4987609114415205593.post-7736434362312027342016-10-12T09:04:00.001+01:002016-10-12T09:04:34.787+01:00Entia et Nomina 2017 CFP<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">The “Entia et Nomina” series features English language workshops for researchers in formally oriented philosophy, in particular in logic, philosophy of science, formal epistemology and philosophy of language. The aim of the workshop is to foster cooperation among philosophers with a formal bent. Previous editions took place at Gdansk University, Ghent University (as part of the Trends in Logic series), Jagiellonian University, and Warsaw University. The sixth conference in the series will take place in Palolem, Goa, India, on 29 January - 5 February 2017. Invited speakers confirmed so far include:</div><br />Krzysztof Posłajko (Jagiellonian University)<br />Katarzyna Kijania-Placek (Jagiellonian University)<br />Tomasz Placek (Jagiellonian University)<br />Nina Gierasimczuk (Danish Technical University)<br />Cezary Cieślinski (Warsaw University)<br />Marcello Dibello (City University of New York)<br /><br /><br /><div style="text-align: justify;">Authors of contributed papers are requested to submit short (up to 2 normalized pages) and extended (up to 6 pages) abstracts, prepared for blind-review, in PDF format, by 30.10.2016. Decisions about acceptance will be communicated by 20.11.2016.</div><br /><div style="text-align: justify;">Authors of accepted papers will have 40 minutes to present their work. Each paper will be followed by a 10 minute commentary prepared beforehand by another participant. Accepted participants might also be asked to comment on at least one talk. Commentaries will be followed by 10-15 minutes of discussion. Applications can be made also for the role of commentator only, in which case only a short CV is requested. We aim to make the short versions of accepted papers available to the participants ahead of the conference.</div><br />Please send your abstracts, questions and any inquiries to both Rafal Urbaniak <rfl.urbaniak@gmail.com> and Juliusz Doboszewski <jdoboszewski@gmail.com>.<br /><div><br /></div></div>Rafal Urbaniakhttp://www.blogger.com/profile/10277466578023939272noreply@blogger.com2tag:blogger.com,1999:blog-4987609114415205593.post-72966249631765313042016-10-04T22:25:00.001+01:002016-10-04T22:52:45.094+01:00CFA: The Fifth Reasoning Club Conference<span class="gmail_msg" style="background-color: white; color: #212121; font-family: "garamond" , serif; font-size: 13px;"><b class="gmail_msg">Call for Abstracts:<span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-Apple-converted-space gmail_msg"> </span><b class="gmail_msg"><span class="gmail_msg" style="color: black;">The Fifth<span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-Apple-converted-space gmail_msg"> </span>Reasoning<span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-Apple-converted-space gmail_msg"> </span>Club<span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-Apple-converted-space gmail_msg"> </span>Conferenc<wbr></wbr>e</span></b><br class="gmail_msg" /><span class="gmail_msg" style="font-weight: normal;">University of<span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-Apple-converted-space gmail_msg"> </span></span></b><span class="gmail_msg">Torino</span><b class="gmail_msg"><span class="gmail_msg" style="font-weight: normal;">,<span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-Apple-converted-space gmail_msg"> </span></span><b class="gmail_msg"><span class="gmail_msg" style="font-weight: normal;">18-19 May 2017</span></b></b></span><br /><div class="gmail_msg" style="color: #212121; font-family: "helvetica neue", helvetica, arial, sans-serif; font-size: 13px;"><span class="gmail_msg" style="background-color: white; font-family: "garamond" , serif;">Keynote speakers: </span></div><div class="gmail_msg" style="color: #212121; font-family: "helvetica neue", helvetica, arial, sans-serif; font-size: 13px;"><span class="gmail_msg" style="background-color: white; font-family: "garamond" , serif;"><a class="gmail_msg" data-saferedirecturl="https://www.google.com/url?q=http://fitelson.org/&source=gmail&ust=1475702617593000&usg=AFQjCNE49tclPR6DX2P6toA1ZBrDnlsaaA" href="http://fitelson.org/" style="border: 0px; color: #2c67d1; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; position: relative; text-decoration: none; vertical-align: baseline; z-index: 0;" target="_blank">Branden FITELSON</a><span class="m_3414633616540520217inbox-inbox-Apple-converted-space gmail_msg" style="color: #1c2024; font-size: 14px;"> </span><span class="gmail_msg" style="color: #1c2024; font-size: 14px;">(Northeastern University, Boston)</span></span><br /><div class="gmail_msg" style="border: 0px; color: #1c2024; font-size: 14px; font-stretch: inherit; font-variant-numeric: inherit; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><span class="gmail_msg" style="background-color: white; font-family: "garamond" , serif;"><a class="gmail_msg" data-saferedirecturl="https://www.google.com/url?q=http://www.rug.nl/staff/jeanne.peijnenburg/research&source=gmail&ust=1475702617593000&usg=AFQjCNEaPkK9wQTKl-Y38856_i-E5VzICw" href="http://www.rug.nl/staff/jeanne.peijnenburg/research" style="border: 0px; color: #2c67d1; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; position: relative; text-decoration: none; vertical-align: baseline; z-index: 0;" target="_blank">Jeanne PEIJNENBURG</a><span class="m_3414633616540520217inbox-inbox-Apple-converted-space gmail_msg"> </span>(University of Groningen)</span></div><div class="gmail_msg" style="border: 0px; color: #1c2024; font-size: 14px; font-stretch: inherit; font-variant-numeric: inherit; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><span class="gmail_msg" style="background-color: white; font-family: "garamond" , serif;"><a class="gmail_msg" data-saferedirecturl="https://www.google.com/url?q=https://www5.unitn.it/People/en/Web/Persona/PER0003393%23INFO&source=gmail&ust=1475702617593000&usg=AFQjCNFgyVHBcE4u-QXJOTw9EXmR5gxatQ" href="https://www5.unitn.it/People/en/Web/Persona/PER0003393#INFO" style="border: 0px; color: #2c67d1; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; position: relative; text-decoration: none; vertical-align: baseline; z-index: 0;" target="_blank">Katya TENTORI</a><span class="m_3414633616540520217inbox-inbox-Apple-converted-space gmail_msg"> </span>(University of Trento)</span></div><div class="gmail_msg" style="border: 0px; color: #1c2024; font-size: 14px; font-stretch: inherit; font-variant-numeric: inherit; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><span class="gmail_msg" style="background-color: white; font-family: "garamond" , serif;"><a class="gmail_msg" data-saferedirecturl="https://www.google.com/url?q=http://paulegre.free.fr/&source=gmail&ust=1475702617593000&usg=AFQjCNFjcHfMvpewLIFitoAIA4KVpTOqBg" href="http://paulegre.free.fr/" style="border: 0px; color: #2c67d1; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; position: relative; text-decoration: none; vertical-align: baseline; z-index: 0;" target="_blank">Paul EGRÉ</a><span class="m_3414633616540520217inbox-inbox-Apple-converted-space gmail_msg"> </span>(Institut Jean Nicod, Paris)</span></div><span class="gmail_msg" style="background-color: white; font-family: "garamond" , serif;">Please visit<span class="m_3414633616540520217inbox-inbox-Apple-converted-space gmail_msg"> </span><a class="gmail_msg" data-saferedirecturl="https://www.google.com/url?q=http://www.llc.unito.it/notizie/fifth-reasoning-club-meeting-llc-2017&source=gmail&ust=1475702617593000&usg=AFQjCNEhaR6mWnFRUM4O8OoGj5ScRuw9XA" href="http://www.llc.unito.it/notizie/fifth-reasoning-club-meeting-llc-2017" style="color: #7e57c2; position: relative; z-index: 0;" target="_blank">http://www.llc.unito.it/<wbr></wbr>notizie/fifth-reasoning-club-<wbr></wbr>meeting-llc-2017</a><span class="m_3414633616540520217inbox-inbox-Apple-converted-space gmail_msg"> </span>for further information.<br class="gmail_msg" /><br class="gmail_msg" />Submissions for the<span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-Apple-converted-space gmail_msg"> </span>Fifth Reasoning Club Conference<span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-Apple-converted-space gmail_msg"> </span>are now open.<span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-Apple-converted-space gmail_msg"> </span>All PhD candidates and early career researchers with interests in reasoning and inference, broadly construed, are encouraged to submit an abstract of up to 500 words (prepared for blind review) via Easy Chair at<span class="m_3414633616540520217inbox-inbox-Apple-converted-space gmail_msg"> </span><a class="gmail_msg" data-saferedirecturl="https://www.google.com/url?q=https://easychair.org/conferences/?conf%3Drcc17&source=gmail&ust=1475702617593000&usg=AFQjCNFXCFfJRWrHU-qka0X6NSZ1HkkjUQ" href="https://easychair.org/conferences/?conf=rcc17" style="color: #7e57c2; position: relative; z-index: 0;" target="_blank">https://easychair.org/<wbr></wbr>conferences/?conf=rcc17</a></span></div><div class="gmail_msg" style="color: #212121; font-family: "helvetica neue", helvetica, arial, sans-serif; font-size: 13px;"><span class="gmail_msg" style="background-color: white; font-family: "garamond" , serif;"><span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-Apple-converted-space gmail_msg"> </span><br class="gmail_msg" />We especially welcome members of groups that are underrepresented in philosophy to submit. We are committed to promoting diversity in our final programme.</span></div><div class="gmail_msg" style="color: #212121; font-family: "helvetica neue", helvetica, arial, sans-serif; font-size: 13px;"><span class="gmail_msg" style="background-color: white; font-family: "garamond" , serif;"><br /></span><span class="gmail_msg" style="background-color: white; font-family: "garamond" , serif;">The deadline for submissions is<span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-Apple-converted-space gmail_msg"> </span><b class="gmail_msg">1 February 2017</b>. The final decision on submissions will be made by 15 March 2017.<span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-Apple-converted-space gmail_msg"> </span><br class="gmail_msg" /><br class="gmail_msg" />Grants will be available to help cover travel costs for contributed speakers. To apply for a travel grant, please send a CV and a short travel budget estimate in a single pdf file by 1 February 2017 to<span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-Apple-converted-space gmail_msg"> </span><a class="gmail_msg" href="mailto:reasoningclubconference2017@gmail.com" style="color: #7e57c2; position: relative; z-index: 0;" target="_blank">reasoningclubconference2017<wbr></wbr>@gmail.com</a>.</span></div><div class="gmail_msg" style="color: #212121; font-family: "helvetica neue", helvetica, arial, sans-serif; font-size: 13px;"><span class="gmail_msg" style="background-color: white; font-family: "garamond" , serif;"><br /></span><span class="gmail_msg" style="background-color: white; font-family: "garamond" , serif;">For any queries please contact<span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-Apple-converted-space gmail_msg"> </span><a class="gmail_msg" href="mailto:vcrupi@unito.it" style="color: #7e57c2; position: relative; z-index: 0;" target="_blank"><span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-lG gmail_msg" style="outline: transparent dashed 1px;"><span class="m_3414633616540520217inbox-inbox-lG gmail_msg" style="outline: transparent dashed 1px;">Vincenzo</span></span><span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-Apple-converted-space gmail_msg"> </span><span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-lG gmail_msg" style="outline: transparent dashed 1px;"><span class="m_3414633616540520217inbox-inbox-lG gmail_msg" style="outline: transparent dashed 1px;">Crupi</span></span></a><span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-Apple-converted-space gmail_msg"> </span>or<span class="m_3414633616540520217inbox-inbox-m_6107395369616152815inbox-inbox-Apple-converted-space gmail_msg"> </span><a class="gmail_msg" href="mailto:jpkonek@gmail.com" style="color: #7e57c2; position: relative; z-index: 0;" target="_blank">Jaso<wbr></wbr>n Konek</a>.</span></div><div class="gmail_msg" style="color: #212121; font-family: "helvetica neue", helvetica, arial, sans-serif; font-size: 13px;"><span class="gmail_msg" style="background-color: white; font-family: "garamond" , serif;"><br class="gmail_msg" /></span></div><div class="gmail_msg" style="color: #212121; font-family: "helvetica neue", helvetica, arial, sans-serif; font-size: 13px;"><div class="gmail_msg" style="border: 0px; color: #1c2024; font-size: 14px; font-stretch: inherit; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><span class="gmail_msg" style="background-color: white; font-family: "garamond" , serif;">The<span class="m_3414633616540520217inbox-inbox-Apple-converted-space gmail_msg"> </span><a class="gmail_msg" data-saferedirecturl="https://www.google.com/url?q=https://www.kent.ac.uk/secl/researchcentres/reasoning/club/index.html&source=gmail&ust=1475702617593000&usg=AFQjCNE03BNARO81UcWvbvwX6utNeNxKeg" href="https://www.kent.ac.uk/secl/researchcentres/reasoning/club/index.html" style="border: 0px; color: #2c67d1; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; position: relative; text-decoration: none; vertical-align: baseline; z-index: 0;" target="_blank">Reasoning Club</a><span class="m_3414633616540520217inbox-inbox-Apple-converted-space gmail_msg"> </span>is a network of institutes, centres, departments, and groups addressing research topics connected to reasoning, inference, and methodology broadly construed. It issues the monthly gazette<span class="m_3414633616540520217inbox-inbox-Apple-converted-space gmail_msg"> </span><a class="gmail_msg" data-saferedirecturl="https://www.google.com/url?q=http://blogs.kent.ac.uk/thereasoner/about/&source=gmail&ust=1475702617593000&usg=AFQjCNEe8S-E3RtygHYOKsqSYpZCGJs4Sg" href="http://blogs.kent.ac.uk/thereasoner/about/" style="border: 0px; color: #2c67d1; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; position: relative; text-decoration: none; vertical-align: baseline; z-index: 0;" target="_blank">The Reasoner</a>.</span></div><div class="gmail_msg" style="border: 0px; color: #1c2024; font-size: 14px; font-stretch: inherit; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background-color: white;"><br /></span></div><div class="gmail_msg" style="border: 0px; color: #1c2024; font-size: 14px; font-stretch: inherit; line-height: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><span class="gmail_msg" style="background-color: white; font-family: "garamond" , serif;">Earlier editions of the meeting were held in<span class="m_3414633616540520217inbox-inbox-Apple-converted-space gmail_msg"> </span><a class="gmail_msg" data-saferedirecturl="https://www.google.com/url?q=http://www.vub.ac.be/CLWF/RC2012/&source=gmail&ust=1475702617593000&usg=AFQjCNGlXCWL4mxqpkTvgZvR9pYj2vOjSA" href="http://www.vub.ac.be/CLWF/RC2012/" style="border: 0px; color: #2c67d1; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; position: relative; text-decoration: none; vertical-align: baseline; z-index: 0;" target="_blank">Brussels</a>,<span class="m_3414633616540520217inbox-inbox-Apple-converted-space gmail_msg"> </span><a class="gmail_msg" data-saferedirecturl="https://www.google.com/url?q=http://reasoningclubpisa.weebly.com/&source=gmail&ust=1475702617593000&usg=AFQjCNHEPqPbksj_q-JYtKylOepi_I0IjA" href="http://reasoningclubpisa.weebly.com/" style="border: 0px; color: #2c67d1; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; position: relative; text-decoration: none; vertical-align: baseline; z-index: 0;" target="_blank">Pisa</a>,<span class="m_3414633616540520217inbox-inbox-Apple-converted-space gmail_msg"> </span><a class="gmail_msg" data-saferedirecturl="https://www.google.com/url?q=https://reasoningclubkent.wordpress.com/&source=gmail&ust=1475702617594000&usg=AFQjCNFfz-I220zZzEnHzNWF6VYpInl2mw" href="https://reasoningclubkent.wordpress.com/" style="border: 0px; color: #2c67d1; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; position: relative; text-decoration: none; vertical-align: baseline; z-index: 0;" target="_blank">Kent</a>, and<span class="m_3414633616540520217inbox-inbox-Apple-converted-space gmail_msg"> </span><a class="gmail_msg" data-saferedirecturl="https://www.google.com/url?q=http://www.maths.manchester.ac.uk/news-and-events/events/fourth-reasoning-club-conf/&source=gmail&ust=1475702617594000&usg=AFQjCNFQ6h2JKlmcZZXsaM2CF9besNLimA" href="http://www.maths.manchester.ac.uk/news-and-events/events/fourth-reasoning-club-conf/" style="border: 0px; color: #2c67d1; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; position: relative; text-decoration: none; vertical-align: baseline; z-index: 0;" target="_blank">Manchester</a>. </span></div></div>Jason Konekhttp://www.blogger.com/profile/01750769966011528630noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-89760174349528694162016-06-30T13:16:00.002+01:002016-06-30T13:16:20.207+01:00Probabilistic and logical approaches in formal epistemology - Interview in The Reasoner<span style="background-color: white; color: #1d2129; font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 19.32px;">The latest issue of <a href="http://www.thereasoner.org/">The Reasoner</a> has an interview with my colleagues Jan-Willem Romeijn and </span><a class="profileLink" data-hovercard="/ajax/hovercard/user.php?id=100006499369008" href="https://www.facebook.com/barteld.kooi.7" style="background-color: white; color: #365899; cursor: pointer; font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 19.32px; text-decoration: none;">Barteld Kooi</a><span style="background-color: white; color: #1d2129; font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 19.32px;"> by </span><a class="profileLink" data-hovercard="/ajax/hovercard/user.php?id=100009896624697" href="https://www.facebook.com/profile.php?id=100009896624697" style="background-color: white; color: #365899; cursor: pointer; font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 19.32px; text-decoration: none;">Rohan French</a><span style="background-color: white; color: #1d2129; font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 19.32px;"> and myself. The topic is probabilistic and logical approaches in formal epistemology. <a href="http://www.thereasoner.org/">Go check it out!</a></span>Catarinahttp://www.blogger.com/profile/03277956118114314573noreply@blogger.com1tag:blogger.com,1999:blog-4987609114415205593.post-49007124114788587392016-05-17T12:54:00.001+01:002016-05-17T12:54:05.410+01:00CFA: Foundations of Mathematical Structuralism<u>CFA: Foundations of Mathematical Structuralism</u><br /><br />12-14 October 2016, Munich Center for Mathematical Philosophy, LMU Munich<br /><br />In the course of the last century, different general frameworks for the foundations of mathematics have been investigated. The orthodox approach to foundations interprets mathematics in the universe of sets. More recently, however, there have been other developments that call into question the whole method of set theory as a foundational discipline. Category-theoretic methods that focus on structural relationships and structure-preserving mappings between mathematical objects, rather than on the objects themselves, have been in play since the early 1960s. But in the last few years they have found clarification and expression through the development of homotopy type theory. This represents a fascinating development in the philosophy of mathematics, where category-theoretic structural methods are combined with type theory to produce a foundation that accounts for the structural aspects of mathematical practice. We are now at a point where the notion of mathematical structure can be elucidated more clearly and its role in the foundations of mathematics can be explored more fruitfully.<br /><br />The main objective of the conference is to reevaluate the different perspectives on mathematical structuralism in the foundations of mathematics and in mathematical practice. To do this, the conference will explore the following research questions: Does mathematical structuralism offer a philosophically viable foundation for modern mathematics? What role do key notions such as structural abstraction, invariance, dependence, or structural identity play in the different theories of structuralism? To what degree does mathematical structuralism as a philosophical position describe actual mathematical practice? Does category theory or homotopy type theory provide a fully structural account for mathematics?<br /><br /><u>Confirmed Speakers:</u><br /><br /><a href="https://www.andrew.cmu.edu/user/awodey/">Prof. Steve Awodey</a> (Carnegie Mellon University)<br /><a href="http://findresearcher.sdu.dk:8080/portal/da/person/jessica">Dr. Jessica Carter</a> (University of Southern Denmark)<br /><a href="http://poincare.univ-lorraine.fr/fr/membre-titulaire/gerhard-heinzmann">Prof. Gerhard Heinzmann</a> (Université de Lorraine)<br /><a href="http://philosophy.umn.edu/people/FacultyProfile.php?UID=hellm001">Prof. Geoffrey Hellman</a> (University of Minnesota)<br /><a href="http://www.bristol.ac.uk/school-of-arts/people/james-a-ladyman/">Prof. James Ladyman</a> (University of Bristol)<br /><a href="http://philosophy.ucdavis.edu/people/emlandry">Prof. Elaine Landry</a> (UC Davis)<br /><a href="http://www.mcmp.philosophie.uni-muenchen.de/people/faculty/hannes_leitgeb/index.html">Prof. Hannes Leitgeb</a> (LMU Munich)<br /><a href="https://www.york.ac.uk/natural-sciences/staff-profiles/mary-leng/">Dr. Mary Leng</a> (University of York)<br /><a href="http://www.hf.uio.no/ifikk/english/people/aca/oysteinl/">Prof. Øystein Linnebo</a> (University of Oslo)<br /><a href="http://www.faculty.ucr.edu/~reck/">Prof. Erich Reck</a> (UC Riverside)<br /><br /><u>Call for Abstracts:</u><br /><br />We invite the submission of abstracts on topics related to mathematical structuralism for presentation at the conference. Abstracts should include a title, a brief abstract (up to 100 words), and a full abstract (up to 1000 words), blinded for peer review. Authors should send their abstracts (in pdf format), together with their name, institutional affiliation and current position to<a href="mailto:mathematicalstructuralism2016@lrz.uni-muenchen.de">mathematicalstructuralism2016@lrz.uni-muenchen.de</a>. We will select up to five submissions for presentation at the conference. The conference language is English.<br /><br /><u>Dates and Deadlines:</u><br /><br />Submission deadline: 30 June, 2016<br />Notification of acceptance: 31 July, 2016<br />Registration deadline: 1 October, 2016<br />Conference: 12 - 14 October, 2016<br /><br />For further details on the conference, please visit: <a href="http://www.mathematicalstructuralism2016.philosophie.uni-muenchen.de/index.html">http://www.mathematicalstructuralism2016.philosophie.uni-muenchen.de/index.html</a>Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com3tag:blogger.com,1999:blog-4987609114415205593.post-75147984398753588822016-04-24T11:36:00.000+01:002016-04-24T11:36:42.367+01:00CFA: workshop on argument strength<div dir="ltr" style="text-align: left;" trbidi="on">When: <span class="aBn" data-term="goog_174262163" tabindex="0"><span class="aQJ">December 1-2, 2016</span></span><br /><div style="font-size: 12.8px; text-align: justify;">Where: Institute of Philosophy II, Ruhr-University Bochum</div><div style="font-size: 12.8px; text-align: justify;"><br /></div><div style="font-size: 12.8px; text-align: justify;"><<a href="http://homepages.ruhr-uni-bochum.de/defeasible-reasoning/Argument-Strength-2016.html" target="_blank">http://homepages.ruhr-uni-<wbr></wbr>bochum.de/defeasible-<wbr></wbr>reasoning/Argument-Strength-<wbr></wbr>2016.html</a>></div><div style="font-size: 12.8px; text-align: justify;"><br /></div><div style="font-size: 12.8px; text-align: justify;">Description:</div><div style="font-size: 12.8px; text-align: justify;">Arguments vary in strength. The strength of an argument is affected by e.g. the plausibility of its premises, the nature of the link between its premises and conclusion, and the prior acceptability of the conclusion.</div><div style="font-size: 12.8px; text-align: justify;"><br /></div><div style="font-size: 12.8px; text-align: justify;">The aim of this workshop is to bring together experts from the fields of artificial intelligence, philosophy, logic, and argumentation theory to discuss questions related to the strength of arguments. Such questions include:</div><div style="font-size: 12.8px; text-align: justify;"><br /></div><div style="font-size: 12.8px; text-align: justify;">-Which factors influence the strength of an argument?</div><div style="font-size: 12.8px; text-align: justify;">-What are the pros and cons of different formal representations of argument strength?</div><div style="font-size: 12.8px; text-align: justify;">-How to formally model qualifiers on the conclusions of arguments?</div><div style="font-size: 12.8px; text-align: justify;">-How does argument strength propagate when inferences are chained?</div><div style="font-size: 12.8px; text-align: justify;">-How do arguments accrue?</div><div style="font-size: 12.8px; text-align: justify;">-Can weaker arguments defeat and/or defend stronger arguments?</div><div style="font-size: 12.8px; text-align: justify;">-When do more specific arguments defeat more general arguments and vice versa?</div><div style="font-size: 12.8px; text-align: justify;">-How do formal and informal approaches to argument strength relate?</div><div style="font-size: 12.8px; text-align: justify;">-How do preferences assigned to premises influence the evaluation of arguments?</div><div style="font-size: 12.8px; text-align: justify;"><br /></div><div style="font-size: 12.8px; text-align: justify;">Keynote speakers:</div><div style="font-size: 12.8px; text-align: justify;">Gerhard Brewka (University of Leipzig)</div><div style="font-size: 12.8px; text-align: justify;">Gabriele Kern-Isberner (TU Dortmund)</div><div style="font-size: 12.8px; text-align: justify;">Beishui Liao (Zhejiang University)</div><div style="font-size: 12.8px; text-align: justify;">Henry Prakken (Utrecht University)</div><div style="text-align: justify;"><span style="font-size: 12.8px;"></span><br /><div><span style="font-size: 12.8px;">Leon Van Der Torre (University of Luxembourg)</span></div><span style="font-size: 12.8px;"></span></div><div style="text-align: justify;"><br /><span style="font-size: 12.8px;"></span></div><div style="font-size: 12.8px; text-align: justify;">Abstract submission:</div><div style="font-size: 12.8px; text-align: justify;">Authors are invited to submit an abstract (500-1000 words) related to the above or any other questions on the topic of argument strength to <a href="mailto:argumentstrength2016@gmail.com" target="_blank">argumentstrength2016@gmail.<wbr></wbr>com</a> by <span class="aBn" data-term="goog_174262164" tabindex="0"><span class="aQJ">August 1, 2016</span></span>.</div><div style="font-size: 12.8px; text-align: justify;"><br /></div><div style="font-size: 12.8px; text-align: justify;">Important dates:</div><div style="font-size: 12.8px; text-align: justify;">submission deadline: <span class="aBn" data-term="goog_174262165" tabindex="0"><span class="aQJ">August 1, 2016</span></span></div><div style="font-size: 12.8px; text-align: justify;">notifications: <span class="aBn" data-term="goog_174262166" tabindex="0"><span class="aQJ">September 1, 2016</span></span></div><div style="font-size: 12.8px; text-align: justify;">workshop: <span class="aBn" data-term="goog_174262167" tabindex="0"><span class="aQJ">December 1-2, 2016</span></span></div><div style="font-size: 12.8px; text-align: justify;"><br /></div><div style="font-size: 12.8px; text-align: justify;">Organizing committee:</div><div style="font-size: 12.8px; text-align: justify;">Mathieu Beirlaen</div><div style="font-size: 12.8px; text-align: justify;">AnneMarie Borg</div><div style="font-size: 12.8px; text-align: justify;">Jesse Heyninck</div><div style="font-size: 12.8px; text-align: justify;">Dunja Šešelja</div><div style="font-size: 12.8px; text-align: justify;">Christian Straßer</div><div style="font-size: 12.8px; text-align: justify;"><br /></div></div>Rafal Urbaniakhttp://www.blogger.com/profile/10277466578023939272noreply@blogger.com1tag:blogger.com,1999:blog-4987609114415205593.post-39206589544361806732016-03-18T07:27:00.000+00:002016-03-18T07:29:50.500+00:00Five Years MCMP: Quo Vadis, Mathematical Philosophy?The Munich Center for Mathematical Philosophy invites participation to the following event:<br /><br /><b>Five Years MCMP: Quo Vadis, Mathematical Philosophy?</b><br /><br />MCMP, LMU Munich<br />June 2-4, 2016<br /><a href="http://www.lmu.de/5yearsmcmp" target="_blank">www.lmu.de/5yearsmcmp</a><br /><br />On the one hand, the workshop will celebrate the five years of existence of the <a href="http://www.mcmp.philosophie.uni-muenchen.de/index.html" target="_blank">Munich Center for Mathematical Philosophy (MCMP)</a>. On the other hand, and much more importantly, the workshop will be devoted to the question of where we should be heading in the future: what next, mathematical philosophy?<br /><br />The workshop will consist of:<br /><br />— a brief look back at five years MCMP;<br />— 16 short talks by young mathematical philosophers;<br />— three evening lectures on the logical empiricist background to mathematical philosophy; <br />— three general discussion sessions;<br />— and an "Ideas Session" in which the participants will be asked to contribute new ideas for the application of logical and mathematical methods to philosophical problems and questions.<br /><br />Registration deadline: <b>May 1st 2016</b><br /><br />Organizers:<br />Prof. Dr. <a href="http://www.mcmp.philosophie.uni-muenchen.de/people/faculty/hannes_leitgeb/index.html" target="_blank">Hannes Leitgeb</a><br />Prof. Dr. <a href="http://www.mcmp.philosophie.uni-muenchen.de/people/faculty/hartmann/index.html" target="_blank">Stephan Hartmann</a><br /> Vincenzo Crupihttp://www.blogger.com/profile/08069145846190162517noreply@blogger.com5tag:blogger.com,1999:blog-4987609114415205593.post-45311247350944152362016-02-29T20:30:00.001+00:002016-02-29T20:30:35.781+00:00Rationality Summer School: Call for applications<span style="font-family: inherit;"><b>Call for applications: International Rationality Summer Institute 2016 </b></span><br /><br /><span style="font-family: inherit;"><u><b>40 full stipends</b></u><br /> </span><br /><span style="font-family: inherit;">We invite applications for the first International Rationality Summer Institute (IRSI), which will take place from September 4-16, 2016, in Aurich (Germany). The topic of the Summer Institute is human rationality from a psychological, philosophical, and cognitive (neuro)science perspective.</span><br /><span style="font-family: inherit;"><br />Topics of the courses are: <i>Rationality and normativity</i>, <i>Norms </i>vs. <i>evidence in reasoning research</i>, <i>Rational belief change</i>, <i>Inductive reasoning</i>, <i>Causal cognition</i>, <i>Probabilistic reasoning and argumentation</i>, <i>Language and reasoning</i>, <i>Mental models and rationality</i>, <i>Probabilities and conditionals</i>, <i>Bounded rationality</i>, <i>Neural bases of reasoning</i>, <i>Development of reasoning</i>, <i>Logical and probabilistic approaches to rationality</i>, <i>Intuition and analytic thinking</i>, <i>Scientific objectivity and inductive inference</i>.</span><br /><span style="font-family: inherit;"><br />Faculty members are: John Broome, Vincenzo Crupi, Igor Douven, Aidan Feeney, York Hagmayer, Stephan Hartmann, Konstantinos Katsikopoulos, Martin Monti, David Over, Arthur Paul Pedersen, Jérôme Prado, Eva Rafetseder, Marco Ragni, Hans Rott, Jan Sprenger, Jakub Szymanik, and Valerie Thompson. In addition to the courses, we will have two keynote speakers: Gerd Gigerenzer and Johan van Benthem.</span><br /><span style="font-family: inherit;"><br />We invite applications by doctoral students and early-stage postdocs interested in human rationality and with a background in psychology, philosophy, cognitive (neuro)science, or related fields. Advanced Master’s students with a Bachelor’s degree in one of the disciplines and with an outstanding interest in the topic are also encouraged to apply.</span><br /><span style="font-family: inherit;"><br />The IRSI is generously funded by the Volkswagen Stiftung. Successful applicants will get a full stipend that covers the participation fee, board and lodging, and the reimbursement of traveling costs.</span><br /><br /><span style="font-family: inherit;"><span style="color: red;"><b>Applications close on April 15, 2016</b></span></span><br /><span style="font-family: inherit;"><br />The IRSI is organized by Markus Knauff, Patricia Garrido-Vásquez, and Marco Ragni (Giessen). Advisory board: Ralph Hertwig (Berlin), Gabriele Kern-Isberner (Dortmund), Gerhard Schurz (Düsseldorf), Wolfgang Spohn (Konstanz), and Michael Waldmann (Göttingen).</span><br /><span style="font-family: inherit;"><br />Please find more information on the Summer Institute and on how to apply at <a href="http://www.irsi2016.de/">http://www.irsi2016.de</a>. For inquiries, please send an e-mail to <a href="mailto:info@irsi2016.de">info@irsi2016.de</a>.</span><br /><br />Vincenzo Crupihttp://www.blogger.com/profile/08069145846190162517noreply@blogger.com6tag:blogger.com,1999:blog-4987609114415205593.post-3125025326563169762016-02-26T19:14:00.002+00:002016-02-26T19:14:59.695+00:00Swamplandia 2016 - schedule and abstracts<div dir="ltr" style="text-align: left;" trbidi="on">Submission deadline for Swamplandia 2016 is approaching. Meanwhile, tentative schedule with keynote speakers' titles and abstracts is available online. <a href="http://www.swamplandia2016.ugent.be/program/">Here</a>.</div>Rafal Urbaniakhttp://www.blogger.com/profile/10277466578023939272noreply@blogger.com1tag:blogger.com,1999:blog-4987609114415205593.post-42346990841459174262016-02-10T09:45:00.000+00:002016-02-10T09:45:01.033+00:00On the adoption of logical principles<p>Two weeks ago I had the pleasure of attending a one-day workshop on <a href="http://thenatureoflogic.weebly.com/">The Nature of Logic</a> organized by the University of York. The focus of the day was Saul Krikpe's unpublished works on the 'adoption problem', an interpretation of Lewis Carroll's "What the Tortoise Said to Achilles". "What the Tortoise Said to Achilles" is probably my favorite piece of philosophy, ever; York is a day-trip away from Durham; and it was a chance to hear Kripke speak in the flesh, all three reasons to expect a very interesting and enjoyable day, and the workshop did not disappoint.</p><p>The talks were all thought-provoking, but it was the first, by Romina Padró, that set the stage for the day and also triggered the thoughts that I want to try to articulate here today. Padró recently completed her dissertation on <a href="http://academicworks.cuny.edu/gc_etds/603/"><i>What the Tortoise Said to Kripke: the Adoption Problem and the Epistemology of Logic</i></a>. The "Adoption Problem" is detailed in S. 2.2, but the basic issue of this: Suppose you are confronted with someone, call him Harry, who has "no notion of the principles in question [modus ponens and universal instantiation] and has <i>never</i> inferred in accordance with them" (p. 31). Surely Harry has an impoverished reasoning ability and it would be useful to introduce him to these logical principles, such that he accepts them and can henceforth go on to reason according to them. This is the adoption of a logical principle:</p><blockquote><p>By 'adopt' here we mean that the subject, Harry in this case, picks up a way of inferring according to, say, UI, something he wasn't able to do before, <i>on the basis</i> of the <i>acceptance</i> of the corresponding logical principle (p. 31, emphasis in the original).</p></blockquote><p>The adoption problem is then whether such principles as MP and UI can be adopted. Padró's talk at the workshop was directed at arguing that they cannot: That in order to apply MP after it has been accepted, one must already be able to appeal to a notion of modus ponens. This is precisely what the Tortoise is pointing out to Achilles in Carroll's classic piece.</p><p>I remain unconvinced by Padró's argument, in part because it seems to me that Harry can accept a principle without applying it, and that once he has accepted it, he can then go on to apply it -- if he cannot apply it, then I would argue he hasn't in fact accepted it, contrary to assumption. But I will leave this point aside, and assume that there are some principles which cannot be adopted, and that MP and UI are, if anything are, prime candidates for such principles. The questions that I had -- and they are only questions, I don't have any idea how one would go about answering them, which is part of why I'm writing this, in case the collective power of the internet is smarter than me (it almost certainly is) -- stem from generalising the issue.</p><p>Padró's talk focussed on whether or not MP and UI are adoptable, and mentioned briefly other logical principles that may be similar, such as &I and &E, as well as some that likely can be adopted, such as disjunctive syllogism. This raises a general methodological point: <b>How does one determine if a principle is adoptable?</b> If every logical principle is adoptable, then we have no problem; if no logical principle is adoptable, then we have no problem. But if some are and some are not, then it would be useful to have a principled way of identifying them, preferably in advance. The argument for MP and UI is that in order to apply them, one must invoke the principles themselves:</p><blockquote><p>If someone who never inferred in accordance with MPP were to be told that "For any A and B, if A then B, and A, then B," the subject wouldn’t be in a better position to perform a MPP inference. For the principle to be of use with any particular inference, she will need to infer in accordance with the MPP pattern that she does not use in the first place: in any particular case, she will only get to B from her premises by performing a MPP inference on the instantiation of 'For all A and B, if A, and if A then B, then B,' but that is exactly what she couldn't do to begin with (p. 36).</p></blockquote><p>It seems then that one could argue that &I and &E cannot be adopted, since one must already have a concept of conjunction in order to introduce or eliminate conjunctions. But surely this is a matter of how the rule is formulated: With sufficient cleverness, I'm sure I could define &I and &E in a way that doesn't use 'and' at all, but only 'or' and 'not'. Would the principle then be adoptable, because it is formulated without appeal to the notion it is purporting to introduce?</p><p>If the answer is yes, then it immediately raises this question: <b>If whether a principle can be adopted depends on how it is formulated, how do we know that MP and UI cannot be reformulated in a way that doesn't invoke them?</b> For example, surely one could formulate MP in such a way that all Harry needs to know is disjunction and negation. If one wishes to maintain that MP-formulated-with-conditionals is not adoptable while MP-formulated-with-disjunction-and-negation is, then there is good reason to think that one must maintain that these are <i>distinct</i> logical principles. In that case, we're left with what I suspect is an extremely difficult question to answer: <b>What are the identity conditions of logical principles?</b></p><p>At this point, I have no good intuitions about how to begin answering these questions.</p><hr/><p>© 2016 <a href="mailto:s.l.uckelman@durham.ac.uk">Sara L. Uckelman</a>.</p>Sara L. Uckelmanhttp://www.blogger.com/profile/14716054827293611237noreply@blogger.com3tag:blogger.com,1999:blog-4987609114415205593.post-53433124130456417822016-01-30T14:36:00.000+00:002016-01-30T14:36:16.528+00:00Meta-arithmetic and philosophy CFP (Swamplandia 2016)<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">Swamplandia 2016<br />Meta-arithmetical results and their philosophical meaning<br />Ghent, May 30 - June 1, 2016<br /><br />Logicians and mathematicians devoted considerable effort to investigate the properties and limitations of arithmetical theories. Unfortunately, philosophical motivations and implications of some of these results are either not known or not clear. The main aim of the workshop is to present philosophically relevant meta-arithmetical results and discuss their philosophical implications in more depth. The workshop is focused on, but not restricted to formal theories of truth, theories of provability in arithmetic, logic of provability and philosophically relevant results about complexity or computability. Keynote speakers will deliver invited lectures and give extended tutorials. The title of the workshop comes from the fact that philosophical approaches to mathematical results are rather tricky.<br /><br />Keynote speakers<br />Diderik Batens (Ghent University)<br />Cezary Cieśliński (University of Warsaw)<br />Jeffrey Ketland (University of Oxford)<br />Lavinia Picollo (Munich Center for Mathematical Philosophy)<br />Saeed Salehi (University of Tabriz)<br />Peter Verdée (Université Catholique de Louvain)<br />Albert Visser (Utrecht University)<br /><br />Submissions <br />We welcome submissions of papers that strike a balance between technical developments and philosophical discussion. If you’re interested in presenting at the workshop, please send your extended abstract (1000-1500 words) prepared for double-blind review in PDF format to<br /><br />swamplandia2016@gmail.com <br /><br />by March 1, 2016. Authors of accepted papers will have 30-45 minutes to present their work.<br /><br />Publication<br />A Studia Logica volume on the philosophical aspects of meta-arithmetical and set-theoretic results will be edited by the organizers. Participants are welcome to submit papers for the volume some time after the conference. Details TBA. <br /><br />Presentation abstract submission: March 1, 2016<br />Acceptance notification: April 15, 2016<br />Fee payment deadline: May 1, 2016<br />Workshop: May 30, 2016 - June 1, 2016<br /><br />Fees<br />Faculty: 60 EUR<br />Students: 40 EUR<br />Late fee: 30 EUR + basic fee<br />If your attendance will not be covered by any grant or if you are a student with financial difficulties, please include a statement saying so at the end of your extended abstract, so we can consider you for a conference fee waiver.<br /><br />Organizers: Rafal Urbaniak, Pawel Pawlowski and Erik Weber </div></div>Rafal Urbaniakhttp://www.blogger.com/profile/10277466578023939272noreply@blogger.com1tag:blogger.com,1999:blog-4987609114415205593.post-84011561114247763722016-01-27T11:23:00.000+00:002016-01-27T11:28:26.736+00:00Two doctoral fellowships at the MCMPYou would like to write a PhD thesis at the Munich Center for Mathematical Philosophy (MCMP) on paradoxes of truth and/or vagueness, and on the metaphilosophical question about how to handle diverging solutions to such paradoxes? Great! Then please consider applying for one of the<br /><br />*** Two Doctoral Fellowships at the MCMP *** <br /><br />which we are advertising right now (as part of the European Training Network DIAPHORA that includes philosophers from Barcelona, Munich, Neuchatel, Stirling, Stockholm, Edinburgh, Paris).<br /><br />More information can be found at:<br /><br /><a href="http://www.ub.edu/grc_logos/files/user77/1453661865-DIAPHORA_call%20for%20applications.pdf" target="_blank">http://www.ub.edu/grc_logos/files/user77/1453661865-DIAPHORA_call%20for%20applications.pdf</a><br /><br /><br />Vincenzo Crupihttp://www.blogger.com/profile/08069145846190162517noreply@blogger.com2tag:blogger.com,1999:blog-4987609114415205593.post-65060316087033658072015-12-08T19:49:00.000+00:002015-12-12T19:28:02.365+00:00Why I don't care what possible worlds are<p>This afternoon, I lectured to my 2nd year students on Lewis and Stalnaker on possible worlds (with a bit of Kripke thrown in since we'd done the 1st lecture of <i>Naming and Necessity</i> two weeks ago). I included these two papers in the syllabus for the same reason I did last year -- because they are pieces of work of historical importance for their role in the debate on realism w.r.t. possible worlds. And like last year, I found both pieces difficult to lecture on, not because they are especially difficult, or especially problematic, but because, as a modal logician, <i>I simply don't care</i>. Resolving this debate -- whether possible worlds really are "out there" like Lewis thinks or whether they're more of a pragmatic tool as Stalnaker thinks -- will not change my practice one whit.</p> <p>I try not to let my students know that I feel this way (I try to keep my philosophical "politics" out of the classroom -- except when the opportunity to rant on why I think "not philosophical enough" is a <i>horrible</i> criticism, but that is not apropos here), but I do try to let them know that there is more to the issue than resolving the debate, there is the question of whether the debate needs to be resolved before modal logicians can go about their business with impunity. Last year I put it as an essay question, but I don't remember if anyone took it up. This year, in yesterday's tutorial I divided the group into two and randomly told one "You prepare a case in favor of realism", and the other group "You guys get anti-realism", and during the ensuing discussion, I heard someone sort of whisper to someone else "Does it matter?", which I thought an appropriate to revisit the issue. We discussed it some in the tutorial, with one person feeling quite strongly that if one didn't properly settle the 'foundational' issues, then there would be no guarantee that the modal logician wouldn't one day be led astray. At the end of lecture today I posted two questions hoping to get people's gut feelings -- who thinks Lewis is right, who thinks he's not, and who thinks the question has to be resolved, and who thinks it doesn't. As expected, I got roughly equal hands for each, and was lucky enough to have two people willing to articulate their gut feelings. One (on the side of "yes, we do") argued from the basis of metaphysical possibility: If we're going to use possible worlds for analysing metaphysical possibility, we're sure going to want to know if they are metaphysically possible! The other said that you might need to look at reality to determine which axioms you adopted, but after that, it shouldn't matter <i>what</i> possible worlds in fact are when you start using them as a tool in modal reasoning.</p> <p>All of this set me up to spend some more time thinking this afternoon about <i>why</i> I don't care. It's a rather scientific, rather than philosophical, position to take -- scientists don't care what the "real nature" of particles are (well, except for the foundationalists, i.e., the physicists), mathematicians don't care what numbers "really" are, modal logicians don't care what possible worlds "really" are, etc. The foundational issue raised in the tutorial yesterday gave rise to an apt comparison with mathematics: Mathematicians don't really care about what numbers are, because <i>whatever they are</i>, they sure work really really well, and by now it seems highly unlikely that we could discover something about what numbers are that would cast the results that we've derived using them into doubt. Modal logic isn't in quite the same position with respect to possible worlds, but it seems similar.</p> <p>I also thought about what a situation in which it mattered what possible worlds were, metaphysically, would look like -- in what sort of situation would the metaphysical nature of possible worlds make a difference? Well, when discussing metaphysical possibility/necessity, as noted above. I happen to find that concept a highly dubious one (on extra-logical grounds), so I'm happy to simply put up my hands and say "that is not a modal concept I am interested in explicating". But as I tried to come up with concrete scenarios in which modal logic is applied, rather than simply theorized about, in each of these cases, the notion of possible world was interpreted as something quite concrete: For example, states of a computer programme. Then I thought about the other student's comment about needing to hash out what the right axioms were, and that possibly being when it was necessary to know something about the metaphysical status of possible worlds. But what is it that axioms specify? Do they specify anything about the worlds themselves? No: What modal axioms do is specify <i>how the worlds are related to each other</i>, and these axioms will hold (or not) in virtue of the relations between the worlds -- whatever the worlds may be. They may be Lewisian possible worlds, they may be states of a computer, they may be moments in time, they may be pebbles, they may be fruitcakes. The axioms -- that which really is the meat of modal reasoning -- are all about how the worlds are related to each other, and not about how the worlds are composed [1].</p> <p>And that is at least part of the reason why I, as a modal logician, don't really care about what possible worlds are.</p> <hr/> <p>[1] At this point, I realize that everything that I've been saying is about <i>propositional</i> modal logic, and that if what you're interested in is quantified modal logic, then you might object that how the worlds are composed, i.e., what objects are in them and what properties those objects have, is of crucial importance, AND that the axioms adopted will have consequences for the internal composition, e.g., whether the Barcan or Converse Barcan formulas are axioms. To which I would reply: Hmmm, this is very interesting, I will have to think on the case of quantified modal logic further.</p> <hr/><p>© 2015, Sara L. Uckelman</p>Sara L. Uckelmanhttp://www.blogger.com/profile/14716054827293611237noreply@blogger.com3tag:blogger.com,1999:blog-4987609114415205593.post-47195834288712192152015-11-02T11:09:00.002+00:002015-11-02T11:32:17.089+00:00The beauty (?) of mathematical proofs -- empirical predictions <div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;">By <b>Catarina Dutilh Novaes</b></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><br /></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;">This is the final post in my series on<span style="color: #222222; font-family: "georgia" , "utopia" , "palatino linotype" , "palatino" , serif; font-size: 15.4px; line-height: 21.56px;"> beauty, function, and explanation in mathematical proofs (</span><a href="http://m-phi.blogspot.nl/2015/10/the-beauty-of-mathematical-proofs.html" style="color: #274e13; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 15.4px; line-height: 21.56px; text-decoration: none;">Part I is here</a><span style="color: #222222; font-family: "georgia" , "utopia" , "palatino linotype" , "palatino" , serif; font-size: 15.4px; line-height: 21.56px;">; </span><a href="http://m-phi.blogspot.nl/2015/10/the-beauty-of-mathematical-proofs_6.html" style="color: #274e13; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 15.4px; line-height: 21.56px; text-decoration: none;">Part II is here</a><span style="background-color: #eefcff; color: #222222; font-family: "georgia" , "utopia" , "palatino linotype" , "palatino" , serif; font-size: 15.4px; line-height: 17.71px;">; </span><a href="http://m-phi.blogspot.nl/2015/10/the-beauty-of-mathematical-proofs_7.html" style="color: #274e13; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 15.4px; line-height: 17.71px; text-decoration: none;">Part III is here</a><span style="background-color: #eefcff; color: #222222; font-family: "georgia" , "utopia" , "palatino linotype" , "palatino" , serif; font-size: 15.4px; line-height: 17.71px;">; </span><a href="http://m-phi.blogspot.nl/2015/10/the-beauty-of-mathematical-proofs_8.html" style="color: #274e13; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 15.4px; line-height: 17.71px; text-decoration: none;">Part IV is here</a><span style="background-color: #eefcff; color: #222222; font-family: "georgia" , "utopia" , "palatino linotype" , "palatino" , serif; font-size: 15.4px; line-height: 17.71px;">; </span><a href="http://m-phi.blogspot.nl/2015/10/the-beauty-of-mathematical-proofs_9.html" style="color: #274e13; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 15.4px; line-height: 17.71px; text-decoration: none;">Part V is here</a><span style="background-color: #eefcff; color: #222222; font-family: "georgia" , "utopia" , "palatino linotype" , "palatino" , serif; font-size: 15.4px; line-height: 17.71px;">; </span><a href="http://m-phi.blogspot.nl/2015/10/the-beauty-of-mathematical-proofs-proof.html" style="color: #274e13; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 15.4px; line-height: 17.71px; text-decoration: none;">Part VI is here</a>; P<a href="http://m-phi.blogspot.nl/2015/10/the-beauty-of-mathematical-proofs_14.html">art VII is here</a><span style="color: #222222; font-family: "georgia" , "utopia" , "palatino linotype" , "palatino" , serif; font-size: 15.4px; line-height: 21.56px;">). Here I tease out some empirical predictions of the account developed in the previous posts, according to which beauty and explanatoriness will largely (though not entirely) coincide in mathematical proofs. I also comment on how the account, based on a dialogical conception of mathematical proofs, could be made more palatable for those who would prefer a non-relative, absolute analysis of beauty and explanatoriness.</span></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><br /></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;">=====================</div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><br /></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;">To summarize, the present account defends the thesis that when mathematicians employ aesthetic vocabulary to describe proofs, both positively (‘beautiful’, ‘elegant’) and negatively (‘ugly’, ‘clumsy’), they are by and large (though not exclusively) tracking the epistemic property of explanatoriness (or lack thereof) of a proof. Up to this point, the account is compatible with both subjective (agent-relative) and objective understandings of beauty and explanation, so long as the two dimensions go together (i.e. both understood as either subjective or as objective). However, on the basis of a dialogical conception of mathematical proofs, I’ve also argued that both explanation and beauty are essentially relative notions with respect to proofs: an explanation is not explanatory <i style="mso-bidi-font-style: normal;">an sich</i>, but rather explanatory for its intended audience; and if a proof is deemed beautiful to the extent that it fulfills this explanatory function, then beauty too emerges as a relative notion.<o:p></o:p></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><br /></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;">I’ve also suggested ways in which the present account can be made more palatable for those who strongly prefer objective accounts of explanatoriness and beauty. By maximally expanding the range of Skeptics who will deem a proof explanatory – and so aiming towards the notion of a <i style="mso-bidi-font-style: normal;">universal audience</i> – in the limit (idealized) case a proof may be deemed explanatory by all (i.e. those who have the required expertise to understand it in the first place). On this conception then, a proof may also be understood to be beautiful in an absolute sense, i.e. insofar it fulfills its explanatory function towards any potential (suitably qualified) audience. The conception of <i style="mso-bidi-font-style: normal;">beauty as fit</i> defended by Raman-Sundström (2012), which relies on an objectively conceived notion of fit,<a href="https://www.blogger.com/blogger.g?blogID=4987609114415205593#_ftn1" name="_ftnref1" style="mso-footnote-id: ftn1;" title=""><span class="MsoFootnoteReference"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]--><span class="MsoFootnoteReference"><span style="font-family: "cambria"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: "ＭＳ 明朝"; mso-fareast-language: NL; mso-fareast-theme-font: minor-fareast; mso-hansi-theme-font: minor-latin;">[1]</span></span><!--[endif]--></span></span></a> may be viewed as an example of such an account, and indeed her description of fit bears a number of similarities with concepts typically associated with explanatoriness.<a href="https://www.blogger.com/blogger.g?blogID=4987609114415205593#_ftn2" name="_ftnref2" style="mso-footnote-id: ftn2;" title=""><span class="MsoFootnoteReference"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]--><span class="MsoFootnoteReference"><span style="font-family: "cambria"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: "ＭＳ 明朝"; mso-fareast-language: NL; mso-fareast-theme-font: minor-fareast; mso-hansi-theme-font: minor-latin;">[2]</span></span></span></span></a></div><a name='more'></a><!--[endif]--><o:p></o:p><br /><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><br /></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;">Besides the objective (absolute) vs. subjective (relative) divide, we’ve also discussed the reductive vs. literal divide. Now, while with respect to the former I take a clear stance favoring a subjective, agent-relative account of explanation and beauty, with respect to the latter the present proposal defends a conciliatory position through the key concept of functional beauty. If the primary function of a proof is to produce explanatory persuasion in a suitably equipped audience, and if beauty arises (among other reasons) from something performing its function well, then a proof will be beautiful to the extent that it performs its epistemic function(s) well. And thus, we can say that mathematicians are both tracking epistemic properties <i style="mso-bidi-font-style: normal;">and</i> truly making aesthetic judgments at the same time when using aesthetic vocabulary to describe proofs. But I’ve also suggested that there may well remain a residue of non-epistemic beauty, for example in the experience of beauty that may emerge with a proof that proceeds in unexpected ways.<o:p></o:p></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><br /></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;">At any rate, explanation itself seems to be a multi-dimensional notion (as also suggested by Hafner and Mancosu (2005)), and a proof may be deemed explanatory for a variety of different reasons: conceptual clarification, generalization or unification, purity,<a href="https://www.blogger.com/blogger.g?blogID=4987609114415205593#_ftn3" name="_ftnref3" style="mso-footnote-id: ftn3;" title=""><span class="MsoFootnoteReference"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]--><span class="MsoFootnoteReference"><span style="font-family: "cambria"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: "ＭＳ 明朝"; mso-fareast-language: NL; mso-fareast-theme-font: minor-fareast; mso-hansi-theme-font: minor-latin;">[3]</span></span><!--[endif]--></span></span></a> computational effectiveness etc. And thus, via a multi-faceted conception of explanation, we arrive at a multi-faceted conception of the function(s) of a mathematical proof, which in turn yields a multi-faceted conception of the beauty (or ugliness) of a proof. This is a desirable result in view of the pluralistic, non-monolithic conception of mathematical proof that emerges from careful examination of the varieties of mathematical practices (in particular, with respect to proofs) – what Wittgenstein in the <i style="mso-bidi-font-style: normal;">Remarks on the Foundations of Mathematics</i> (III-46A, III-48) describes as the ‘motley’ of mathematical techniques of proof.<o:p></o:p></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><br /></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;">The present account also gives rise to fairly precise empirical predictions, which means that it can be put to empirical test, perhaps along the lines of the work of Inglis and Aberdein (for some of these predictions, at least some preliminary results already exist):<o:p></o:p></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><br /></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;">- <i style="mso-bidi-font-style: normal;">Beauty and explanatoriness are used as comparative, graded notions.</i> The claim is that these are notions predominantly used to express predilections for certain proofs over others, and thus are essentially comparative. The results by Inglis and Aberdein seem to support this interpretation of these notions.<o:p></o:p></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><br /></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;">- <i style="mso-bidi-font-style: normal;">Mathematicians disagree in their assessments of the explanatoriness of proofs.</i> This follows straightforwardly from the claim that the explanatoriness of a proof is audience-relative. Colyvan (2012, 79) suggests (possibly based on personal observation) that there is indeed quite some disagreement among mathematicians on which proofs are explanatory and which are not (or less). Inglis and Aberdein’s (ms.) results on diversity in proof appraisal suggest this much, but the questions posed in their study did not address explanatoriness specifically. So further research on this specific issue is required.<o:p></o:p></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><br /></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;">- <i style="mso-bidi-font-style: normal;">Mathematicians disagree in their aesthetic judgments about proofs</i>. Again, this follows from the claim that explanatoriness (and thus beauty) is an audience-relative notion, though it may well be that certain kinds of proofs (e.g. ‘brute force’ combinatorial proofs) are unanimously seen as ‘ugly’.<a href="https://www.blogger.com/blogger.g?blogID=4987609114415205593#_ftn4" name="_ftnref4" style="mso-footnote-id: ftn4;" title=""><span class="MsoFootnoteReference"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]--><span class="MsoFootnoteReference"><span style="font-family: "cambria"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: "ＭＳ 明朝"; mso-fareast-language: NL; mso-fareast-theme-font: minor-fareast; mso-hansi-theme-font: minor-latin;">[4]</span></span><!--[endif]--></span></span></a> Inglis and Aberdein’s results on diversity in proof appraisal also suggest this much, but hitherto they only worked with a small range of proofs. We need further investigation of mathematicians’ appraisals of a much larger range of proofs to be able to draw more general conclusions on how systematic the phenomenon is.<o:p></o:p></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><br /></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;">- <i style="mso-bidi-font-style: normal;">Judgments of functionality and explanatoriness will strongly correlate</i>. This follows from the idea that the primary function of a proof is to produce explanatory persuasion. Inglis and Aberdein (2015) found a reasonably strong correlation between ‘explanatory’ and the utility dimension, though less strong than between ‘explanatory’ and the precision dimension, and the negative correlation between ‘explanatory’ and intricacy. However, given that utility and function are not exactly equivalent notions (utility may also be understood in the sense of the practical applicability of the proof), here too further work seems to be required.<o:p></o:p></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><br /></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;">- <i style="mso-bidi-font-style: normal;">Judgments of beauty and explanatoriness will correlate, but not coincide</i>. This follows from the idea that the beauty (or ugliness) of a proof can be largely but not entirely understood in epistemic terms related to explanatoriness. In the dataset used in (Inglis and Aberdein 2015), the correlation between beauty and explanatoriness is greater than zero, but it is not that high (admittedly, lower than the present account would predict). Here too we might need further research, for example with a more graded scale of appraisal, to establish the extent to which mathematicians tend to view what they consider to be explanatory proofs as also beautiful, or whether a proof can be viewed as beautiful without being viewed as explanatory, or yet whether two proofs may be deemed equally explanatory but not equally beautiful, which would suggest that there is more to beauty than explanatoriness.<o:p></o:p></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><br /></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;">- <i style="mso-bidi-font-style: normal;">Judgments of beauty and persuasiveness will correlate</i>. This follows from the claim that to provoke persuasion is one of the main functions of a proof, and that the beauty of a proof can be largely understood in functional terms. Indeed, some of the proofs that are typically viewed as ‘ugly’ such as probabilistic or computer-assisted proofs tend to be viewed as unpersuasive as well, perhaps because they fail to transfer to the audience the reason(s) why the theorem must be true (Easwaran 2009). Inglis and Aberdein (2015) did not include ‘persuasive’ among the adjectives on their list, so at this point a systematic investigation of the relations between attributions of beauty and attributions of persuasiveness with respect to proofs remains to be done.<o:p></o:p></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><br /></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;">Regarding convergence or dissent in proof appraisal on the aesthetic and on the explanatory dimensions, the crucial prediction of the present account is that they will display the same pattern: if mathematicians largely (dis)agree in their judgments on the explanatoriness of proofs, then they should largely (dis)agree in their judgments on the aesthetic properties of proofs, and vice-versa. If this prediction is not corroborated, then this will constitute a major blow for the present account (which means, in good old Popperian terms, that the account is falsifiable). At any rate, the fact that the philosophical analysis presented here gives rise to a number of fairly precise empirical predictions is, I take it, overall an advantage of the proposal.<o:p></o:p><br /><br /><div style="line-height: 18.4px;">FULL SERIES: </div><div style="line-height: 18.4px;"><a href="http://m-phi.blogspot.nl/2015/10/the-beauty-of-mathematical-proofs.html" style="background-color: #eefcff; color: #274e13; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 15.4px; line-height: 21.56px; text-decoration: none;">Part I is here</a></div><div style="line-height: 18.4px;"><a href="http://m-phi.blogspot.nl/2015/10/the-beauty-of-mathematical-proofs_6.html" style="background-color: #eefcff; color: #274e13; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 15.4px; line-height: 21.56px; text-decoration: none;">Part II is here</a></div><div style="line-height: 18.4px;"><a href="http://m-phi.blogspot.nl/2015/10/the-beauty-of-mathematical-proofs_7.html" style="background-color: #eefcff; color: #274e13; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 15.4px; line-height: 17.71px; text-decoration: none;">Part III is here</a></div><div style="line-height: 18.4px;"><a href="http://m-phi.blogspot.nl/2015/10/the-beauty-of-mathematical-proofs_8.html" style="background-color: #eefcff; color: #274e13; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 15.4px; line-height: 17.71px; text-decoration: none;">Part IV is here</a></div><div style="line-height: 18.4px;"><a href="http://m-phi.blogspot.nl/2015/10/the-beauty-of-mathematical-proofs_9.html" style="background-color: #eefcff; color: #274e13; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 15.4px; line-height: 17.71px; text-decoration: none;">Part V is here</a></div><div style="line-height: 18.4px;"><a href="http://m-phi.blogspot.nl/2015/10/the-beauty-of-mathematical-proofs-proof.html" style="background-color: #eefcff; color: #274e13; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 15.4px; line-height: 17.71px; text-decoration: none;">Part VI is here</a></div><div style="line-height: 18.4px;"><span style="background-color: #eefcff; color: #222222; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 15.4px; line-height: 17.71px;">P</span><a href="http://m-phi.blogspot.nl/2015/10/the-beauty-of-mathematical-proofs_14.html" style="background-color: #eefcff; color: #274e13; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 15.4px; line-height: 17.71px; text-decoration: none;">art VII is here</a><br /><a href="http://m-phi.blogspot.nl/2015/11/the-beauty-of-mathematical-proofs.html">Part VIII is here</a></div></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><br /></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><b>References</b></div><div class="MsoNormal" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><br /></div><div class="MsoNormal" style="text-align: justify;"><span style="line-height: 18.4px;">Easwaran, Kenny (2009). Probabilistic proofs and transferability. <i>Philosophia Mathematica</i> 17 (3):341-362.</span></div><div class="MsoNormal" style="text-align: justify;"><span style="line-height: 18.4px;">Detlefsen, Michael & Arana, Andrew (2011). Purity of Methods. <i>Philosophers' Imprint</i> 11 (2).</span><br /><span style="background-color: #eefcff; color: #222222; font-family: "georgia" , "utopia" , "palatino linotype" , "palatino" , serif; font-size: 15.4px; line-height: 18.4px; text-indent: -24px;">Hafner, Johannes & Mancosu, Paolo (2005). The Varieties of Mathematical Explanation. <i>In Visualization, Explanation and Reasoning Styles in Mathematics</i>. Edited by Paolo Mancosu, et al., 215–250. Berlin: Springer.</span></div><div class="MsoNormal" style="text-align: justify;"><span style="line-height: 18.4px;">Inglis, Matthew & Aberdein, Andrew (2015). Beauty Is Not Simplicity: An Analysis of Mathematicians' Proof Appraisals. <i>Philosophia Mathematica</i> 23 (1):87-109.</span></div><div class="MsoNormal" style="text-align: justify;"><span style="line-height: 18.4px;">Inglis, Matthew & Aberdein, Andrew (ms.). Diversity in Proof Appraisal. Available at </span><span style="line-height: 18.4px;">https://www.academia.edu/8488060/Diversity_in_Proof_Appraisal</span></div><div class="MsoNormal" style="text-align: justify;"><span style="line-height: 18.4px;">Raman-Sundström, Manya (2012). Beauty as Fit: An Empirical Study of Mathematical Proofs. </span></div><div class="MsoNormal" style="text-align: justify;"><span style="line-height: 18.4px;"><i>Proceedings of the British Society for Research into Learning Mathematics </i>32(3):156-160. Available at http://www.bsrlm.org.uk/IPs/ip32-3/BSRLM-IP-32-3-27.pdf</span></div><div class="MsoNormal" style="text-align: justify;"><span style="line-height: 18.4px;">Raman-Sundström, Manya & Öhman, Lars-Daniel (ms.). Mathematical fit: a case study.</span></div><div class="MsoNormal" style="text-align: justify;"><span style="line-height: 18.4px;">Wittgenstein, Ludwig (1978). <i>Remarks on the Foundations of Mathematics</i>. B. Blackwell.</span></div><!--[if gte mso 9]><xml> <o:DocumentProperties> <o:Revision>0</o:Revision> <o:TotalTime>0</o:TotalTime> <o:Pages>1</o:Pages> <o:Words>1354</o:Words> <o:Characters>7398</o:Characters> <o:Company>N</o:Company> <o:Lines>119</o:Lines> <o:Paragraphs>20</o:Paragraphs> <o:CharactersWithSpaces>8732</o:CharactersWithSpaces> <o:Version>14.0</o:Version> </o:DocumentProperties> <o:OfficeDocumentSettings> <o:AllowPNG/> </o:OfficeDocumentSettings></xml><![endif]--> <!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:TrackMoves/> <w:TrackFormatting/> <w:HyphenationZone>21</w:HyphenationZone> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:DoNotPromoteQF/> <w:LidThemeOther>NL</w:LidThemeOther> <w:LidThemeAsian>JA</w:LidThemeAsian> 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QFormat="true" Name="TOC Heading"/> </w:LatentStyles></xml><![endif]--> <!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:Standaardtabel; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:Cambria; mso-ascii-font-family:Cambria; mso-ascii-theme-font:minor-latin; mso-hansi-font-family:Cambria; mso-hansi-theme-font:minor-latin; mso-ansi-language:NL;} </style><![endif]--> <!--StartFragment--> <!--EndFragment--><br /><div style="mso-element: footnote-list;"><!--[if !supportFootnotes]--><br clear="all" /><hr align="left" size="1" width="33%" /><!--[endif]--> <br /><div id="ftn1" style="mso-element: footnote;"><div class="MsoFootnoteText" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><a href="https://www.blogger.com/blogger.g?blogID=4987609114415205593#_ftnref1" name="_ftn1" style="mso-footnote-id: ftn1;" title=""><span class="MsoFootnoteReference"><span style="font-size: 10.0pt; line-height: 115%;"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]--><span class="MsoFootnoteReference"><span style="font-family: "cambria"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: "ＭＳ 明朝"; mso-fareast-language: NL; mso-fareast-theme-font: minor-fareast; mso-hansi-theme-font: minor-latin;">[1]</span></span><!--[endif]--></span></span></span></a><span style="font-size: 10.0pt; line-height: 115%;"> “Making a distinction between whether a proof is beautiful and whether a person can grasp that beauty can help explain phenomena such as why mathematicians judge different proofs to be beautiful, or why mathematicians and non-mathematicians do the same, without drawing a necessary conclusion that mathematical beauty is subjective. Moreover, the metaphor of ‘fit’ suggests a more objective view of beauty might be warranted— whether a proof is appreciated as beautiful is a subjective claim, but whether a proof fits a theorem, which relies more on the nature of the proof than our perception of it, is a more objective one.” (Raman-Sundström 2012, 160)<o:p></o:p></span></div></div><div id="ftn2" style="mso-element: footnote;"><div class="MsoFootnoteText" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><a href="https://www.blogger.com/blogger.g?blogID=4987609114415205593#_ftnref2" name="_ftn2" style="mso-footnote-id: ftn2;" title=""><span class="MsoFootnoteReference"><span style="font-size: 10.0pt; line-height: 115%;"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]--><span class="MsoFootnoteReference"><span style="font-family: "cambria"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: "ＭＳ 明朝"; mso-fareast-language: NL; mso-fareast-theme-font: minor-fareast; mso-hansi-theme-font: minor-latin;">[2]</span></span><!--[endif]--></span></span></span></a><span style="font-size: 10.0pt; line-height: 115%;"> She presents two kinds of fit: internal fit as the idea “that the proof directly illuminates what the theorem is about, providing a sense of <i style="mso-bidi-font-style: normal;">why</i>the theorem is true” (emphasis added); and external fit as the idea “that the proof derives its beauty from the way it is connected to a family of other theorems”. (Raman-Sundström 2012, 159) In both cases, the connections with the notion of explanatoriness as described in the literature are evident. In recent work (Sundström & Öhman ms.), Sundström further explores the connections between fit, beauty, and explanation.<o:p></o:p></span></div></div><div id="ftn3" style="mso-element: footnote;"><div class="MsoFootnoteText" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><a href="https://www.blogger.com/blogger.g?blogID=4987609114415205593#_ftnref3" name="_ftn3" style="mso-footnote-id: ftn3;" title=""><span class="MsoFootnoteReference"><span style="font-size: 10.0pt; line-height: 115%;"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]--><span class="MsoFootnoteReference"><span style="font-family: "cambria"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: "ＭＳ 明朝"; mso-fareast-language: NL; mso-fareast-theme-font: minor-fareast; mso-hansi-theme-font: minor-latin;">[3]</span></span><!--[endif]--></span></span></span></a><span style="font-size: 10.0pt; line-height: 115%;"> A proof is said to be pure if it only relies on concepts already contained in the statement of the theorem, i.e. when it does not make use of concepts alien to the theorem itself. See (Detlefsen and Arana 2011).<o:p></o:p></span></div></div><div id="ftn4" style="mso-element: footnote;"><div class="MsoFootnoteText" style="line-height: 115%; text-align: justify; text-justify: inter-ideograph;"><a href="https://www.blogger.com/blogger.g?blogID=4987609114415205593#_ftnref4" name="_ftn4" style="mso-footnote-id: ftn4;" title=""><span class="MsoFootnoteReference"><span style="font-size: 10.0pt; line-height: 115%;"><span style="mso-special-character: footnote;"><!--[if !supportFootnotes]--><span class="MsoFootnoteReference"><span style="font-family: "cambria"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: "ＭＳ 明朝"; mso-fareast-language: NL; mso-fareast-theme-font: minor-fareast; mso-hansi-theme-font: minor-latin;">[4]</span></span><!--[endif]--></span></span></span></a><span style="font-size: 10.0pt; line-height: 115%;"> Similarly, and as noted by Raman-Sundström (2012), the fact that mathematicians do not converge in their attributions of beauty to proofs still does not warrant the conclusion that ‘beauty is (merely) in the eyes of the beholder’, as there may be a gap “between whether a proof is beautiful and whether a person can grasp that beauty”. In other words, disagreement on this dimension would not be sufficient to confirm a subjective understanding of the beauty of proofs, though convergence on this dimension may well be sufficient to <i style="mso-bidi-font-style: normal;">dis</i>confirm it.<o:p></o:p></span></div></div></div>Catarinahttp://www.blogger.com/profile/03277956118114314573noreply@blogger.com5